Integrable deformations of strings
Habilitation Thesis
Linus Wulff
Brno 2021
Abstract
This thesis deals with deformations of string sigma models which have the property that
they preserve integrability. This means that starting from an integrable string sigma
model and deforming it one obtains a new integrable model, which reduces to the original
one when the deformation parameter is taken to zero. There are different types of such
deformations but a large class, which we will focus on here, are the so-called Yang-Baxter
deformations. They are defined by a constant matrix R which solves the classical YangBaxter
equation. After introducing these deformations in the simplest setting of the
Principal Chiral Model we will describe their close relation to the transformation known
as non-abelian T-duality. In the case of string theory there are additional conditions on
the sigma model. In particular, it must be Weyl invariant. We show that Yang-Baxter
deformations preserve the Weyl invariance to at least two loop order in the sigma model
perturbation theory, provided R satisfies a so-called unimodularity condition. The proof
of this important fact is greatly simplified by working in a formalism with an enlarged
symmetry group, known as Double Field Theory. This also allows us to find the first
quantum (α ) correction to these deformed models.
1
List of included papers
This thesis is based on the following papers:
[I] S. Hronek and L. Wulff, “Relaxing unimodularity for Yang-Baxter deformed strings,”
JHEP 10 (2020), 065 [arXiv:2007.15663 [hep-th]].
[II] R. Borsato and L. Wulff, “Quantum Correction to Generalized T Dualities,” Phys.
Rev. Lett. 125 (2020) no.20, 201603 [arXiv:2007.07902 [hep-th]].
[III] R. Borsato, A. Vilar L´opez and L. Wulff, “The first α -correction to homogeneous
Yang-Baxter deformations using O(d, d),” JHEP 07 (2020) no.07, 103 [arXiv:2003.05867
[hep-th]].
[IV] R. Borsato and L. Wulff, “Two-loop conformal invariance for Yang-Baxter deformed
strings,” JHEP 03 (2020), 126 [arXiv:1910.02011 [hep-th]].
[V] R. Borsato and L. Wulff, “Marginal deformations of WZW models and the classical
Yang–Baxter equation,” J. Phys. A 52 (2019) no.22, 225401 [arXiv:1812.07287
[hep-th]].
[VI] R. Borsato and L. Wulff, “Non-abelian T-duality and Yang-Baxter deformations of
Green-Schwarz strings,” JHEP 08 (2018), 027 [arXiv:1806.04083 [hep-th]].
[VII] R. Borsato and L. Wulff, “On non-abelian T-duality and deformations of supercoset
string sigma-models,” JHEP 10 (2017), 024 [arXiv:1706.10169 [hep-th]].
[VIII] R. Borsato and L. Wulff, “Integrable Deformations of T-Dual σ Models,” Phys.
Rev. Lett. 117 (2016) no.25, 251602 [arXiv:1609.09834 [hep-th]].
[IX] R. Borsato and L. Wulff, “Target space supergeometry of η and λ-deformed strings,”
JHEP 10 (2016), 045 [arXiv:1608.03570 [hep-th]].
Reprints can be found at the end of this thesis. All authors made an equal contribution
to the papers.
Acknowledgments
I wish to thank my co-authors, in particular Riccardo Borsato, for enjoyable collaborations
on the fascinating topic of integrable deformations. I thank Stanislav Hronek for helpful
comments on a draft of this thesis.
2
Contents
1 Introduction 4
2 Yang-Baxter deformations and non-abelian T-duality 6
2.1 Yang-Baxter deformations of the
Principal Chiral Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Non-abelian T-duality for PCM . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 NATD with respect to a subgroup . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 YB deformations from NATD . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 String sigma model and Weyl invariance conditions 15
3.1 Homogeneous Yang-Baxter deformations . . . . . . . . . . . . . . . . . . . 17
3.2 O(D, D) invariant formulation of string effective action . . . . . . . . . . . 18
3.3 Frame-like formulation of DFT . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 One-loop Weyl invariance and unimodularity
condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Two-loop Weyl invariance and α -correction 27
4.1 α -correction to double Lorentz transformation . . . . . . . . . . . . . . . . 27
4.2 α -corrected DFT action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 α -correction to YB deformations . . . . . . . . . . . . . . . . . . . . . . . 31
5 Conclusions & further developments 33
3
Chapter 1
Introduction
This thesis summarizes work done over the past five years on the topic of integrable deformations
of string sigma models. It consists of nine papers together with an introductory
part explaining some of the important results in those papers and giving important background.
To keep things as simple as possible we focus on the Principal Chiral Model for
introducing the Yang-Baxter deformation and non-abelian T-duality (Chapter 2), and on
the bosonic string when we discuss deformations of string sigma models and Weyl invariance
(Chapters 3 & 4). Many extensions of the results, as well as a more complete list of
references, can be found in the papers.
Two-dimensional non-linear sigma models have many applications in physics. Here we
will mainly be interested in the application to string theory where certain two-dimensional
sigma models describe the dynamics of the string itself. The action of the non-linear sigma
model takes the form
S = d2
ξ ∂+xm
∂−xn
(Gmn(x) + Bmn(x)) , (1.1)
where the 2d coordinates are ξ0
, ξ1
and we have defined the light-cone derivatives ∂± =
∂/∂ξ0
± ∂/∂ξ1
. This is a theory of D scalar fields, xm
= xm
(ξ), m = 0, . . . , D − 1, from
the 2d point of view. The reason we call them x is that these fields can be interpreted
as coordinates of a D-dimensional space, which we will often refer to as the background
(or target space). The matrices Gmn(x) and Bmn(x) are symmetric and anti-symmetric
respectively and the first is required to be non-degenerate and therefore can be interpreted
as a metric on the D-dimensional background. For the applications we have in mind both
the 2d metric and the background metric Gmn will be taken to be Lorentzian. The
simplest example being Gmn = ηmn = diag(−1, 1, . . . , 1), the D-dimensional Minkowski
metric and Bmn = 0. In that case we have a theory of D free scalars which we can easily
solve. However, for general functions Gmn(x) and Bmn(x) we don’t know how to solve the
theory, particularly at the quantum level. But it turns out that there exist special choices
of Gmn and Bmn for which the theory can be essentially solved, in particular the energy
spectrum can be fully determined. This happens when the theory has hidden symmetries
which lead to infinitely many conserved charges. Such theories are called integrable.1
1
In general an integrable theory should have as many conservation laws as degrees of freedom. In our
case we are dealing with a field theory which has degrees of freedom living at each point in spacetime so
we need infinitely many conservation laws.
4
An example where this happens is when Gmn is the metric on a group manifold while
Bmn = 0, as we will see in the next chapter. In this case the model is known as the
Principal Chiral Model (PCM). Such integrable models are very useful since they are
solvable, at least in a certain sense, and therefore one can learn a lot about them and
they can serve as toy models for more complicated systems. However, finding models that
are integrable is very hard. A strategy which has proven very useful is to ask instead the
following question: suppose we have an integrable sigma model such as the PCM, is there
a way to modify it a little bit while still preserving the integrability of the model. This
turns out to be possible in many cases and the resulting model is then called an integrable
deformation of the original model. A large class of such integrable deformations are the
so-called Yang-Baxter (YB) deformations, which we will introduce in the next chapter
and which will be the main topic of this thesis.
The rest of the introductory part of this thesis is organized as follows. In chapter 2
we will introduce the YB deformation of the PCM. Then we will introduce an operation
known as non-abelian T-duality (NATD), which is a certain non-local field redefinition
in the sigma model. Finally we will see how the so-called homogeneous YB deformations
can be constructed using NATD. The PCM is not a string sigma model but it is useful as
a simple toy model to introduce the basic ideas.
In chapter 3 we will introduce the bosonic string and the requirement of Weyl invariance
which forces the background fields to solve a generalization of Einstein’s equations.
Then we will generalize the YB deformation to the case of the bosonic string. To verify
that the deformed model solves the Weyl invariance conditions directly turns out to be
very hard. For this reason a reformulation of these conditions, featuring a larger O(D, D)
symmetry, is introduced. This formulation, known as Double Field Theory (DFT), makes
it easy to see that the deformed model solves the correct equations at the classical level.
We then turn to the problem of including quantum corrections in the 2d theory, which
are parametrized by α , the inverse string tension. At one-loop order one finds that the
deformed background solves the correct equations if a certain algebraic condition, known
as the unimodularity condition, is fulfilled. Finally, in chapter 4 we analyze what happens
at the two-loop order, which corresponds to including the first α -correction in the
equations for the background. This means that the background should solve Einstein’s
equations with a correction involving the Riemann tensor squared. It turns out to be
possible to incorporate this correction in the DFT description, which makes it possible to
determine what happens to the YB deformation at this order. We will see that a certain
correction to the YB deformation is required to solve the equations at first order in α .
Finally we end with some conclusions and a list of important further developments.
5
Chapter 2
Yang-Baxter deformations and
non-abelian T-duality
In this chapter we will introduce the integrable Yang-Baxter deformation [1, 2] in the
setting of the Principal Chiral Model. We will then show that in the homogeneous case,
which we will define, this deformation can be constructed using a transformation known
as non-abelian T-duality [3].
2.1 Yang-Baxter deformations of the
Principal Chiral Model
Consider a 2d sigma model with target space a group manifold. This is known as the
Principal Chiral Model (PCM) and the action is
SPCM = d2
ξ tr(g−1
∂+gg−1
∂−g) = d2
ξ tr(∂+gg−1
∂−gg−1
) , (2.1)
for the field g(ξ) ∈ G valued in (some representation of) the group G. This model has
a global G × G symmetry corresponding to the fact that multiplying g by a constant
element of G from the left or from the right leaves the PCM action invariant. Because of
this the equations of motion of the model take the form of a conservation equation for a
symmetry current
∂+j− + ∂−j+ = 0 , (2.2)
where we have introduced the right-invariant Maurer-Cartan form valued in the Lie algebra
of G
j± = ∂±gg−1
. (2.3)
By construction j satisfies a flatness condition
∂+j− − ∂−j+ − [j+, j−] = 0 . (2.4)
A very important property of this model is that it is integrable, in particular it possesses
infinitely many conserved quantities. This follows from the fact that the equations of
6
motion can be expressed as the flatness of a one-parameter family of connections known
as a Lax connection,
∂+L− − ∂−L+ + [L+, L−] = 0 , (2.5)
where
L±(z) = −
1
1 ± z
j± , (2.6)
as is easily verified using the flatness of j. Here z ∈ C is an auxiliary parameter known
as the spectral parameter. We can now argue that this leads to infinitely many conserved
quantities as follows. For simplicity we take the theory to be defined on a compact spatial
circle. Computing the holonomy of the connection around this circle
M(z) = Pe S1 dξ1L1(z)
, (2.7)
one finds that it is conserved. To see this consider deforming the integration contour by
translating it forward in time by a little bit. The change in the holonomy is given by an
integral of the curvature of L, but this vanishes by the flatness condition (2.5). Therefore
the eigenvalues of M(z) are conserved and by Taylor expanding in z we obtain an infinite
set of conserved quantities. This demonstrates the integrability of the model.1
Now consider deforming the PCM action by introducing a constant operator acting
on the second current factor in the PCM Lagrangian in (2.1). Without loss of generality
we can write this operator as (1 + ηR)−1
where η is the deformation parameter. The
deformed action is
SYB = d2
ξ tr j+
1
1 + ηR
j− , (2.8)
where we take R : g → g to be an arbitrary constant linear operator on the Lie algebra of
G. The original PCM is clearly recovered when the deformation is removed, i.e. when we
take η → 0. For a general R this deformation will break the integrability of the model,
leading to a model over which one has much less control. However, it is interesting to
ask whether special choices of R exist which preserve the integrability of the model. To
analyze this question we look at the equations of motion of the deformed model. Let us
first note that the deformation we have introduced generically breaks the left G symmetry
but always preserves the right-acting copy of G since the action is written in terms of the
right-invariant current. Defining the deformed currents2
J+ =
1
1 + ηRT
∂+gg−1
, J− =
1
1 + ηR
∂−gg−1
, (2.9)
the equations of motion again take the form of a conservation equation
∂+
˜J− + ∂−
˜J+ = 0 , (2.10)
where ˜J± = Adg J± = g−1
J±g is the Noether current for the right-acting G symmetry.
We now want to ask when this model can be integrable. Let us assume that the Lax
connection can again be expressed in terms of the components of the conserved currents,
so that
L± = a−1
±
˜J± , (2.11)
1
Strictly speaking one should also show that they are independent and mutually Poisson commuting.
This requires a bit more work.
2
The transpose of R is defined in the standard way as tr(XR(Y )) = tr(RT
(X)Y ).
7
for some coefficients a± depending on η and the spectral parameter z. The flatness
condition for L then reads
a+∂+
˜J− − a−∂−
˜J+ + [ ˜J+, ˜J−] = 0 . (2.12)
From the flatness condition for j in (2.4) we can find a deformed flatness condition for ˜J
by writing j+ = Ad−1
g
˜J+ + ηRT
J+ and similarly for j− but with R instead of RT
. One
finds, after multiplying with Adg,
∂+
˜J− −∂−
˜J+ +[ ˜J+, ˜J−]+η Adg R∂+J− −η Adg RT
∂−J+ −η2
Adg[RT
J+, RJ−] = 0 . (2.13)
Using this equation to eliminate the commutator term in (2.12) we find, keeping terms to
first order in η,
(a+ − 1)∂+
˜J− − (a− − 1)∂−
˜J+ − η Adg R∂+j− + η Adg RT
∂−j+ + O(η2
) = 0 . (2.14)
Noting that a± = 1 ± z + O(η) 3
and using the equations of motion (2.10) the vanishing
of the terms linear in η implies
R + RT
+ (a+ + a−)1 [j+, j−] = 0 , (2.15)
where (· · · )1 denotes the term of first order in η. This condition says that the symmetric
part of R should be proportional to the identity operator. But it is evident from (2.8)
that a contribution to R proportional to the identity can be absorbed into a redefinition
of η and a rescaling of the action. Doing this we find that R should be anti-symmetric,
RT
= −R, and a+ + a− should not have a term linear in η. At the second order in η a
similar calculation gives the condition
[Rj+, Rj−] − R[Rj+, j−] − R[j+, Rj−] = −
1
2
(a+ + a−)2[j+, j−] . (2.16)
This requires that R satisfy the (modified) classical Yang-Baxter equation
[RX, RY ] − R([RX, Y ] + [X, RY ]) = c[X, Y ] ∀X, Y ∈ g , (2.17)
for some constant c, and that a+ + a− = 2 − 2cη2
+ O(η3
). The homogeneous case
c = 0 is the classical Yang-Baxter equation and the inhomogeneous case c = 0 is the
modified classical Yang-Baxter equation. When c = 0 we can rescale R to set c = ±1.
We have found that, at least assuming the Lax connection is expressed in terms of the
components of the conserved Noether current, R must be an anti-symmetric R-matrix
solving the (modified) classical Yang-Baxter equation. In fact these conditions on R are
also sufficient for the deformed model to be integrable, namely it is not hard to see that
flatness of the connection
L± =
1 + cη2
1 ± z
˜J± , (2.18)
is equivalent to the equations of motion of the deformed model (2.8). The deformed
flatness condition (2.13) is useful when checking this. Using the YB equation it takes the
simpler form
∂+
˜J− − ∂−
˜J+ + (1 + cη2
)[ ˜J+, ˜J−] = −ηRg(∂+
˜J− + ∂−
˜J+) , (2.19)
where we have defined Rg = Adg R Ad−1
g . Note that the RHS is proportional to the
equations of motion, (2.10).
3
The sign difference compared to (2.6) comes from the fact that for η = 0 the two Lax connections
differ by a gauge transformation.
8
2.2 Non-abelian T-duality for PCM
There exists a very interesting change of variables that one can perform for sigma models
like the PCM. This change of variables is actually non-local and leads to a model whose
Lagrangian looks completely different, but which is nevertheless (classically) equivalent to
the original model. This change of variables goes under the name of non-abelian T-duality
(NATD). The simplest example of NATD for the PCM goes as follows. We start with the
action for the PCM
S = d2
ξ tr(g−1
∂+gg−1
∂−g) . (2.20)
But now we rewrite this action in first order form as
SF = d2
ξ tr(A+A− + νF+−(A)) , (2.21)
where A is a gauge field and F(A) is its field strength, i.e. F+−(A) = ∂+A− − ∂−A+ +
[A+, A−]. The field ν ∈ g is a Lagrange multiplier enforcing the constraint that the field
strength of A vanish. The general solution to this constraint is that A is pure gauge, i.e.
(locally) A± = g−1
∂±g, and plugging this into the action gives back the original action
(2.20). This shows that the two models are equivalent. The remarkable thing is now that
there is another way to get a second order action from (2.21). Noting that the action is
actually quadratic in A we may instead integrate out A. The equations of motion for A
give
A± ∂±ν ± [ν, A±] = 0 ⇒ Ar
± = ± (1 ± ν · f)−1 r
s∂±νs
, (2.22)
where (ν · f)r
s = νt
fts
r
and we have written A = Ar
Tr, ν = νr
Tr with Tr generators of g
satisfying [Tr, Ts] = frs
t
Tt and tr(TrTs) = κrs with κrs non-degenerate but not necessarily
positive(negative) definite. Plugging this solution for A± into the action (2.21) we obtain
the NATD action
SNATD = d2
ξ ∂+νr κ
1 − ν · f rs
∂−νs
. (2.23)
Example. The simplest example is when G is abelian (in this case we are really dealing
with abelian T-duality). Let us take it to be two-dimensional with g generated by T1, T2.
Since the structure constants vanish the duality becomes trivial in this case, taking g =
exrTr
we have
S = d2
ξ ∂+xr
∂−xs
κrs , → SNATD = d2
ξ ∂+νr
∂−νs
κrs . (2.24)
The two models are clearly the same, at least locally. Note that global properties of the
original model are lost under NATD. For example if we started from a torus where x1,2
have finite range the duality process does not tell us what the range of ν1,2
is, it could be
taken finite or infinite. The models could only be globally equivalent in the former case.
We can make this example more interesting by noting that we can add a total derivative
term to the original action before dualizing. Following the same steps are before one then
finds
S = d2
ξ ∂+xr
∂−xs
(κrs + ζεrs) , → SNATD = d2
ξ ∂+νr κ
1 + ζκ−1ε rs
∂−νs
.
(2.25)
9
We have introduced the parameter ζ multiplying the total derivative. Taking ζ → 0 gives
back the previous case. Note that the term we have added is locally a total derivative,
but is only a total derivative globally if the x1
or x2
-direction is not compact. The NATD
model simplifies to
SNATD =
det κ
det κ + ζ2
d2
ξ ∂+νr
∂−νs
(κrs − ζεrs) , (2.26)
which, upon rescaling νr
, is again locally equivalent to the original model. While this
example is still too trivial to be really interesting it does suggest that including the possibility
of adding total derivatives to the action before dualizing could lead to interesting
effects. We will come back to this after considering more general NATD on a subgroup.
2.3 NATD with respect to a subgroup
We started with a model with a global symmetry group G × G and ended up with a
dual model whose symmetry group has only one factor of G.4
In fact NATD typically
breaks some of the global symmetry. What we considered so far is the simplest example
of NATD, but a NATD model can be constructed for any subgroup of the isometry group
G × G. The example we considered above corresponds to dualizing with respect to the
left copy of G ⊂ G × G. Let us now describe how to dualize with respect to a subgroup
K of the left copy of G, K ⊂ G ⊂ G × G.5
We start by writing the group element as g = kf with k ∈ K and f ∈ G so that
g−1
dg = Adf (k−1
dk) + f−1
df . (2.27)
To avoid a redundant description we take k to depend on the coordinates we want to
dualize x and f to depend on the remaining ’spectator’ coordinates y. Setting j = dff−1
the first order action (2.21) becomes
SF = d2
ξ tr((A+ + j+)(A− + j−) + νF+−(A)) , A± ∈ k . (2.28)
As before, integrating out ν implies that A is pure gauge, i.e. A = k−1
dk for some k ∈ K,
recovering the original PCM action. Since A ∈ k it follows that ν should belong to its
dual (with respect to the metric on g), denoted k∗
. If k is generated by {Tr} then k∗
is
generated by {Tr
} with tr(Tr
Ts) = δr
s. The equations of motion for A± imply that
A+ = (O−1
)T
(∂+ν − PT
j+) , A− = O−1
(−∂−ν − PT
j−) , (2.29)
where
O = PT
(1 − adν)P . (2.30)
Here P is a projection operator from g down to k. From the definition of the transpose,
tr(XP(Y )) = tr(PT
(X)Y ), it follows that PT
projects on the dual Lie algebra k∗
. Note
4
It acts as ν → Adg(ν) = g−1
νg with g a constant element of the group G.
5
More generally one could consider a subgroup of the full G×G, but this is more conveniently described
using a (G × G)/G coset sigma model.
10
also that the definition of the adjoint action, adν X = [ν, X], implies that adT
ν = − adν.
The inverse of O refers to the subspace where it is defined, in particular we have O−1
O = P
and OO−1
= PT
. This (partial) inverse must exist for the dual model to be well-defined.
Substituting the solution for A± into the Lagrangian gives the NATD model
SNATD = d2
ξ tr j+j− + (∂+ν − j+)O−1
(∂−ν + j−) . (2.31)
For K = G the j’s are absent and this reduces to our previous result (2.23).
Now we are ready to generalize this by including total derivative terms before dualization.
It might seem that this cannot lead to anything interesting but as we will see,
due to the fact that NATD is a non-local field redefinition, this can lead to dual models
which are different even locally.
2.4 YB deformations from NATD
When we considered the example of abelian T-duality in the previous section we noted
that we could include a total derivative term in the action before dualizing. We will now
analyze this possibility more carefully and see that it leads to a surprising connection to
Yang-Baxter models. We therefore generalize our starting point to
S = d2
ξ tr(g−1
∂+gg−1
∂−g) − ζω(g−1
∂+g, g−1
∂−g) , (2.32)
where ζ is a free parameter and ω : g ⊕ g → R is anti-symmetric and linear in the two
arguments. In order not to change the local physics of the original model we require the
extra term to be a total derivative (locally). Thinking of this term as a two-form the
condition is that it should be closed, i.e.
0 = dω(g−1
dg, g−1
dg) = −2ω(g−1
dg ∧ g−1
dg, g−1
dg) , (2.33)
which requires that
ω([X, Y ], Z) + ω([Y, Z], X) + ω([Z, X], Y ) = 0 ∀X, Y, Z ∈ g . (2.34)
This is in fact the definition of a Lie algebra 2-cocycle. We therefore learn that it is
possible to modify our starting point in this way whenever such a 2-cocycle exists. We
may write ω(Tr, Ts) = ωrs and in terms of these components the 2-cocycle condition reads
ωr[sftu]
r
= 0 . (2.35)
Suppose we have a subgroup K ⊂ G which admits a 2-cocycle ω. We can then add the
term
− ζ ω(k−1
dk, k−1
dk) , (2.36)
to the PCM action, or, equivalently, the term
− ζ ω(A, A) = −ζ ωrsAr
∧ As
, (2.37)
11
to the first order action (2.28). We can now perform the NATD on K following the same
steps as in section 2.3. The result is the “Deformed T-Dual” (DTD) action
SDTD = d2
ξ tr j+j− + (∂+ν − j+)O−1
(∂−ν + j−) , (2.38)
where now6
O = PT
(1 − adν +ζω)P . (2.39)
Setting the parameter ζ to zero we recover the NATD action (2.31). We have therefore
constructed a deformation of the NATD action controlled by the deformation parameter
ζ. However, it may happen that this is not an actual deformation but just a rewriting
of the NATD action. This happens if it is possible to absorb the ζω piece into adν by a
shift in ν. This in turn is possible precisely if ω takes the form ωrs = Xtfrs
t
for some Xt,
i.e. if the 2-cocycle ω is exact. Therefore the non-trivial deformations are characterized
by 2-cocycles modulo exact ones, i.e. by elements of the second Lie algebra cohomology
group H2
(k). A standard result in Lie algebra theory is that elements of H2
(k) classify
non-trivial central extensions of k. Indeed, an equivalent way of arriving at (2.38) is to
perform NATD on a central extension of k, characterized by ω, as originally suggested in
[4]. Let us also note that since we started from the PCM, which is integrable, the model in
(2.38) will also be integrable. This follows from the fact that adding the cocycle (closed Bfield)
term does not affect the local physics of the model, while the NATD transformation
is a canonical transformation [5] and so must preserve the property of integrability.
So far this DTD model looks nothing like the YB deformation in (2.8). However, if
ω is non-degenerate they are essentially the same, as we will now show. A Lie algebra
admitting a non-degenerate 2-cocycle is called quasi-Frobenius (or symplectic). Therefore
we will now consider the case where g has a quasi-Frobenius subalgebra k ⊂ g generated
by {Tr} with 2-cocycle ω. In this case ωrs is invertible as a matrix and we set R = ω−1
.
By multiplying the free indices of the 2-cocycle condition (2.35) by R’s we find that R
satisfies the equation
R[r|u|
Rs|v|
fuv
t]
= 0 . (2.40)
This is nothing but the component form of the classical Yang-Baxter equation (2.17)
with zero RHS (c = 0). Furthermore, R trivially extends from the subalgebra k to all
of g by taking its remaining components to vanish. Therefore we have learned that nondegenerate
2-cocycles on a Lie algebra are in one-to-one correspondence with R-matrices
solving the (non-modified) classical Yang-Baxter equation. This observation suggests a
possible connection between the DTD model (2.38) and the YB deformation of the PCM
(2.8). However, it is not straightforward to identify them as the former involves fields
f ∈ G (the spectators) and ν ∈ k∗
while the latter involves only g ∈ G. To relate them we
need to somehow re-express the Lie algebra valued field ν in terms of a group valued field
h and then identify g with hf. It is not too hard to find a map that does the job working
to the first few orders in ν. With a bit of work one finds that the correct all order map
6
Note that we are thinking here of ω as a map from the Lie algebra k to its dual k∗
by setting
tr(Y ω(X)) = ω(X, Y ) ∀X, Y ∈ k.
12
is essentially given by the derivative of the exponential map, namely
ν = − ζPT 1 − Ad−1
h
log Adh
ω(log h) = −ζPT 1 − e− adX
adX
ω(X) (2.41)
= − ζPT
ω(X) − 1
2
[X, ω(X)] + 1
6
[X, [X, ω(X)]] + . . . ,
where we have written h = eX
and expanded in powers of X ∈ k.
We will now show that performing this field redefinition in the DTD action (2.38)
indeed leads to the YB action (2.8). The first step is to understand how ∂±ν and adν
transform. To do this we note that the 2-cocycle condition on ω : k → k∗
implies that7
ω([X, Y ]) = PT
[ω(X), Y ] + PT
[X, ω(Y )] , X, Y ∈ k . (2.42)
Up to the projection by PT
this says that ω acts as a derivation with respect to the Lie
bracket. If we formally extend the action of ω to act as a derivation also on the universal
enveloping algebra, not just the Lie algebra, we may write (2.41) more simply as8
ν = −ζPT
(e−X
ω(eX
)) = −ζPT
(h−1
ω(h)) , (2.43)
which is not hard to verify for the first few terms in the expansion of the exponential
using the cocycle condition on ω. Using this expression we find
dν = ζPT
(h−1
dhh−1
ω(h)) − ζPT
(h−1
ω(dh)) = −ζω(h−1
dh) + ζPT
[h−1
dh, h−1
ω(h)]
= (PT
− O)(h−1
dh) . (2.44)
Furthermore we have for Y ∈ k
PT
adν Y = −ζPT
[PT
(h−1
ω(h)), Y ] = −ζPT
[h−1
ω(h), Y ] = −ζPT
(ωh − ω)Y , (2.45)
where ωh = Adh ω Ad−1
h . Using this we can write
O−1
= [PT
(1 + ηRh)ζωh]−1
= ηRh[PT
(1 + ηRh)]−1
= ηRh(1 + ηRh)−1
= 1 − (1 + ηRh)−1
,
(2.46)
where η = ζ−1
, Rh : k∗
→ k is the inverse of ωh and we used the fact that Rh(1−PT
) = 0.
Using these facts we find that the DTD Lagrangian in (2.38) becomes
tr (h−1
∂+h + j+)(1 + ηRh)−1
(h−1
∂−h + j−) − ζh−1
∂+hωh(h−1
∂−h)
= tr ∂+gg−1 1
1 + ηR
∂−gg−1
+ η−1
ω(∂+hh−1
, ∂−hh−1
) , (2.47)
7
This follows from the fact that for any Z ∈ k we have
tr(ω([X, Y ])Z) = ω([X, Y ], Z) = −ω([Y, Z], X) − ω([Z, X], Y ) = − tr(ω([Y, Z])X) − tr(ω([Z, X])Y )
= tr([Y, Z]ω(X)) + tr([Z, X]ω(Y )) = − tr(Z[Y, ω(X)]) + tr(Z[X, ω(Y )]) .
8
This works in spite of the extra PT
on the RHS of (2.42) because of the fact that PT
[PX, (1−PT
)Y ] =
0 for all X, Y ∈ g. This in turn follows by noting that for any Z ∈ g we have
tr(ZPT
[PX, (1 − PT
)Y ]) = tr([PZ, PX](1 − PT
)Y ) = tr((1 − P)[PZ, PX]Y ) = 0 ,
since P(g) = k is a subalgebra.
13
where g = hf. The first term on the RHS is precisely the Lagrangian of the YB model
(2.8). The second term represents a closed B-field. Since this term is locally a total
derivative it does not affect the local physics and can be dropped for most purposes.
Therefore we have shown that
SPCM − ζ ω(k−1
dk, k−1
dk)
NATD on K
−−−−−−−−−→ SYB + η−1
ω(dhh−1
, dhh−1
) ,
(2.48)
with the parameters related by ζ = η−1
. Since the local physics is independent of the extra
ω terms we have shown that homogeneous YB deformations, i.e. those with R matrix
solving the classical YB equation (2.17) with c = 0, can be constructed as a NATD of the
PCM. This observation has some important consequences, in particular
ˆ Integrability of the YB deformation is now obvious since NATD preserves integra-
bility.
ˆ Since NATD can be performed for general 2d sigma models with isometries we can
use it to give a more general definition of homogeneous YB deformations.
ˆ We can classify the possible deformations by classifying quasi-Frobenius subalgebras
of the isometry algebra.
Regarding the last point of classification there are several useful mathematical results. In
particular a standard result is that a quasi-Frobenius subalgebra of a compact Lie algebra
must be abelian. In this case, when the subalgebra that is dualized is abelian, one can
show that the YB deformation coincides with another deformation that has been studied a
lot in the string theory literature and that goes by the name of T-duality–shift–T-duality
(TsT) [6]. Therefore another way of thinking about (homogeneous) YB deformations is
as a non-abelian generalization of TsT transformations.
14
Chapter 3
String sigma model and Weyl
invariance conditions
So far we have dealt with the PCM because it offers a simple setting to introduce the
ideas of NATD and YB deformations. But as we have emphasized this is only a toy
model. The PCM does not represent a consistent string sigma model since it fails to be
conformal invariant at the quantum level. In string theory we start from a general model
of 2d scalars coupled to 2d gravity1
of the form (for simplicity we will consider only the
bosonic string here)
S = −
1
4πα
d2
ξ
√
−g ∂ixm
∂jxn
(gij
Gmn + εij
Bmn) , (3.1)
where Gmn = Gnm and Bmn = −Bnm are functions of xm
while gij(ξ) is the 2d worldsheet
metric. The factor in front of the action T = 1
4πα
is the string tension. This action
is manifestly invariant under 2d diffeomorphisms. This gauge invariance can be used to
eliminate two of the three components of the worldsheet metric gij. A very important
fact about the above action is that it has another gauge symmetry which can eliminate
also the last component of gij, so that the worldsheet metric is not dynamical. This is
the invariance under Weyl transformations
gij → eλ
gij , (3.2)
with λ an arbitrary function of the worldsheet coordinates ξi
. A cosmological constant
term in the action would break this symmetry, which is why we did not add such a term
in (3.1). The crucial point is that for the action (3.1) to make sense also at the quantum
level we must make sure that the quantum theory remains invariant under the Weyl
transformation (3.2). Using the diffeomorphisms to gauge fix gij = eσ
ηij, with ηij the 2d
Minkowski metric, this turns into conformal symmetry of the 2d theory. In particular this
implies that the theory looks the same on all length (or energy) scales. This requirement
is very restrictive. Taylor expanding the functions Gmn and Bmn we have a Lagrangian
with infinitely many couplings, corresponding to the coefficients in the Taylor expansion.
But in a quantum field theory all couplings will typically depend on the energy scale at
which we probe the theory. This running of the couplings is captured by their so-called
1
The usual Einstein-Hilbert term is a total derivative in two dimensions so we don’t include it.
15
β-functions. But, since energy scale is inversely related to distance scale, having couplings
that depend on the energy scale is not consistent with conformal symmetry. Therefore in
a conformal theory all the β-functions must vanish, which in our case leads to infinitely
many constraints — one for each coefficient in the Taylor expansion of Gmn and Bmn.
These conditions can be summarized as differential equations that Gmn and Bmn must
satisfy. In fact, this turns out to not quite be enough. One must also modify the original
action by a quantum counter term involving a new (scalar) function of x — the dilaton
Φ. The quantum corrected action takes the form
S = −
1
4πα
d2
ξ
√
−g ∂ixm
∂jxn
(gij
Gmn + εij
Bmn) + α R(2)
Φ , (3.3)
where R(2)
is the Ricci scalar of the worldsheet metric gij. Note that the term we added is
not Weyl invariant (unless Φ is a constant). It is precisely its non-zero Weyl transformation
that compensates the non-invariance of the first term at the quantum level. A reflection
of this is that the dilaton term comes with an extra factor of α , the quantum mechanical
expansion parameter in this model. A quantum mechanical calculation, e.g. [7], shows
that this model preserves Weyl invariance at the quantum level provided that the following
β-function equations are satisfied
0 = Rmn + 2 m nΦ −
1
4
HmopHn
op
+ O(α ) , (3.4)
0 = m
Hmno − 2 m
ΦHmno + O(α ) , (3.5)
0 = 2
Φ − 2 mΦ m
Φ +
1
12
HmnoHmno
+ O(α ) , (3.6)
where H is the field strength of B, i.e. Hmno = 3∂[mBno] and Rmn is the Ricci tensor constructed
from the metric Gmn. In addition one finds that the dimension of spacetime must
take the precise value D = 26, referred to as the critical dimension of the bosonic string.
Remarkably, we recognize the first equation to be Einstein’s equation (with particular
matter sources given by Φ and Bmn). Therefore string theory contains Einstein’s general
relativity in a quantum mechanically consistent framework. The presence of higher
quantum corrections, the O(α ) terms, shows that string theory gives Einstein’s theory
plus corrections. These α -corrections will be important for us later. These β-function
equations provide the dynamical equations of motion for the background fields Gmn, Bmn
and Φ. They can be summarized in an action for the spacetime fields which takes the
form
S =
1
2κ2
0
d26
x
√
−Ge−2Φ
R −
1
12
HmnoHmno
+ 4∂mΦ∂m
Φ + O(α ) . (3.7)
This is the low-energy effective action for the bosonic string. In fact, the more realistic
supersymmetric string theories have the same terms in their low-energy effective actions
(although in that case the spacetime dimension is D = 10), plus additional terms involving
other fields not present for the bosonic string. In addition to the α -corrections there are
also string loop corrections organized in powers of eΦ
. We will not have anything to say
about those here.
16
3.1 Homogeneous Yang-Baxter deformations
We have seen in the previous chapter how, in the case of the PCM, homogeneous YB
deformations can be constructed using non-abelian T-duality. But NATD can be carried
out for a general 2d sigma model with isometries and therefore this construction is not
limited to the PCM. At the classical level NATD can be shown to act as a canonical
transformation, which means that it gives rise to a classically equivalent model. Adding
the closed B-field term represented by the 2-cocycle on the subalgebra of the isometry
algebra we are dualizing we obtain a YB deformed model. Applying this procedure to the
classical bosonic string action (3.1) we obtain a new, classically equivalent model, which
is a deformation of the one we started with. Let us call the new metric and B-field ˜G and
˜B. With a little bit of work one finds that they are related to the original ones by the
equation
˜G + ˜B = (G + B)(1 + ηΘ(G + B))−1
, (3.8)
where we are suppressing the indices m, n and we have introduced the object
Θmn
= km
r kn
s Rrs
, (3.9)
where km
r are the components of the Killing vectors corresponding to the isometries dualized.
As before Rrs
is the R-matrix satisfying the CYBE, (2.17) with c = 0, and η is
the deformation parameter. Let us show that this reproduces what we found in the case
of the PCM. Writing the principal chiral model action in terms of G and B we read off
Gmn = tr(g−1
∂mgg−1
∂ng) , Bmn = 0 . (3.10)
The isometries we are dualizing are of the form g → hg for a constant h ∈ K ⊂ G. The
corresponding Killing vectors are read off from
g−1
δg = g−1
g = Adg = δxm
g−1
∂mg , δxm
= r
km
r , (3.11)
and we find that
kr
m
(∂mgg−1
)s
= δs
r . (3.12)
Using this fact (3.8) becomes
( ˜G + ˜B)mn = tr(g−1
∂mgg−1
∂ng) − η tr(g−1
∂mgg−1
∂og)Θop
tr(g−1
∂pgg−1
∂ng) + . . .
= tr(g−1
∂mgg−1
∂ng) − η tr(∂mgg−1
R∂ngg−1
) + . . .
= tr(∂mgg−1 1
1 + ηR
∂ngg−1
) , (3.13)
in agreement with (2.8).
So far we considered only what happens at the level of the classical sigma model.
In order to define the YB deformation in string theory we have to understand it at the
quantum level. It is important to note first of all that, while at the classical level NATD
is a symmetry of the theory, this can no longer be the case at the quantum level. At best
NATD can be a map between in-equivalent string CFTs [8]. If this is the case NATD
or YB deformations constructed from it would map solutions of the β-function equations
(3.4)–(3.6) to new solutions. In particular we must be able to find also a formula for the
17
dilaton ˜Φ such that the fields of the deformed background solve the same equations. As a
first step we will neglect the O(α ) terms, which corresponds to working to one loop order
in sigma model perturbation theory. The most straightforward approach would be to
simply plug our expressions for the deformed metric and B-field in (3.8) into (3.4)–(3.6)
and try to find a ˜Φ such that they are satisfied. However, this is very difficult to do in
practice due to the non-linearity of the map in (3.8). Just trying to compute the Riemann
tensor from ˜G is quite a bit of work. Of course one can work out the first few orders in
the η-expansion, but this is not very satisfactory. It turns out that a much more efficient
way to address these questions is to use a reformulation of the string low-energy effective
action which linearizes the map (3.8).
3.2 O(D, D) invariant formulation of string effective
action
The YB map (3.8) is complicated and non-linear which makes it hard to work with for
the purposes we are interested in. Let us see if we can use different variables to make the
map simpler. To simplify things we will start by assuming that B = 0. In the following
we will also absorb the deformation parameter η into the definition of Θ. Separating (3.8)
into the symmetric and anti-symmetric part we then have
˜G = G(1 − Θ2
G)−1
, ˜B = −GΘG(1 − Θ2
G)−1
, (3.14)
where Θ2
= ΘGΘ. From these expressions we see that
˜G−1
= G−1
− Θ2
, ˜G − ˜B ˜G−1 ˜B = G , ˜B ˜G−1
= −GΘ . (3.15)
These expressions look much more promising. Allowing for non-zero B they become
˜G−1
= G−1
+ G−1
BΘ + ΘBG−1
− Θ(G − BG−1
B)Θ , ˜G − ˜B ˜G−1 ˜B = G − BG−1
B ,
˜B ˜G−1
= BG−1
− (G − BG−1
B)Θ . (3.16)
From this we conclude that if we could use instead of G and B two of the variables G−1
,
G−BG−1
B and BG−1
the map (3.8) would become linear. We can group these variables
into a symmetric 2D × 2D-matrix
HMN
=
(G − BG−1
B)mn (BG−1
)m
n
−(G−1
B)m
n (G−1
)mn , (3.17)
in terms of which the YB deformation (3.8) becomes simply
˜HMN
= OP
M
OQ
N
HPQ
, OM
N
=
δm
n Θmn
0 δn
m
. (3.18)
In fact the matrix OM
N
belongs to the group O(D, D) since it preserves the split signature
metric
ηMN =
0 δm
n
δn
m 0
, OM
P
ON
Q
ηPQ = ηMN . (3.19)
18
In fact it is easy to see that HMN
(or rather HM
N = HMP
ηPN ) itself belongs to O(D, D)!
Therefore there is a natural action of O(D, D) when we group the metric and B-field into
the ”generalized metric” H given by
H → OT
HO O ∈ O(D, D) . (3.20)
We see that if we could reformulate the low-energy effective action (3.7) for the string in
terms of H, rather that G and B, it would be much simpler to check that the equations of
motion are preserved by the YB map (3.18). Remarkably it turns out that this is indeed
possible.
To explain this we must go back to the notion of (abelian) T-duality. When string
theory is compactified on an n-dimensional torus, Tn
, T-dualities on the torus directions
lead to an O(n, n; Z) symmetry group (e.g. [7]). If we restrict our attention to the massless
fields, as in the low-energy effective action, this becomes and O(n, n; R) symmetry [9].
This symmetry is there in tree-level string theory (i.e. ignoring string loop corrections)
but to all orders in the inverse string tension α [10]. It has been suggested that if one
could formulate a theory in D dimensions (recall that D = 26 for the bosonic string) with
O(D, D) symmetry its compactification on Tn
would automatically have the T-duality
symmetry. In this way the T-duality symmetry of string theory would be made manifest.
This idea goes by the name of Double Field Theory (DFT) [11, 12, 13]. The reason for
the name is that to have manifest O(D, D) symmetry one must double the number of
spacetime dimensions, replacing xm
by an O(D, D) vector XM
= (˜xm, xm
). However, the
physical spacetime is still supposed to be D-dimensional and this is ensured by imposing
the O(D, D) invariant section condition (a.k.a. strong constraint)
∂M Y ∂M
Z = 0 , ∂M ∂M
Z = 0 , (3.21)
for all Y, Z built out of fields of the theory. Note that doubled indices M, N, . . . are raised
and lowered with the O(D, D) metric η (3.19). The standard solution to this constraint
is to simply set ˜xm = 0 so that effectively ∂M = (0, ∂m). In DFT the metric and B-field
are combined precisely as we argued above into the generalized metric HMN
(3.17). This
is quite natural since as we saw O(D, D) acts in a natural way on this object. In the
low energy description of string theory there is also the dilaton and in DFT it appears
combined with the determinant of the metric as the generalized dilaton d (c.f. (3.7))
e−2d
=
√
−Ge−2Φ
. (3.22)
The string low energy effective Lagrangian (3.7), at lowest order in α , can be written in
terms of these fields as
L = e−2d
R , (3.23)
where the generalized Ricci scalar is defined as
R = 4∂M (HMN
∂N d) − ∂M ∂N HMN
− 4HMN
∂M d∂N d +
1
8
HMN
∂M HKL∂N HKL
−
1
2
HMN
∂M HKL
∂KHLN . (3.24)
Imposing the standard solution to the section condition ∂M = (0, ∂m) this reduces to
L =
√
−Ge−2Φ
R −
1
12
HmnoHmno
+ 4∂mΦ∂m
Φ + 4∂m
√
−Ge−2Φ
∂m
Φ , (3.25)
19
which, upon dropping the total derivative term, reproduces precisely the Lagrangian in
(3.7). The fact that the low energy dynamics of the bosonic string can be cast in the form
of (3.23) is remarkable – the global O(D, D) symmetry is completely unexpected. This
symmetry acts as
XM
→ X M
= XN
ON
M
, HMN
(X) → OP
M
OQ
N
HPQ
(X ) , d(X) → d(X ) ,
(3.26)
for OM
N
a constant element of O(D, D). Of course this is in the doubled space, before
solving the section condition. But even after solving the section condition there is still an
extra global symmetry that remains, namely the transformations above which preserve
the choice XM
= (0, xm
). These transformations are of the form
1 Θ
0 1
, (3.27)
i.e. of the same form as we found for YB in (3.18), but now with Θ an arbitrary constant
anti-symmetric matrix. Note that this implies a very non-obvious symmetry of the lowenergy
effective action (3.7), namely that it is invariant under the YB map (3.8) (plus
transformation of the dilaton read off from the fact that d = Φ − 1
2
log
√
−G is invariant)
but with Θ any constant anti-symmetric matrix.
This formulation has already taught us something interesting, but to understand how
YB deformations preserve the β-function equations we must do a little more work. The
reason is that for YB deformations Θ is not constant, so they are not simply a global
O(D, D) transformation. In fact Θ is constructed out of Killing vectors of the background
in question and so depends on the background we are deforming. For this reason it cannot
be a symmetry of the action. Instead it should map solutions of the equations of motion
(β-function equations) into new solutions. To see how this happens, and also to address
what happens when α -corrections are included, it turns out to be convenient to work
with a generalized vielbein rather than a generalized metric. We will now introduce this
“frame-like” formulation of DFT.
3.3 Frame-like formulation of DFT
We are familiar with the fact that we can formulate ordinary Riemannian geometry either
in terms of the metric or in terms of the vielbein. The fact that the vielbein contains
more degrees of freedom than the metric is compensated by an extra gauge invariance,
invariance under local Lorentz transformations with gauge group O(D − 1, 1). In analogy
with this we introduce a generalized vielbein for the generalized metric
HMN
= EA
M
EB
N
HAB
. (3.28)
The “flat” metric HAB
is the analog of the Minkowski metric in the ordinary Riemannian
case and since we are just doubling the dimension we take it to have the form
HAB
=
¯ηab 0
0 ¯ηab , (3.29)
20
where ¯η = diag(−1, 1, . . . , 1) is the D-dimensional Minkowski metric. But we must remember
that we also have another metric, namely the constant O(D, D) metric ηMN
(ηMN ) in (3.19). We will demand that
ηAB = EA
M
EB
N
ηMN =
¯ηab
0
0 −¯ηab
. (3.30)
The point of this constraint is that the analog of the local Lorentz symmetry in the
standard case is EA
M
→ ΛA
B
EB
M
where ΛA
B
must preserve both HAB
in (3.29) and ηAB
in (3.30) which means that it reduces to two copies of the Lorentz group, i.e.
ΛA
B
=
(Λ(+)
)a
b 0
0 (Λ(−)
)a
b , (3.31)
with Λ(±)
two independent D-dimensional Lorentz transformations. It is now easy to see
that we may take the generalized vielbein to have the form
EA
M
=
1
√
2
e(+)a
m − e(+)an
Bnm e(+)am
−e
(−)
am − e
(−)n
a Bnm e
(−)m
a
. (3.32)
Here the two sets of vielbeins e(±)
, with e
(±)m
a e
(±)n
b ηab
= Gmn
, transform under the two
copies of the Lorentz group as e
(±)m
a → Λ
(±)b
a e
(±)m
b . We can use the double Lorentz
symmetry to fix the gauge e(+)
= e(−)
= e. With this gauge fixing only the diagonal
of the double Lorentz group, Λ(+)
= Λ(−)
= Λ, survives and becomes the usual Lorentz
group. This gauge fixing is needed whenever we want to make contact with the usual
gravity description which has only a single copy of the Lorentz group.
In ordinary Riemannian geometry the vielbein transforms under both diffeomorphisms
and local Lorentz transformations. But we are familiar with the fact that we can work with
objects which transform as scalars under diffeomorphisms, but transform non-trivially
under local Lorentz transformations, namely the spin connection components ωc
ab
=
ec
m
ωm
ab
and derivatives with ’flat’ indices ∂a = ea
m
∂m. In fact, the spin connection is the
only diffeomorphism scalar that can be constructed from the derivative of the vielbein.2
In the present case we have a similar situation. The generalized vielbein transforms under
both generalized diffeomorphisms and double Lorentz transformations. The generalized
diffeomorphisms act on a doubled vector field by the generalized Lie derivative
V M
→ L V M
= N
∂N V M
+ (∂M
N − ∂N
M
)V N
. (3.33)
Looking at the transformation of the generalized vielbein one sees that (with the standard
solution of the section condition) m
is identified with the diffeomorphism and m with the
B-field gauge parameters, δBmn = 2∂[m n]. The analog of the spin connection in this case
would be the objects constructed from the derivative of the generalized vielbein which are
generalized diffeomorphism scalars. It is not hard to see that the only combination that
is a generalized diffeomorphism scalar is
FABC = 3∂[AEB
M
EC]M , (3.34)
2
Since em
a
(x) → em
a
(x + ) − ∂m
n
en
a
under an infinitesimal diffeomorphism the requirement that
cc
ab
ea
m
eb
n
∂nem
c
transform as a scalar becomes cc
ab
ea
m
eb
n
∂n∂m
k
ek
c
= 0 which requires cc
ab
= cc
[ab]
leading to the spin connection.
21
where we have defined ∂A = EA
M
∂M . Note that FABC is also manifestly invariant under
O(D, D), since it just rotates the M-indices by a constant O(D, D) matrix. Of course we
also have the generalized dilaton, which is not a scalar due to the
√
−G in (3.22), but its
derivative can be combined with the derivative of the generalized vielbein to produce a
second generalized diffeomorphism scalar
FA = ∂B
EB
M
EAM + 2∂Ad . (3.35)
The objects FABC and FA are usually referred to as generalized fluxes, and this formulation
of DFT is referred to as the flux formulation [14]. The advantage of working with the
generalized fluxes is that both diffeomorphisms and B-field gauge transformations as well
as global O(D, D) symmetry is now manifest. The price we pay for this is that the
double Lorentz transformations are not manifest. Instead the generalized fluxes transform
similarly to connections as
δFABC = 3∂[AλBC] + 3λ[A
D
FBC]D , δFA = ∂B
λBA + λA
B
FB , (3.36)
under an infinitesimal transformation δEA
M
= λA
B
EB
M
. In the usual Riemannian case we
would deal with this by constructing the gauge covariant curvature of the spin connection,
i.e. the Riemann tensor, and writing actions in terms of that. In that way all the
symmetries can be made manifest. This is not possible in the present case. The reason is
that there is no analog of the Riemann tensor. For example, looking at the transformations
it seems that one should consider the field strength 4∂[AFBCD], which is invariant to leading
order in fields. However, from the definition of FABC in terms of the generalized vielbein
we have the Bianchi identity
4∂[AFBCD] = 3F[AB|
E
FCD]E , (3.37)
so this would-be field strength is not an independent field. Similarly we have
2∂[AFB] = −(∂C
− FC
)FABC , 2∂[A∂B] = FABC∂C
. (3.38)
The absence of a Riemann tensor makes it more difficult to construct an action (and
especially higher derivative corrections [15]). However, we will see that an action can
nevertheless be constructed without too much effort. Before we do this we need one more
fact. The presence of the constant O(D, D) metric ηAB
in (3.30) as well as the doubled
metric HAB
in (3.29), which both square to one, means that we can construct constant
projection operators
(P±)AB
=
1
2
(η ± H)AB
. (3.39)
This means that we have a canonical splitting of the index A = (a, a), where the Ddimensional
index a(a) corresponds to P+(P−) projection. This means that the two
generalized fluxes really contain six fields
Fabc , Fabc , Fabc , Fabc , Fa , Fa . (3.40)
Their double Lorentz transformations are
δFabc =3∂[aλbc] + 3λ[a
d
Fbc]d ,
δFabc =∂aλbc + λa
d
Fdbc + 2λ[b
d
F|ad|c] ,
δFa =∂b
λba + λa
b
Fb ,
δFabc =3∂[aλbc] + 3λ[a
d
Fbc]d ,
δFabc =∂aλbc + λa
d
Fdbc + 2λ[b
d
F|ad|c] ,
δFa =∂b
λba + λa
b
Fb ,
(3.41)
22
where λab(λab) are the non-zero components λ(+)
(λ(−)
) of λAB. The only two-derivative
action we can construct using the fields in (3.40) consists of their squares plus the total
derivative terms ∂a
Fa and ∂a
Fa. It is not hard to check that the only combination invariant
under the double Lorentz transformations is3
R = 4∂a
Fa − 2Fa
Fa + Fabc
Fabc + 1
3
Fabc
Fabc . (3.42)
With a bit of work one can rewrite this in terms of the generalized metric and dilaton
and show that it coincides with our previous expression for the generalized Ricci scalar
(3.24). Therefore the Lagrangian
L = e−2d
R , (3.43)
reproduces again (after solving the section condition in the standard way and fixing the
gauge e(+)
= e(−)
) the low energy effective Lagrangian of the bosonic string in (3.7). The
equations of motion following from this action are easily derived using the variation of
the generalized fluxes
δFABC = 3∂[AδEBC] + 3δE[A
D
FBC]D δFA = ∂B
δEBA + δEA
B
FB + 2∂Aδd , (3.44)
where we have defined δEAB = δEA
M
EBM = δE[AB] and one finds
R = 0 and Rab = 0 , (3.45)
where the generalized Ricci tensor is given by4
Rab = ∂aFb + (∂c
− Fc
)Fabc − FcdaFdc
b . (3.46)
These equations are equivalent to the string beta function equations (3.4)–(3.6) to lowest
order in α when we plug in the form of the generalized vielbein (3.32) and gauge fix the
double Lorentz transformations by setting e(+)
= e(−)
. However, this rewriting is much
more convenient for discussing generalizations of T-duality and in particular Yang-Baxter
deformations, as we will now see.
3.4 One-loop Weyl invariance and unimodularity
condition
Recall that our motivation for introducing the O(D, D) covariant formalism in the previous
section was to make it easier to understand YB deformations. Indeed we saw that
the generalized metric transforms as in (3.18) which means that the generalized vielbein
transforms as
˜EA
M
= EA
N
ON
M
, ON
M
= δM
N + ΘN
M
=
δn
m Θnm
0 δm
n
. (3.47)
This is a very simple linear transformation compared to the original non-linear transformation
(3.8). While O ∈ O(D, D) it is, in general, not constant. In fact Θ is constructed
3
It can also be written as minus the same thing with the projections reversed.
4
We can also define Rab by reversing the projections. Then Rab = Rba, due to the Bianchi identities.
23
in terms of Killing vectors of the background (3.9) and these are in general not constant.5
Since the generalized fluxes are only guaranteed to be invariant under constant O(D, D)
transformations we need to check how they transform under this transformation. From
the definition in (3.34) we find
˜FABC = 3˜∂[A
˜EB
M ˜EC]M = FABC + 3E[A
N
Θ|N
K
∂K|EB
M
EC]M + 3˜∂[AΘ|N|
M
EB
N ˜EC]M .
(3.48)
Let us look at the terms involving Θ on the RHS. First we note that since the only
non-zero component of ΘM
N
is Θmn
, the contraction (with the O(D, D) metric) of two
Θ’s will vanish. Therefore we may replace ˜ECM in the last term with ECM . Using
˜∂A = ∂A + EA
M
ΘM
N
∂N are left with terms linear and quadratic in Θ. Let us look first
at the linear terms
3E[A
N
Θ|N
K
∂K|EB
M
EC]M + 3∂[AΘ|N|
M
EB
N
EC]M . (3.49)
These terms actually cancel. To see this we recall that Θ is built from Killing vectors and
the fact that G and B are invariant translates to the generalized Lie derivative (3.33) of
EA
M
along these being zero
0 = Lkr EA
M
= kN
r ∂N EA
M
+ (∂M
krN − ∂N kM
r )EA
N
, kN
r = (0, kn
r ) . (3.50)
Using this the first term becomes
3E[A
N
Θ|N
K
∂K|EB
M
EC]M = −3Rrs
krN (∂M
ksK − ∂KkM
s )E[A
N
EB
K
EC]M
= − 3∂KΘMN E[A
N
EB
K
EC]
M
= −3∂[AΘ|N|
M
EB
N
EC]M , (3.51)
completing the proof. This leaves only the term quadratic in Θ. However, this terms also
vanishes due to the YB equation (2.40). Indeed we have
3E[A
K
EB
N
EC]
M
ΘK
L
∂LΘNM = 6E[A
K
EB
N
EC]
M
Rrs
Rtu
krKkL
s ∂LktN kuM
= 6E[A
K
EB
N
EC]
M
Rrs
Rtu
krKkuM kl
[s∂lkt]N = 3E[A
K
EB
N
EC]
M
Rrs
Rtu
krKkuM fst
v
kvN
= 3E[A
K
EB
N
EC]
M
krKkuM kvN Rrs
Rtu
fst
v
= 0 . (3.52)
In the third step we used the commutation relation of the Killing vectors 2kl
[r∂lkm
s] =
frs
t
km
t . We have learned the remarkable fact that despite the O(D, D) transformation
involved being non-constant the generalized flux FABC is still invariant. What about the
flux FA? We have
˜FA =˜∂B ˜EB
M ˜EAM + 2˜∂Ad = −∂M ˜EAM + 2˜∂Ad
=FA − ∂M EA
N
ΘN
M
− EA
N
∂M ΘN
M
+ 2EA
N
ΘN
M
∂M d . (3.53)
Note that we have assumed that the generalized dilaton does not transform, ˜d = d, which
is natural since this is what happens for ordinary T-duality. Using the invariance of EA
M
under the isometries involved in Θ the first term involving Θ becomes
1
2
Rrs
frs
t
ktM EA
M
− Rrs
krN ∂AkN
s = 1
2
Rrs
frs
t
ktM EA
M
, (3.54)
5
If all the Killing vectors involved commute we can pick coordinates where they are constant. In this
case the YB deformation becomes equivalent to a so-called TsT transformation.
24
since kM
r = (0, km
r ). Using the fact that the generalized dilaton transforms as a scalar
density, kN
r ∂N d = 1
2
∂N kN
r , the last term cancels against a piece of the second term and
the result is
˜FA = FA + EA
M
ktM Rrs
frs
t
. (3.55)
Unlike FABC we find that FA is in general shifted by something proportional to the
contraction of the R-matrix with the structure constants. Conversely, this extra term will
vanish leaving also FA invariant if the algebraic condition
Rrs
frs
t
= 0 (3.56)
is satisfied. We call this the unimodularity condition because it is equivalent to the
structure constants of the algebra of the Killing vectors involved in Θ having vanishing
trace, frs
s
= 0. To see this one contracts the YB equation (2.40) with ωrs = (R−1
)rs
leading to
Ruv
fuv
t
= −Rtu
fus
s
. (3.57)
If the unimodularity condition is satisfied we have seen that both generalized fluxes are
unchanged by the YB deformation
˜FABC = FABC , ˜FA = FA . (3.58)
This is quite remarkable when contrasted with the complicated transformation of the
metric and B-field in (3.8). But not only are the fluxes invariant, their derivatives are as
well. For example we have
˜∂A
˜FB = ˜∂AFB = ∂AFB + EA
M
ΘM
N
∂N FB = ∂AFB , (3.59)
where we used the fact that kM
r ∂M FA = 0 since kM
r generate isometries of the original
background. Clearly this observation extends to any number of derivatives of the fluxes.
But this observation means that the equations of motion (3.45), (3.42) and (3.46) are
invariant under the YB deformation. Therefore, if we start from a solution of these
equations and apply a unimodular YB deformation we obtain another solution. Since we
have seen that these equations coincide with the string β-function equations to lowest
order in α we find that the YB deformation maps consistent string backgrounds to other
consistent backgrounds, at least to leading order in the α expansion (corresponding to
one loop order in sigma model perturbation theory).
What about non-unimodular deformations? In this case ˜FA includes the extra shift in
(3.55) so that the generalized fluxes are not invariant. In fact this extra shift proportional
to the trace of the structure constants fits with what is known about non-abelian T-duality
at one loop. In that case there is an anomaly which spoils the one-loop Weyl-invariance
for non-unimodular groups [16, 17]. This fact would suggest that the non-unimodular YB
models will not solve the one-loop β-function equations. However, a direct calculation,
plugging in ˜FABC = FABC and ˜FA into the equations (3.45), shows that it can nevertheless
happen that the extra terms coming from the shift in FA can decouple, giving again an
admissible string background. This can happen in particular if G + B is a degenerate
matrix, and several examples are known. These examples are very special however and
typically non-unimodular deformations fail to lead to admissible backgrounds at one-loop
order.
25
Before we go on to consider what happens at two loops let us note an important fact.
Starting from some admissible string background we construct the YB deformation in the
doubled language by transforming the generalized vielbein (3.32) according to (3.47). But
if we start with a generalized vielbein with e(+)
= e(−)
= e, which we need to do to connect
to the standard gravity description, we obtain a deformed one with ˜e(+)
= ˜e(−)
. Therefore,
to read off the deformed background we should perform one more step – a double Lorentz
transformation which sets ˜e(+)
= ˜e(−)
= ˜e. Assuming we start from e(+)
= e(−)
= e one
finds that after the transformation (3.47) the required double Lorentz transformation may
be taken as
Λ(+)
= 1 , (Λ(−)
)a
b
= ˜Λa
b
= ea
m
eb
n (1 + (G − B)Θ) (1 − (G + B)Θ)−1
m
n
, (3.60)
corresponding to picking the deformed vielbein to be
˜eam
= ean
(1 + (G − B)Θ)n
m
. (3.61)
Of course, this extra double Lorentz transformation can be ignored at this stage since it
is a symmetry of the theory. However, it will play an important role in the next section.
26
Chapter 4
Two-loop Weyl invariance and
α -correction
So far we have seen that the YB deformation maps solutions of the β-function equations
for the bosonic string (3.4)–(3.6) to new solutions when α -corrections are ignored. What
happens when we take α -corrections into account? We will now answer this question
for the first α -correction, ignoring O(α 2
)-terms. The first correction to the β-function
equations for the bosonic string arises from a correction to the effective action involving
the square of the Riemann tensor
S = S0 + α S1 + O(α 2
) (4.1)
where S0 was given in (3.7) and [18]
S1 =
1
2κ2
0
d26
x
√
−Ge−2Φ 1
4
RmnopRmnop
− 1
8
RmnopHmnq
Hop
q
+ 1
96
HmnoHm
pqHnpr
Hoq
r − 1
32
(H2
)mn(H2
)mn
. (4.2)
Checking directly that the YB map (3.8) maps solutions of the equations of motion
corresponding to this corrected action to new solutions seems hopelessly complicated.
But if we are again able to rewrite things in a manifestly O(D, D) covariant form the
proof becomes trivial. Remarkably, the α -corrected action can again be written in an
O(D, D) invariant form. This is quite surprising given the fact that there is no O(D, D)
covariant analog of the Riemann tensor [19].1
Instead this correction appears through a
modification of the double Lorentz transformations [22]. To see why this happens it is
instructive to consider first the case of the heterotic string.
4.1 α -correction to double Lorentz transformation
In addition to the fields of the bosonic string the heterotic string also has gauge fields and
fermions. We will set these to zero here for simplicity. Then the lowest order effective
1
However, it can be understood from the fact that the correction can be generated [20] using a trick
originally due to Bergshoeff and de Roo [21].
27
action is the same as for the bosonic string (3.7). However, the α -corrections differ. In
particular, a famous fact is that the Bianchi identity for the NSNS three-form dH = 0 is
corrected in the heterotic string to (e.g. [23])
dH =
α
4
tr(R ∧ R) . (4.3)
This correction is required by the Green-Schwarz anomaly cancellation mechanism [24].
We may solve this condition by taking
H = dB +
α
4
Ω3 , dΩ3 = tr(R ∧ R) , (4.4)
where Ω3 = tr(ω∧dω)+ 2
3
tr(ω∧ω∧ω) is a gravitational Chern-Simons three-form. But H
is by definition invariant under Lorentz transformations while Ω3 is not, and this requires
a non-standard transformation for B. From the transformation of the spin connection,
ω → ω = Λ−1
dΛ + Λ−1
ωΛ, we find
dB → dB = H −
α
4
Ω3 = dB +
α
4
d tr(dΛΛ−1
∧ ω) + 1
3
tr(dΛΛ−1
∧ dΛΛ−1
∧ dΛΛ−1
) ,
(4.5)
or
B = B +
α
4
[tr(dΛΛ−1
∧ ω) + BWZW
(Λ)] , (4.6)
where
dBWZW
(Λ) = 1
3
tr(dΛΛ−1
∧ dΛΛ−1
∧ dΛΛ−1
) . (4.7)
This modification of the gauge-transformations of B is not part of the O(D, D) covariant
formalism described in the previous section. Therefore it is clear that the transformations
need to be modified to account for the first α -correction to the heterotic string in
that formalism. To see how we should modify the transformations let us consider the
infinitesimal version (Λ = 1 + λ) of the above transformation which reads
δB =
α
4
tr(dλ ∧ ω) . (4.8)
To generalize this to the O(D, D) covariant form we note that a change in the transformation
of B under Lorentz transformations implies a change in the transformation of the
generalized vielbein under double Lorentz transformations
δEA
M
EBM = λAB + ˆλAB , (4.9)
where ˆλ is of order α . Since the lowest order transformation parameters λAB have nonzero
components λab and λab, any non-zero ˆλab or ˆλab can be absorbed into these. We may
therefore take the components of ˆλ to have only mixed projections ˆλab = −ˆλba. Looking
at (4.8) we see that ˆλ should involve a derivative of λab or λab and the doubled analog of
the spin connection. In fact the closest analog of the spin connection is Fabc and Fabc, as
is easily seen from the first term in the transformations in (3.41). This suggests that we
should take
ˆλab = −ˆλba =
a
2
∂bλcd
Facd +
b
2
∂aλcd
Fbcd = −
a
2
tr(∂bλ(−)
Fa) −
b
2
tr(∂aλ(+)
Fb) , (4.10)
28
for some a, b of order α . Note that we suppressed the contracted indices in the last
expression. Of course, for this to make sense, the corrected transformations (4.9) must
close (to first order in α ). We find
[δ , δ]Ea
M
EbM = λa
cˆλcb − λa
cˆλcb + ˆλacλ c
b − ˆλacλc
b + δ ˆλab − δˆλab
+ O(α 2
) (4.11)
and plugging in the expression for ˆλ we find
[δ , δ]Ea
M
EbM =
a
2
tr(∂bλ (−)
∂aλ(−)
) − tr(∂bλ(−)
∂aλ (−)
) + tr(∂bλ (−)
[λ(−)
, Fa]) − tr(∂bλ(−)
[λ (−)
, Fa])
−
b
2
tr(∂aλ(+)
∂bλ (+)
) − tr(∂aλ (+)
∂bλ(+)
) + tr(∂aλ(+)
[λ (+)
, Fb]) − tr(∂aλ (+)
[λ(+)
, Fb])
+ O(α 2
)
= 2Ea
M
Eb
N
∂[M YN] −
a
2
tr(∂b[λ(−)
, λ (−)
]Fa) −
b
2
tr(∂a[λ(+)
, λ (+)
]Fb) + O(α 2
) (4.12)
with
YN =
a
2
tr(λ(+)
∂N λ (+)
) −
b
2
tr(λ(+)
∂N λ (+)
) . (4.13)
The first term is a generalized diffeomorphism with parameter YN and the second is precisely
of the form of the corrected Lorentz transformation with parameter [λ, λ ]. Therefore
the transformations indeed close to the required order in α .
We have found that it is possible to correct the double Lorentz transformations at order
α . The correction depends on two parameters, a and b. How does this correction compare
to the one required for the heterotic string, (4.8)? Fixing the gauge e(+)
= e(−)
= e in
(3.32) one finds from the definition of the generalized fluxes (3.34) that
Fa
bc
= −
1
√
2
ω(+)
a
bc
, Fa
bc =
1
√
2
ω(−)a
bc , ω(±)ab
m = ωm
ab
± 1
2
Hm
ab
(4.14)
and using these in the transformation (4.9) one finds2
δ ¯Gmn = −
a
2
∂(mλ(−)cd
ω(−)
n)cd −
b
2
∂(mλ
(+)
|cd|ω
(+)
n)
cd
, (4.15)
δ ¯Bmn = −
a
2
∂[mλ(−)cd
ω(−)
n]cd +
b
2
∂[mλ
(+)
|cd|ω
(+)
n]
cd
. (4.16)
Of course only the transformations with λ(+)
= λ(−)
= −λ preserve our gauge choice
e(+)
= e(−)
and reduce to the usual Lorentz transformations,3
but the more general case
will be needed below. Note that we have denoted the fields G, B with a bar. The reason
is that these fields, which are natural from the O(D, D) covariant formulation, are not the
ones we would usually consider since they are not Lorentz invariant. But we can define
new fields
Gmn = ¯Gmn −
a
4
ω(−)cd
m ω
(−)
ncd −
b
4
ω(+)cd
m ω
(+)
ncd , Bmn = ¯Bmn +
a + b
4
ω[m
cd
Hn]cd , (4.17)
2
Note that the components of F and λ are expressed relative to their defining index structures, FABC
and λA
B
.
3
The sign of λ is fixed by requiring the usual lowest order transformation δωab
= dλab
.
29
which transform as
δGmn = 0 , δBmn =
a − b
2
∂[mλcd
ωn]cd , (4.18)
under local Lorentz transformations. We see that taking a = −α and b = 0 we reproduce
the correct heterotic string transformation (4.8), while taking a = b we get the correct
transformation for the bosonic string where B does not transform.
4.2 α -corrected DFT action
We have seen that there is freedom to modify the double Lorentz transformations at order
α by adding the terms in (4.10). The correction has two free parameters, a and b, and to
match the correction to the heterotic string we need to set one of them to zero while for the
bosonic string we must set them equal. Modifying the double Lorentz transformations the
lowest order action given by (3.43), (3.42) is no longer invariant, since its transformation
will produce terms of order α . However, one can show that the terms of order α can be
canceled by adding certain terms of order α to the lowest order action [22]. The resulting
DFT action to order α takes the form
S = dX e−2d
R + aR(−)
+ bR(+)
, (4.19)
where we have defined
R(−)
= − (∂a
− Fa
) (∂b
− Fb
) FacdFb
cd
− 1
2
RabcdRabcd
+ ∂a
Fb
FacdFb
cd
(4.20)
+ FabC
Fa
de
∂CFbde − 2
3
Fabc
Fad
e
Fbe
f
Fcf
d
+ Fabc
Fabd + 1
2
Fabc
Fabd Fcef Fdef
,
where the first term is a total derivative and in the second we have introduced the object
Rabcd = 2∂[aFb]cd − FabeFe
cd − 2F[a|c|
e
Fb]ed , (4.21)
which may be considered a doubled analog of the Riemann tensor4
(although it is noncovariant
under double Lorentz transformations). The expression for R(+)
is simply obtained
by exchanging over- and underlined indices in R(−)
. This α -corrected DFT action
can be shown to reproduce (after suitable field redefinitions) the first α -correction to the
bosonic string effective action (4.2) upon setting a = b = −α . Setting a = −α and b = 0
it also reproduces the correction to the heterotic string effective action [22]. Note that for
this to work one has to take into account the α terms in the relation between ¯G, ¯B, the
natural fields from the DFT point of view, and G, B, the usual Lorentz invariant fields
in (4.17).
4
Indeed, solving the section condition and fixing the gauge e(+)
= e(−)
one finds that its components
are
Rab
cd =
1
2
R(−)ab
cd + ω(+)eab
ω
(−)
ecd .
30
4.3 α -correction to YB deformations
The existence of this α -corrected DFT action has interesting consequences for YangBaxter
deformations of strings. Because the action (4.19) is expressed in terms of the
generalized fluxes it follows that the equations of motion following from this action are
also expressed in terms of these. But we have seen that (unimodular) Yang-Baxter deformations
leave the generalized fluxes invariant and so it follows that the YB deformation
maps solutions of the DFT equations to solutions, at least up to and including order α .5
It would seem therefore that the YB deformed background should solve the β-function
equations also at order α without any need of modifying it by terms of order α . This is
not correct however, in general the background will receive a correction of order α . The
reason is that while this correction is not needed in the DFT language the natural fields of
that formulation, which we denoted ¯G and ¯B, are not the usual Lorentz covariant G and
B but rather related to these by (4.17). In addition we have seen that to go to the usual
fields from the doubled formulation we must set the two vielbeins e(+)
and e(−)
equal, but
since the YB deformation does not preserve this relation a compensating double Lorentz
transformation (3.60) is needed. Since the fields do not transform covariantly under double
Lorentz transformations this induces an α -correction for G and B. In fact we can
easily write down the explicit form of this correction as follows.
From (4.16) we see that setting λ(+)
= 0 we have
δ ¯Gmn = −
a
2
∂(mλ(−)cd
ω
(−)
n)cd , δ ¯Bmn = −
a
2
∂[mλ(−)cd
ω
(−)
n]cd . (4.22)
Of course this is the correction one gets if one starts from the gauge e(+)
= e(−)
. But in
our case we have e(+)
= e(−)
and we want to perform the double Lorentz transformation
which sets them equal. But if we think of it the other way around this is precisely the
inverse of the transformation needed to go from e(+)
= e(−)
to the e(+)
= e(−)
we started
from. Therefore the transformation we are looking for is as above but with the sign of
the RHS changed. In order to find the finite transformation rather than the infinitesimal
one we first note that (cf. (4.17))
δ ¯Gmn −
a
4
ω(−)cd
m ω
(−)
ncd = 0 , δ ¯Bmn +
a
4
ω[m
cd
Hn]cd = −
a
2
∂[mλ(−)cd
ωn]cd . (4.23)
The last transformation is now of the form (4.8), for which the finite transformation is
given by (4.6). Therefore we find the finite transformation to be
δ ¯Gmn = −
a
2
[∂(mΛ(−)
(Λ(−)
)−1
]cd
ω
(−)
n)cd −
a
4
(∂mΛ(−)
)cd
(∂nΛ(−)
)cd , (4.24)
δ ¯Bmn = −
a
2
[∂[mΛ(−)
(Λ(−)
)−1
]cd
ω
(−)
n]cd +
a
4
BWZW
mn (Λ(−)
) , (4.25)
where
dBWZW
(Λ) = 1
3
tr(dΛΛ−1
∧ dΛΛ−1
∧ dΛΛ−1
) . (4.26)
To find the α -correction to YB deformations all we need to do is take Λ(−)
= ˜Λ, defined
in (3.60), and recall that we are performing the transformation in the opposite direction,
5
The same conclusion can be shown to apply to non-abelian T-duality and its generalization known
as Poisson-Lie T-duality [25, 26, 27].
31
going from e(+)
= e(−)
to e(+)
= e(−)
, rather than the other way around, so the sign of the
RHS is opposite. In addition we must take into account the relation between the barred
fields and the usual unbarred ones (4.17). The correction then takes the form
δ(G − B)mn =
α
2
ω(−)cd
m ω
(+)
ncd + [∂n
˜Λ˜Λ−1
]cd +
α
4
∂m
˜Λcd
∂n
˜Λcd +
α
4
BWZW
mn (˜Λ) , (4.27)
where we have set a = b = −α as appropriate for the bosonic string (for the heterotic
string the correction vanishes) and also included a redefinition of the metric G → G −
α
4
Hm
cd
Hncd to go to the scheme of [28]. Finally, the correction to the dilaton is derived
from the fact that the generalized dilaton d defined in (3.22) is uncorrected, since it does
not transform under double Lorentz transformations. Therefore δΦ = 1
4
Gmn
δGmn (in a
suitable scheme). Note that we must of course also take into account any possible α
corrections to the undeformed background, which have to be determined by other means.
32
Chapter 5
Conclusions & further developments
We introduced the Yang-Baxter deformations in the simple case of the PCM and showed
that they preserve the integrabilty of the model. In the homogeneous case, when the
R matrix involved solves the classical rather than the modified classical Yang-Baxter
equation, we showed that the deformation can be constructed using non-abelian T-duality.
Since the notion of NATD exists for more general 2d sigma models this allowed us to define
a notion of YB deformation for bosonic strings in general (in backgrounds with isometries).
We saw that these deformations indeed make sense in string theory, at least when R is
unimodular, since they preserve the one-loop Weyl invariance. This was made possible by
working in the O(D, D) invariant formalism of DFT where the YB deformation greatly
simplifies. Finally, we saw that these observations extend to first order in α , where one
however finds that the YB deformation must be corrected in a particular way. Again this
result relied crucially on the existence of an O(D, D) invariant formalism.
There are many important developments in this subject which we did not have time
to mention here. We end by listing some of these:
ˆ We have considered mainly the homogeneous YB deformations here. The inhomogeneous
ones are also interesting and were in fact the first to be considered. In that
case there is a connection [29], via a generalization of T-duality known as PoissonLie
T-duality [30], to another integrable deformation known as the λ-deformation
[31].
ˆ We have considered only the bosonic string here, but for applications in string
theory the superstring is more relevant. Yang-Baxter deformations of the AdS5 ×S5
superstring were introduced in [32, 33] and a general definition for homogeneous
deformations was given in [34]. The original inhomogeneous deformation violated
unimodularity, but later it was realized how to construct unimodular examples [35].
ˆ It appears to be possible to use DFT to describe also the correction at the next
order in α , i.e. α 2
[20, 36]. However, recently we showed that the DFT description
breaks down at order α 3
[15]. How to deal with Yang-Baxter deformations at that
order is an open problem.
ˆ It has been argued that in the context of the AdS/CFT holographic duality YB
deformations should correspond to non-commutative deformations of the dual field
33
theory [37, 38], though no precise checks of this idea have been made.
ˆ Recently these deformations have also been embedded in the general language of
affine Gaudin models [39] and holomorphic 4d Chern-Simons theory [40].
34
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37
Reprints of articles
38
JHEP10(2020)065
Published for SISSA by Springer
Received: August 14, 2020
Accepted: September 10, 2020
Published: October 12, 2020
Relaxing unimodularity for Yang-Baxter deformed
strings
Stanislav Hronek and Linus Wulff
Department of Theoretical Physics and Astrophysics, Faculty of Science,
Masaryk University, 611 37 Brno, Czech Republic
E-mail: 436691@mail.muni.cz, wulff@physics.muni.cz
Abstract: We consider so-called Yang-Baxter deformations of bosonic string sigmamodels,
based on an R-matrix solving the (modified) classical Yang-Baxter equation. It is
known that a unimodularity condition on R is sufficient for Weyl invariance at least to two
loops (first order in α ). Here we ask what the necessary condition is. We find that in cases
where the matrix (G + B)mn, constructed from the metric and B-field of the undeformed
background, is degenerate the unimodularity condition arising at one loop can be replaced
by weaker conditions. We further show that for non-unimodular deformations satisfying
the one-loop conditions the Weyl invariance extends at least to two loops (first order in α ).
The calculations are simplified by working in an O(D, D)-covariant doubled formulation.
Keywords: Bosonic Strings, Conformal Field Models in String Theory, String Duality
ArXiv ePrint: 2007.15663
Open Access, c The Authors.
Article funded by SCOAP3
.
https://doi.org/10.1007/JHEP10(2020)065
JHEP10(2020)065
Contents
1 Introduction 1
2 Doubled (flux) formulation 3
2.1 The first α -correction 6
3 Yang-Baxter deformations and one-loop Weyl invariance 8
3.1 First order terms 10
3.2 Second order terms 11
4 Two-loop Weyl invariance 12
5 Conclusions 15
1 Introduction
Yang-Baxter (YB) deformations of 2D σ-models were introduced by Klimˇc´ık in [1]. The
name comes from the fact that the deformation is constructed using an R-matrix which
solves the (modified) classical Yang-Baxter equation. It was later realized that these deformations
preserve the classical integrability of the σ-model [2]. Delduc, Magro and Vicedo
constructed the YB deformation for symmetric spaces in [3], and then for the AdS5 × S5
superstring in [4], based on the Drinfeld-Jimbo R-matrix solving the modified classical
YB equation. Shortly thereafter it was shown in [5] that essentially the same construction
works also for R-matrices solving the ordinary (non-modified) classical YB equation.
The latter are often referred to as homogenous YB deformations and have an interesting
realization in terms of non-abelian T-duality [6–8].
Surprisingly it was found, starting with the paper [9], that the backgrounds corresponding
to these deformed string σ-models did not always satisfy the equations of supergravity,
but a certain generalization of these [10, 11]. When this is the case the deformed σ-model
is only scale invariant, but not Weyl invariant, at one loop and cannot be interpreted as
a consistent string. For supercoset models such as the AdS5 × S5 superstring a condition
was found on the R-matrix that leads to a viable, i.e. one-loop Weyl invariant, deformed
string σ-model. The R-matrix should be unimodular, i.e. its trace with the Lie algebra
structure constants should vanish, Rrsfrs
t = 0 [12].
Subsequently, using the realization via non-abelian T-duality, homogeneous YB deformations
were defined for a general Green-Schwarz superstring with isometries [13].1
Interestingly, examples were found where a non-unimodular R-matrix nevertheless gave
1
In the abelian case these deformations are equivalent to so-called TsT-transformations, consisting of
T-duality, a coordinate shift and a T-duality back [14].
– 1 –
JHEP10(2020)065
rise to a good (super)gravity background [15, 16]. Therefore, while the unimodularity condition
is sufficient, it is not necessary to solve the one-loop Weyl invariance conditions, i.e.
the background (super)gravity equations.
Here we will determine the precise conditions for (bosonic) YB deformations to respect
one-loop Weyl-invariance. We find that, at least for deformations of symmetric spaces,
the only exceptions to the unimodularity condition occur when the matrix (G + B)mn,
where G, B are the metric and B-field of the undeformed background, is degenerate.2,3 In
that case, the unimodularity condition is no longer necessary and is replaced by weaker
conditions which we give. This is consistent with the examples found in [15, 16] since the
AdS3 × S3 background considered there has degenerate G + B.
We then go on to analyze what happens at two loops, i.e. when we include the first
α -correction to the (super)gravity equations. We find that the conditions at two loops are
weaker and only a subset of the one-loop conditions are needed.
These calculations are simplified enormously by working with the O(D, D)-covariant
formulation known as Double Field Theory (DFT). In DFT a manifestly O(D, D)-covariant
formulation is achieved by doubling the coordinates to XM = (˜xm, xm). One then imposes
a “section condition” which effectively removes half of them, leaving the right number of
physical coordinates. Here we will work only with the standard choice of section, XM =
(0, xm) or ∂M = (0, ∂m), and therefore the coordinates are not doubled. However, the
tangent space is effectively doubled and there are two copies of the Lorentz group. Therefore
there are two sets of vielbeins e(+) and e(−) which transform independently under each
Lorentz group factor. Fixing the gauge e(+) = e(−) = e breaks the doubled Lorentz
group down to its diagonal, which becomes the standard Lorentz group. With this gauge
fixing the action and equations of motion of the doubled formulation reduce to those of
standard (super)gravity. The reason the doubled formulation is useful is that the YB
deformation becomes equivalent to a coordinate dependent O(D, D)-transformation which
is easy to analyze. In fact the so-called generalized fluxes, the basic fields of the so-called
flux formulation we are using [18], transform very simply under the YB deformation. The 3form
flux is invariant while the 1-form acquires a shift. This shift vanishes in the unimodular
case and the generalized fluxes are simply invariant, from which one can immediately
conclude that such YB deformations preserve Weyl invariance at least to two loops [19]. In
the present case, we are interested in non-unimodular R-matrices and we have to take the
shift into account. Provided that this shift satisfies certain conditions, which we determine,
the Weyl-invariance is preserved at least up to two loops. It is interesting that it is possible
to shift the 1-form generalized flux in certain ways and still preserve the equations of the
doubled formulation including the first α -correction. This should have an interpretation
in gauged DFT [20], but we will not pursue this here.
In [19] the doubled formulation was used to determine the first α -correction to the
2
In the homogeneous case we prove this only for rank R < 8 for technical reasons.
3
Gauge-transformations of B, which could affect this, are severely restricted by the fact that B is required
to be invariant under the isometries involved in the deformation. This is required in the homogeneous
case [13]. In the inhomogeneous case with a WZ-term [17] it is not required and our analysis is incomplete
in that case.
– 2 –
JHEP10(2020)065
deformed background for unimodular R. This correction arises because the fields of the
doubled formulation are not Lorentz-covariant once α -corrections are included and a double
Lorentz transformation is needed to go to the gauge e(+) = e(−) = e and reduce to
the standard (super)gravity fields, thus leading to a correction to the background.4 Our
analysis here shows that no additional corrections are needed in the non-unimodular case,
so the correction to the deformed background is still given by the expressions found in [19].
The outline of this paper is as follows. First we review the elements we need of the
flux formulation of DFT and how the α -correction to the double Lorentz transformations
determine the action to the first order in α . In section 3 we derive the conditions for a YB
deformation to lead to a (super)gravity background, i.e. the conditions needed for one-loop
Weyl-invariance. The situation at two loops is analyzed in section 4 where we find weaker
conditions than at one loop. We end with some conclusions.
2 Doubled (flux) formulation
The O(D, D)-covariant formulation of (super)gravity used in DFT [22–24] turns out to be
very powerful for the kinds of questions we are interested in here. In particular we will
work with a frame-like formulation of DFT [25–27] where the structure group consists of
two copies of the Lorentz group O(1, D−1)×O(D−1, 1). In particular we use the so-called
flux formulation of [18, 28] where the first α -correction to the bosonic and heterotic string
can also be nicely incorporated. We will always assume that the section condition is solved
in the standard way ∂M = (0, ∂m) so that we are really just working with a rewriting of
(super)gravity.
The starting point is to introduce a generalized (inverse) vielbein parametrized as
EA
M
=
1
√
2


e(+)a
m − e(+)anBnm e(+)am
−e
(−)
am − e
(−)
a
nBnm e
(−)
a
m

 . (2.1)
Here e(±) are two sets of vielbeins for the metric Gmn which transform independently as
Λ(±)e(±) under the two Lorentz-group factors. To go to the standard supergravity picture
one fixes a gauge e(+) = e(−) = e, leaving only one copy of the Lorentz-group. The dilaton
Φ is encoded in the generalized dilaton d defined as
e−2d
= e−2Φ
√
−G . (2.2)
There are two constant metrics, the O(D, D)-metric ηAB and the generalized metric HAB
which take the form
ηAB
= ηAB =
¯η 0
0 −¯η
, HAB
=
¯η 0
0 ¯η
, (2.3)
where ¯η = (−1, 1, . . . , 1) is the usual Minkowski metric. The flat tangent space indices
A, B, . . . are raised and lowered with ηAB, ηAB. The generalized vielbein is used to convert
4
The correction agrees with what is found by a much more involved calculation using standard (super)gravity
[21].
– 3 –
JHEP10(2020)065
between these indices and coordinate indices M, N, . . .. In particular we have the usual
expressions for the O(D, D)-metric and the generalized metric in a coordinate basis
ηMN
= EA
M
ηAB
EB
N
=
0 δm
n
δm
n 0
, (2.4)
HMN
= EA
M
HAB
EB
N
=
Gmn − BmkGklBln BmkGkn
−GmkBkn Gmn . (2.5)
We also define
∂A = EA
M
∂M , (2.6)
where ∂M = (0, ∂m) is the ordinary derivative.
The basic fields of the flux formulation are the generalized fluxes. These are constructed
from the generalized vielbein as
FABC = 3∂[AEB
M
EC]M , FA = ∂B
EB
M
EAM + 2∂Ad . (2.7)
The importance of these objects comes from the fact that they transform as scalars under
generalized diffeomorphisms implemented by the generalized Lie derivative defined as
LXY M
= XN
∂N Y M
+ (∂M
XN − ∂N XM
)Y N
. (2.8)
The generalized diffeomorphisms contain the usual diffeomorphisms and B-field gaugetransformations.
The generalized fluxes satisfy the following Bianchi identities
4∂[AFBCD] = 3F[AB
E
FCD]E , 2∂[AFB] = −(∂C
− FC
)FABC . (2.9)
Note also that
[∂A, ∂B] = FABC ∂C
. (2.10)
The bosonic/heterotic5 string low-energy effective action can be cast in doubled form as
S = dX e−2d
R , (2.11)
where the generalized Ricci scalar is defined as6
R = −4∂A
F
(−)
A + 2FA
F
(−)
A − F
(−)
ABCF(−)ABC
−
1
3
F
(−−)
ABC F(−−)ABC
. (2.12)
Here we have defined certain projections of the generalized fluxes using the natural projection
operators
P± =
1
2
(η ± H) , (2.13)
5
Setting the gauge fields and fermions of the heterotic string to zero.
6
The last two terms are often written instead as 1
4
FACDFB
CD
HAB
− 1
12
FABC FDEF HAD
HBE
HCF
−
1
6
FABC FABC
. In terms of the generalized metric we have instead
R = 4∂M (HMN
∂N d) − ∂M ∂N HMN
− 4HMN
∂M d∂N d
+
1
8
HMN
∂M HKL
∂N HKL −
1
2
HMN
∂M HKL
∂K HLN .
– 4 –
JHEP10(2020)065
as follows
F
(±)
A = (P±)A
B
FB , (2.14)
and
F
(±)
ABC = (P )A
D
(P±)B
E
(P±)C
F
FDEF , F
(±±)
ABC = (P±)A
D
(P±)B
E
(P±)C
F
FDEF .
(2.15)
Setting e(+) = e(−) = e in the generalized vielbein (2.1) this can be shown to reduce to the
correct low-energy effective (super)gravity action.
We will be interested in whether certain transformations of the generalized fluxes map
a solution to another solution, so we will need the equations of motion following from the
action (2.11). These can be easily found using the variations of the generalized fluxes with
respect to the generalized vielbein and dilaton
δEFABC = 3∂[AδEBC] + 3δE[A
D
FBC]D , δEFA = ∂B
δEBA + δEA
B
FB , δdFA = 2∂Aδd ,
(2.16)
where δEAB = δEA
M EBM is anti-symmetric by construction. The equations of motion
become
R = 0 , ∂
(+)
A F
(−)
B + (∂C
− FC
)F
(−)
ABC − F
(+)
CDAF(−)DC
B = 0 . (2.17)
Here we have defined the projected derivatives ∂
(±)
A = (P±)A
B∂B. The second equation
of motion can equivalently be written with the opposite projections by exchanging + and
− superscripts. Setting e(+) = e(−) = e they reduce to correct (super)gravity equations
of motion.
The action (2.11) is invariant under three important symmetries. The first is generalized
diffeomorphisms, which encode regular diffeomorphisms and B-field gauge transformations.
In the flux formulation we are working with here the generalized diffeomorphism
invariance is manifest since the fluxes and the derivative ∂A transform as scalars. The
second symmetry is that of global O(D, D)-transformations
XM
→ XN
hN
M
, EA
M
→ EA
N
hN
M
with hM
N
∈ O(D, D) (2.18)
and hM
N constant. Again the action is manifestly invariant under these transformations
since the fluxes are invariant. In our case we are always imposing the standard section
condition ∂M = (0, ∂m) so this symmetry is (partially) broken.
Finally, the most important symmetry for us will be the invariance under double
Lorentz transformations
δEA
M
EBM = δEAB = λAB with (P+)A
C
(P−)B
D
λCD = 0 . (2.19)
The parameters of the infinitesimal double Lorentz transformation λAB commute with the
projectors P± so their non-trivial components are λ
(+)
AB and λ
(−)
AB, corresponding to the
two copies of the Lorentz group. These two copies rotate the two vielbeins e(±) in (2.1)
independently. The (double) Lorentz invariance of the action (2.11) is not manifest. It can
– 5 –
JHEP10(2020)065
be verified with a bit of algebra using the variations of the fluxes (2.16). In particular it
follows from these expressions that under a double Lorentz transformation
δF
(±)
ABC = λ
( )
A
D
F
(±)
DBC + 2λ
(±)
[B
D
F
(±)
|AD|C] + ∂
( )
A λ
(±)
BC , (2.20)
which, except for the projections, is precisely the transformation of a connection. Indeed,
suppressing the last two indices we have7
δF
(±)
M = ∂
( )
M λ(±)
+ [λ(±)
, F
(±)
M ] , (2.21)
so that F
(±)
M behave very much like connections. In fact, fixing the double Lorentz transformations
by setting e(+) = e(−) = e the non-zero components of F(±) are [28]
F
(+)
M
ab
=
1
2


Gmnω
(+)ab
n
−(1 − BG)m
nω
(+)ab
n

 , F
(−)
Mab =
1
2


Gmnω
(−)
nab
(1 + BG)m
nω
(−)
nab

 , (2.22)
where ω
(±)cd
m = ωm
cd ± 1
2 Hm
cd. These expressions will be useful later.
A very important point is that the double Lorentz transformations receive α corrections.
In fact, this is a good thing since it allows us to derive the first α -correction
to the action (2.11) from the knowledge of the correction to the Lorentz transformations.
We will now see how this works.
2.1 The first α -correction
At the first order in α the double Lorentz transformations get corrected to [28]
δEAB = λAB + a tr ∂
(−)
[A λF
(−)
B] − b tr ∂
(+)
[A λF
(+)
B] , (2.23)
where a = b = −α for the bosonic string and a = −α , b = 0 for the heterotic string (a =
b = 0 for type II). The correction involves the connection-like objects F
(±)
ABC (note the trace
over the last two indices) and is therefore of the form of a Green-Schwarz transformation.
The knowledge of the correction to the Lorentz transformation can be used to find the
α -correction to the action [28], as we will now review. For simplicity we will set b = 0 in
the derivation and restore b at the end. The variation (2.23) is then of the form δ = δ0 +aδ1
and a short calculation gives for the projections of the generalized fluxes appearing in the
lowest order action (2.11), (2.12)
δ1
F
(−)
A = −
1
2
(∂B
− FB
) tr ∂
(−)
A λF
(−)
B , (2.24)
δ1
F
(−−)
ABC =
3
2
tr ∂
(−)
[A λF(−)D
F
(−)
D|BC] (2.25)
and
δ1
F
(−)
ABC = (P−)[C
D
tr ∂
(−)
B] λR
(−)
AD +
1
2
tr F
(−)
A ∂D
λ F
(−)
DBC +
1
2
tr ∂
(−)
[B λF(−)D
F
(+)
C]AD .
(2.26)
7
We will try to be clear about when we suppress the last two indices to avoid possible confusion with
the generalized flux with one index FA.
– 6 –
JHEP10(2020)065
In the last expression we have defined the ‘curvature’ of the ‘connection’ F
(−)
ABC as (suppressing
the last two indices which are projected by P−)
R
(−)
AB = 2∂[A F
(−)
B] − (P+)[B
D
FA]DEF(−)E
− [F
(−)
A , F
(−)
B ] . (2.27)
This object will be useful later. In particular when we project the indices A and B with P+
we have, writing ¯R
(−)
AB = (P+)A
C(P+)B
DR
(−)
CD, (again the last two indices are suppressed)
δ0 ¯R
(−)
AB = 2λ
(+)
[A
C ¯R
(−)
|C|B] + [λ(−)
, R
(−)
AB] + F
(+)
CAB∂C
λ(−)
− ∂C
λ
(+)
ABF
(−)
C , (2.28)
which apart from the last two terms is the expected transformation of a curvature.
At lowest order in α the action is Lorentz invariant. At the next order we find
δ1
R = − 4(∂A
− FA
)δ1
F
(−)
A − 2∂B
FA
tr ∂
(−)
A λF
(−)
B
−
2
3
F(−−)ABC
δ1
F
(−−)
ABC − 2F(−)ABC
δ1
F
(−)
ABC . (2.29)
Using the expressions for the δ1-variations (2.24), (2.25) and (2.26) as well as the Bianchi
identity for FA (2.9), (2.10) and the section condition this becomes
δ1
R = δ0
−∂A
(∂B
− FB
) tr F
(−)
A F
(−)
B + (∂A
− FA
) FB
tr F
(−)
A F
(−)
B
− FABC
tr ∂A∂
(+)
B λF
(−)
C + FABC
tr ∂
(+)
A λ∂BF
(−)
C − 2FABC
tr ∂Cλ∂AF
(−)
B
+ 2F(−)ABC
tr ∂BλR
(−)
CA + FABC
∂CλAD tr F(−)D
F
(−)
B
+ ∂B
λAC
∂A tr F
(−)
C F
(−)
B + FABC
FBCD tr ∂D
λF
(−)
A
− F(−−)ABC
F
(−)
DBC tr ∂AλF(−)D
− F(−)ABC
F
(−)
DBC tr ∂D
λF
(−)
A
− F(−)ABC
F
(+)
CAD tr ∂BλF(−)D
. (2.30)
We must now find terms of order α whose lowest order Lorentz transformation cancels the
terms on the r.h.s. . The first term on the second line must be canceled by the variation
of a term of the form FABC tr ∂AF
(−)
B F
(−)
C and we find
δ1
R = δ0
−∂A
(∂B
− FB
) tr F
(−)
A F
(−)
B + (∂A
− FA
) FB
tr F
(−)
A F
(−)
B
− δ0
FABC
tr ∂AF
(−)
B F
(−)
C − FABC
tr ∂
(−)
C λ ¯R
(−)
AB + ∂C
λAB
tr F
(−)
C
¯R
(−)
AB
+ 2∂C
λAB
tr F
(−)
A F
(−)
B F
(−)
C + 2FABC
tr F
(−)
A F
(−)
B ∂
(+)
C λ
+ F
(++)
ABE ∂C
λAB
tr F
(−)
C F(−)E
+ 2FABE
∂BλCA tr F(−)C
F
(−)
E
+ FABC
FDBC tr ∂(+)D
λF
(−)
A − F(−)ABC
F
(−)
DBC tr ∂D
λF
(−)
A , (2.31)
where we used the definition of the ‘curvature’ in (2.27). Using (2.28) we see that the
last two terms on the second line come from the variation of tr ¯R(−)AB ¯R
(−)
AB and the
– 7 –
JHEP10(2020)065
remaining terms are also easy to write as the variation of something. When the dust has
settled one finds, reinstating b, that the corrected action
S = dX e−2d
R + aR(−)
+ bR(+)
(2.32)
is invariant under Lorentz transformations up to and including order α where
R(−)
= ∂A
(∂B
− FB
) tr F
(−)
A F
(−)
B − (∂B
− FB
) FA
tr F
(−)
A F
(−)
B
+
1
2
tr ¯R(−)AB ¯R
(−)
AB +
1
6
FABC
C
(−)
ABC . (2.33)
In the last term we have introduced the ‘Chern-Simons’ form
C
(−)
ABC = 6 tr F
(−)
[A ∂BF
(−)
C] + 3(F
(−)
D[AB − FD[AB) tr F
(−)
C] F(−)D
− 4 tr F
(−)
[A F
(−)
B F
(−)
C] .
(2.34)
The expression for R(+) is obtained by reversing the projections in an obvious way. These
expressions agree with the ones written in [29] but are much more compact.
3 Yang-Baxter deformations and one-loop Weyl invariance
Yang-Baxter deformations are closely related to a generalization of T-duality known as
Poisson-Lie (PL) T-duality. In particular homogeneous YB deformations can be constructed
using non-abelian T-duality [6, 7]. It is therefore not surprising that they have a
natural formulation in terms of DFT. In the flux formulation we are working with they are
described as a coordinate dependent O(D, D)-transformation [13, 30, 31]
EA
M
→ ˜EA
M
= EA
N
(1 + Θ)N
M
. (3.1)
The only non-zero components of ΘN
M are Θmn = km
r kn
s Rrs where km
r are Killing vectors
belonging to some Lie algebra g indexed by (r, s, t, . . .) and Rrs is a constant anti-symmetric
matrix satisfying, in the homogeneous case, the classical YB equation
[RX, RY ] − R([RX, Y ] + [X, RY ]) = 0 , ∀X, Y ∈ g , (3.2)
which implies the ‘Jacobi identity’ for Θ8
ΘN[K
∂N ΘLM]
= 0 . (3.3)
If we start from a symmetric space σ-model we can also define the inhomogeneous deformation
[3] where R satisfies the modified classical YB equation
[RX, RY ] − R([RX, Y ] + [X, RY ]) = [X, Y ] , ∀X, Y ∈ g . (3.4)
8
Conversely, if we don’t impose any condition on R, this condition follows by requiring that we get a
(super)gravity solution [32].
– 8 –
JHEP10(2020)065
The canonical solution is the Drinfeld-Jimbo R-matrix defined to annihilate elements of the
Cartan subalgebra and to multiply generators corresponding to positive(negative) roots by
+i(−i). We can define again Θmn = km
r kn
s Rrs which also satisfies (3.3).9
Note that letting R be multiplied by a small parameter, usually called η, these become
deformations of the original background. It is not hard to show, using the definitions (2.7),
that these deformations preserve the form of the generalized fluxes up to a shift of FA [19]
˜FABC = FABC , ˜FA = FA − 2KA . (3.5)
In addition derivatives of FABC are invariant, e.g. ˜∂A
˜FBCD = ∂AFBCD. Because of the
shift this is in general not true for FA, instead
˜∂A
˜FB = ∂AFB − 2∂AKB − 2EA
N
ΘN
M
∂M KB . (3.6)
The shift of FA is given by a certain distinguished Killing vector namely KM = (0, Km)
with
Km
= nΘmn
= nkm
r kn
s Rrs
= −1
2 Rrs
frs
t
km
t , (3.7)
where the third step involves using the algebra of the Killing vectors. This shift vanishes
precisely when R is unimodular, i.e. when Rrsfrs
t = 0.10 In this case the generalized fluxes
and their derivatives are invariant under the deformation and this directly implies that the
deformation preserves Weyl-invariance at least up to order α (2 loops) [19]. If we drop the
unimodularity condition we will generically get a scale-invariant but not Weyl-invariant σmodel
at one loop. This is reflected in the background solving the generalized supergravity
equations [10, 11] instead of the usual ones, the extra Killing vector appearing in these
equations being given by Km.
Here we want to ask what happens if you don’t require unimodularity but still require
the deformed model to preserve one-loop Weyl invariance.11 We will argue that, at least
in the case of symmetric spaces, it is possible to find such non-unimodular R-matrices (at
least of low enough rank to be interesting) only if the combination of metric and B-field
of the original model G ± B is a degenerate matrix. An example where this happens is
for AdS3 × S3 and indeed in that case several non-unimodular R-matrices that lead to
(super)gravity solutions have been found [15, 16].
The requirement that the equations of motion (2.17) remain invariant under the deformation,
which is equivalent to preservation of one-loop Weyl-invariance, becomes, us-
9
This was first noted in special examples in [33]. We thank S. van Tongeren for pointing this out to
us. The fact that the r.h.s. in the modified YB equation does not contribute can be seen as follows. For
a symmetric space g is generated by Pa, Mab with commutators of the form [P, P] ∼ M, [M, P] ∼ P and
[M, M] ∼ M. The Killing vectors are given by kr
m
= a
m
( ˆPAdg)a
r (see for example [13]), where a
m
are inverse vielbeins of the left-invariant one-forms and ˆP projects on the Lie algebra generators Pa. Now
since the structure constants are Ad-invariant and since they have no component corresponding to three
Pa generators it follows that the r.h.s. in the modified YB equation does not contribute.
10
It is easy to see that the Drinfeld-Jimbo R-matrix of the inhomogeneous deformation is not unimodular.
11
This corresponds to having a solution of the generalized supergravity equations which also solves the
standard supergravity equations. Such ‘trivial’ solutions were analyzed in [34].
– 9 –
JHEP10(2020)065
ing (3.5) and (3.6)
∂
(+)
A K
(−)
B + (P+)A
C
EC
N
ΘN
M
∂M K
(−)
B − KC
F
(−)
ABC = 0 , (3.8)
∂A
K
(−)
A + EA
N
ΘN
M
∂M K(−)A
− KA
F
(−)
A + KA
K
(−)
A = 0 . (3.9)
Since we should think of Θ as being multiplied by a small deformation parameter these
equations contain terms of first and second order in this parameter (note that K (3.7) is
of first order). These contributions then need to vanish separately.
3.1 First order terms
At the lowest order in the deformation we find the conditions
∂
(+)
A K
(−)
B − KC
F
(−)
ABC = 0 , ∂A
K
(−)
A − KA
F
(−)
A = 0 . (3.10)
Using the form of the generalized vielbein (2.1) with e(+) = e(−) = e, the fact that KM =
(0, Km) and the form of F
(−)
ABC in (2.22) the first equation becomes
a[(1 + B)b
c
Kc] −
1
2
Habc(1 + B)c
dKd
= 0 . (3.11)
Symmetrizing in a, b and using the fact that K is Killing we find that ˜K = iKB is also a
Killing vector. Anti-symmetrizing we find, using LKB = 0, that
dK + i ˜KH = 0 . (3.12)
This equation implies that H is invariant under ˜K since L ˜KH = di ˜KH = −ddK = 0. We
also have the same equation with K and ˜K exchanged since d ˜K = diKB = −iKH from
the invariance of the B-field under isometries, which we have assumed here.12 From the
dilaton equation we get, using the fact that K and ˜K are Killing vectors, the condition
˜Km
∂mΦ = 0 , (3.13)
i.e. the dilaton is invariant under the isometry generated by ˜K. To summarize, the conditions
we find at this order are that ˜K = iKB generates isometries of the background fields
G, H, Φ and satisfies (3.12).
For our later discussion of two-loop conformal invariance it will be useful to express
these conditions in terms of the generalized fluxes. The fact that K and ˜K generate
symmetries of the original background implies that under YB deformations
˜F
(±)
A
˜∂A
(something invariant) = F
(±)
A ∂A
(something invariant) . (3.14)
In addition we have
KA
F
(−)
ABC =
1
2
(Km
+ ˜Km
)ω
(−)
mbcδb
Bδc
C
= −
1
2
( bKc + b
˜Kc)δb
Bδc
C −
1
4
(Km
+ ˜Km
)Hmbcδb
Bδc
C
= 0 , (3.15)
12
This seems to be required in the construction of the general homogeneous deformations [13]. In the
inhomogeneous case this should be relaxed [17], but we will not try to do this here since it would take us
too far afield.
– 10 –
JHEP10(2020)065
where we used invariance of the vielbein under K, ˜K which implies iKωab = − aKb and
similarly for ˜K as well as the equation (3.12) and the same with K and ˜K exchanged. The
same is true with the opposite projection and therefore we have
˜FA ˜F
(±)
ABC = FA
F
(±)
ABC . (3.16)
3.2 Second order terms
At second order in the deformation the conditions (3.8) and (3.9) read
(P+)A
C
EC
N
ΘN
M
∂M K
(−)
B = 0 , EA
N
ΘN
M
∂M K(−)A
+ KA
K
(−)
A = 0 . (3.17)
We need to evaluate
EA
N
ΘN
M
∂M KB = EA
N
EB
L
ΘN
M
∂M KL + EA
N
ΘN
M
∂M EB
L
KL
= Rrs
EA
N
EB
M
krN kL
s ∂LKM − KL
∂LksM , (3.18)
where we used the fact that Θmn = km
r kn
s Rrs and the isometry of the generalized vielbein,
i.e. its generalized Lie derivative (2.8) along kr vanishes, in the second step. Now we use
the form of Km in (3.7) and the algebra of the Killing vectors to reduce this to
EA
N
ΘN
M
∂M KB = −
1
2
krAkwBRrs
fsv
w
Rtu
ftu
v
. (3.19)
Using the Jacobi identity and the (modified) classical YB equation this expression can be
seen to be symmetric in the indices A and B. The second order conditions now become
(G − B)ankn
r (G + B)bmkm
w Rrs
fsv
w
Rtu
ftu
v
= 0 , K2
+ ˜K2
= 0 . (3.20)
The first condition can be expressed as
kn
r km
w Rrs
fsv
w
Rtu
ftu
v
= vm
+ vn
+ + vm
− vn
− , (3.21)
where v± are zero-eigenvectors of G ± B, i.e. (G ± B)v± = 0. When G ± B is degenerate
precisely one such vector v± exists (up to rescaling). When G ± B is non-degenerate, for
example if B vanishes, then the r.h.s. is zero and we get the condition
kn
r km
w Rrs
fsv
w
Rtu
ftu
v
= 0 . (3.22)
This condition is very strong and in fact it seems to imply the unimodularity condition,
at least for deformations of symmetric spaces. In that case the condition becomes (see
footnote 9)
( ˆPAdg)a
r( ˆPAdg)b
wRrs
fsv
w
Rtu
ftu
v
= 0 , (3.23)
and taking g = e
aPa and expanding in this leads to
Rrs
fsv
w
Rtu
ftu
v
= 0 . (3.24)
It is easy to see from the form of the Drinfeld-Jimbo R-matrix that this rules out the
inhomogeneous deformations. For the homogeneous deformations R is invertible on the
– 11 –
JHEP10(2020)065
subalgebra where it is defined and this condition is equivalent to the condition that the
distinguished Lie algebra element Rrsfrs
tTt must lie in the center of the algebra.13 While
we have not found a general proof that this implies unimodularity one can easily verify
that this is true for R-matrices of rank< 8. In the rank 2 case this is trivial to see.
For rank 4 the relevant algebras are classified in [36] and it is easy to check that only
unimodular examples satisfy the condition. For rank 6 the relevant algebras are classified
in [37] (nilpotent algebras are automatically unimodular) and again only unimodular ones
satisfy the condition. In addition we note that for AdS5, corresponding to the isometry
group SO(2, 4), the maximum rank of R is 8 [12], however it is easy to see that the 8dimensional
algebras in question have a trivial center and can therefore not lead to any
exception to the unimodularity condition. This rules out non-unimodular deformations of
AdSn with n ≤ 5 if G + B is invertible.
Therefore we conclude that for deformations of symmetric spaces non-unimodular Rmatrices
can lead to one-loop Weyl invariant σ-models only if G ± B of the undeformed
model is degenerate (with the caveat that we checked this only up to rank 6). In that
case they must satisfy (3.20) as well as the conditions we found at first order, namely
that ˜K = iKB generates isometries of G, H, Φ and equation (3.12).14 Examples of such
backgrounds were found in [15, 16].
We will now turn to the question of what happens at two loops, i.e. including the first
α -correction to the (super)gravity equations of motion. We will find that the conditions
at two loops as actually weaker. We will only need to satisfy the conditions we found at
first order in the deformation to solve also the two-loop equations.
4 Two-loop Weyl invariance
Here we will show that the α -correction to the equations of motion can be cast in a form
that is manifestly invariant under non-unimodular YB deformations satisfying the one-loop
Weyl invariance conditions of the previous section. In fact our calculation will be more
general. We will assume only that the following remain invariant under the transformation
in question
FABC , ∂A1 · · · ∂An (anything invariant) ,
FA
F
(±)
ABC , F
(±)
A ∂A
(anything invariant) . (4.1)
However, FA and its derivatives need not be invariant. As we have seen this is true for
any YB deformation that is one-loop Weyl invariant (3.14), (3.16) (it is trivially true for
unimodular deformations since in that case also FA is invariant under the deformation).
13
It also implies that the algebra can be constructed as a so-called symplectic double extension of a
lower-dimensional symplectic, or quasi-Frobenius, Lie algebra [35]. The question is then if the symplectic
double extension of a unimodular Lie algebra is always unimodular, in which case this condition would
imply unimodularity.
14
For general inhomogeneous deformations with WZ-term a more careful analysis, where the condition
of invariance of B is dropped, is required.
– 12 –
JHEP10(2020)065
To get the equations of motion at order α we must vary the corrected action (2.32)
using the expressions for the variations of the fluxes in (2.16). The variation with respect
to the generalized dilaton is easy and gives just the vanishing of the Lagrangian itself
R + aR(−)
+ bR(+)
= 0 . (4.2)
In the following we will set b = 0 to simplify the calculations. In the end our results will
apply also for b = 0. Displaying only the order α -terms that are not trivially invariant
under the YB deformation we have from (2.33)
R(−)
= −2∂A
FB
tr F
(−)
A F
(−)
B + FA
FB
tr F
(−)
A F
(−)
B + . . . (4.3)
where the ellipsis denotes terms involving only FABC, which are trivially invariant. Using
the invariance of the expressions in (4.1) we see that the r.h.s. is invariant. Therefore the
dilaton equation remains satisfied to order α for such deformations.
Varying the action (2.32) with respect to the generalized vielbein using (2.16) the terms
involving FA, i.e. the first two terms, in R(−) (2.33) give the following contributions to the
equations of motion
2(∂C
− FC
) (∂D
F
(+)
A + ∂
(+)
A FD
)F
(−)
DCB − (∂C − FC) (∂D
F(+)C
+ ∂(+)C
FD
)F
(−)
DAB
+ (∂
(−)
A FC
− ∂C
F
(+)
A ) tr F
(−)
C F
(−)
B − (∂C
F
(+)
A + ∂
(+)
A FC
) tr F
(−)
C F
(−−)
B
+ 2(∂C
FD
+ ∂D
FC
)F(+)E
CAF
(−)
DEB − (A ↔ B) + . . . , (4.4)
where we suppress terms that are manifestly invariant, i.e. constructed form the invariant
combinations in (4.1). The variation of the R2
AB-term in (2.33) gives rise to the terms
4∂C
∂
(+)
B FD
F
(−)
DCA + 4FC
(∂D
− FD
) ¯R
(−)
DBCA − 4∂C
FD
F
(++)
DBEF(−)E
CA
+ 2∂
(+)
A FC
tr F
(−)
C F
(−−)
B − 4∂(+)C
FD
F(+)E
CAF
(−)
DEB
+ 2FC
F
(++)
ACD tr F(−)D
F
(−−)
B + 4FCF(++)CEF
F(+)D
EAF
(−)
FDB
− 2FC
F(++)DE
A
¯R
(−)
DECB − 4FC
F(−)DE
B
¯R
(−)
ADEC
+ 4FCF(−)DEC ¯R
(−)
ADEB − (A ↔ B) + . . . , (4.5)
where we have noted that using the definition (2.27) we have
FA ¯R
(−)
ABCD = ∂
(+)
B FA
F
(−)
ACD − FA
F
(++)
ABE F(−)E
CD + . . . . (4.6)
Finally the variation of the CABC-term in (2.33) gives
∂AFC
tr F
(−)
B F
(−)
C − F
(+)
C FCDE
R
(−)
DEAB − 2FC
F(++)DE
B
¯R
(−)
DECA
− 4FC
F(+)DE
BR
(−)
DECA − ∂C FDF(++)CDE
F
(−)
EAB
+ 2(∂C
− FC
) FDF(++)DE
AF
(−)
ECB + 2FC∂D
F
(+)
DBEF(−)EC
A
– 13 –
JHEP10(2020)065
+ 2FC∂D
F
(++)
DBEF(−)EC
A − F
(+)
C ∂DFCDE
F
(−)
EAB − FCFDF(+)CDE
F
(−)
EAB
− 2FC
FD
F
(+)
CEAF(−)E
DB + 2FC
F
(+)
ACD tr F(−)D
F
(−)
B + FC
F
(++)
ACD tr F(−)D
F
(−)
B
− FC
F
(++)
ACD tr F(−)D
F
(−−)
B + 2FCF(++)CDE
FDA
F
F
(−)
EBF
+ FCF(+)EFC
F
(+)
EFDF(−)D
AB + 2FC
F
(+)
EFBF(+)EFD
F
(−)
DCA
− (A ↔ B) + . . . . (4.7)
Now we need to add together these three potentially non-invariant contributions to the
equations of motion.
Using the Bianchi identity for FA (2.9) and noting also that the second term in (4.4)
can be written
2∂
(+)
C ∂(C
FD)
F
(−)
DAB
= ∂C
FD
F
(++)
CDE F(−)E
AB − 2∂
(−)
C FD
F
(−)
DAB + 2FC
∂CFD
F
(−)
DAB + . . .
= ∂C
FD
[F
(++)
CDE + 2F
(+)
CDE]F(−)E
AB + 2FC
∂CFD
F
(−)
DAB + . . . (4.8)
we find, after a bit of algebra, that all terms involving only F
(+)
A can be eliminated leaving
the terms
8FC
∂D
∂
(+)
[A F
(−)
D]CB − 4FC
∂D
[F
(−)
AC
E
F
(−)
DEB] − 4FC
∂D
[F
(−)
AB
E
F
(−)
DCE]
− 4FC
F
(++)
AD
E
∂D
F
(−)
ECB − 8FC
F(−)DE
B∂
(+)
[A F
(−)
D]EC + 8FCF(−)DEC
∂
(+)
[A F
(−)
D]EB
+ 4FC
∂D
[F
(+)
DEAF(−)E
CB] − 8FC
FD
∂
(+)
[A F
(−)
D]CB − 4FC
∂
(+)
A FD
F
(−)
DCB
− 4FC
FD
F
(+)
CEAF(−)E
DB + 4FC
FD
F
(−)
AC
E
F
(−)
DEB + 4FCFDF
(−)
ABEF(−)CDE
− 8FCF(−)DEC
F
(++)
AD
F
F
(−)
FEB − 8FCF(−)DEC
F
(−)
AE
F
F
(−)
DFB
− 4FC
F
(−)
DECF(−)DEF
F
(−)
ABF − 4FC
F
(−)
AFCF
(−)
DEBF(−)DEF
− 4FC
F
(−)
FCBF
(+)
DEAF(+)DEF
− 2FC ∂(−)C
F(+)D
+ (∂E − FE)F(+)CDE
− F(−)EFC
F
(+)
FE
D
F
(−)
DAB
+ 2 ∂
(+)
C F
(−)
A − FE
F
(−)
CAE tr F
(−)
B F(−)C
− (A ↔ B) + . . . (4.9)
The last two terms drop out using the lowest order equations of motion (2.17). We now
rewrite the first term as
8FD
∂C
∂
(−)
[B F
(+)
D]AC − 8FD
∂C
∂
(+)
[A F
(−)
C]BD + ∂
(−)
[B F
(+)
D]AC
= − 8FD
∂
(−)
[D ∂
(−)
B] F
(+)
A + (∂C
−FC
)F
(+)
B]AC −F
(−)
EFBF(+)FE
A − 4FD
F
(−−)
BDE∂E
F
(+)
A
− 4FD
F
(−)
EBD∂E
F
(+)
A + 8FD
F
(+)
[D|AC|∂
(−)
B] F(+)C
− 8FC
FD
∂
(+)
[A F
(−)
C]BD
+ 4FDF
(+)
FEA∂
(−)
B F(−)EFD
+ 8FDF(−)EFD
∂
(+)
[A F
(−)
E]BF + 4FD
F
(+)
BC
E
∂C
F
(+)
DAE
− 4FD
F(−)E
CB∂C
F
(+)
DAE − 8FD
(∂C
− FC
) ∂
(+)
[A F
(−)
C]BD + ∂
(−)
[B F
(+)
D]AC
− 8FEF(−)CDE
∂
(+)
[A F
(−)
C]BD + ∂
(−)
[B F
(+)
D]AC + . . . (4.10)
– 14 –
JHEP10(2020)065
The first term vanishes by the lowest order equations of motion (2.17). In the last two
terms we can use the Bianchi identity for FABC (2.9), which implies in particular that
2∂
(+)
[A F
(−)
C]BD + 2∂
(−)
[B F
(+)
D]AC =F
(−)
AB
E
F
(−)
CED − F
(+)
BA
E
F
(+)
DCE + F
(−)
AD
E
F
(−)
CBE
− F
(+)
DA
E
F
(+)
BEC + F
(++)
AC
E
F
(−)
EBD + F(+)E
ACF
(−−)
BDE . (4.11)
After a bit of algebra we are left with
8FD
F
(+)
[D|A|
C
∂
(−)
B] F
(+)
C + (∂E
− FE
)F
(+)
B]CE − F(−)EF
B]F
(+)
FEC
− 4FD
F
(−−)
BD
E
∂
(−)
E F
(+)
A + (∂C
− FC
)F
(+)
EAC − F
(−)
CFEF(+)FC
A
− 4FEF(−)CDE
F
(−)
AD
F
F
(−)
CFB − F
(+)
DA
F
F
(+)
BFC + F
(++)
AC
F
F
(−)
FDB + F(+)F
ACF
(−−)
BDF
− 8FD
F(−)CE
B ∂
(+)
[A F
(−)
C]ED + ∂
(−)
[E F
(+)
D]AC + 4FD
F
(−−)
BDEF(−)CEF
F
(+)
FCA
− 4FC
F
(−)
AFCF
(−)
DEBF(−)DEF
+ 4FC
F(+)DE
A ∂EF
(−−)
BCD − ∂
(−)
B F
(−)
ECD + ∂DF
(−)
ECB
− 4FC
F
(−)
FCBF
(+)
DEAF(+)DEF
− (A ↔ B) + . . . . (4.12)
The first two terms vanish by the lowest order equations of motion and the remaining terms
cancel using the Bianchi identity for FABC. This completes the proof that the α -correction
to the equations of motion can be cast in a manifestly invariant form provided that the
expressions in (4.1) are invariant. In particular this implies that if a YB deformation
preserves Weyl invariance at one-loop it also preserves it at two loops.
5 Conclusions
We have analyzed the conditions for a YB deformation of the bosonic/heterotic string
sigma-model to be Weyl-invariant at one loop, i.e. for the corresponding background to be
a (super)gravity solution. When (G + B)mn of the undeformed background is invertible
one finds no solution in the inhomogeneous case (although our analysis for the YB model
with WZ-term is not quite complete). For a homogeneous deformation of a symmetric
space one finds that the distinguished Lie algebra element Rrsfrs
tTt must belong to the
center of the algebra. We showed that, at least for rank R < 8, this in fact implies the
usual unimodularity condition Rrsfrs
t = 0 of [12]. When (G + B)mn of the undeformed
background is non-invertible instead the unimodularity condition is replaced by the weaker
conditions (3.12), (3.20) together with the condition that ˜K = iKB generate isometries of
the undeformed background G, H, Φ. This is consistent with what has been seen in specific
examples [15, 16] and the conditions we find agree with those coming from an analysis of
generalized supergravity, see appendix E of [16], when specifying to YB deformations. We
have also seen that when these conditions are satisfied the deformation in fact preserves
Weyl-invariance at least to two loops, i.e. the background solves the low-energy effective
string equations including the first α -correction.
Interestingly, while in the case of unimodular deformations the fact that the two-loop
equations are satisfied is trivial in the doubled formulation we are using, this is not the
– 15 –
JHEP10(2020)065
case for non-unimodular ones due to the shift of FA by the generalized Killing vector KA.
In fact it took quite a bit of work to show that the equations of motion can be cast in
a form where it is easy to see that they are invariant under the deformation. It would
be interesting to understand if one can improve the formulation so that the invariance is
manifest also in the non-unimodular case and, if so, what this implies for the structure of
higher-derivative corrections. Perhaps the natural starting point to analyzing this question
is the gauged version of DFT [20].
For unimodular YB deformations the first α -correction to the deformed background
was derived in [19], also by using the doubled formulation. The same correction is valid
also for the non-unimodular examples discussed here.
It would be interesting to extend our analysis to the general case of inhomogeneous
YB deformation with WZ-term by relaxing the requirement that B is invariant under the
isometries. The conditions must become essentially the same in that case since they are
mostly fixed by the generalized supergravity analysis. It would be interesting to understand
if there exist any non-unimodular Weyl-invariant examples in that case. It seems unlikely
to be the case since R is much more constrained than in the homogeneous case.
Finally, it would be interesting to extend the present analysis to the case of Poisson-Lie
T-duality, for which the first α -correction was recently found [38–40] using essentially the
same approach as for YB.
Acknowledgments
We thank R. Borsato, D. Marqu´es and S. van Tongeren for interesting discussions and
R. Borsato for comments on the manuscript. The work of LW is supported by the grant
“Integrable Deformations” (GA20-04800S) from the Czech Science Foundation (GAˇCR).
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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Quantum Correction to Generalized T Dualities
Riccardo Borsato *
Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
Linus Wulff †
Department of Theoretical Physics and Astrophysics, Masaryk University, 611 37 Brno, Czech Republic
(Received 22 July 2020; accepted 29 September 2020; published 13 November 2020)
Poisson-Lie duality is a generalization of Abelian and non-Abelian T duality, and it can be viewed as a
map between solutions of the low-energy effective equations of string theory, i.e., at the (super) gravity
level. We show that this fact extends to the next order in α0
(two loops in σ-model perturbation theory)
provided that the map is corrected. The α0 correction to the map is induced by the anomalous Lorentz
transformations of the fields that are necessary to go from a doubled OðD; DÞ-covariant formulation to the
usual (super)gravity description.
DOI: 10.1103/PhysRevLett.125.201603
Introduction.—The notion of T duality [1,2] is central in
string theory. It says that a closed string on a background
with Abelian isometries has another description as a string
on a dual background. In the simplest case of T duality on a
circle, the duality acts by inverting the radius of the circle.
More generally, backgrounds may have non-Abelian isometry
groups, and at least at the classical level there is indeed
a generalization to a non-Abelian version of T duality [3].
Unlike in the Abelian case, non-Abelian T duality (NATD)
does not generically preserve the isometries of the background,
and it is therefore not obvious how to invert the
transformation. This problem was overcome by Klimčík
and Ševera in [4,5]. They realized that the map can be made
invertible by relaxing the notion of isometry. One requires
the background to have instead so-called Poisson-Lie (PL)
symmetry, namely to possess vector fields vi, with
½vi; vjŠ ¼ −fij
k
vk, under which the metric and B field of
the σ-model transform as
Lvi
Mmn ¼ −˜fjk
ivj
p
vk
q
MmpMqn; ð1Þ
where Mmn ¼ Gmn − Bmn and ˜fjk
i are structure constants
of a dual Lie algebra. This more general notion of
symmetry allows to define a dual background (see below).
This construction became known as “Poisson-Lie T duality”
since the group structure underlying it is that of a PL
group. The σ models on the original and dual backgrounds
are classically equivalent being related by a canonical
transformation [6]. When the dual structure constants ˜f
vanish, vi generate standard isometries, and one recovers
(N)ATD.
At the world sheet quantum level, i.e., including string α0
corrections, things are more subtle. While Abelian T
duality remains a symmetry of the world sheet conformal
field theory to all orders in α0
, it was quickly realized that
NATD cannot be a symmetry at the quantum level [7]. At
best it can map one world sheet conformal field theory to
another—inequivalent—one. It can therefore be used to
generate new string backgrounds from old ones. Except for
an anomaly when dualizing nonunimodular groups [8,9],
this has been shown to work to zeroth order in α0
, i.e., at the
low-energy (supergravity) level of the string effective
equations, which corresponds to one loop order in σ-model
perturbation theory. Similar results are known for PL
duality, see, e.g., [10,11]. It has been a long-standing
problem whether PL and NATD can be extended beyond
this lowest order.
Here, we show that PL duality can be extended to order
α0
, i.e., two loops in the σ-model perturbation theory,
provided that the map is corrected. A special case of our
results gives the corrections to NATD. When specifying to
the Abelian case we recover the results of [12].
To find this correction we exploit a powerful formulation
of the string effective equations inspired from double field
theory (DFT). It has long been known that the bosonic
string compactified on a d torus has an Oðd; dÞ T duality
symmetry [13]. DFT is a field theory where this symmetry
is made manifest form the start [14–18] and is therefore
well suited to working with T duality. This is achieved by
doubling the dimension of the physical manifold, and by
imposing a “section condition” which effectively eliminates
the dependence on half of the coordinates, giving the
correct dimension in the end (D ¼ 26 for the bosonic string
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Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
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.
PHYSICAL REVIEW LETTERS 125, 201603 (2020)
0031-9007=20=125(20)=201603(6) 201603-1 Published by the American Physical Society
and D ¼ 10 for the superstring). Here, we always work
with the standard choice of section, so that the background
depends only on the physical coordinates. In this formulation
it is rather the dimension of the tangent space that is
doubled, and we have two copies of the Lorentz group
instead of one [19]. The standard Lorentz group is the
diagonal of the doubled one, and under this breaking the
equations of DFT reduce to the standard string effective
equations, at lowest order in α0
. A crucial point is that at the
quantum level it is impossible to preserve both the OðD; DÞ
and the Lorentz covariance of the fields [20–23]. If we
insist on fields which transform nicely under T duality and
OðD; DÞ, they must transform noncovariantly under
Lorentz transformations [24] (see [25] for another manifestation
of this fact). The fact that the Lorentz transformation
of the fields receives corrections at order α0
makes the discussion of the Lorentz invariance of the
theory nontrivial. But this can be turned into a virtue rather
than a shortcoming. In fact the α0
correction to the Lorentz
transformation fixes the correction to the DFT action [24].
Remarkably, this α0-corrected OðD; DÞ-covariant action
correctly reproduces the α0
corrections to the bosonic
and heterotic string effective actions [24].
Our strategy is to use the rewriting of PL duality in the
doubled language, where it takes a natural form, see, e.g.,
[26–30]. The basic fields of the formulation we use, the
“generalized fluxes,” turn out to be invariant under PL
duality. Since the string effective equations, including the
first α0
correction, can be written in terms of the generalized
fluxes [24,31], at least to this order PL duality maps
solutions of the doubled equations to solutions. At the
standard (super)gravity level there are in fact explicit
corrections to the PL duality rules. They arise from the
noncovariance of the doubled fields under the double
Lorentz transformation needed to gauge fix down to the
diagonal subgroup and to go to the standard (nondoubled)
description. See Fig. 1 for a summary. This strategy was
used in [32] to find the α0
correction to the “homogeneous
Yang-Baxter deformations” (related to NATD [33,34]) and
it works for any OðD; DÞ transformation leaving the
generalized fluxes invariant.
Poisson-Lie duality.—In PL duality, fij
k
and ˜fij
k are
interpreted as structure constants of Lie groups denoted by
G and ˜G. These are combined into a “Drinfel’d double” D
whose Lie algebra is generated by TI ¼ ðTi; ˜Ti
Þ, where Ti
are generators of LieðGÞ, and ˜Ti
of Lieð ˜GÞ. Obviously
LieðGÞ and Lieð ˜GÞ are subalgebras of D but there are also
mixed commutation relations
½Ti; TjŠ ¼ fij
kTk; ½ ˜Ti
; ˜Tj
Š ¼ ˜fij
k
˜Tk
;
½Ti; ˜Tj
Š ¼ ˜fjk
iTk − fik
j ˜Tk
: ð2Þ
Importantly, D is endowed with the invariant symmetric
bilinear form hTI; TJi defined by
hTi; Tji ¼ h ˜Ti
; ˜Tj
i ¼ 0; hTi; ˜Tj
i ¼ δj
i : ð3Þ
Having introduced D we can now present PL duality as
an invertible map between an “original” background
(specified by a metric Gmn, a Kalb-Ramond field Bmn,
and a dilaton Φ) and another “dual” background (with
fields ˜Gmn; ˜Bmn, and ˜Φ). We split the coordinates of the
original background as xm
¼ ðyσ
; xμ
Þ, where yσ
are coordinates
on the group G to be dualized, and xμ
are
coordinates that play the role of spectators under the
dualization. Similarly, for the dual background we have
˜xm ¼ ð˜yσ; xμÞ with ˜yσ coordinates on ˜G. The y and ˜y
dependence is in fact encoded in the group elements gðyÞ ∈
G and ˜gð˜yÞ ∈ ˜G featuring below. To present the map
between the original and dual backgrounds we first need
the fact that the condition (1) implies that Mmn ≡ Gmn −
Bmn is of the form
M ¼ U _Mð1 þ Π _MÞ−1
UT
; ð4Þ
where we suppressed matrix indices for readability. The
matrix Um
r depends only on y and it is of block form with
nonvanishing components Uμ
ν
¼ δμ
ν
and Uσ
i
¼ uσ
i
, the
latter being the components of the Maurer-Cartan form u ¼
g−1vg ¼ g−1dg ¼ dyσuσ
iTi [35]. The matrix Πrs depends
only on y and its only nontrivial components are
Πij
¼ hAd−1
g ∘P∘Adg
˜Ti
; ˜Tj
i; ð5Þ
where AdgX ¼ gXg−1
and P is the projector on LieðGÞ.
Notice that in general Π ≠ 0 thanks to the mixed commutation
relation of D if ˜fij
k ≠ 0. The map between Mmn and
˜Mmn is achieved by relating both of them to _Mrs, a matrix
depending only on spectators xμ
and on which no other
condition is imposed [36]. The dual background ˜Mmn is
obtained by [37]
˜M ¼ ˜U½ð _M þ ˜ΠÞP þ ¯PŠ−1
ð _M ¯P þPÞ ˜UT
; ð6Þ
FIG. 1. Starting with the PL duality map for the doubled fields
ðE; dÞ, the map for the standard (super)gravity fields ðG; B; ΦÞ is
obtained after a double Lorentz transformation ðΛðþÞ
; Λð−Þ
Þ ¼
ðΛ; 1Þ to set eðþÞ
¼ eð−Þ
, thus breaking the double Lorentz group
down to its diagonal subgroup. The α0
corrections to the PL
duality map follow from the anomalous Lorentz transformations
of the fields.
PHYSICAL REVIEW LETTERS 125, 201603 (2020)
201603-2
where ˜Uμ
ν
¼ δμ
ν
, ˜Uσ
i
¼ ˜uσjδji
and P, ¯P project on indices
i, j and μ, ν, respectively. As previously ˜u ¼ ˜g−1
d˜g ¼
d˜yσ ˜uσi
˜Ti
is a Maurer-Cartan form, and now ˜Πij ¼
hAd−1
˜g ∘ ˜P∘Ad˜gTi; Tji, where ˜P projects on Lieð ˜GÞ.
Finally, the dilatons of the two backgrounds are related
by [38]
expð−2ΦÞ
ðdet GÞ1=2
det u
¼ expð−2 ˜ΦÞ
ðdet ˜GÞ1=2
det ˜u
: ð7Þ
Taking ˜G Abelian (˜fij
k ¼ 0) implies Π ¼ 0 and Eq. (4)
simplifies to M ¼ U _MUT
, encoding the usual consequences
of having isometries for M. Then parametrizing
the Abelian group as ˜g ¼ expð˜yi
˜Ti
Þ with ˜yi ¼ ˜yσ
δσi it
follows that ˜uσi ¼ δσi, ˜Πij ¼ ˜ykfij
k
, and from (6) and (7)
we recover the rules of NATD in the presence of spectators
[39]. Even simpler is the case when also G is Abelian
(fij
k
¼ 0) so that M is invariant under dim(G) Uð1Þ
isometries. Then also ˜Π ¼ 0 and Eq. (6) implements dim
(G) factorized T dualities, reducing to the celebrated
Buscher rules when only one isometry is dualized.
Double formulation.—The nonlinear maps in (4) and (6)
admit a much simpler and linear formulation in the doubled
language, where one works with matrices OM
N
of dimension
2D × 2D. These are elements of the group OðD; DÞ,
meaning that OM
P
ON
Q
ηPQ ¼ ηMN where
ηMN ¼

0 δm
n
δm
n
0

: ð8Þ
In fact let us construct the (inverse) “generalized vielbein”
which we parametrize as
EA
M
¼
1
ffiffiffi
2
p
eðþÞan
Mnm eðþÞam
−e
ð−Þn
a Mnm e
ð−Þm
a
!
; ð9Þ
where A is a flat index and M curved. We use similar
parametrizations for ˜EA
M
and _EA
R
, adding tildes and dots.
Above, eðÆÞ are two possible vielbeins for the metric Gmn.
They are not necessarily equal and in general they are
related by a nontrivial Lorentz transformation. Each of
them transform under only one of the two copies of the
Lorentz group [distinguished by the (þ) and (−)] arising in
the doubled formulation. The generalized vielbein is one of
the main ingredients of the “framelike formulation” of
DFT, and it will be important for our derivation of the
“unimodularity condition” (18) and the α0
corrections to PL
duality. It is straightforward to check that the relations (4)
and (6) are equivalent to the relations
E ¼ _Eð1 þ ΠÞU; ˜E ¼ _Eð1 þ ˜ΠÞ ˜U; ð10Þ
where we suppressed indices. In our notation all dotted
quantities only depend on xμ
. The nonvanishing
components of ΠR
S
and ˜ΠR
S
are again only Πij
and ˜Πij
and the antisymmetry properties Πij
¼ −Πji
and ˜Πij ¼
− ˜Πji imply that ð1 þ ΠÞ; ð1 þ ˜ΠÞ are elements of OðD; DÞ.
The matrices U; ˜U are also elements of OðD; DÞ with
Ui
σ
¼ ui
σ
, Ui
σ ¼ uσ
i
, ˜Uiσ
¼ ˜uiσ
, ˜Uiσ ¼ ˜uσi, Uμ
ν
¼ Uμ
ν ¼
˜Uμ
ν
¼ ˜Uμ
ν ¼ δμ
ν
. We are using a notation so that uσ
i and
˜uiσ
are the inverses of uσ
i
and ˜uσi, respectively. To match
(4) and (6) with (10) one finds that the (þ) and (−)
vielbeins must transform differently
e
ðÆÞm
a ¼ _e
ðÆÞs
a OðÆÞs
r
ðU−1
Þr
m
;
˜e
ðÆÞm
a ¼ _e
ðÆÞs
a ˜OðÆÞs
r
ð ˜U−1
Þr
m
; ð11Þ
where
OðþÞ ¼ 1 þ _MΠ; ˜OðþÞ ¼ ¯P þ ð ˜Π þ _MÞP;
Oð−Þ ¼ 1 − _MT
Π; ˜Oð−Þ ¼ ¯P þ ð ˜Π − _MT
ÞP: ð12Þ
In both cases the (þ) and (−) vielbeins are then related by
Lorentz transformations as e
ð−Þm
a ¼ Λa
b
e
ðþÞm
b and ˜e
ð−Þm
a ¼
˜Λa
b
˜e
ðþÞm
b where
Λ ¼ _e−1
Oð−ÞO−1
ðþÞ _e; ˜Λ ¼ _e−1 ˜Oð−Þ
˜O−1
ðþÞ _e; ð13Þ
if we fix _e ¼ _eðþÞ
¼ _eð−Þ
. Finally, the transformation (7) is
translated into
d þ
1
2
log det u ¼ _d ¼ ˜d þ
1
2
log det ˜u; ð14Þ
where d; _d; ˜d are called “generalized dilatons” and are
parametrized as in d ¼ Φ − 1
4 log det G [41].
PL duality as a map between string backgrounds.—The
double formulation is very useful because in this language
it is very simple to prove that the PL duality transformation
is a solution generating technique in string
theory, at least to leading and subleading order in the α0
expansion. From EA
M
and d one can construct the
generalized fluxes
FABC ¼ 3E½A
M
∂MEB
N
ECŠN;
FA ¼ EBM∂MEB
NEAN þ 2EA
M∂Md; ð15Þ
that are the dynamical fields of the framelike formulation
of DFT. In fact the DFTequations of motion can be written
only in terms of the above fluxes and their flat derivatives
∂AF ¼ EA
M∂MF, both at leading and subleading order in
the α0
expansion [31]. Under the transformation (10) we
have
FABC ¼ 3 _E½A
½A
∂μ
_EB
N _ECŠN
þ 3 _E½A
i _EB
j _ECŠkfij
k
þ 3 _E½A
i _EBj
_ECŠk
˜fjk
i; ð16Þ
PHYSICAL REVIEW LETTERS 125, 201603 (2020)
201603-3
FA ¼ _EBμ
∂μ
_EB
N _EAN þ 2 _EA
μ
∂μ
_d
þ _EA
i
fij
j
− _EAið˜fij
j þ fjk
i
Πjk
Þ: ð17Þ
For the reader’s convenience we give the details of the
computation in the Supplemental Material [37]. The
results for the dual background are analogous upon
exchanging tilded and untilded quantities, and appropriately
raising or lowering i, j, k indices. Because of the
symmetry of (16) under this transformation, it immediately
follows that FABC ¼ ˜FABC. Equation (17) instead is
not symmetric under this transformation, but it becomes
symmetric if we impose the tracelessness of the structure
constants
fij
j
¼ 0; ˜fij
j ¼ 0; ð18Þ
as detailed in the Supplemental Material. When this
“unimodularity condition” holds we have simply
FA ¼ _EBμ
∂μ
_EB
N _EAN þ 2 _EA
μ
∂μ
_d; ð19Þ
and FA ¼ ˜FA immediately follows. Notice that not
only both fluxes but also their flat derivatives are
invariant under the PL transformation. In fact, since
they only depend on spectators xμ
it follows
that EA
M
∂MF ¼ EA
μ
∂μF ¼ ˜EA
μ
∂μ
˜F ¼ ˜EA
M ˜∂M
˜F.
If we start from a string background, or in other words
given a model with EA
M
and d of the PL form (10) and (14)
that satisfies the doubled equations of motion to zeroth and
first order in α0
, we conclude that the dual model given by
˜EA
M
and ˜d also satisfies the same equations, at least when
(18) holds. This observation extends to higher orders under
the assumption that there exists a formulation of the string
effective action in terms of the generalized fluxes and their
flat derivatives [31] also at higher orders in α0
. This in turn
should be true as long as it is possible to make diffeomorphisms,
B-field gauge transformations, and OðD; DÞ
symmetry manifest.
This is a proof that PL duality is a solution generating
technique in string theory at least when both structure
constants are traceless, as found already to lowest order in
[42]. When ˜G is Abelian this condition reduces to the
unimodularity condition for NATD [8,9].
α0
corrections.—So far, we have shown that in the
doubled formulation the PL duality transformation works
and remains uncorrected at least to order α0
. Note that the
assumption is that the DFT equations are satisfied without
the need of correcting the OðD; DÞ form (10) of the PL
transformation, and therefore only _M and _d in (10) and (14)
can depend on α0.
The description of the two models in terms of standard
(i.e., nondoubled) fields ðG; B; ΦÞ and ð ˜G; ˜B; ˜ΦÞ is different,
and the PL duality transformation between these does
receive α0
corrections. The reason is that when going from a
doubled to a standard (super)gravity formulation we must
first perform a double Lorentz transformation to set the two
vielbeins eðþÞ
and eð−Þ
equal [24]. At order α0
the fields of
the doubled formulation transform noncovariantly under
local Lorentz transformations, and this induces extra α0
corrections also for the standard fields. The situation is
illustrated in Fig. 1. Because of the noncovariance even
under the diagonal of the double Lorentz group, we say that
the reduction from the doubled to the standard formulation
picks a specific noncovariant “scheme,” which we call the
scheme of DFT. To translate our results into the covariant
schemes of [43–46] one must implement α0
-dependent field
redefinitions. We provide a dictionary [47] in the
Supplemental Material [37].
The first α0
correction to Mmn induced by the compensating
double Lorentz transformation with ΛðþÞ
and Λð−Þ
is [48]
aΔ
ð−Þ
Λð−Þ M
ðDFTÞ
nm þ bΔ
ðþÞ
ΛðþÞ M
ðDFTÞ
mn ; ð20Þ
where a ¼ b ¼ −α0
for the bosonic string and a ¼ −α, b ¼
0 for the heterotic string (and a ¼ b ¼ 0 for type II). The
finite form of the anomalous transformations is [37]
Δ
ðÆÞ
Λ M
ðDFTÞ
mn ¼
1
2
trð∂mΛΛ−1
ω
ðÆÞ
n Þ − B
WZW;ðΛÞ
mn
þ
1
4
trð∂mΛΛ−1
∂nΛΛ−1
Þ; ð21Þ
where ω
ðÆÞb
ma ¼ ωma
b
Æ 1
2 Hma
b
and ω is the spin connection
for the vielbein e after the diagonal gauge fixing. The
WZW-like contribution to B is defined by
dBWZW;ðΛÞ
¼ −
1
12
trðdΛΛ−1
dΛΛ−1
dΛΛ−1
Þ: ð22Þ
The α0
corrections to the original model can be obtained,
for example, after choosing e ¼ eð−Þ
and doing the double
Lorentz transformation on eðÆÞ
to achieve the diagonal
gauge with ðΛðþÞ
; Λð−Þ
Þ ¼ ðΛ; 1Þ and Λ given in (13) [49].
Then the correction is
ΔMðDFTÞ
¼ bΔ
ðþÞ
Λ MðDFTÞ
þ α0
Uð1 þ _MΠÞ−1
Δ _Mð1 þ Π _MÞ−1
UT
; ð23Þ
where the second term comes when expanding (4) with the
α0
corrections _M → _M þ α0
Δ _M. Notice that for the heterotic
string (b ¼ 0) the PL map is uncorrected in the DFT
scheme in the gauge e ¼ eð−Þ
[50]. For the dual background
the same reasoning applies, and choosing ˜e ¼ ˜eð−Þ
PHYSICAL REVIEW LETTERS 125, 201603 (2020)
201603-4
Δ ˜MðDFTÞ
¼ bΔ
ðþÞ
˜Λ
˜MðDFTÞ
þα0 ˜Uð _MPþ ˜Πþ ¯PÞ−1
Δ _Mð−P ˜U−1 ˜M þ ¯P ˜UT
Þ:
ð24Þ
The transformation of the dilatons follows from the fact
that the generalized dilaton (14) is not anomalous under
Lorentz [24] and that the parametrization in terms of
standard metric and dilaton holds to α0
order. Then
ΔΦðDFTÞ
¼ α0
Δ_d þ
1
4
Gmn
ΔG
ðDFTÞ
mn ;
Δ ˜ΦðDFTÞ
¼ α0Δ_d þ
1
4
˜Gmn
Δ ˜G
ðDFTÞ
mn ; ð25Þ
where we allowed an α0
correction _d → _d þ α0
Δ_d.
We refer to the Supplemental Material for an example of
a computation of such α0
corrections, and for Refs. [51,52].
When specifying the map to a single Uð1Þ T duality,
Eqs. (24) and (25) reproduce the rules written in [12] by
Kaloper and Meissner, as proved in [32].
Conclusions.—In this Letter, we employed the framelike
formulation of DFT to show that [when the conditions in
(18) hold] PL duality is a map between solutions of the lowenergy
effective string equations at least to first order in α0
and quite possibly to all orders. We did this for a twoparameter
family of theories interpolating between the
bosonic and the heterotic string (when the gauge fields
and fermions of the latter are set to zero). It would be
interesting to generalize these results to the case in which
G, for example, is replaced by the coset G=H.
The importance of Eqs. (23), (24), and (25) is twofold.
First, they provide the necessary quantum corrections to the
PL duality transformation rules in order to extend the map
to order α0
. Second, they imply that the form of the α0
corrections of backgrounds admitting PL symmetry is
strongly constrained by the PL symmetry itself [53]. In
particular, Eqs. (23) and (25) can be interpreted as an
efficient way to compute α0
corrections for PL symmetric
backgrounds, since the only unknowns are Δ _M and Δ_d,
and they can be found by imposing the order α0
equations of
motion. This is much simpler than trying to compute the
corrections directly for M and Φ. It would be interesting to
see if, when considering nonconformal σ models, the α0
corrections that we find preserve the form of the β
functions.
The work of R. B. is supported by the fellowship of “la
Caixa Foundation” (ID 100010434) with code LCF/BQ/
PI19/11690019, by Agencia Estatal de InvestigaciónSpain
(FPA2017-84436-P and Unidad de Excelencia
María de Maetzu MDM-2016-0692), by Xunta de
Galicia-Consellería de Educación (Centro singular
de investigación de Galicia accreditation 2019–2022,
ED431C-2017/07 and ED431G2019/05), and by
FEDER. The work of L. W. is supported by the grant
“Integrable Deformations” (GA20-04800S) from the Czech
Science Foundation (GACR).
Note added.—When this work was being written up we
learned of the closely related independent work [54], and
shortly after of [55].
*
riccardo.borsato@usc.es
†
wulff@physics.muni.cz
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[35] We will use indices r, s where r ¼ ði; μÞ to distinguish them
from indices m, n where m ¼ ðσ; μÞ.
[36] We obviously have to assume invertibility of the inverse
matrices in (4) and (6).
[37] See the Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.125.201603 for a rewriting
of this formula more familiar to the usual form of
the (N)ATD rules. In the case of no spectators (and
assuming invertibility of _M) Eq. (4) reduces to M ¼
uð _M−1
þ ΠÞ−1
uT
and Eq. (6) to ˜M ¼ ˜uð _M þ ˜ΠÞ−1 ˜uT
,
which are more symmetric and more familiar in the PL
duality literature. Also, we review how to obtain the finite
form of the transformation used in (21) from the infinitesimal
one of [24], as explained in [32].
[38] R. Von Unge, J. High Energy Phys. 07 (2002) 014.
[39] Compared to [40] here ˜yi ¼ −νi and we swap ˜eðÆÞ
appearing later.
[40] R. Borsato and L. Wulff, J. High Energy Phys. 08 (2018) 027.
[41] Because _d depends only on xμ
it follows that ∂σΦ ¼
1
2 ∂σ log det M − ∂σ log det u and similarly for ˜Φ.
[42] A. Bossard and N. Mohammedi, Nucl. Phys. B619, 128
(2001).
[43] E. Bergshoeff and M. de Roo, Phys. Lett. B 218, 210 (1989).
[44] E. A. Bergshoeff and M. de Roo, Nucl. Phys. B328, 439
(1989).
[45] R. R. Metsaev and A. A. Tseytlin, Nucl. Phys. B293, 385
(1987).
[46] C. M. Hull and P. K. Townsend, Nucl. Phys. B301, 197
(1988).
[47] R. Borsato and L. Wulff, J. High Energy Phys. 03 (2020)
126.
[48] Notice the first transposition.
[49] Notice that it is not possible to set directly eðþÞ
¼ eð−Þ
without the help of a Lorentz transformation, as this would
be incompatible with the assumption that _eðÆÞ
depend only
on xμ
. Therefore we used the gauge _e ¼ _eðþÞ
¼ _eð−Þ
.
[50] Alternatively, one may choose e ¼ eðþÞ
and the anomalous
terms would be aΔ
ð−Þ
Λ−1 M
ðDFTÞ
nm .
[51] A. Eghbali, Phys. Rev. D 99, 026001 (2019).
[52] A. Eghbali, R. Naderi, and A. Rezaei-Aghdam,
arXiv:2002.00675.
[53] This is under the assumption that the PL symmetry is not
anomalous.
[54] F. Hassler and T. Rochais, arXiv:2007.07897.
[55] T. Codina and D. Marques, arXiv:2007.09494.
PHYSICAL REVIEW LETTERS 125, 201603 (2020)
201603-6
JHEP07(2020)103
Published for SISSA by Springer
Received: April 23, 2020
Accepted: June 27, 2020
Published: July 16, 2020
The first α -correction to homogeneous Yang-Baxter
deformations using O(d, d)
Riccardo Borsato,a Alejandro Vilar L´opeza and Linus Wulffb
a
Instituto Galego de F´ısica de Altas Enerx´ıas (IGFAE), Universidade de Santiago de Compostela,
Santiago de Compostela, Spain
b
Department of Theoretical Physics and Astrophysics, Masaryk University,
611 37 Brno, Czech Republic
E-mail: riccardo.borsato@usc.es, alejandrovilar.lopez@usc.es,
wulff@physics.muni.cz
Abstract: We use the O(d, d)-covariant formulation of supergravity familiar from Double
Field Theory to find the first α -correction to (unimodular) homogeneous Yang-Baxter
(YB) deformations of the bosonic string. A special case of this result gives the α -correction
to TsT transformations. In a suitable scheme the correction comes entirely from an induced
anomalous double Lorentz transformation, which is needed to make the two vielbeins
obtained upon the YB deformation equal. This should hold more generally, in particular
for abelian and non-abelian T-duality, as we discuss.
Keywords: Bosonic Strings, String Duality
ArXiv ePrint: 2003.05867
Open Access, c The Authors.
Article funded by SCOAP3
.
https://doi.org/10.1007/JHEP07(2020)103
JHEP07(2020)103
Contents
1 Introduction 1
2 O(d, d) covariant formulation of supergravity 3
3 Yang-Baxter deformations in O(d, d) language 5
4 The α -correction to YB deformations 7
4.1 Compensating anomalous transformation 8
4.2 The correction to Yang-Baxter deformations 9
4.3 Manifestly covariant form of the correction 11
4.4 Tests on examples 12
5 T-duality and TsT transformations 13
5.1 Corrections to TsT transformations 17
6 Concluding comments 19
A Field redefinitions between different schemes 21
1 Introduction
Yang-Baxter deformations were originally constructed as deformations of the Principal Chiral
Model and (super)coset sigma models with the interesting property that they preserve
integrability [1–4]. The deformations are built using an R-matrix which solves the classical
Yang-Baxter equation (CYBE)
[RX, RY ] − R([RX, Y ] + [X, RY ]) = c2
[X, Y ] , ∀X, Y ∈ g , (1.1)
where c = 0 gives the standard CYBE equation and c = 0 corresponds to the modified
CYBE. We will consider only the c = 0 case here, for which the deformed models are often
called homogeneous YB models. It was shown in [5, 6] that homogeneous deformations
can be generated using non-abelian T-duality. One simply adds a closed, non-degenerate,
B-field defined on a subalgebra of the isometry algebra and dualizes on that subalgebra.1
This construction means that these deformations can be defined for a general sigma model
as long as it admits isometries that can be dualized. In particular the YB deformation of
the Green-Schwarz superstring was constructed in [7]. A special case of this deformation
1
From this construction one obtains a deformation of non-abelian T-duality, but it is possible to show
that a local field redefinition (i.e. a diffeomorphism in target space) plus a shift of the B-field permit to
rewrite the result as a homogeneous YB deformation. See [5, 6] for more details.
– 1 –
JHEP07(2020)103
is when the isometries are abelian and in that case the deformed model is simply a Tduality-shift-T-duality
(TsT) transformation [8], which are usually called β-shifts or βtransformations
in the context of O(d, d).
Just as in non-abelian T-duality [9, 10], these models may in principle have a Weyl
anomaly. When the anomaly is present the target space fields do not solve the standard
supergravity equations but a generalization of these [11, 12]. Similar to the non-abelian Tduality
case this anomaly is absent if one requires the R-matrix to satisfy a unimodularity
condition [13]. This is the case we consider here although unimodularity is not a necessary
condition to avoid a Weyl anomaly [14–16].
The realization of homogeneous YB models using T-duality makes it natural to try to
describe these models using the O(d, d)-covariant language of Double Field Theory (DFT),
as was done starting with the work of [17]. In fact the YB deformations take the form
of a so-called β-transformation [14, 18] in O(d, d) language. This language is particularly
useful since, as we will show in this paper, unimodular homogeneous YB deformations leave
the generalized fluxes — the basic building blocks in the O(d, d)-covariant formalism —
invariant (see also [18, 19]). With this observation it becomes very simple to prove that the
deformed model solves the low-energy field equations, since those have an O(d, d)-covariant
formulation in terms of the generalized fluxes. In fact the same is true for the first α correction
to these equations as shown in [20]. Therefore it is also straightforward to argue
that YB-deformed bosonic strings2 are Weyl invariant at least up to two loops. The fact
that all higher derivative corrections should respect the O(d, d) structure suggest that this
should even be true to all orders in α , although a complete proof that the string effective
action can be written only in terms of generalized fluxes is not known to the authors.
Naively this argument may seem to suggest that the YB-deformed backgrounds should
not receive any α -corrections beyond those coming from the intrinsic α -dependence of
the original (i.e. undeformed) background. But this is at odds with the results of [21],
where non-trivial corrections were found working to second order in the expansion in the
deformation parameter. When focusing on the class of TsT transformations, it is also at
odds with the fact that abelian T-duality is known to receive α -corrections, as was shown
in various works starting from [22–28], which would be expected to lead to corrections to
TsT. As we will explain in more detail in the rest of the paper, the resolution is that while
in the doubled formalism there is indeed no correction, corrections appear when one wants
to go from the doubled formalism to a standard (super)gravity formulation. In order to do
that one has to fix the double Lorentz gauge-invariance in such a way that the two vielbeins
that naturally exist in the doubled formulation are set equal. This requires a certain double
Lorentz transformation and — given that the fields of the doubled formulation have an
anomalous transformation3 under double Lorentz transformations [33] — this induces an
2
Note that while here we consider only the bosonic string for definiteness, very similar results hold for
the heterotic string. In fact they can both be treated at the same time by introducing parameters that
interpolate between the two as in for example [20]. Note that the relevant equations for the target space
fields for the bosonic string are the type II supergravity equations with RR fields set to zero. Therefore we
will often loosely refer to them as the (super)gravity equations.
3
The fact that manifest O(d, d) symmetry requires the fields to transform non-covariantly was clarified
in the works [29–32].
– 2 –
JHEP07(2020)103
extra α -correction to the deformed background whose form we determine. A special case
of our formula gives the α -correction to TsT transformations.
This discussion naturally connects also to the identification of α -corrections to abelian
T-duality transformations, as mentioned above. We will discuss also this and comment on
the comparison to the results of [26]. Starting from the corrections to T-duality we will
be able to provide an independent way to obtain α -corrections to TsT transformations,
which does not make use of the double O(d, d) formulation.
The outline of the paper is as follows. First we give a very brief introduction to the
concepts needed from the O(d, d)-covariant formulation as used in DFT. In section 3 we
describe what Yang-Baxter deformations are in this language and show that they leave
the generalized fluxes invariant. The α -correction to these deformations induced by the
compensating anomalous Lorentz transformation is described in section 4. Section 5 focuses
on abelian T-duality and TsT transformations and we show that the results agree with those
obtained using the O(d, d)-covariant formulation. We end with some concluding comments.
2 O(d, d) covariant formulation of supergravity
We will take inspiration from DFT and use the O(d, d) covariant formulation of (super)gravity.
In particular we will work with the so-called frame-like formulation of DFT [34–
36] where the structure group is taken as two copies of the Lorentz group O(1, d − 1) ×
O(d − 1, 1). More details and references can be found in the reviews [37–39]. However,
unlike in DFT, we will always assume that the section condition is solved in the standard
way ∂M = (0, ∂m) so that we are really just working with a rewriting of supergravity. Here
we will actually consider only the NSNS sector as appropriate for the bosonic string.
In the frame-like formulation one writes the generalized metric in terms of generalized
(inverse) vielbeins
HMN
= EA
M
HAB
EB
N
, (2.1)
where HAB is block diagonal with the usual Minkowski metric ¯η = (−1, 1, . . . , 1) in each
block. Coordinate indices are raised and lowered with the O(d, d) metric
ηMN
= ηMN =
0 1
1 0
, (2.2)
and flat indices with the metric
ηAB
= ηAB =
¯η 0
0 −¯η
. (2.3)
The generalized metric can be parameterized in the form
HMN
=
Gmn − BmkGklBln BmkGkn
−GmkBkn Gmn , (2.4)
in terms of the usual metric G and B-field. We take the generalized (inverse) vielbein to be
EA
M
=
1
√
2
e(+)a
m − e(+)anBnm e(+)am
−e
(−)
am − e
(−)
a
nBnm e
(−)
a
m
. (2.5)
– 3 –
JHEP07(2020)103
Here e(±) are two sets of vielbeins which transform independently as Λ(±)e(±) under the
two Lorentz-group factors. To go to the standard supergravity picture one fixes a gauge
e(+) = e(−) = e leaving only one copy of the Lorentz-group.
An important object is the so-called generalized Weitzenb¨ock connection, defined in
terms of the generalized vielbeins as
ΩABC = EA
M
∂M EB
N
ECN . (2.6)
From this the generalized fluxes are constructed as
FABC = 3Ω[ABC] , FA = ΩB
BA + 2EA
M
∂M
ˆd , (2.7)
where ˆd is the generalized dilaton related to the standard one as e−2 ˆd = e−2Φ
√
−G. The
importance of these objects comes from the fact that the generalized fluxes are scalars under
generalized diffeomorphisms. This follows from the fact that a generalized diffeomorphism
is implemented by the generalized Lie derivative which acts on a vector field as
LXY M
= XN
∂N Y M
+ (∂M
XN − ∂N XM
)Y N
. (2.8)
The NSNS sector supergravity equations, or bosonic string low-energy effective equations,
can be expressed in terms of the generalized fluxes only. To do this we first introduce the
projectors
P± =
1
2
(η ± H) . (2.9)
Defining the following projections of the generalized fluxes
F
(±)
ABC = (P )A
D
(P±)B
E
(P±)C
F
FDEF , F
(±)
A = (P±)A
B
FB , (2.10)
they take the form4
(P+)A
C
(P−)B
D
∂CFD −(FE
−∂E
)F
(−)
CDE +
1
4
FC
EF
FDEF −
1
4
(F2
)CD = 0, (2.11)
R = −4∂AF(−)A
+2FAF(−)A
+
1
4
FA
CD
FBCDHAB
−
1
12
F2
−
1
6
FABCFABC
= 0, (2.12)
where (F2)AB = FACDHCEHDF FBEF and F2 = HAB(F2)AB. The last line defines the
generalized Ricci scalar and these equations of motion can be derived from the action
S = dX e−2 ˆd
R . (2.13)
Let us emphasize again that for us this is just a convenient rewriting of the usual bosonic
string effective action and equations of motion at lowest order in α .
4
Note that eq. (3.78) in [37] is not correct, since for example the P+P+ projection does not vanish.
– 4 –
JHEP07(2020)103
3 Yang-Baxter deformations in O(d, d) language
We first need to show how to write YB deformations in O(d, d) language at leading order in
α , which will be needed later when discussing their α -corrections. Under a YB deformation
we have (e.g. [7, 40])5
G − B → ˜G − ˜B = (G − B)(1 + Θ(G − B))−1
. (3.1)
The transformation of the dilaton is such that the generalized dilaton ˆd is invariant. The
transformation of G and B is equivalent to the following transformation of the generalized
metric (2.4)
H → ˜H = hT
Hh , hM
N
= δM
N
+ ΘM
N
, (3.2)
where
ΘM
N
=
0 Θmn
0 0
, Θmn
= km
r kn
s Rrs
, (3.3)
where km
r are Killing vectors of the undeformed background6 and Rrs is a constant antisymmetric
matrix satisfying (1.1) with c = 0 (r, s are Lie algebra indices). Later we
will show that if we just impose that R is constant and anti-symmetric, the additional
property of satisfying the CYBE (1.1) will have a natural interpretation. The generalized
vielbein (2.5) then transforms as
EA
M
→ ˜EA
M
= EA
N
hN
M
. (3.4)
Note that the two sets of vielbeins in (2.5) transform differently, namely
˜e(±)m
a = e(±)n
a δm
n − (Bnk Gnk)Θkm
. (3.5)
This means that if we start from an undeformed background in a gauge such that e(+) =
e(−) = e, we will need to accompany the YB deformation by a generalized double Lorentz
transformation. We will keep ˜e(+) invariant and transform ˜e(−) by
(Λ(−)
)a
b
= ˜Λa
b
= [1 + (G − B)Θ]a
c
([1 − (B + G)Θ]−1
)c
b
, (3.6)
in order to preserve the gauge ˜e(+) = ˜e(−). At the (super)gravity level this is of no
concern since all objects transform covariantly, but when one considers α -corrections this
transformation becomes important due to anomalous transformations of the fields, as we
will discuss in the next section.
Note that this is a reformulation of YB deformations in the form of an O(d, d) transformation,
in fact h has the form of a so-called β-transformation or β-shift. It is not a
standard O(d, d) transformation, such as the ones under which the DFT action is invariant,
5
We use a tilde to denote quantities after doing the deformation. We absorb the deformation parameter
(usually denoted by η) into Θ to simplify the expressions.
6
The assumption is that the Lie derivatives along km
r of the metric, the B-field and the dilaton of the
original background vanish. One could in principle relax the isometry condition on B by demanding only
that the Lie derivative of H vanishes, but we will not consider this generalization here.
– 5 –
JHEP07(2020)103
though. This is first of all because Θmn is (in general) not constant and second, and more
importantly, because Θmn depends on the background itself since it is constructed using
Killing vectors. This is therefore not a symmetry but a map of a background to another
background, which is in fact a deformation of the first if we take Θ to be multiplied by a
small parameter.
It follows from the transformation of the generalized vielbein that the generalized
Weitzenb¨ock connection (2.6) transforms as
˜ΩABC = EA
L
hL
M
∂M EB
N
ECN + EA
L
hL
M
∂M hK
N
(h−1
)N
P
EB
K
ECP
= ΩABC + EA
L
ΘL
M
∂M EB
N
ECN + EA
L
hL
M
∂M ΘKN EB
K
EC
N
, (3.7)
where we used the fact that any expression with two Θ’s contracted (with ηMN ) vanishes.
Now we use the fact that kr generate isometries, i.e. the generalized Lie derivative of EA
M
and ˆd along kr vanish7
kL
r ∂LEA
M
+ (∂M
krL − ∂LkM
r )EA
L
= 0 , kK
r ∂K
ˆd =
1
2
∂KkK
r . (3.8)
Using this fact one finds that the change of the generalized flux FABC is proportional to
the YB equation for R in the form
ΘM[K
∂M ΘLN]
= 0 . (3.9)
Therefore FABC is invariant under a YB deformation (see also [18, 19]8). For FA we find
˜FA = FA − ΘL
K
∂KEA
L
− ∂KΘL
K
EA
L
+ 2EA
N
ΘN
M
∂M
ˆd , (3.10)
and using (3.8) we find
˜FA = FA + EA
M
∆FM , ∆FM =
−2 nΘmn
0
. (3.11)
Therefore FA is invariant precisely when the R-matrix is unimodular, since nΘmn ∝
ft
rsRrs, and ft
rsRrs = 0 is the unimodularity condition of [13].
We have therefore shown that the generalized fluxes are invariant under unimodular
YB deformations. In fact their derivatives are also invariant since for example
∂AFB = EA
M
∂M FB → ∂AFB − EA
N
ΘN
M
∂M FB = ∂AFB , (3.12)
because kM
r ∂M FB = Lkr FB = 0 by isometry.
Since the (NSNS sector) supergravity equations of motion can be cast in terms of the
generalized fluxes and their derivatives, this is enough to conclude that they are invariant
7
Recall that ˆd is a density rather than a scalar, hence the non-zero r.h.s. in the second equation. Note
also that we are assuming the vielbeins and not just the metric to be invariant. This assumption was also
made in [7], whose derivation we rely on, but it should be possible to relax it. We comment more on this
in the next section.
8
This is however at odds with [41].
– 6 –
JHEP07(2020)103
under unimodular YB deformations. In other words such YB deformations map SUGRA
solutions to SUGRA solutions. Moreover, also the first α -correction to the bosonic string
equations can be cast in terms of the generalized fluxes and their derivatives, and therefore
our argument shows that in fact the YB deformation preserves Weyl invariance at least
to two loops.9 In fact one would expect that all α -corrections to the equations can be
expressed in O(d, d) covariant form, which probably means they can be written only in
terms of the generalized fluxes and their derivatives. If this is the case then our argument
implies that YB deformations of the bosonic string preserve Weyl-invariance to all loops, i.e.
they map a consistent bosonic string to another consistent bosonic string to all orders in α .
4 The α -correction to YB deformations
Our general argument above has shown that YB deformations preserve two-loop Weyl invariance
for the bosonic string. In fact they seem to require no additional α -corrections to
the background besides those that are induced from the corrections to the original background.
Here we want to understand how this fits with the results of [21] where additional
α -corrections were found for YB deformations. The resolution is that the additional α corrections
are indeed absent in the O(d, d) covariant approach, but when one goes down to
a standard supergravity formulation one has to fix the double Lorentz symmetry by fixing
e(+) = e(−) = e. The double Lorentz transformation required to do this induces, via the
anomalous transformation of the generalized vielbein at order α , additional α -corrections
to the YB deformed model. Let us now see how this works.
It was shown in [33] that at order α the generalized vielbein acquires an anomalous
transformation under (double) Lorentz transformations. The transformation of the vielbein
is given by10
δEA
M
= −λA
B
EB
M
+ α ˆδλEA
M
, ˆδλEA
M
= ∂
(−)
[A λC
D
F
(−)
B]D
C
− ∂
(+)
[A λC
D
F
(+)
B]D
C
EBM
,
(4.1)
where λC
D are parameters of an infinitesimal double Lorentz transformation and the second
term is the anomalous piece. Note that we have defined the projected derivatives ∂
(±)
A =
(P±)A
B∂B. After fixing the gauge e(+) = e(−) = e the non-zero components of F(±) are [33]
F
(+)
M
ab
=
1
2
Gmnω
(+)ab
n
−(1 − BG)m
nω
(+)ab
n
, F
(−)
Mab =
1
2
Gmnω
(−)
nab
(1 + BG)m
nω
(−)
nab
, (4.2)
where ω
(±)cd
m = ωm
cd ± 1
2 Hm
cd, and the spin-connection is related to the vielbein and the
Christoffel symbols Γp
mn as
ωmc
d
= ec
n
∂men
d
− Γp
mnec
n
ep
d
. (4.3)
9
The action was written in terms of the fluxes in [20]. But the variation of the generalized fluxes are again
expressed in terms of the generalized fluxes which shows that the equations of motion are also expressed in
this way, which is all we need.
10
We are specifying here to the case of the bosonic string by setting a = b = −α in the formulas of [33].
– 7 –
JHEP07(2020)103
This leads to the anomalous infinitesimal transformations11
ˆδ ¯Gmn = −
1
2
∂(mλ(+)cd
ω
(+)
n)cd −
1
2
∂(mλ(−)cd
ω
(−)
n)cd , (4.4)
ˆδ ¯Bmn =
1
2
∂[mλ(+)cd
ω
(+)
n]cd −
1
2
∂[mλ(−)cd
ω
(−)
n]cd . (4.5)
Of course, after fixing the gauge e(+) = e(−) = e only the transformations with
λ(+) = λ(−) = λ remain and the anomalous Lorentz transformations of the fields become
ˆδ ¯Gmn = −∂(mλcd
ωn)cd , (4.6)
ˆδ ¯Bmn =
1
2
∂[mλcd
Hn]cd . (4.7)
We see from these expressions that we can define new fields that transform non-anomalously
by12
G(MT)
mn = ¯Gmn + α
1
2
ωmcdωn
cd
+
3
8
HmklHn
kl
, (4.8)
B(MT)
mn = ¯Bmn +
α
2
Hcd[mωn]
cd
. (4.9)
The explicit non-covariant terms are constructed such as to cancel the anomalous Lorentz
transformations. Notice that the above redefinitions also fix the finite form of the anomalous
Lorentz transformations of ¯G, ¯B.
4.1 Compensating anomalous transformation
When we are dealing with the YB deformation it is crucial to remember the compensating
double Lorentz transformation needed to make ˜e(+) = ˜e(−) given by (3.6). Setting λ(+) = 0
and λ(−) = ˜λ in (4.5) we find that this induces an extra transformation of the fields at
order α given by13
ˆδ ¯Gmn = −
1
2
∂(m
˜λcd
˜ω
(−)
n)cd , ˆδ ¯Bmn = −
1
2
∂[m
˜λcd
˜ω
(−)
n]cd . (4.10)
We now need the finite form of the transformation since we are doing a finite transformation
˜Λ = e
˜λ given by (3.6). To find it we use the same strategy as above. We redefine G and
B by terms involving the spin connection in such a way that the new fields do not have
11
The bar on the fields is to emphasize that these are the fields coming from the doubled formulation
and which have an anomalous Lorentz transformation. Below we will define unbarred fields that transform
covariantly.
12
We have included an extra shift of Gmn by H2
mn to go to the scheme of Metsaev and Tseytlin (MT),
see [33] and appendix A.
13
In (4.5) we assumed e(+)
= e(−)
= e and we were doing a double Lorentz transformation from that
starting point. Here we can use the same logic, assuming that we start from the gauge ˜e(+)
= ˜e(−)
= ˜e for
a YB deformation and go back to the situation where ˜e(+)
= ˜e and ˜e(−)
= ˜ΛT
˜e as in (3.5). In this way we
construct the inverse of the anomalous transformation we want. We remind that ω
(±)cd
m = ωm
cd
± 1
2
Hm
cd
,
so that the (±) on the torsionful spin-connection should not be confused with the (±) on the two vielbeins
coming from DFT. Setting λ(+)
= 0 means that for the deformed model we take ˜e = ˜e(+)
.
– 8 –
JHEP07(2020)103
any anomalous transformation. From this one can then read off the finite form of the
transformation.
For ¯Gmn this is easily done by noting that ¯Gmn + α
4 ˜ω
(−)cd
m ˜ω
(−)
ncd is invariant under the
above transformation and so the finite transformation for ¯G is14
δcomp
¯Gmn = −
1
2
[˜Λ∂(m
˜ΛT
]cd
˜ω
(−)
n)cd +
1
4
[˜Λ∂m
˜ΛT
]cd
[˜Λ∂n
˜ΛT
]cd . (4.11)
For ¯Bmn things are more subtle because a similar term 1
4 ˜ω
(−)cd
[m ˜ω
(−)
n]cd vanishes by antisymmetry.
The part involving H in ω(−) can be integrated as before, while the part
involving ω can be found by the following trick. Consider the anomalous transformation
of H = dB instead. One finds that H transforms like the Chern-Simons form for ω
ˆδ ¯H = −
1
4
δCS(ω) = −
1
4
δtr ωdω +
2
3
ωωω . (4.12)
The finite transformation of the CS form is
δCS(ω) = d(˜Λd˜ΛT
ω) −
1
3
tr(˜ΛT
d˜Λ˜ΛT
d˜ΛΛT
d˜Λ) . (4.13)
This implies that the transformation of B can be taken to be
δcomp
¯Bmn = −
1
2
[˜Λ∂[m
˜ΛT
]cd
˜ω
(−)
n]cd + BWZW
mn , (4.14)
where BWZW is defined by
dBWZW
= −
1
12
tr(˜ΛT
d˜Λ˜ΛT
d˜Λ˜ΛT
d˜Λ) . (4.15)
Now that we have found the pieces induced by the compensating double Lorentz transformation
we are ready to write the α -correction to the YB-transformed metric and B-field.
4.2 The correction to Yang-Baxter deformations
Putting everything together the α -correction to the YB-deformed background in the
scheme of Hull and Townsend is15
δ( ˜G − ˜B)(HT)
mn =
1
2
˜ω
(−)
mcd ˜ω(+)
n
cd
− [˜Λ∂n
˜ΛT
]cd
+
1
4
∂m
˜Λcd
∂n
˜Λcd − BWZW
mn + δ ( ˜G − ˜B)mn ,
(4.16)
δ˜Φ(HT)
=
1
4
˜Gkl
δ ˜Gkl +
1
48
( ˜H2
− H2
) . (4.17)
The correction to the dilaton follows from the fact that in the HT scheme when
Φ = Φ(HT) − 1
48 α H2 the combination e−2Φ
√
−G is invariant under YB deformations, up
14
Recall that we are computing minus the anomalous transformation we are after.
15
See appendix A for the field redefinitions connecting all schemes. Here we set the parameter q of Hull
and Townsend to zero.
– 9 –
JHEP07(2020)103
to order α included16 [21]. The term δ ( ˜G − ˜B) takes into account the scheme-change of
the undeformed background17
δ(G − B)mn = −
1
2
ω(−)
m
cd
ω
(+)
ncd , (4.18)
needed to relate the HT scheme to the O(d, d) covariant scheme (see appendix A) and it
takes the form
δ ( ˜G − ˜B)mn = (1 + (G − B)Θ)−1
δ(G − B)(1 + Θ(G − B))−1
mn
. (4.19)
Note that in addition to this, one has the α -corrections to the original background, which
will need to be included in (3.1) and will therefore induce a term of the same form — where
now δ(G − B) is the correction to the original background.
It is important to stress that our derivation assumes that the B-field and vielbein of the
undeformed background are invariant under the isometries generated by the Killing vectors
entering Θ. When there is no gauge where this is possible, equations (4.16) and (4.17) do
not necessarily lead to a background solving the α -corrected supergravity equations. See
however the next subsection.
The spin connection for the YB deformed background entering these expressions is
computed using the vielbein ˜e = ˜e(+) defined in (3.5) and is given by
˜ωm
ab
(˜e(±)
) = ωm
ab
+ m[(B G)Θ][a|k|
([1−(B G)Θ]−1
)k
b]
−˜e(±)[a|k|
˜e(±)b]l
k
˜Glm . (4.20)
To see that (4.16) and (4.17) reproduces the results found in [21] one sets B = 0 and
expands to order Θ2 obtaining
δ ˜Gmn = − mΘcd
cΘdn − nΘcd
cΘdm + O(Θ4
) (4.21)
δ ˜Bmn = 2∂[m ωn]
cd
Θcd − ΘcdRmn
cd
+ O(Θ3
) (4.22)
δ˜Φ =
1
16
m
Θcd mΘcd −
3
8
m
Θcd
cΘdm + O(Θ4
) , (4.23)
which, up to a diffeomorphism and B-field gauge transformation, is the same as in [21].
Note that one has to use the fact that the isometry of the vielbein implies that
ikωab
= − a
kb
. (4.24)
It is worth noting that in the case of a single TsT transformation the correction simplifies.
Recall that, given two isometric coordinates y1, y2, a TsT transformation is implemented
by the sequence of T-duality y1 → T(y1) followed by a shift y2 → y2−ηT(y1) and by
another T-duality T(y1) → y1. It is understood as a special case of YB with Θ = ηk1 ∧ k2,
where ki = ∂yi are Killing vectors. The above correction simplifies in the TsT case since
BWZW vanishes. This follows by noting that ˜Λ = 1 + 2Θ([1 − (B + G)Θ]−1) which means
that when Θ has rank 2 the Lorentz transformation is only non-trivial in a 2 × 2 block. In
this block it is eλ with λ an anti-symmetric 2 × 2 matrix. Since such a matrix only has one
independent component, the r.h.s. of (4.15) vanishes.
16
Notice that Φ is in fact the dilaton in the HT scheme at q = 1/6.
17
For the same reason we have also a 1
2
˜ω
(−)
m
cd
˜ω
(+)
ncd term in the correction above, generated by the schemechange
after the deformation.
– 10 –
JHEP07(2020)103
4.3 Manifestly covariant form of the correction
The expression (4.16) for the α -correction is not manifestly covariant but one can show
that it is nevertheless covariant. We start by noting that18
˜ω (±)ab
m = −˜e[a|k|
˜eb]l
k( ˜G ± ˜B)ml +
1
2
˜e[a|k|
˜eb]l
m( ˜G ± ˜B)kl − m[(G − B)Θ][a|k|
([1 + (G − B)Θ]−1
)k
b]
(4.25)
where ˜ω (±) = ˜ω(±) − ω. With a bit of algebra one finds
˜ω
(+)
mcd =
1
2
mBcd−
1
2
[(G−B) mΘ(G+B)]cd+[1+(G−B)Θ][c|k|( k
B(1+Θ(G−B))−1
)d]m
+[1+(G−B)Θ][c|k|((G−B) k
Θ(1+(G−B)Θ)−1
(G−B))d]m
=
1
2
Hmcd−X
(+)
kcd [(G−B)(1+Θ(G−B))−1
]k
m , (4.26)
where we have defined19
X
(±)
kcd =
1
2
kΘcd − [cΘd]k ±
1
2
HcdlΘl
k (4.27)
and we used the YB equation in the last term of the first expression and also the isometry
of B in the next to last term. A similar calculation gives
[˜ΛT
˜ω (−)
m
˜Λ + ˜ΛT
m
˜Λ]cd = −
1
2
Hmcd + X
(−)
kcd [(G + B)(1 − Θ(G + B))−1
]k
m . (4.28)
Using these expressions we find that (4.16) can be written instead as
δ( ˜G − ˜B)mn = −
1
4
m
˜Λcd
n
˜Λcd − Bcov-WZW
mn +
1
2
[ m
˜Λ˜ΛT
]cd
X
(+)
kcd ( ˜G − ˜B)k
n −
1
2
Hncd
+
1
2
[˜ΛT
n
˜Λ]cd
X
(−)
kcd ( ˜G − ˜B)m
k
−
1
2
Hmcd (4.29)
+
1
4
˜Λc
e
˜Λd
f − δc
eδd
f ( ˜G − ˜B)m
k
X
(−)ef
k Hncd + ( ˜G − ˜B)k
nX
(+)
kcd Hm
ef
− 2( ˜G − ˜B)m
k
X
(−)ef
k X
(+)
lcd ( ˜G − ˜B)l
n −
1
2
Hm
ef
Hncd ,
where ˜G − ˜B is given by (3.1) and we have defined
Bcov-WZW
mn = BWZW
mn −
1
2
tr ω[m
˜Λ n]
˜ΛT
+
1
2
tr ω[m
˜ΛT
∂n]
˜Λ . (4.30)
The correction to the dilaton is still given by (4.17). All terms except Bcov-WZW are now
manifestly covariant. For the latter the identity
tr [˜ΛT ˜Λ]3
− tr [˜ΛT
d˜Λ]3
= −
3
2
dtr ω[d˜Λ˜ΛT
+ ˜ΛT
d˜Λ] −
3
2
tr ω[ ˜Λ˜ΛT
+ ˜ΛT ˜Λ]
+ 3tr R[ ˜Λ˜ΛT
+ ˜ΛT ˜Λ] , (4.31)
18
Here and in the following the covariant derivative is the one for the undeformed metric G. Moreover,
unless written explicitly otherwise, one should use the undeformed vielbein to go from curved to flat indices.
19
When the vielbeins are invariant under the isometries, (4.24) gives ω
(±)
lcd Θl
k = X
(±)
kcd .
– 11 –
JHEP07(2020)103
where R = dω + ω ∧ ω is the curvature 2-form, implies
dBcov-WZW
= −
1
12
tr [˜ΛT ˜Λ]3
+
1
4
tr R[ ˜Λ˜ΛT
+ ˜ΛT ˜Λ] . (4.32)
Therefore also the transformation of B is covariant (up to B-field gauge transformations).
The manifestly covariant form of the correction given by (4.29) is actually more useful
than the original form (4.16). The reason is that our derivation has assumed that the
vielbeins are invariant under the isometries used to construct Θ, and therefore (4.16) is
valid only in this case. Being covariant, (4.29) is valid also when the vielbeins are not
invariant under the isometries, as long as there exists a gauge in which they are invariant.
In fact, even though it is not guaranteed by our construction, these expressions can be
valid more generally, i.e. even in cases where it is not possible to find a gauge in which the
vielbeins are invariant. We mention one such example below.
4.4 Tests on examples
We have tested the formulas (4.16), (4.17) for α -corrections to YB deformations on a
number of examples, to check that they generate backgrounds solving the α -corrected
supergravity equations. First we worked out deformations of a Bianchi II background first
considered in [21]. We tested our results both on the abelian deformations Θ = k1 ∧ k4
and Θ = k2 ∧ k3, and on the non-abelian deformation Θ = k1 ∧ k4 + k2 ∧ k3. We refer
to [21] for the α -correction of the undeformed background and for the definition of the
Killing vectors ki, whose non-trivial commutation relations are just [k1, k2] = k3. On this
Bianchi II example we find that BWZW is trivial even when considering the non-abelian
deformation.
We worked out also deformations of the pure NSNS AdS3 × S3 background.20 Its
YB deformations were classified in [16]. We worked out various abelian deformations
corresponding to TsT transformations on the sphere, on AdS, or mixing the two spaces.
We worked out also the non-abelian deformation generated by Θ = (k0 +¯k0)∧ks +k+ ∧¯k−.
Here ks is a Killing vector on the sphere and we refer to [16] for the definitions we use for
the AdS Killing vectors. In this case we cannot immediately apply (4.16) because it is not
possible to find a vielbein for the AdS3 metric that is invariant under all the isometries
entering Θ. We can anyway obtain α -corrections for this non-abelian deformation if we use
the covariant formula (4.29). Alternatively, we can interpret this particular deformation
as a non-commuting sequence of TsT transformations. Doing so, we can first work out the
α -corrected abelian deformation generated by Θ = k+ ∧ ¯k−, and after doing that we can
work out the abelian deformation Θ = (k0 + ¯k0) ∧ ks.21
20
The α corrections of the undeformed background are simply obtained by multiplying metric and B-field
by 1 + 2α on the AdS part and by 1 − 2α on the sphere part.
21
After doing the first abelian deformation, and before applying the second one, one has to carefully
choose the vielbein such that it is invariant under the k0 + ¯k0 isometry. At this stage it is not necessary
anymore to impose the invariance under k+, ¯k−, which is what saves the day in this approach.
– 12 –
JHEP07(2020)103
5 T-duality and TsT transformations
Abelian T-duality transformations are another class of O(d, d) transformations and we can
follow exactly the reasoning in section 4 to obtain their α -corrections. When we remain
in the non-covariant scheme that comes from DFT, the corrections to the dualized metric
and B-field will be given again by the formula (a hat on the field is used to denote the
T-dualization)
δ(G − B)mn = −
1
2
ˆω
(−)
mcd(ˆΛ∂n
ˆΛT
)cd
+
1
4
∂m
ˆΛcd
∂n
ˆΛcd − BWZW
mn , (5.1)
where now the Lorentz matrix is
ˆΛa
b
= δa
b
− 2G−1
yy eyaey
b
. (5.2)
We are assuming that we are dualising along the coordinate y and expressions for the corrections
in other schemes will be obtained by implementing the relevant field redefinitions,
see appendix A.
In [42] α -corrections to the T-duality rules from the DFT formulation were also discussed.
There however instead of writing the generic form of the corrections in terms of
the finite form of the Lorentz transformation as above, it was noted that ˆΛ reduces to a
constant22 when choosing a specific gauge for the vielbein23
eµ
α
= eµ
α
, ey
α
= 0 , eµ
ι
= eσ
Vµ , ey
ι
= eσ
. (5.3)
Here we are rewriting the fields in terms of fields of a dimensional reduction
ds2
= Gmndxm
dxn
= gµνdxµ
dxν
+ e2σ
(dy + V )2
,
B =
1
2
Bmndxm
∧ dxn
=
1
2
bµνdxµ
∧ dxν
+
1
2
W ∧ V + W ∧ dy ,
Φ = φ +
1
2
σ.
(5.4)
Since ˆΛ is constant the anomalous Lorentz transformation is trivial in this gauge, and also
the α -corrections to T-duality will be trivial.24 (Note that while it is possible to avoid
corrections for a single T-duality it is not possible in general for more than one T-duality,
as shown in [43].) In [42] this observation was used to obtain the α -corrections to the Tduality
rules in the scheme of Bergshoeff and de Roo (BR) [44, 45]. We use this result as a
22
It is ˆΛ = diag(−1, 1, . . . , 1) where the dualized coordinate is placed first.
23
For curved indices we take m = y, µ and similarly we also have flat indices a = ι, α. We denote by eµ
α
the vielbein for the reduced metric gµν appearing below.
24
Importantly, this statement is gauge dependent, in accordance with the fact that the scheme under
discussion is not Lorentz-covariant. Covariant schemes such as HT or MT will not have this type of
gauge ambiguity.
– 13 –
JHEP07(2020)103
starting point to write below the T-duality rules to 2 loops in a family of different schemes.
ˆσ = −σ+ a1−
a4
2
+2a5+2γ+ (Dσ)2
−
1
8
(a1+4a2−a5−2γ+)
× e2σ
V λρ
Vλρ+e−2σ
Wλρ
Wλρ −
1
2
(γ−−a6)V λρ
Wλρ , (5.5)
ˆVµ = Wµ+
1
2
(γ+−b3+a5)Wβ
α
wµα
β
+
e2σ
4
(−4a2+2b1+b3+γ+)hµλρV λρ
+
+
1
4
(6a1−a4+4a5+4b1−2b2+4b3+4γ+)WµpDp
σ+
1
2
(a4−2b2)WµρDρ
φ−
−
1
2
(a1+2b1)Dρ
Wµρ−
1
2
(γ−−a6) e2σ
Vβ
α
wµα
β
+
1
2
hµλρWλρ
−2e2σ
VµρDρ
σ , (5.6)
ˆWµ = Vµ−
1
2
(γ+−b3+a5)Vβ
α
wµα
β
−
e−2σ
4
(−4a2+2b1+b3+γ+)hµλρWλρ
+
+
1
4
(6a1−a4+4a5+4b1−2b2+4b3+4γ+)VµρDρ
σ−
1
2
(a4−2b2)VµρDρ
φ+
+
1
2
(a1+2b1)Dρ
Vµρ+
1
2
(γ−−a6) e−2σ
Wβ
α
wµα
β
+
1
2
hµλρV λρ
+2e−2σ
WµρDρ
σ ,
(5.7)
ˆφ = φ−
1
16
(a1−4a2−a5+4c1+48c2) e2σ
VλρV λρ
−e−2σ
WλρWλρ
+
+
1
2
(a1−8c1+2c4)D2
σ−
1
2
(a4−4c3−4c4)DρσDρ
φ, (5.8)
ˆgµν = gµν −
1
2
(a1+4a2+a5) e2σ
VµρVν
ρ
−e−2σ
WµρWν
ρ
+
+(−2a1+a4)DµDνσ+2a3D(µσDν)φ, (5.9)
ˆbµν = bµν −
1
2
(γ+−b3+a5) Vβ
α
w[µα
β
Wν]−Wβ
α
w[µα
β
Vν] +
+
1
2
(a1+2b1) Dρ
Wρ[µVν]−Dρ
Vρ[µWν] +
+
1
4
(4a2−2b1−b3−γ+) e2σ
V[µhν]λρV λρ
−e−2σ
W[µhν]λρWλρ
+(2b1+b2)hµνρDρ
σ+
+
1
4
(−6a1+a4−4a5−4b1+2b2−4b3−4γ+) V[µWν]ρDρ
σ+W[µVν]ρDρ
σ −
−
1
2
(a4−2b2) V[µWν]ρDρ
φ−W[µVν]ρDρ
φ −2b3V ρ
[µWν]ρ+
+
1
2
(γ−−a6) e−2σ
Wβ
α
w[µα
β
Wν]−e2σ
Vβ
α
w[µα
β
Vν]−
1
2
Wλρ
hλρ[µVν]+
+
1
2
V λρ
hλρ[µWν]−2e−2σ
W[µWν]ρDρ
σ−2e2σ
V[µVν]ρDρ
σ . (5.10)
Setting α → 0 they reduce to the Buscher rules that in terms of these fields read simply
as σ → −σ and V ↔ W. Here D denotes the covariant derivative with respect to
the reduced metric gµν, and wµα
β is the reduced spin-connection. We have also defined
Vµν = ∂µVν − ∂νVµ, Wµν = ∂µWν − ∂νWµ and hµνρ = 3(∂[µbνρ] − 1
2 W[µνVρ] − 1
2 V[µνWρ]) =
Hµνρ − 3W[µνVρ]. Apart from the order-α parameters γ± needed to interpolate between
the bosonic and the heterotic strings (see appendix A), the T-duality rules depend on co–
14 –
JHEP07(2020)103
efficients ai, bi, ci (that are also of order α ) so that they are valid for any scheme related
to the one of BR by these field redefinitions
Gmn = G(BR)
mn −a1Rmn−a2H2
mn−a3 mΦ nΦ−a4 m nΦ−a5ωmb
a
ωna
b
−a6ω(m
ab
Hn)ab ,
Bmn = B(BR)
mn −b1
p
Hmnp−b2Hmnp
p
Φ−b3ω[m
ab
Hn]ab , (5.11)
Φ = Φ(BR)
−c1R−c2H2
−c3 pΦ p
Φ−c4
2
Φ.
By turning on these coefficients we can cover all schemes typically considered in the literature,
see appendix A for the field redefinitions relating them.25
As expected, it is possible to tune the coefficients in order to set to zero all corrections
to the T-duality transformations. For generic γ± it is enough to set
a2 = −
a1
4
+
γ+
4
, a3 = 0, a4 = 2a1 , a5 = −γ+ , a6 = γ− , b1 = −
a1
2
,
b2 = a1 , b3 = 0, c2 = −
a1
24
−
c1
12
, c3 = a1−4c1 , c4 = 4c1−
a1
2
(5.12)
and T-duality reduces to the Buscher rules even to 2 loops. We will denote the fields in this
(gauge-fixed) scheme by G , B , Φ . When specifying to the bosonic string (γ+ = α /2, γ− =
0), they are related to the HT scheme by26
Gmn = G(HT)
mn −
1
2
α ω
(−)ab
(m ω
(+)
n)ab = G(HT)
mn + α −
1
2
ωmabωab
n +
1
8
H2
mn ,
Bmn = B(HT)
mn +
1
2
α ω
(−)ab
[m ω
(+)
n]ab = B(HT)
mn −
1
2
α Hab[mωab
n] ,
Φ = Φ(HT)
+ α
1 + 3q
24
H2
.
(5.13)
This matches with the field redefinitions that we would write for ¯G, ¯B, ¯Φ as expected. The
difference is that here we are also imposing the specific gauge (5.3) and for that reason we
denote the fields differently.
The rules above can be compared to the ones first derived by Kaloper and Meissner
in [26] for the bosonic string (γ+ = α /2, γ− = 0). The scheme used is obtained setting the
coefficients to
a1 = α a2 = −
α
4
, b1 = −
α
2
, b3 =
α
2
, c1 =
α
8
c2 = −
5α
96
c3 = −
α
2
, (5.14)
25
Writing the rules for generic ai, bi, ci coefficients as above, or in other words translating them into new
schemes starting from a given one, is straightforward although it requires work to compute all tensors in
the dimensional reduction. After that is done we can start from scheme A where ˆσ(A)
= −σ(A)
+ α ξ, for
some ξ. To obtain the rules in scheme B related as σ(B)
= σ(A)
+ α s for some s, we just have to compute
ˆσ(B)
= ˆσ(A)
+ α ˆs = −σ(A)
+ α (ξ + ˆs) = σ(B)
+ α (ξ + ˆs + s). Notice that the fields themselves may have
some explicit α -dependence. In this example the field σ is odd under Buscher rules, and then the shift in
the corrections ˆs + s is even. Fields even under Buscher receive corrections that are odd.
26
Here we are further setting a1 = c1 = 0. Turning on a1, c1 would introduce terms that vanish by means
of 1-loop equations.
– 15 –
JHEP07(2020)103
and the rest of them equal to zero. To match results, one has to take into account the
possibility of transforming the reduced fields by doing diffeomorphisms and gauge transformations.
Under such symmetries, the T-dual reduced fields transform as:
ˆV → ˆV + α (LξW + dv) , (5.15)
ˆW → ˆW + α (LξV + dw) , (5.16)
ˆb → ˆb + α Lξb + dβ +
1
2
V ∧ dv +
1
2
W ∧ dw , (5.17)
and the remaining fields transform normally under diffemorphisms. We are restricting to
transformations which are first order in α , both for diffeomorphims and gauge transformations.
The dw and dβ terms come from gauge transformations of the B field with parameter
βµdxµ + w dy, while v appears when including diffeomorphisms of the form y → y + α v.
Choosing the following set of parameters
ξµ
= Dµ
σ, w = −VνDν
σ, v = −WνDν
σ, βµ = bµν −
1
2
VµWν −
1
2
WµVν Dν
σ, (5.18)
we obtain the following set of rules
ˆσ = −σ +
α
2
e2σ
4
VλρV λρ
+
e−2σ
4
WλρWλρ
+ 2 (Dσ)2
, (5.19)
ˆVµ = Wµ +
α
2
e2σ
2
hµλρV λρ
+ 2WµρDρ
σ , (5.20)
ˆWµ = Vµ −
α
2
e−2σ
2
hµλρWλρ
− 2VµρDρ
σ , (5.21)
ˆbµν = bµν + α V[µ
ρ
Wν]ρ − V[µWν]ρ + W[µVν]ρ Dρ
σ
−
e2σ
4
V[µhν]λρV λρ
+
e−2σ
4
W[µhν]λρWλρ
, (5.22)
and both gµν and φ remain invariant. These match with the rules given by Kaloper and
Meissner in [26] up to the sign of the α correction of the b field.27
Diffeomorphisms and gauge transformations of the reduced fields can also be used to
simplify the rules and to obtain some nice expressions for the T-duality rules without the
need of the dimensional reduction. We do this in a Lorentz-covariant scheme, the HT
scheme for the bosonic string introduced in (5.13) where we fix q = −1/3. Using the same
parameters for the transformations presented in the previous paragraph, it is possible to
27
The fact that this is a typo in [26] is confirmed by the fact that there (4.9) and (4.11) are not compatible.
For the field H of [26] (here h) which is even under T-duality at leading order in α , the correction to the
T-duality transformation should rather be −2 the expression in (4.9). For odd fields the same contribution
would be instead multiplied by +2. This easily follows from the first calculation they do to remove by a field
redefinition the part of the action that is odd under T-duality, which is later reinterpreted as a correction
to the T-duality transformation. Since the expressions in [46] agree with those in [26] we disagree also with
that paper.
– 16 –
JHEP07(2020)103
obtain the following rules for the T-duality transformation28
ˆMyy =
1
Myy
, ˆMyµ =
Myµ
Myy
, ˆMµy = −
Mµy
Myy
, (5.23)
ˆMµν = Mµν −
MµyMyν
Myy
− α
1
Myy
R(−)
µyνy −
1
ˆMyy
ˆR(−)
µyνy , (5.24)
ˆΦ = Φ −
1
2
log Myy −
α
8
R(−)
− ˆR(−)
. (5.25)
In these expressions
R(−)
mna
b
= 2∂[mω
(−)b
n]a + 2ω
(−)c
[ma ω
(−)b
n]c (5.26)
is the Riemann tensor constructed from the torsionful connection ω(−), R(−) the corresponding
Ricci scalar and Mmn = Gmn − Bmn. Note also that the dual appears explicitly
in the α corrections but, to the order needed, it can be calculated using just the standard
Buscher rules.
5.1 Corrections to TsT transformations
The fact that there exists a (gauge-fixed) scheme — for the sake of the discussion we will
call it the “Buscher scheme” — such that T-duality is given just by the Buscher rules is
useful. Here we use it to obtain an expression for α -corrections to TsT transformations
that does not necessarily use all the knowledge of DFT. TsT transformations are a special
case of YB deformations, and we will show that the result agrees with that in section 4.
In order to do the TsT transformation we assume that there are two U(1) isometries
with corresponding coordinates y1 and y2, and to avoid burdening the notation we will
continue labelling by xµ the rest of the coordinates.29 We will do a T-duality y1 →
T(y1) followed by a shift y2 → y2 − ηT(y1) and by another T-duality T(y1) → y1. TsT
transformations are special cases of YB if we take Θ = ηk1 ∧ k2, where ki = ∂yi are Killing
vectors. Each step will be performed in the scheme that is most convenient. Therefore,
when doing T-duality we will prefer to move to the Buscher scheme, while when doing the
shift we will prefer to go to a covariant scheme. We will show that the α -corrections to TsT
transformations can be understood as arising from these shifts coming from the scheme
changes. Because these scheme-changing shifts arise at intermediate steps, we will have to
look at how they are further modified by the remaining steps in the TsT transformation.
Suppose we start from the HT scheme. In order to do the first T-duality on y1 we
find convenient to first go to the Buscher scheme. This is achieved by implementing the
redefinitions (5.13) after taking care of choosing the vielbein as in (5.3). This effectively
shifts the fields at order α as δ1(Gmn−Bmn) = −1
2 ω
(−)
mabω
(+)ab
n . We can immediately account
for this contribution in the final result: because we will have to do a TsT transformation
including this contribution δ1 (and we only care about the order α ) we are essentially
shifting the original metric and B-field as G − B → G − B + δ1(G − B) appearing in the
28
Here 1-loop equations of motion were used to simplify the form of the corrections.
29
The reader should be careful, since when doing T-duality along y1 the coordinate y2 should be treated
on the same footing as xµ
when using the T-duality rules (5.5).
– 17 –
JHEP07(2020)103
map (3.1). After expanding to first order in α we obtain the first contribution to the α
correction of the final result
−
1
2
[(1 + (G − B)Θ)−1
]m
p
ω
(−)
pab ω(+)ab
q [(1 + Θ(G − B))−1
]q
n . (5.27)
While in the Buscher scheme we can easily do the first T-duality on y1 because we just need
to use the Buscher rules. Notice that under Buscher the gauge choice (5.3) is preserved.
To perform the shift it is more convenient to go back to the HT scheme, which is
covariant. That means that we will have to use (5.13) again, although now it will be done
using the data of the T-dual background δ2( ˆGmn − ˆBmn) = +1
2 ˆω
(−)
mab ˆω
(+)ab
n . A hat is used
to denote that the first T-dualization has already been done. Notice that under the first
T-duality and shift, the vielbein em
a (in matrix form) changes as



eσ 0 0
eσVy2 ey2
2 ey2
α
eσVµ eµ
2 eµ
α



T
−→



e−σ 0 0
e−σWy2 ey2
2 ey2
α
e−σWµ eµ
2 eµ
α



s
−→



e−σ(1 − ηWy2 ) −ηey2
2 −ηey2
α
e−σWy2 ey2
2 ey2
α
e−σWµ eµ
2 eµ
α


 .
(5.28)
The shift is spoiling the choice (5.3) for the vielbein, and that is an important point
because we will want to restore this gauge before going back to the Buscher scheme and
implementing the last T-duality. To achieve it we implement the Lorentz transformation
em
a → em
bLb
a where
Lb
a
=



1−ηWy2
D1/2
ηeσ√
gy2y2
D1/2 0
−
ηeσ√
gy2y2
D1/2
1−ηWy2
D1/2 0
0 0 δb
a


 , where D = 1 − ηWy2 (2 − ηWy2 ) + η2
e2σ
gy2y2 .
(5.29)
At this point one wants to go to the Buscher scheme, in order to perform the last T-duality,
which will produce a new correction δ3(
ˆˆGmn −
ˆˆBmn) = −1
2
ˆˆω
(−)
mab
ˆˆω
(+)ab
n . Now a double hat
is used to denote that a T-duality and a shift (followed by the compensating Lorentz
transformation) have been implemented. The contribution δ2 (on which we implement the
effect of the shift) and δ3 can be considered together. In fact all expressions from covariant
terms cancel out and we are left with
1
2
−ˆˆω
(−)
mab(L−1
∂nL)ab
− ˆˆω
(+)
nab(L−1
∂mL)ab
+ (L−1
∂mL)ab(L−1
∂nL)ab
. (5.30)
In order to account for the effect of the last T-duality on the above expression one uses:
the fact that in the first two terms only (mn) = (yiyj) contribute, that in the summation
of a, b only 1, 2 contribute, the fact that under T-duality
ˆω
(±)
ιιβ = −ω
(±)
ιιβ , ˆω
(±)
αιβ = ±ω
(±)
αιβ, ˆω
(±)
ιαβ = ω
(±)
ιαβ, ˆω
(±)
αβγ = ω
(±)
αβγ, (5.31)
and finally that the last term in (5.30) vanishes if m or n are yi, so that it actually remains
the same after T-duality. After taking everything into account the result after the T-duality
is simply
1
2
˜ω
(−)
mab(L−1
∂nL)ab
− ˜ω
(+)
nab(L−1
∂mL)ab
+ (L−1
∂mL)ab(L−1
∂nL)ab
. (5.32)
A tilde denotes the quantities of the TsT-transformed background.
– 18 –
JHEP07(2020)103
After the last T-duality has been performed, we go back from Buscher to the HT scheme
using (5.13) obtaining the final contribution to the α corrections which is δ4( ˜Gmn− ˜Bmn) =
+1
2 ˜ω
(−)
mab ˜ω
(+)ab
n .
Collecting together all contributions we obtain the α correction to the TsT deformed
background in the HT scheme
δ( ˜G − ˜B)mn =
1
2
˜ω
(−)
mab − (L−1
∂mL)ab ˜ω(+)ab
n + (L−1
∂nL)ab
+ (L−1
∂mL)ab(L−1
∂nL)ab
−
1
2
[(1 + (G − B)Θ)−1
]m
p
ω
(−)
pab ω(+)ab
q [(1 + Θ(G − B))−1
]q
n. (5.33)
Because of the steps of TsT, the vielbein used to construct the above spin-connection of
the deformed model is defined as ˜ea
m = La
beb
n(1 − (G + B)Θ)n
m, where the undeformed
vielbein must respect (5.3), and one can check that the Lorentz transformation used here
is related to the one in (3.6) simply as L2 = ˜Λ. To compare to the result (4.16) we need
to use the same deformed vielbein used there, meaning that we should rather take ˜ea
m =
(L2)a
beb
n(1−(G+B)Θ)n
m. After taking into account this extra Lorentz transformation we
match with (4.16) in the case of TsT if we remember that BWZW can be taken to be zero,
and if we use that we for TsT we can write L−1dL = dLL−1 because here L is essentially
a 2 × 2 anti-symmetric matrix and it commutes with itself.
With a similar reasoning we can obtain the α -corrections to the dilaton of the TsTtransformed
background. The simplification in this case is that the dilaton is insensitive to
the shift, because by assumption it is isometric and the field redefinitions for the dilaton
between the schemes of Buscher and HT are covariant. In HT scheme at generic q we get
Φ = Φ +
1
2
log
Gy1y1
Gy1y1
+ α
1 + 3q
24
(H2
− H2
) −
1
2
δ1Gy1y1
Gy1y1
+
δ4Gy1y1
Gy1y1
. (5.34)
At q = 1/6 on finds
e−2Φ
− det G = e−2Φ
√
− det G , (5.35)
which is in agreement with (4.17), since there the result was written when setting q = 0,
and one therefore has the extra H2-terms.
6 Concluding comments
In this paper we have demonstrated that it is possible to extend the YB deformation as
a solution-generating technique in string theory at least to first order in the α -expansion.
The explicit expression that we found for the corrections allowed us to test successfully our
results on explicit examples. We expect our formula to be useful when addressing specific
questions on the α -corrected YB-deformed backgrounds. For example, it would be interesting
to see whether the singularities that are sometimes introduced by the deformation
procedure are in fact cured by α -corrections. Another point is the computation of physical
observables on the deformed backgrounds — such as entropy calculations in black hole
– 19 –
JHEP07(2020)103
solutions,30 see e.g. [42, 47–49] — for which the explicit corrections are needed. It would
be also interesting to investigate the relation to (quantum) integrability when considering
YB-deformations of integrable 2-dimensional σ-models.
We have seen that the α -correction to YB deformations comes from a compensating
Lorentz transformation under which the O(d, d) covariant metric and B-field transform
anomalously. It is natural to expect that the same should be true also for T-duality. In
fact abelian and non-abelian T-dualities are used to construct the YB deformation and
they can also be obtained as a limit (sending the deformation parameter to infinity) of
YB deformations. In fact we have already argued that for abelian T-duality the correction
is given by precisely the same mechanism. It is therefore very natural to expect the first
α -correction to non-abelian T-duality31 (on a unimodular algebra) to be given by the same
expression, with the Lorentz transformation required for NATD substituted for ˜Λ in (4.16)
and (4.17).
As in previous works on YB deformations and NATD (see e.g. [5–7]) here it was assumed
that the undeformed B-field and vielbein are isometric, i.e. that they have vanishing
Lie derivative with respect to the Killing vectors entering Θ. The covariant form of the
corrections we have found seems to be valid more generally but it would be interesting to
analyze more systematically how to relax these assumptions.
In [16] YB deformations of strings on AdS3 × S3 were studied, and their relation to
marginal deformations of WZW models was analyzed. The results of the current paper show
that marginal deformations of current algebras include (at least to 2 loops and probably to
all loops) also cases which do not solve the “strong version” of the marginality condition of
Chaudhuri and Schwartz [52], see [16] for more details. These additional possibilities arise
when considering algebras that are not compact. Let us also comment that the deformation
generated by the unimodular non-abelian R9 of [16] must be marginal to all loops, since it
can be simply understood as a non-commuting sequence of TsT transformations.
We expect that generalizations of our discussion to a construction in the spirit of the
E-model of Klimˇcik [53–55] will lead to an understanding of the form of α -corrections for
the η-deformation [2, 3], the λ-deformation [56, 57], and to Poisson-Lie T-duality [58].
Another important question we hope to return to is if the structure of the correction
found here persists beyond first order in α or whether novel corrections are required at
order α 2.
Acknowledgments
We thank D. Marqu´es for discussions and A. Tseytlin for comments on the manuscript. The
work of RB is supported by the fellowship of “la Caixa Foundation” (ID 100010434) with
code LCF/BQ/PI19/11690019, by FPA2017-84436-P, and by Xunta de Galicia (ED431C
2017/07). AVL is supported by the Spanish MECD fellowship FPU16/06675. RB and
30
While in this paper we have considered only the case of the bosonic string, it is easy to generalize our
results to generic values of the parameters a, b interpolating between the bosonic and the heterotic string.
31
Using very different arguments NATD has been argued to preserve Weyl invariance at least to 2 loops,
and probably to all orders in [50, 51].
– 20 –
JHEP07(2020)103
AVL are also supported by the European Regional Development Fund (ERDF/FEDER
program), by the “Mar´ıa de Maeztu” Units of Excellence program MDM-2016-0692 and by
the Spanish Research State Agency. The work of LW is supported by the grant “Integrable
Deformations” (GA20-04800S) from the Czech Science Foundation (GACR).
A Field redefinitions between different schemes
In this appendix we collect the field redefinitions needed to relate — to first order in the α
expansion — the schemes we use in this paper and others relevant in the literature. These
are the schemes of Hull and Townsend (HT) [59], Metsaev and Tseytlin (MT) [60], Kaloper
and Meissner (KP) [26, 61], Bergshoeff and de Roo (BR) [44, 45]. From [33] we read
G(BR)
mn = G(MT)
mn −
1
2
γ+H2
mn,
B(BR)
mn = B(MT)
mn − γ+
p
Hmnp − 2Hmnp
p
Φ + H[m
ab
ωn]ab
B(MT)
mn − γ+H[m
ab
ωn]ab,
Φ(BR)
= Φ(MT)
−
1
8
γ+H2
.
(A.1)
The symbol is used when the expressions are simplified by means of the 1-loop equations
of motion. We relate the parameters γ± = (a ± b)/4 to a, b used in [33]. The bosonic
string is obtained at γ+ = α /2, γ− = 0 and the heterotic string at γ± = ±α /4. In the
following we will specify to the case of the bosonic string. To relate HT and MT schemes
we use
G(HT)
mn = G(MT)
mn −
1
2
α H2
mn,
B(HT)
mn = B(MT)
mn ,
Φ(HT)
= Φ(MT)
+
1
8
α −1 +
1
6
(1 − 6q) H2
.
(A.2)
The parameter q appears in [59], and we normally set q = 0 in the rest of the paper as
in [21]. Notice that the sign of the correction to the metric differs from what one would
read in [59]. We have checked that this is the correct sign in order to have the correct
α -corrections for T-duality and YB deformations. From [61] we read that
G(MT)
mn = G(KM)
mn + α Rmn,
B(MT)
mn = B(KM)
mn − α Hmnp
P
Φ,
Φ(MT)
= Φ(KM)
+ α
1
8
R −
1
2
(∂Φ)2
+
1
96
H2
.
(A.3)
The fields of the non-covariant scheme that follows from the DFT formulation are denoted
simply with a bar ¯G, ¯B, ¯Φ. They are related to the fields in the HT scheme as
¯Gmn = G(HT)
mn −
1
2
α ω
(−)ab
(m ω
(+)
n)ab = G(HT)
mn + α −
1
2
ωmabωab
n +
1
8
H2
mn ,
¯Bmn = B(HT)
mn +
1
2
α ω
(−)ab
[m ω
(+)
n]ab = B(HT)
mn −
1
2
α Hab[mωab
n] .
(A.4)
– 21 –
JHEP07(2020)103
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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JHEP03(2020)126
Published for SISSA by Springer
Received: December 10, 2019
Revised: February 12, 2020
Accepted: March 6, 2020
Published: March 23, 2020
Two-loop conformal invariance for Yang-Baxter
deformed strings
Riccardo Borsatoa and Linus Wulffb
a
Instituto Galego de F´ısica de Altas Enerx´ıas (IGFAE), Universidade de Santiago de Compostela,
15782 Santiago de Compostela, Spain
b
Department of Theoretical Physics and Astrophysics, Masaryk University,
611 37 Brno, Czech Republic
E-mail: riccardo.borsato@usc.es, wulff@physics.muni.cz
Abstract: The so-called homogeneous Yang-Baxter (YB) deformations can be considered
a non-abelian generalization of T-duality-shift-T-duality (TsT) transformations. TsT
transformations are known to preserve conformal symmetry to all orders in α . Here we
argue that (unimodular) YB deformations of a bosonic string also preserve conformal symmetry,
at least to two-loop order. We do this by showing that, starting from a background
with no NSNS-flux, the deformed background solves the α -corrected supergravity equations
to second order in the deformation parameter. At the same time we determine the
required α -corrections of the deformed background, which take a relatively simple form.
In examples that can be constructed using, possibly non-commuting sequences of, TsT
transformations we show how to obtain the first α -correction to all orders in the deformation
parameter by making use of the α -corrected T-duality rules. We demonstrate this on
the specific example of YB deformations of a Bianchi type II background.
Keywords: Bosonic Strings, Conformal Field Models in String Theory, String Duality
ArXiv ePrint: 1910.02011
Open Access, c The Authors.
Article funded by SCOAP3
.
https://doi.org/10.1007/JHEP03(2020)126
JHEP03(2020)126
Contents
1 Introduction and summary of results 1
1.1 First α -correction to deformed backgrounds 3
2 Two-loop conformal invariance conditions 4
3 Expansion in the deformation parameter 5
3.1 First order in the deformation parameter 5
3.2 Second order in the deformation parameter 7
4 α -corrections from T-duality rules at two loops 11
5 Examples 14
5.1 Solvable pp-wave 14
5.2 Bianchi type II background 16
6 Conclusions 21
A Killing identities 22
B Relations needed for second order calculation 23
B.1 Additional identities 26
1 Introduction and summary of results
Yang-Baxter (YB) deformations were first introduced by Klimˇcik in [1]. It was later understood
that they have the remarkable property of preserving integrability [2]. If one starts
from an integrable sigma model and performs a YB deformation the resulting model is
also integrable. This made people interested in applying them in string theory, which was
done for the AdS5 × S5 superstring in [3, 4]. The YB deformation is based on an R-matrix
for which there are two basic possibilities — R can solve either the classical Yang-Baxter
equation (CYBE) or the modified classical Yang-Baxter equation (mCYBE). The former
case is often referred to as homogeneous YB deformations and is the case we consider here.
It was shown in [5] that these models typically have a Weyl-anomaly1 unless the R-matrix
is unimodular, i.e. its contraction with the structure constants of the isometry algebra of
the original model vanishes RIJ fIJ
K = 0. This is similar to the anomaly encountered in
non-abelian T-duality (NATD) [8] on a non-unimodular group [9–11]. Indeed it was argued
in [12] that homogeneous YB deformations should have a realization in terms of NATD and
this was then proven in [13] (see also [14]). While the original YB deformations were defined
only for sigma models of the symmetric space type, the realization of the homogeneous
1
This manifests itself, in the superstring case, as a target space solving the generalized supergravity
equations [6, 7] rather than the standard ones.
– 1 –
JHEP03(2020)126
models using NATD meant that they could be defined for a general string sigma model
with isometries. This was carried out for the Green-Schwarz superstring in [15] and rules
for writing the supergravity background directly in terms of the R-matrix were derived.2
The simplest class of such YB deformations is when R is defined on an abelian
subalgebra of the isometry algebra. In this case the deformation is equivalent to a
T-duality-shift-T-duality (TsT) transformation [19]. These are also known as O(d, d)transformations
[20, 21] and they have been argued to map a consistent string background
to another consistent string background, i.e. there should exist corrections to the background
fields such that the corrected background solves the α -corrected supergravity equations
to all orders in α [22–27].3 Here we want to ask what happens for YB deformations in
general at the quantum level.4 Unimodular YB deformations are known to give a conformal
theory at one loop, i.e. the background solves the (super)gravity equations. Here we will
analyze the two-loop equations in the bosonic string case. For simplicity we will restrict
to deformations of backgrounds with vanishing NSNS-flux. We will show, to second order
in the deformation parameter, that the deformed background can be corrected so that it
solves the 2-loop equations. Furthermore the correction to the background fields can be
cast in a relatively simple form. Using the knowledge of the full corrections in special cases
derived using T-duality (see below), we write an expression to all orders in the deformation
parameter, which works in some simple cases but not in general.
Since the homogeneous YB deformations can be constructed using NATD, our results
indicate that also NATD should preserve conformality at two loops, and possibly all orders
in α . A convincing argument for the preservation of conformality for NATD would
follow from a generic analysis to all orders in the deformation parameter η, since NATD
is recovered in a η → ∞ limit. Another piece of evidence for this comes from the recent
analysis of renormalizability of deformed sigma models with two-dimensional target space
in [30], and very recently [31] (see also [32]). Some of the deformations considered have a
limit where they reduce to NATD and it was found that the models behave nicely beyond
lowest order in α suggesting that things should work out to all orders in α .
For YB deformations of TsT-type we can also exploit another method to obtain explicit
α -corrections and to promote those backgrounds to two-loop solutions. We can in fact use
the known α -corrections to the T-duality rules when doing the chain of T-duality-shiftT-duality.
This strategy will automatically bring in the needed α -dependence into the
deformed background, and will make sure that the deformed background is a solution to
the two-loop equations. The interplay between T-duality and higher α -corrections was
studied in various works [26, 33–37]. In this paper we will use the α -corrections for the
T-duality rules of [34], to obtain explicit α -corrections for YB deformed models. This
strategy allows us to start from any background with isometries (it is not necessary to set
the NSNS-flux to zero), and to keep the dependence on the deformation parameter exact.
2
These rules were first guessed, at the supergravity level and restricted to the case of vanishing NSNS
flux, in [16] (see also [17] and [18]).
3
Note however that the form of the α -corrections are only known in special cases and to low loop order,
e.g. [26].
4
Homogeneous YB deformations also have an O(d, d) interpretation as so called β-shifts [28, 29].
– 2 –
JHEP03(2020)126
Certain YB deformations, while they cannot be understood as simple TsT transformations,
can still be obtained as a non-commuting sequence of TsT’s [5]. The noncommutativity
is related to the fact that certain isometries needed to perform one TsT
transformation may be broken by the application of another TsT. Therefore, in certain
cases a sequence of TsT transformations can be implemented only in one precise order.
Non-commuting sequences of TsT transformations are nice examples to study, because we
can obtain explicit results by applying what is known about abelian T-duality and TsT, and
at the same time be able to say something about NATD and more general YB deformations.
In the remaining part of the introduction, we will summarize the main results obtained
when expanding the two-loop equations to second order in the deformation parameter.
1.1 First α -correction to deformed backgrounds
The (homogeneous) Yang-Baxter deformation of a bosonic string background G, B, Φ is
given by [15–18]
˜G − ˜B = (G − B)(1 + ηΘ(G − B))−1
, ˜Φ = Φ −
1
2
ln det (1 + ηΘ(G − B)) . (1.1)
Here η is the deformation parameter and Θ is constructed by taking an anti-symmetric
R-matrix solving the classical Yang-Baxter equation (CYBE), R[I|L|RJ|M|fLM
K] = 0, on
a subalgebra of the isometry algebra of the original background (with structure constants
fIJ
K) and contracting with the corresponding Killing vectors
Θij
= kI
i
RIJ
kJ
j
≡ ki
× kj
, (ikIj) = 0 , (1.2)
where we simplify the notation by introducing the anti-symmetric product ‘×’. Assuming
that G, B, Φ define a one-loop conformal bosonic string sigma model, the same is true of
˜G, ˜B, ˜Φ if R is unimodular, i.e. RIJ fIJ
K = 0 [5].5
Here we want to ask what happens at two loops, i.e. the next order in α . We will
work in an expansion in the deformation parameter up to order η2. To simplify the calculations
we will assume that the starting background has B = 0 which gives the deformed
background
˜Gij = Gij +η2
(Θ2
)ij +O(η4
), ˜Bij = ηΘij +O(η3
), ˜Φ = Φ−
1
4
η2
ΘijΘij
+O(η4
). (1.3)
We find that to this order in η the first α -correction (i.e. two-loop correction) to the
background is given by (in the scheme of Hull and Townsend [40])
δ ˜Gij = δGij + 2η2
(δGΘ2
)(ij) + η2
(ΘδGΘ)ij − 2η2
Θk(iRj)
klm
Θlm + η2
Θmn
i jΘmn ,
δ ˜Bij = 2η(δGΘ)[ij] − ηRijklΘkl
, (1.4)
δ˜Φ = δΦ −
1
2
η2
(δGΘ)mnΘmn
+
1
16
η2 k
Θmn
kΘmn −
3
8
η2 k
Θmn
mΘnk
+
1
4
η2
iΦ i
(Θmn
Θmn) .
5
The unimodularity condition is sufficient but not necessary in general. Relaxing it one finds at order
η, assuming B = 0, the necessary condition dK = 0 where Kn
= mΘmn
. This is equivalent to
mkn
I fJK
I
RJK
= 0 which is in general weaker than the unimodularity condition kn
I fJK
I
RJK
= 0. The
reason for this is that sometimes the anomalous terms generated by a non-unimodular R can be removed
by a field redefinition [38] (see also [39]). Here we will take R to be unimodular for simplicity.
– 3 –
JHEP03(2020)126
Here δG, δΦ denote the α corrections to the undeformed background with B =δB =0. Note
that the terms involving δG just come from correcting the undeformed metric in (1.3), while
the terms involving the Riemann tensor in δ ˜G and δ ˜B are obtained simply by replacing
Θij → Θij − α RijklΘkl in (1.3). The correction to the dilaton does not look nice in this
scheme but by changing the scheme one can arrange it so that
e−2˜Φ
det ˜G = e−2Φ
√
det G , (1.5)
so that the correction to the dilaton just comes from the correction to the determinant of
the metric. This is achieved by the scheme change6
Φ → Φ + α −
1
2
2
Φ + ( Φ)2
−
1
16
HklmHklm
. (1.6)
With a little help from the corresponding expressions derived to all orders in η for
a particular background in (5.30) and (5.31) one can write a completion of (1.4) to all
orders in the deformation. First of all it is natural to expect that one should correct the
undeformed metric and take Θij → Θij − α RijklΘkl in the expressions in (1.1). On top of
this we need to extend the last term in the transformation of the metric and looking at the
example in (5.30) and (5.31) suggests the following form for the corrections to all orders in η
˜Gij − ˜Bij = G(1+η[Θ−α R·Θ])−1
ij
−
1
2
α ∂i lndet(1+ηΘ)∂j lndet(1+ηΘ)
+
1
2
α η G(1+ηΘ)−1
ik
k
jΘmn
+ G(1−ηΘ)−1
jk
k
iΘmn
G(1+ηΘ)−1
nm
(1.7)
with the transformation of the dilaton read off from (1.5) (in the HT scheme after the
shift (1.6)). Here indices are raised and lowered with the undeformed metric including
its α -corrections. We have also defined the contraction of Θ with the Riemann tensor
(R · Θ)ij = RijklΘkl. Note that this expression can be thought of as an α -corrected openclosed
string map, such as appears for example in the work of Seiberg and Witten on
non-commutative gauge theories [41]. While this result works for the rank 2 examples in
section 4 it unfortunately does not work in general.
2 Two-loop conformal invariance conditions
The conditions for two-loop conformal invariance of the bosonic string sigma model were
worked out in [42–44]. Following Hull and Townsend (HT) the conditions in their scheme
are [40]7
FG
ij = FG
0,ij + α FG
1,ij = 0 , FB
ij = FB
0,ij + α FB
1,ij = 0 , FΦ
ij = FΦ
0,ij + α FΦ
1,ij = 0 , (2.1)
6
On-shell this is equivalent to turning on the q parameter in the scheme of Hull and Townsend [40].
7
To go from their conventions to ours one sends Φ → 2Φ and H → 1
2
H.
– 4 –
JHEP03(2020)126
where the one-loop conditions are
FG
0,ij = Rij −
1
4
HiklHj
kl
+ 2 i jΦ ,
FB
0,ij = k
Hijk − 2 k
ΦHijk ,
FΦ
0,ij = 2 2
Φ − 4 iΦ i
Φ +
1
6
HijkHijk
(2.2)
and the two-loop corrections are
FG
1,ij =
1
2
RiklmRj
klm
+
1
4
RikljHkmn
Hl
mn +
1
4
Rklm(iHj)
mn
Hkl
n +
1
24
iHklm jHklm
−
1
8
k
Hlm
i kHlmj +
1
16
HikpHjlqHklm
Hpq
m +
1
16
HikpHjl
p
Hkmn
Hl
mn , (2.3)
FB
1,ij = k
Hlm
[iRj]klm −
1
4
kHlijHkmn
Hl
mn +
1
2
k
Hlm
[iHj]mnHkl
n
, (2.4)
FΦ
1,ij = −
1
4
RijklRijkl
+
1
12
( iHjkl)( i
Hjkl
) +
1
8
Hij
mHklm
Rijkl +
1
4
Rij(H2
)ij
−
5
96
HijkHi
lmHjl
nHkmn
−
3
32
H2
ij(H2
)ij
, (2.5)
where H2
ij = HiklHj
kl. Here we have set to zero the parameter q of [40].
3 Expansion in the deformation parameter
In this section we expand the conditions for two-loop conformal invariance in powers of
the deformation parameter η, and we find the explicit α corrections for the background
such that the conditions hold to the quadratic order in η. Here will not need to impose the
equation for the dilaton. It is known that when the equations for G and B are satisfied
the dilaton equation is satisfied up to a constant [40]. Since we assume the undeformed
background to solve all the two-loop equations and since there is no way to introduce a
constant at higher orders in η,8 the dilaton equation will not add anything.
3.1 First order in the deformation parameter
At order η1 we see, by looking at (1.3), that the metric is not deformed while9
H
(1)
ijk = 3 [iΘjk] . (3.1)
Using this in (2.4) we find
F
B(1)
1,ij = k
H(1)lm
[iRj]klm = k
(H
(1)
lm[iRj]k
lm
) + 2H
(1)
lm[i
l
Rj]
m
=
3
2
k
[i(Rjk]lmΘlm
) −
1
2
k
[Rijlm kΘlm
]
+ 2 k(R[i
klm
|l|Θj]m) − 2 k
Φ H
(1)
lm[iRj]k
lm
, (3.2)
8
The parameter η is always accompanied by Θ and it is not possible to construct a constant from a
general Θ.
9
We indicate the order in η with a superscript in parenthesis. Since it is clear that this refers to the
deformed background we drop the tilde.
– 5 –
JHEP03(2020)126
where we have used the lowest order equations (2.2). Using the two derivative Killing
identity (A.2) we have
k(Ri
klm
lΘjm) = kRi
klm
lΘjm + Ri
klm
k lΘjm
= kRi
klm
lΘjm + 2Ri
klm
k (lΘj)m − Ri
klm
k jΘlm
= −
3
2
k(Ri
klm
jΘlm) + 2Ri
klm
RjklnΘmn − Rim
kl
RjnklΘmn
+ Rklmn
RklmiΘjn + 3Riklm
k
Φ jΘlm
+ 2Ri
klm
kΦ lΘjm . (3.3)
Using this together with the identity (A.9) we find
F
B(1)
1,ij = 3 k
[i(Rjk]lmΘlm
) − 6 k
Φ [i(Rjk]lmΘlm
) + 2Rklmn
Rklm[iΘj]n . (3.4)
Taking into account the α -corrections to the classical background, α δG and α δΦ, and
the B-field at order η1, α (δ ˜B)(1), we have
α −1
FB
ij = 3 k
[i(δ ˜B)
(1)
jk]−6 k
Φ [i(δ ˜B)
(1)
jk]+3 k
[i(Rjk]lmΘlm
)−6 k
Φ [i(Rjk]lmΘlm
)
+3δ( k
) [iΘjk]−6δ( k
Φ) [iΘjk]+2Rklmn
Rklm[iΘj]n . (3.5)
In the case where the metric and dilaton do not receive corrections, δG = δΦ = 0, the
terms in the second line vanish, and the terms in the first line also vanish provided we take
(δ ˜B)
(1)
ij = −RijklΘkl
. (3.6)
In the general case the assumption that the corrected original background solves the twoloop
equations implies that
Rklm
nRklmi = −2δ(Rin + 2 i nΦ) = − k
iδGkn − k
nδGki + Gkl
i nδGkl
+ 2
δGin + 2 k
Φ( iδGkn + nδGki − kδGin) − 4 i nδΦ , (3.7)
where we used the expressions for the variation of the Ricci tensor and Christoffel symbols
(3.10) and (3.13).
Using this it is not hard to see, noting that δΦ must respect the isometries, that the
δΦ-terms cancel without any further correction to B. With a little bit more work one can
show, using the fact that LkδGij = 0, i.e. that the correction to the undeformed metric
does not break any isometries, that all terms cancel if one takes
(δ ˜B)
(1)
ij = 2(δGΘ)[ij] − RijklΘkl
. (3.8)
The first term is simply the correction induced by the correction to the undeformed metric,
i.e. δ(B(1))ij = δΘij, which comes from the fact that the indices on Θij were lowered with
the metric (note that the Killing vectors km
I , with an upper index, are not corrected by
assumption). Thus we have proven that a two-loop Weyl invariant sigma-model remains
two-loop Weyl invariant under a YB deformation, at least to first order in the deformation
parameter. We now consider what happens at second order.
– 6 –
JHEP03(2020)126
3.2 Second order in the deformation parameter
It is easy to see that at order η2 the B-field equation, F
B(2)
1,ij = 0, is trivially satisfied. For
the metric equation we find
F
G(2)
1,ij = R
(2)
(i
klm
Rj)klm−
1
2
R(i
klm
Rj)nlm(Θ2
)k
n
−R(i
klm
Rj)kl
n
(Θ2
)mn
+
1
4
RkijlH(1)kmn
H(1)l
mn+
1
4
Rklm(iH
(1)
j)
mn
H(1)kl
n+
1
24
iH
(1)
klm jH(1)klm
−
1
8
k
H(1)lm
i kH
(1)
lmj . (3.9)
Note that we choose to define all tensors to have lower indices, e.g. Rijkl, and then raise
indices with the undeformed metric Gij.
The last two terms do not involve the Riemann tensor and the calculations can be
simplified somewhat if we remove them by shifting the metric and dilaton. Under a shift
of the metric we have
δ( i jΦ) = −δΓk
ij kΦ = −
1
2
k
Φ( jδGki + iδGkj − kδGij) (3.10)
and
δRijkl = k(δΓilj − Γm
lj δGim) +
1
2
Rm
jklδGim − (k ↔ l) , (3.11)
so that in particular
R
(2)
ijkl = k(Γ
(2)
[ij]l + Γm
l[i(Θ2
)j]m) −
1
2
(Θ2
)m
[iRj]mkl − (k ↔ l)
= − k [i(Θ2
)j]l + l [i(Θ2
)j]k − (Θ2
)m
[iRj]mkl . (3.12)
The variation of the Ricci tensor becomes (symmetrization in ij understood)
δRij = δGkl
Rikjl + Gkl
δRikjl
= δGklRk
ij
l
+ Rk
jδGik + j[Gkl
δΓikl − Gkl
Γm
klδGim] − k
[δΓijk − Γl
jkδGil]
= k
iδGkj −
1
2
Gkl
i jδGkl −
1
2
2
δGij . (3.13)
From this expression we see that the last two terms in (3.9) can be canceled by shifting
the metric and dilaton as
Gij → Gij −
1
8
α HiklHj
kl
, Φ → Φ −
1
32
α HklmHklm
. (3.14)
The two-loop contribution then becomes (symmetrization in ij understood)
F
G(2)
1,ij = R
(2)
i
klm
Rjklm−
1
2
Ri
klm
Rjnlm(Θ2
)k
n
−Ri
klm
Rjkl
n
(Θ2
)mn+
1
8
RkijlH(1)kmn
H(1)l
mn
+
1
2
RklmiH
(1)
j
mn
H(1)kl
n−
1
8
Rklmn
H
(1)
ikl H
(1)
jmn−
1
24
H(1)klm
i jH
(1)
klm . (3.15)
Here we have used the Bianchi identity for H and the lowest order equations of motion,
which in particular imply
2
Hklm = 3 n
[kHlm]n = −3Rnp[klHm]
np
+ 6 n
Φ [kHlm]n . (3.16)
– 7 –
JHEP03(2020)126
Note that terms with two derivatives of H(1) indeed give something involving the Riemann
tensor since they involve three derivatives acting on a product of two Killing vectors giving
at least two derivatives on one Killing vector.
Expressing all terms in terms of the basis defined in appendix B we have (symmetrization
in ij understood)
R
(2)
i
klm
Rjklm = − · (f12 + f20) − (2 ˆf5 − ˆf6) + 2g32 + g34 − g35 + h7 −
1
2
h8
+
1
2
h10 + 2m7 + 2m9 (3.17)
RklmiH
(1)
j
mn
H(1)kl
n = −g3 + 2g4 − 2g6 + g8 − 2g14 + g15 (3.18)
Rklmn
H
(1)
ikl H
(1)
jmn = 4g16 + 4g17 + g19 , (3.19)
H(1)klm
i jH
(1)
klm = 3g3 − 6g4 − 6g6 + 3g8 − 18g14 + 9g15 + 6g28 − 6g29 − 3g31 + 12g32
− 12g33 + 12g34 . (3.20)
While the order η α -correction to ˜B in (3.8) contributes the terms (for the moment
we assume that the undeformed background is not corrected) (symmetrization in ij under-
stood)
−
1
2
(δ ˜H)
(1)
ikl H
(1)
j
kl
=
3
2 [i(Rkl]mnΘmn
)H
(1)
j
kl
= g3 − g8 − g15 + g16 +
1
2
g19 . (3.21)
For the two-loop correction we therefore get 1
8 times
− 8 · (f12 + f20) − 8 (2 ˆf5 − ˆf6) + 3g3 + 10g4 − 6g6 − 5g8 − 2g14 − 7g15 + 4g16 − 4g17
+ 3g19 + 4g30 + 2g31 + 12g32 + 4g33 + 4g34 − 8g35 − 8h8 + 4h10 + 16m7 + 16m9 (3.22)
To this we have to add the terms arising from the α -corrections to ˜G and ˜Φ. We will
ignore the corrections to the undeformed background until the end of the section.
Consider the following possible α -corrections to the metric at order η2 (symmetrization
in ij understood)
δ1
˜Gij = iΘmn jΘmn
, (3.23)
δ2
˜Gij = ki × mkn kj × m
kn
, (3.24)
δ3
˜Gij = iΘmn
mΘnj , (3.25)
δ4
˜Gij = Ri
klm
ΘjkΘlm . (3.26)
Note that we could write also the second one in terms of Θ as
δ2
˜Gij =
1
2
mΘin
m
Θj
n
−
1
2
n
Θim
m
Θjn− iΘmn
mΘnj +
1
4
iΘmn
jΘmn , (3.27)
– 8 –
JHEP03(2020)126
but the above expression is more convenient for the following calculation. Using (3.13)
and (3.10) these variations give rise to the terms
δ1
˜G : − · (2f3 + f28) − ( ˆf1 + 2 ˆf6) + g31 − 4m5 − 4m6 + 2m20
δ2
˜G :
1
2
· (f1 + 2f7 − f14 − 2f17 + f22 + 2f23) + (− ˆf1 + 2 ˆf2 + 2 ˆf3 − ˆf4 + 2 ˆf5) + g28
− g29 − 2g30 +
1
2
g31 − 2m12 + m13 +
3
8
i j(2 k
Θmn
mΘnk − 3 k
Θmn
kΘmn)
δ3
˜G : −
1
2
· (f1 + f3 + f10 − f11 + f22 + f28 + f30 − f31)
+
1
4
( ˆf1 − 2 ˆf2 − 2 ˆf3 + ˆf4 − 2 ˆf5 + 2 ˆf7 − 4 ˆf8) + g30 − m5 − m6 + m7 − m8 − m10
+ m11 − m13 + m20 + m22 − m23
δ4
˜G :
1
2
· (f9 + f14 − f26) −
1
4
(3 ˆf1 + 2 ˆf2 + 2 ˆf3 − 3 ˆf4 − 2 ˆf5 + 4 ˆf6) + h9 − m1 + m2
− m3 − m15 ,
where we used the identity (B.50) in calculating the last variation.
Taking the following correction to the metric and dilaton
(δ ˜G)
(2)
ij =
1
4
(−3δ1+2δ2+2δ3+6δ4) ˜Gij , (δ˜Φ)(2)
= −
3
32
(2 k
Θmn
mΘnk −3 k
Θmn
kΘmn)
(3.28)
and using appendix B we are left with 1
8 times the following order α terms
12g1 + 8g2 + g3 − 6g4 + 4g5 − 6g6 − 12g7 + 3g8 + 12g10 − 9g12
+ 24g13 + 12g14 − 6g15 + 6g16 − 6g19 − 6g20 + 12g21 + 8g22 − 2g23 − 12g24 + 16g25
+ 6h1 + 8h2 − 16h3 − 4h5 + 16h6 − 4h8 + 12h9 + 8h10 − 4h11 + 4 ˆf7 (3.29)
Next we use the Yang-Baxter equation which, in terms of Θ, reads
Θk[l
kΘmn]
= 0 . (3.30)
Hitting this with Ripmn
p we get the identity
0 = Rilmn
l
(Θkj
k
Θmn
) + 2Ri
lmn
l(Θkm
k
Θnj) = · (f19 − 2f11) . (3.31)
Adding −4 times the r.h.s. to our expression we are left with 1
8 times
12g1+8g2−3g3−6g4+12g5−6g6−12g7+3g8+12g10−9g12
+24g13+12g14−6g15+6g16−6g19−6g20+8g21+8g22−6g23−12g24+24g25
+6h1+8h2−16h3+24h6+12h9+8h10−4h11−8(m4−2m10+2m11)+4 ˆf7 , (3.32)
where the m-terms vanish by the Yang-Baxter equation. Using the identities (B.47)–
(B.49), (B.55) and (B.56) this reduces to (symmetrization in ij understood)
h10 −
1
2
h11 +
1
2
ˆf7 = RklmiRklmn
(Θ2
)nj −
1
2
Rklm
n
Rklmp
ΘinΘjp +
1
4
i j( l
Θmn
lΘmn) .
(3.33)
– 9 –
JHEP03(2020)126
The first two terms vanish if the original background does not suffer α -corrections, while
the last term can be canceled by shifting the dilaton.
To summarize we have found that with the following correction to the metric and
dilaton in the HT scheme at order η2, taking into account also (3.14), (symmetrization in
ij understood)
(δ ˜G)
(2)
ij = −
3
4
iΘmn
jΘmn −
1
2
mΘni jΘmn
−
3
2
Ri
klm
ΘlmΘkj , (3.34)
(δ˜Φ)(2)
=
1
16
k
Θmn
kΘmn −
3
8
k
Θmn
mΘnk , (3.35)
the deformed model is Weyl invariant at two loops provided the undeformed model is. The
shift in the metric does not look particularly natural but it can be brought to a nicer form
by noting that (symmetrization in ij understood)
mΘni jΘmn
= mkn × ki jΘmn
+
1
2
iΘmn
jΘmn
= ivj + Ri
klm
ΘkjΘlm +
1
2
iΘmn
jΘmn , (3.36)
where vj = mkn × kjΘmn. The first term represents a diffeomorphism, so it can be
dropped (note that the dilaton does not transform, vi
iΦ = 0, since it is isometric). It
will be convenient to perform a further diffeomorphism generated by vi = 1
2 Θmn iΘmn
after which we have (symmetrization in ij understood)
(δ ˜G)
(2)
ij = −2Ri
klm
ΘlmΘkj + Θmn
i jΘmn , (3.37)
(δ˜Φ)(2)
=
1
16
k
Θmn
kΘmn −
3
8
k
Θmn
mΘnk +
1
4
iΦ i
(Θmn
Θmn) . (3.38)
We will now consider what happens when the undeformed background receives α -
corrections.
Taking into account the lowest order correction to the metric and dilaton as well as
the first order correction to ˜B (3.8) we have (symmetrization in ij understood)
δ(R
(2)
ij −
1
4
H
(1)
ikl H
(1)
j
kl
+2[ i jΦ](2)
)+RklmiRklmn
(Θ2
)nj −
1
2
Rklm
n
Rklmp
ΘinΘjp . (3.39)
Using (3.7) and the variations in (3.13) and (3.10) this becomes, after a tedious calculation,
− 3 k
δGn
i
l
Θ[nkΘj]l − δGinkk
× [kl
, lkj] × kkn
+ 2δGknkk
× [kl
, lkj] × ikn
+ δGkn ikk
× [kl
, lkn
] × kj − δGkn
n
(ki × [kl
, lkj] × kk
)
− 2 kΦ δGinkn
× [kl
, lkk
] × kj + 2 k
Φ δGknki × [kl
, lkn
] × kj . (3.40)
The first term vanishes by the Yang-Baxter equation. Using the fact that kl
I lkn
J −
kl
J lkn
I = fIJ
Kkn
K and the YB equation (i.e. RIJ RKLfJK
M antisymmetrized in ILM
vanishes) this further reduces to
−
1
2
RIJ
RKL
fJK
M
fIL
N
δGinkMjkn
N = RMJ
RKI
fJK
L
fIL
N
δGinkMjkn
N
= −
1
2
RMJ
RKI
fKI
L
fJL
N
δGinkMjkn
N = 0 , (3.41)
– 10 –
JHEP03(2020)126
where we have used first the YB equation, then the Jacobi identity and finally the unimodularity
condition RKIfKI
L = 0.
This shows that the only additional corrections that arise are the ones coming from
correcting the undeformed metric in ˜G(2) and Φ(2) so that
(δ ˜G)
(2)
ij = 2(δGΘ2
)(ij)+(ΘδGΘ)ij −2Θk(iRj)
klm
Θlm+Θmn
i jΘmn , (3.42)
(δ˜Φ)(2)
= −
1
2
(δGΘ)mnΘmn
+
1
16
k
Θmn
kΘmn−
3
8
k
Θmn
mΘnk +
1
4
iΦ i
(Θmn
Θmn).
(3.43)
This completes the proof that, at least to second order in the deformation and when B = 0,
unimodular YB deformations preserve conformality at two loops.
4 α -corrections from T-duality rules at two loops
Homogeneous Yang-Baxter deformations are closely related to non-abelian T-duality [12,
13] and it can be shown that the non-abelian T-dual model is in fact recovered in the
maximally deformed limit η → ∞ [13], see also [14, 15]. The simplest class of YangBaxter
deformations — the “abelian” one — is related to just abelian T-duality, and is
equivalent to doing TsT transformations [45, 46]. In general, a Yang-Baxter deformation
generated by Θ = k1 ∧ k2 where k1 = ∂x1 and k2 = ∂x2 are commuting Killing vectors,
is equivalent to doing first a T-duality x1 → ˜x1, then a shift x2 → x2 + η˜x1, and then a
T-duality back ˜x1 → x1. Some “non-abelian” deformations are non-commuting sequences
of TsT’s [5, 47]. The non-abelian nature is related to the fact that the order in which the
TsT transformations are performed is important, as certain T-dualities would break the
isometries that are needed to perform the other T-dualities in the sequence. In this section
we want to exploit the relation to TsT transformations and combine it with the knowledge
of the first α -corrections of the T-duality rules, to obtain two-loop corrections for all YangBaxter
deformations that are obtainable by TsT transformations, or more generically by
a non-commuting sequence of them. This strategy allows us to obtain backgrounds at two
loops that are exact in the deformation parameter η. Moreover, these tools can be applied
to any starting background with isometries, and it is not needed to restrict to B = 0 as we
assume in most of this paper.
Because at each step all that we are doing is (abelian) T-duality and coordinate transformations,
we are bound to preserve conformal invariance on the worldsheet to the very
end, and we can check explicitly that the solutions we generate do solve the two-loop equations.
This argument can be repeated also to higher orders in the α expansion, and it is
enough to conclude that all Yang-Baxter deformations that are obtainable by a generically
non-commuting sequence of TsT transformations, do not break the conformality of the
original model to all orders in α .
At leading order in α the T-duality rules are given by the Buscher rules [48]. At higher
loops these rules get corrected in α . We will use the α -corrections to the T-duality rules
derived by Kaloper and Meissner in [34]. The rules were obtained by carefully analysing the
two-loop effective action of the bosonic string, and identifying the terms that are symmetric
– 11 –
JHEP03(2020)126
or anti-symmetric under the Buscher rules. The α -corrections of the T-duality rules were
then fixed by requiring that they give a symmetry of the full two-loop effective action,
compensating for the antisymmetry of those terms.10
Already at leading order in α , the T-duality rules are more easily presented in terms
of fields of a dimensional reduction, where we reduce along the direction that we want to
T-dualize. We follow [34] and we rewrite the metric, Kalb-Ramond field and dilaton of the
D-dimensional spacetime in terms of the following (D − 1)-dimensional fields
ds2
= Gijdxi
dxj
= gµνdxµ
dxν
+ e2σ
(dx + V )2
,
B =
1
2
Bijdxi
∧ dxj
=
1
2
bµνdxµ
∧ dxν
+
1
2
W ∧ V + W ∧ dx ,
Φ = φ +
1
2
σ .
(4.1)
Here we are assuming that we have brought the solution in a form such that the isometry
we want to dualize is simply implemented by a shift of a coordinate, that we denote by
x. We use Greek indices for the (D − 1)-dimensional spacetime.11 We have introduced a
(D − 1)-dimensional metric gµν, and antisymmetric bµν, vectors Vµ and Wµ, and scalars φ
and σ. Above we also used form notation V = Vµdxµ, W = Wµdxµ. In components, the
relations to identify the fields of the dimensional reduction are
σ =
1
2
log Gxx , Vµ =
Gµx
Gxx
, gµν = Gµν −
GµxGνx
Gxx
,
φ = Φ −
1
4
log Gxx , Wµ = Bµx , bµν = Bµν +
Gx[µBν]x
Gxx
.
(4.2)
It is also useful to notice that Gµν = gµν, Gµx = −V µ, Gxx = e−2σ +V 2. The combination
hµνρ = 3 ∂[µbνρ] −
1
2
W[µνVρ] −
1
2
V[µνWρ] = Hµνρ − 3W[µνVρ] , (4.3)
is gauge invariant. In terms of these new fields the Buscher rules are simply
σ → −σ, V ↔ W . (4.4)
All other fields remain unchanged under T-duality at leading order in α .
In [34] Kaloper and Meissner derived the corrections to the T-duality rules in a particular
scheme introduced by Meissner in [49]. We will call it the Kaloper-Meissner (KM)
scheme. In order to apply the T-duality rules of KM to our case, we will therefore first need
to implement the field redefinitions to go from the scheme of HT to that of KM. We can
do so by combining the formulas given in [40] (see their equations (61) and (64)) relating
the HT scheme to the Metsaev-Tseytlin (MT) scheme of [43], and those given in [49] (see
10
In [34] the authors claim that their results can be applied also to the heterotic string, but the action they
start with is missing the Chern-Simons terms that are expected there. See [37] for α -corrected T-duality
rules that encompass both the bosonic and the heterotic string.
11
The discussion of the α -corrected T-duality rules and their derivation simplifies if written in terms of
tangent-space indices, but we will not do so here.
– 12 –
JHEP03(2020)126
his equations (3.7), (4.1) and (4.7)) to go from MT to KM.12 The field redefinitions that
we will use are13
G
(HT)
ij = G
(KM)
ij + α Rij −
1
2
H2
ij ,
B
(HT)
ij = B
(KM)
ij + α −Hijk
k
Φ ,
Φ(HT)
= Φ(KM)
+ α −
3
32
H2
+
1
8
R −
1
2
( Φ)2
.
(4.5)
Once we are in the scheme of KM we can use their α -corrected T-duality rules [34]
σ → −σ+α ( σ)2
+
1
8
(e2σ
Z+e−2σ
T)
Vµ → Wµ+α Wµν
ν
σ+
1
4
hµνρV νρ
e2σ
(4.6)
Wµ → Vµ+α Vµν
ν
σ−
1
4
hµνρWνρ
e−2σ
bµν → bµν +α Vρ[µWρ
ν]+ W[µρ
ρ
σ+
1
4
e2σ
h[µρλV ρλ
Vν]+ V[µρ
ρ
σ−
1
4
e−2σ
h[µρλWρλ
Wν]
Indices are always raised/lowered using the (D − 1)-dimensional metric gµν, and the transformations
are written using also the following definitions
Vµν = ∂µVν − ∂νVµ , Zµν = VµρV ρ
ν , Z = Z µ
µ ,
Wµν = ∂µWν − ∂νWµ , Tµν = WµρW ρ
ν , T = T µ
µ .
(4.7)
In general, at higher loops, not only σ, V and W will change under T-duality. In fact, at
two loops in the scheme of KM also bµν gets modified.14 It is important to remark that
already before doing T-duality the fields will in general have an explicit α -dependence. In
particular, σ, V and W that transform according to (4.6) may in general depend on α , and
this must be taken into account already when implementing the leading order T-duality
rules (the Buscher rules).
One could in principle combine the T-duality rules of KM in (4.6) with the field
redefinitions in (4.5), to obtain the α -corrections of the T-duality rules in the scheme of
12
The field redefinitions given in [49] relate the KM and the MT schemes only on-shell, but this is enough
for our purposes, since we just want to make sure that we can generate solutions of the two-loop equations.
13
These are the redefinitions needed when we set the parameter q of [40] to zero. Different values of q
would affect the coefficient of H2
that appears in the redefinition of the dilaton. Importantly, the coefficient
in front of H2
ij that appears in the redefinition of the metric has the opposite sign compared to what one
would expect from formulas in [40] or [49]. We have checked in various examples, some not included in
this paper, that we must have the sign that we use here, as this is fixed by requiring that we want to
have a solution of the two-loop equations after doing T-duality in the KM scheme and going back to the
HT scheme.
14
In [34] the rules were given in terms of transformations of hµνρ. Here we preferred to rewrite them
as a transformation of bµν . Importantly, the α -corrections to the T-duality rules of bµν (or equivalently
hµνρ) differ by an overall sign compared to those given in [34], and our formula corrects the one given there.
We thank A. Vilar L´opez for discussions on this point. A future paper will contain also more details on
this [50].
– 13 –
JHEP03(2020)126
HT. We will not do so here, as the scheme of KM appears to be the minimal scheme for
what concerns the complexity of the corrections to the T-duality rules. In other schemes, all
other fields of the dimensional reduction will in general receive α -corrections. Therefore,
to obtain Yang-Baxter deformations in the scheme of HT we will follow this strategy:
1. Start from a solution of the two-loop equations in the HT scheme. In general that
implies finding α -corrections for this initial solution.
2. Go to the scheme of KM using (4.5).
3. Do TsT or sequences of TsT transformations, using the α -corrected T-duality rules
in (4.6).
4. Go back to the scheme of HT using (4.5).
We have worked out examples to test this method and obtain explicit results for α corrections
of Yang-Baxter deformed models. This also allows us to relate to the results of
section 3 that are perturbative in η. We will provide an example in the next section.
5 Examples
In this section we consider two particularly simple examples.
5.1 Solvable pp-wave
We start with the pp-wave background considered in [51]
ds2
= 2dx+
dx−
−
k
(x+)2
x2
m(dx+
)2
+ dx2
m , Φ = mx+
+
d
2
k ln x+
, (5.1)
where 0 < k < 1
4 is a constant, m is another constant and d is the number of transverse
dimensions. This background is known not to receive α -corrections. This follows from the
fact that the only non-zero component of the Riemann tensor is R+m+n = δmnk(x+)−2.
Consider the following four Killing vectors
k1 = (x+
)ν
∂1 − ν(x+
)ν−1
x1∂− , k3 = (2ν − 1)∂− ,
k2 = (x+
)1−ν
∂1 − (1 − ν)(x+
)−ν
x1∂− , k4 = (x+
)ν
∂2 − ν(x+
)ν−1
x2∂− , (5.2)
where we have defined the parameter
ν =
1 +
√
1 − 2k
2
. (5.3)
They form a Heisenberg algebra of isometries with the only non-trivial Lie bracket [k1, k2] =
k3. From the discussion of R-matrices in [5] we see that we can consider the non-abelian
rank 4 deformation
Θ = k1 ∧ k4 + sk2 ∧ k3 , (5.4)
– 14 –
JHEP03(2020)126
where we introduced the parameter s to keep track of the contribution from the second
term. We will show below that in this case this deformation is equivalent to the abelian
one obtained by setting s = 0. First we construct the matrix
Θij
=





0 0 0 0
0 0 a b
0 −a 0 c
0 −b −c 0





, (5.5)
where
a = ν(x+
)2ν−1
x2 − s(2ν − 1)(x+
)1−ν
, b = −ν(x+
)2ν−1
x1 , c = (x+
)2ν
. (5.6)
The deformed background takes the form
˜ds
2
= 2dx+
dx−
+ η2 ac
1 + η2c2
dx2
− η2 bc
1 + η2c2
dx1
−
k
(x+)2
x2
m + η2 a2 + b2
1 + η2c2
(dx+
)2
+
dx2
1 + dx2
2
1 + η2c2
+ dx2
m .
(5.7)
With the B-field and dilaton given by
˜B = −
η
1 + η2c2
(adx1
+ bdx2
) ∧ dx+
+ cdx2
∧ dx1
, ˜Φ = Φ −
1
2
ln(1 + η2
c2
) . (5.8)
One sees from this that
˜H = 4ην(x+
)2ν−1
dx2
∧ dx1
∧ dx+
, (5.9)
which is independent of the parameter s. The fact that also Φ is independent of s suggests
that it might be possible to remove the s dependence also from the metric. Consider the
change of coordinates x2 → x2 + f and x− → x− + gx2 + h where f, g, h are functions only
of x+. One finds that the choice
f =
s
2
η2
(2ν − 1)(x+
)ν+2
, g = −
s
2
η2
(2ν − 1)ν(x+
)ν+1
,
h =
s2
8
η2
(2ν − 1)2
4(3 − 2ν)−1
(x+
)3−2ν
− η2
ν(x+
)3+2ν
, (5.10)
removes the dependence on s completely and reduces the background to the one obtained
by the TsT with
Θ = k1 ∧ k4 . (5.11)
Explicitly, the metric is
˜ds
2
= 2dx+
dx−
+ νη2
c2
(1 + η2
c2
)−1
(x1dx1 + x2dx2
)/x+
− (x+
)−2
kx2
m + ν2
η2
c2
(1 + η2
c2
)−1
(x2
1 + x2
2) (dx+
)2
+
dx2
1 + dx2
2
1 + η2c2
+ dx2
m .
(5.12)
From (1.4) we find the only correction to the deformed background is given by
δG++ = −4η2
(2ν2
− ν)(x+
)4ν−2
, (5.13)
– 15 –
JHEP03(2020)126
which can be canceled by a diffeomorphism δG++ = +v+. In fact the change of coordinates
x− → x− + νη2c2 x2
1+x2
2
2x+(1+η2c2)
, x1,2 → 1 + η2c2 x1,2 brings the deformed metric to
the form
˜ds
2
= 2dx+
dx−
+(x+
)−2
−kx2
m+η2
c2
[−3ν+5ν2
−kη2
c2
]
x2
1+x2
2
1+η2c2
(dx+
)2
+dx2
m . (5.14)
Therefore this background is exact at two loops, as is easily checked directly, and possibly
to all loops.
5.2 Bianchi type II background
Next we consider the Bianchi type II background [52, 53] (the α -corrections to Bianchi
type I were considered in [54])
ds2
= − cosh(τ)e(a+b+c)τ
dτ2
+
eaτ
cosh(τ)
(dx−zdy)2
+cosh(τ)e(a+b)τ
dy2
+cosh(τ)e(a+c)τ
dz2
,
(5.15)
supported by a dilaton linear in τ
Φ = aτ/2 . (5.16)
This solves the Einstein equations provided that the parameters a, b, c are related as
bc = a2
+ 1 . (5.17)
The solution has three Killing vectors
k1 = −∂z − y∂x , k2 = ∂y , k3 = ∂x , (5.18)
which again satisfy a Heisenberg algebra [k1, k2] = k3.
From now on we will simplify things by taking a = 0 and b = c = 1. The twoloop
equations are not automatically satisfied, and we need to find α -corrections for this
background. It is convenient to introduce a new coordinate system {v, x, y, z} where v = eτ ,
since the metric then has a rational dependence on v
ds2
=
2v(dx − zdy)2
v2 + 1
+
v2 + 1 v dy2 + dz2 − dv2
2v
. (5.19)
We assume that the correction to the metric δGij respects the isometries of the background.
We turn on the diagonal components δGii and δG12 = −zδG11. We also allow for a
correction to the dilaton δΦ that, together with δGii, is allowed to depend only on v. The
two-loop equation for the B-field is already satisfied. First it is simpler to solve the twoloop
equation for the dilaton, because there only the correction δΦ contributes. One finds a
second order differential equation −3v6 +45v4 −45v2 +3− v2 + 1
5
(vδΦ (v) + δΦ (v)) = 0
solved by
δΦ =
v
2 (v2 + 1)
+
2v
(v2 + 1)3 +
1
2
arctan v + cΦ log v , (5.20)
where cΦ is a constant. Looking at the two-loop equations for the metric, one can find a
linear combination of those equations that gives an algebraic constraint imposing δG11 = 0.
– 16 –
JHEP03(2020)126
To find δG00, δG22, δG33, we first identify linear combinations of the equations that give
first order differential equations for δG00 and δG33, and we solve them obtaining results
written in terms of δG22. These are then used to get a third order differential equation for
δG22 only, that we also solve. The final result is
δG00 =
−2v8+20v4+8v2−2 v2−3 v2+1
3
varctanv+6
(v4−1)2
+
v2+1 c00 v2−1
2
+v2(c22−2f22)+c22−2f22−4cΦ v2−3 v2 logv+8cΦ
v(v2−1)2 ,
δG22 =
3v2−1 v2+1
3
arctanv+v v4+2v2+5
2(v2−1)(v2+1)2
+
v2+1 v2(d22−c22)+3c22−d22+2logv f22 v2−1 +4cΦ −4f22+8cΦ
4(v2−1)
,
δG33 = δG22−
1
2
v2
+1 (2c00−2c22+d22+2(f22−6cΦ)logv+2f22). (5.21)
For simplicity in what follows we will set all integration constants cΦ = c00 = c22 = d22 =
f22 = 0. This background admits a non-abelian deformation with
Θ = αk1 ∧ k4 + βk2 ∧ k3 , (5.22)
where α, β are parameters and we have introduced an additional flat direction w so that
we can have a fourth Killing vector k4 = ∂w. If both α and β are non-zero, they can be
reabsorbed by redefining w and the deformation parameter η. For simplicity we set α = 0,
β = 1 and analyze the abelian deformation given by
Θ = k2 ∧ k3 . (5.23)
The Yang-Baxter deformation to lowest order in α yields the following deformed back-
ground15
ds2
=
v2+1
2
+4vz2 dy2−8vzdxdy+4vdx2
2(v2+1)(1+η2v)
−
v2+1 dv2
2v
+
1
2
v2
+1 dz2
,
B =
ηvdx∧dy
1+η2v
,
Φ = −
1
2
log 1+η2
v .
(5.24)
We can obtain the first α -correction exactly in the deformation parameter η if we
follow the strategy outlined in section 4. The deformation generated by Θ = k2 ∧ k3 is
equivalent to doing first a T-duality along x, then shifting y → y − η˜x where ˜x is the dual
coordinate to x, and then T-dualising ˜x back.
We first start from the background given by the metric (5.19) and the α -corrections
(5.21). This background solves the two-loop equations in the HT scheme, and we
15
We remind that in this paper we use the convention B = 1
2
Bijdxi
∧ dxj
.
– 17 –
JHEP03(2020)126
need to apply (4.5) in order to find a solution in the KM scheme. Obviously, since the
corrections in (4.5) are multiplied by an explicit power of α , it is enough to use the uncorrected
background to derive them, which simplifies the calculation. Because B = 0, we
can in principle get a non-trivial modification only for the metric from the Ricci tensor,
and for the dilaton from the Ricci scalar. But the Bianchi II background is also Ricci-flat,
therefore it is the same in the KM scheme and in the HT scheme. The next step is that
of identifying the fields of the dimensional reduction as in (4.1). Because we want to do
T-duality along x here, we are taking x = x. This is a straightforward exercise, and instead
of writing down all fields of the dimensional reduction, we only write those that can
potentially change under the corrected T-duality rules
σ =
1
2
log
2v
1 + v2
, V = −zdy , W = 0 , b = 0 . (5.25)
These particular fields of the dimensional reduction happen not to depend on α in this
particular example. We then implement the α -corrected T-duality rules of KM as in (4.6)
and obtain the fields of the dimensional reduction after T-duality
σ = −
1
2
log
2v
1 + v2
−α
v4 − 6v2 + 1
2v (v2 + 1)3 , V = 0 , W = −zdy , b = 0 . (5.26)
After T-duality the scalar σ does depend explicitly on α . The explicit form of the two-loop
background after performing this first T-duality along x is
ds2
=
1
2
v+v−1
−
α v4−6v2+1
(v3+v)2 d˜x2
+
1
2

1+v2
+
α 3v2−1 v2+1
3
arctanv+v v4+2v2+5
(v2−1)(v2+1)2

(dy2
+dz2
)
+

−
v2+1
2v
−
2α v8−10v4−4v2+ v2−3 v2+1
3
varctanv−3
(v4−1)2

dv2
,
B = zd˜x∧dy,
Φ = −
1
2
log
2v
v2+1
+α
2v6+3v4+16v2−1
4v(v2+1)3 +
1
2
arctanv .
(5.27)
In the T-dual frame the metric is diagonal (even to two loops) at the cost of having a nonvanishing
B-field. We can now do the shift y → y − η˜x, that here will have only the effect
of modifying the metric. To perform another T-duality along ˜x we have to first repeat the
– 18 –
JHEP03(2020)126
identification of the fields of the dimensional reduction. We find in particular
σ =
1
2
log
1
2
η2
v2
+1 +v+v−1
+
1
2
α

−
v4−6v2+1
v(v2+1)3
(1+η2v)
+
η2v 3v2−1 v2+1
3
arctanv+v v4+2v2+5
(v2−1)(v2+1)3
(η2v+1)

 ,
V =
−η dy
(1+η2v)2

v 1+η2
v +
α v 3v2−1 v2+1
2
arctanv+3 v6+v4+v2 −1
(v2−1)(v2+1)2

 ,
W = −zdy, b = 0. (5.28)
At this point we can use again the T-duality rules of KM (4.6). After doing that we obtain
the following background
ds2
= −
v2+1 dv2
2v
+
2(dx−zdy)2
η2 (v2+1)+v+v−1
+
v2+1 dy2
2(1+η2v)
+
1
2
v2
+1 dz2
+α δG00dv2
−4α η2
v2 δG22
(v2+1)2
(1+η2v)2 +2v
v2−1
(v2+1)4
(1+η2v)2 (dx−zdy)2
+α
δG22
(1+η2v)2 −η2 v4−6v2+1
2(v2+1)2
(1+η2v)2 dy2
+α δG22dz2
,
˜B =
α ηdv∧dz
(v2+1)(1+η2v)
+ηvdx∧dy
1
1+η2v
+2α
δG22
(v2+1)(1+η2v)2 +α
2v(v2−3)+η2 3v2−1 v2−1
(v2+1)3
(1+η2v)3 ,
˜Φ = −
1
2
log 1+η2
v +α δΦ−α η2 4v(v2+1)2δG22+5v4−10v2+1
4(v2+1)3
(1+η2v)
, (5.29)
where δGij and δΦ are the corrections to the undeformed background given in (5.20)
and (5.21). This is a TsT of the initial Bianchi II that solves the two-loop equations in
the KM scheme. To go to the HT scheme we use again (4.5). Because of the deformation,
now the dictionary to go to the new scheme is non-trivial, and the background in the HT
scheme reads
ds2
= ˜Gijdxi
dxj
,
˜B =
α ηdv∧dz
(v2+1)(1+η2v)
+ηvdx∧dy
1
1+η2v
+2α
δG22
(v2+1)(1+η2v)2 +2α v
v2−3
(v2+1)3
(1+η2v)2 ,
˜Φ = −
1
2
log 1+η2
v +α δΦ−α η2 vδG22
(v2+1)(1+η2v)
−α η2 3v4−14v2−1
4(v2+1)3
(1+η2v)
,
(5.30)
– 19 –
JHEP03(2020)126
where
˜G00 = −
v2 + 1
2v
+ α δG00 − α η2 η2 + 3η2v2 + 2v
2 (v2 + 1) (1 + η2v)2 ,
˜G11 =
2v
(v2 + 1) (1 + η2v)
− 4α η2
v2 δG22
(v2 + 1)2
(1 + η2v)2 − 4vα η2
v2 v2 − 3
(v2 + 1)4
(1 + η2v)2 ,
˜G22 =
v2 + 1
2 (1 + η2v)
− α η2
v2 v2 − 3
(v2 + 1)2
(1 + η2v)2 + α
δG22
(1 + η2v)2 + z2 ˜G11 ,
˜G33 =
1
2
v2
+ 1 + α δG22 − α η2 v2
(v2 + 1) (1 + η2v)
,
˜G12 = −z ˜G11 . (5.31)
Performing the redefinition of the dilaton given in (1.6) this background agrees precisely
with that obtained from the all order expression (1.7).
When we want to work out a deformation generated by Θ = k1 ∧ k4 following the
strategy of section 4, we first need to find a coordinate system in which k1 acts as a simple
shift of a coordinate. We can redefine
x = x + y z , y = y , z = z , (5.32)
so that in the new coordinate system k1 = −∂z . As should be clear from the discussion at
the beginning of this section, the isometry generated by k1 is not broken by α corrections,
therefore the metric will not depend on z also at two loops. The deformation generated
by Θ = k1 ∧ k4 can be obtained by doing T-duality w → ˜w, then the shift z → z − η ˜w,
and then T-duality back ˜w → w. We will omit the explicit results for this particular
deformation, since they involve very long expressions, and we have already presented our
method in the previous deformation generated by Θ = k2 ∧ k3. We have checked that
the resulting background again agrees with that obtained by the α -corrected open-closed
string map (1.7).
The interesting point is that we can combine these two TsT transformations. We can
first do a TsT involving x and y corresponding to Θ = k2 ∧ k3. At the end of this result
the background is still invariant under isometries generated by k1 and k4, and we can do a
second TsT transformation involving z and w, equivalent to Θ = k1 ∧k4. The composition
of the two deformations is equivalent to the deformation given by Θ = k1 ∧ k4 + k2 ∧ k3, as
explained in [15]. The non-abelian nature of the deformation is related to the fact that if we
had started from Θ = k1 ∧ k4 instead, we would have broken the isometries that we would
need to perform the deformation with Θ = k2 ∧k3. As follows from the results of [15], in the
maximally deformed limit η → ∞ we recover the non-abelian T-dual of the original Bianchi
II solution, where the isometries dualized are those corresponding to the Killing vectors
k1, k2, k3 forming a Heisenberg algebra, and k4. By this argument it follows that nonabelian
T-dual models related to this class of Yang-Baxter deformations remain conformal
on the worldsheet to two loops. Because T-duality remains a symmetry of the string at
higher orders in an α -expansion, we can argue that this is true to all loops. Unfortunately
the all order expression (1.7) turns out not to give the correct answer in this case.
– 20 –
JHEP03(2020)126
6 Conclusions
We have argued that (homogeneous) YB deformed string σ-models that are conformal at
one loop remain conformal at two loops,16 i.e. including the first correction in α . We showed
this to second order in the deformation parameter η for a generic unimodular deformation
of a background with vanishing B-field. We also argued that using the α -corrected Tduality
rules of [34] one can verify this to all orders in the deformation parameter for the
cases that can be built from TsT transformations, and we explained that this strategy can
be used also for the non-abelian YB deformations that are equivalent to a non-commuting
sequence of TsT transformations.17 We exemplified our results in the case of a deformation
of a Bianchi type II background.
Our findings suggest that one-loop conformal YB σ-models should in fact remain conformal
to first order in α , and likely all orders. Since these models can be thought of as a
generalization of non-abelian T-duality [12, 13, 15] (which can be recovered in an appropriate
η → ∞ limit) our findings suggest that the same should be true for NATD. This was
also argued recently from a different perspective in [30, 31], studying renormalizability of
a different type of integrable deformation of σ-models.18 To test this idea one should start
from a model which is conformal to all orders in α and then deform it. A good candidate
is therefore the unimodular deformation of AdS3 × S3 constructed in [39].
We saw that the expression (1.7) for the all order in η form of the first α -correction
to YB deformations works in simple cases but fails in general. It is an important problem
to fix it so that it holds in general. If a simple solution exists for the corrections, it is
also interesting in the special case of TsT transformations, whose corrections have, to our
knowledge, not been analyzed before. If, further more, this continues to work to higher
orders in α it could even help in determining the structure of higher α -corrections to the
target space equations of motion. This approach could be said to be an example of using
O(d, d) symmetry to determine/constrain higher α -corrections.
We plan to address some of these questions in the near future.
Acknowledgments
We thank B. Hoare, N. Levine and A. Tseytlin for useful and interesting discussions. LW
wishes to thank the participants of the workshop “New frontiers of integrable deformations”
in Villa Garbald, Castasegna for interesting discussions. RB is grateful to J. Edelstein,
J.A. Sierra-Garcia, and in particular to A. Vilar L´opez for very useful discussions. The work
of RB is supported by the fellowship of “la Caixa Foundation” (ID 100010434) with code
LCF/BQ/PI19/11690019. He is also supported by the Maria de Maeztu Unit of Excellence
MDM-2016-0692, by FPA2017-84436-P, by Xunta de Galicia (ED431C 2017/07), and by
FEDER.
16
Provided, of course, the undeformed background is conformal to two loops.
17
See e.g. [5, 47].
18
Early work on α -corrections in NATD include [55–58].
– 21 –
JHEP03(2020)126
A Killing identities
The Killing vectors satisfy the equations (suppressing the Lie algebra index)
(ikj) = 0 i jkl = Rljinkn
. (A.1)
Using this and the expression for Θ in (1.2) we can derive the useful two-derivative identity
2 k (iΘj)l = 2 kk(i × j)kl + 2Rknl(iΘj)
n
= − (i j)Θkl + 2Rknl(iΘj)
n
− Rk(ij)nΘl
n
+ Rl(ij)nΘk
n
. (A.2)
A special case of this is
2
Θij = −RijklΘkl
+ RikΘj
k
− RjkΘi
k
. (A.3)
In addition we have the unimodularity condition, which in terms of Θ, takes the form
kΘkl
= 0 . (A.4)
We also know that the dilaton respects the isometries so that
ki
iΦ = 0 . (A.5)
Using these facts we can prove the useful identity
k(RijlmΘlm
) = −
1
2
Riklm jΘlm
+ Rimkl
m
Θj
l
− Rilmk
m
Θj
l
− (i ↔ j) . (A.6)
This follows by noting that
2Rijlm kΘlm
= −4 m jki × kkm
= −4 m( jki × kkm
) + 4Rkl jki × kl
= −2 m j(ki × kkm
) + 2 m(RmkjlΘi
l
) + 2Rkl jki × kl
− (i ↔ j)
= − m
j kΘim + m
j mΘik + m
j iΘmk + 2 m
(RmkjlΘi
l
)
+ 2Rkl jki × kl
− (i ↔ j)
=
1
2
Rijlm kΘlm
−
1
2
kRijlmΘlm
−
1
2
Riklm jΘlm
+ Rimkl
m
Θj
l
− Rilmk
m
Θj
l
− (i ↔ j) , (A.7)
where we have used the fact that
l
Φ kΘlj + l
Φ lΘkj + l
Φ jΘkl = 0 , (A.8)
as is easily verified. Acting with k, and using also kΦ times the above identity, one
finds
4 k
(Rijlm kΘlm
) = 3 k
[i(Rjk]lmΘlm
) − 2RimklRjn
kl
Θmn
+ 4Ri
klm
RjklnΘm
n
+ 2Rijlm
k
Φ kΘlm
+ 4Riklm
k
Φ jΘlm
− (i ↔ j) . (A.9)
– 22 –
JHEP03(2020)126
B Relations needed for second order calculation
For the second order calculations we define the following ‘basis’ of terms (for readability
we write all indices as lower indices)
f1 = Rilmn jΘmnΘkl f12 = Rimkn mΘljΘln f23 = Rklmn mΘinΘjl
f2 = Rilmn jΘklΘmn f13 = Rimkn lΘmjΘln f24 = Rklmn lΘimΘjn
f3 = Rikmn jΘmlΘln f14 = Rilmn kΘmnΘlj f25 = lRikmnΘjlΘmn
f4 = Rimnk jΘmlΘln f15 = Rilmn lΘmnΘkj f26 = kRilmnΘjlΘmn
f5 = Rilmn lΘmjΘnk f16 = Rilmn lΘkmΘnj f27 = i jΘmn kΘmn
f6 = Rilmn mΘljΘnk f17 = Rilmn mΘknΘlj f28 = i kΘmn jΘmn (B.1)
f7 = Rilmn mΘnjΘlk f18 = Rikmn mΘlnΘlj f29 = i jΘmn mΘnk
f8 = Rilmn lΘkjΘmn f19 = Rikmn lΘmnΘlj f30 = i kΘmn mΘnj
f9 = Rilmn kΘljΘmn f20 = Rimkn mΘlnΘlj f31 = i mΘnk jΘmn
f10 = Rikmn mΘljΘln f21 = Rklmn iΘjlΘmn f32 = i mΘnk mΘnj
f11 = Rikmn lΘmjΘln f22 = Rklmn iΘmnΘjl f33 = i mΘnk nΘmj
where we suppress the free indices ijk and assume symmetry in ij throughout. We also
define the terms with only one free index
ˆf1 = Rklmn jΘklΘmn
ˆf4 = Rjlmn kΘmnΘkl
ˆf7 = j lΘmn lΘmn
ˆf2 = Rklmn kΘljΘmn
ˆf5 = Rjlmn mΘnkΘkl
ˆf8 = j lΘmn mΘnl (B.2)
ˆf3 = Rklmn mΘklΘnj
ˆf6 = Rjlmn lΘkmΘkn
We will denote for example kf1ijk as ·f1, again suppressing the indices, and similarly
for example (i
ˆf1j) as ˆf1. Using the Killing vector identities, unimodularity and isometry
of the dilaton one finds
· f1 =
1
2
g12 + g23 − 2h6 (B.3)
· f2 =
1
2
g12 + g15 −
1
2
h1 + 2m1 (B.4)
· f3 = g13 − g25 − h5 − h7 − 2m5 − 2m6 (B.5)
· f4 = g14 − g24 + g25 +
1
2
h5 + h7 −
1
2
h8 + 2m6 (B.6)
· f5 = −
1
2
g1 +
1
2
g23 − g25 +
1
2
h3 − h5 − h6 −
1
2
h8 (B.7)
· f6 = −
1
2
g1 −
1
2
g10 −
1
2
g23 + g24 − g25 −
1
2
h3 − h5 +
1
2
h8 (B.8)
· f7 = −
1
2
g10 − g23 + g24 − h3 + h6 + h8 (B.9)
· f8 = g1 + g3 +
1
2
h1 + 2m2 (B.10)
· f9 = g1 + g8 + g10 + h1 + 2m1 − 2m2 (B.11)
– 23 –
JHEP03(2020)126
· f10 = g4 + g7 + g24 − 2g25 + h4 −
1
2
h5 − h6 − h7 +
1
2
h8 + 2m7 − 2m8 (B.12)
· f11 = g5 −
1
2
g23 + g25 − h4 +
1
2
h5 + h6 +
1
2
h8 + 2m10 − 2m11 (B.13)
· f12 = g4 + g24 − 2g25 −
1
2
h3 + h4 −
1
2
h5 − h6 − h7 + 2m7 (B.14)
· f13 = g6 +
1
2
g23 − g24 + g25 +
1
2
h3 − h4 + h5 − h6 +
1
2
h8 + 2m10 (B.15)
· f14 = g2 + g8 + h2 + 2m3 (B.16)
· f15 = g2 − g11 + g21 (B.17)
· f16 = −
1
2
g2 − g5 +
1
2
g11 −
1
4
h2 + 2m16 (B.18)
· f17 =
1
2
g2 + g6 +
1
2
h2 − m3 (B.19)
· f18 = g4 −
3
2
g21 + h3 + h4 − h10 + 2m9 + 2m17 (B.20)
· f19 = g3 + g21 − 2h4 + 2m4 (B.21)
· f20 = g4 + g7 −
3
2
g21 +
3
2
h3 + h4 −
1
2
h10 + 2m9 (B.22)
· f21 = g9 −
1
2
g20 −
1
2
h2 + h9 + 2m14 (B.23)
· f22 = −g16 − g22 + 2h4 − 2m13 (B.24)
· f23 = g17 +
3
2
g22 − h3 − h4 − h11 + 2m12 (B.25)
· f24 = −g18 −
1
2
h3 +
1
2
h11 + 2m18 (B.26)
· f25 = g11 − g1 − g2 −
1
2
h2 − 2h4 − 4m2 + 2m19 (B.27)
· f26 = −g1 − g2 − g10 − h2 − 4h4 − 2m15 (B.28)
· f27 = 2g5 − 2g6 + 2g7 − 2g13 − 2g14 − g20 + 2g28 + 2g32 + 4m21 (B.29)
· f28 = −g12 + 2g13 − g19 − 2g25 + g26 + 2m20 (B.30)
· f29 =
1
2
g3 − g4 − g5 − g7 +
1
2
g8 + g13 +
1
2
g15 +
1
2
g20 − g29 + g30 − g33 + g34 + 2m21
(B.31)
· f30 = g4 − g5 + g7 − g10 − g16 − g22 + g23 − g25 + g26 − g27 + 2m22 (B.32)
· f31 =
1
2
g12 − g13 − g14 +
1
2
g15 +
1
2
g19 −
1
2
g23 − g24 + g25 + g27 + 2m23 (B.33)
· f32 = −g7 +
1
2
g8 +
1
2
g10 +
1
2
g16 +
1
2
g22 − g23 + g24 + g25 − g26 + g27 + 2m24 (B.34)
· f33 =
1
2
g3 − g5 + g6 −
1
2
g10 −
1
2
g16 −
1
2
g22 −
1
2
g23 + 2m25 (B.35)
and
ˆf1 = g12+g19+g20 (B.36)
ˆf2 = g10+g16+
1
2
g20+h2 (B.37)
– 24 –
JHEP03(2020)126
ˆf3 = −g9+g11+g22 (B.38)
ˆf4 = g15+g23−2g33 (B.39)
ˆf5 = −g14+g24+g32+g34 (B.40)
ˆf6 = g13+g25+g34+g35 (B.41)
ˆf7 = −g4−g6−3g14+g26+g28−g29−g30+2g32+2g34 (B.42)
ˆf8 =
1
2
(g3−g4−g6+g8−3g14+3g15+2g27+g28−g29+g30−g31+2g32−4g33+2g34)
(B.43)
where we have defined the 2RΘ2-terms
g1 = kRilmn lΘkjΘmn g13 = Rilmn jΘmk lΘkn g25 = Rilmn j lΘkmΘkn
g2 = kRilmn kΘmnΘlj g14 = Rilmn jΘkl mΘkn g26 = i kΘmn j kΘmn
g3 = Rilmn lΘkj kΘmn g15 = Rilmn jΘkl kΘmn g27 = i kΘmn j mΘnk
g4 = Rilmn lΘkj mΘkn g16 = Rklmn iΘmn kΘlj g28 = Rkijl mΘnk mΘnl
g5 = Rilmn kΘmj lΘkn g17 = Rklmn kΘli mΘnj g29 = Rkijl mΘnk nΘml
g6 = Rilmn kΘlj mΘkn g18 = Rklmn kΘmi lΘnj g30 = Rkijl kΘmn mΘnl (B.44)
g7 = Rilmn mΘkj nΘkl g19 = Rklmn iΘkl jΘmn g31 = Rkijl kΘmn lΘmn
g8 = Rilmn kΘlj kΘmn g20 = Rklmn i jΘklΘmn g32 = mRkijl mΘnkΘnl
g9 = Rklmn iΘjl kΘmn g21 = Rilmn k lΘmnΘkj g33 = mRkijl nΘmkΘnl
g10 = iRklmn kΘljΘmn g22 = Rklmn i kΘmnΘlj g34 = mRkijl kΘmnΘnl
g11 = iRklmn kΘmnΘlj g23 = Rilmn j kΘmnΘkl g35 = mRkijl kΘlnΘmn
g12 = iRklmn jΘklΘmn g24 = Rilmn j mΘnkΘkl
the R2Θ2-terms
h1 = RipklRjpmnΘklΘmn h5 = RilmnRjlkpΘmkΘnp h9 = RkijpRklmnΘmnΘpl
h2 = RilmnRmnkpΘkpΘjl h6 = RikmpRjlnpΘklΘmn h10 = RklmiRklmn(Θ2
)nj (B.45)
h3 = RilmnRmnkpΘklΘjp h7 = RilmnRjlmk(Θ2
)nk h11 = RklmnRklmpΘinΘjp
h4 = RilmnRklmpΘnpΘjk h8 = RilmnRjkmn(Θ2
)lk
and the terms involving the dilaton
m1 = Rilmn kΦ jΘklΘmn m10 = Rilmn mΦ kΘljΘkn m19 = kRilmn lΦ ΘkjΘmn
m2 = Rilmn kΦ lΘkjΘmn m11 = Rilmn mΦ kΘnjΘkl m20 = kΦ i kΘmn jΘmn
m3 = Rilmn kΦ kΘmnΘlj m12 = Rklmn kΦ mΘniΘlj m21 = kΦ i jΘmn mΘnk
m4 = Rilmn lΦ kΘmnΘkj m13 = Rklmn kΦ iΘmnΘlj m22 = kΦ i kΘmn mΘnj
m5 = Rilmn mΦ jΘnkΘkl m14 = Rklmn kΦ iΘjlΘmn m23 = kΦ i mΘnk jΘmn
m6 = Rilmn mΦ jΘklΘkn m15 = kRilmn kΦ ΘmnΘlj m24 = kΦ i mΘnk mΘnj
m7 = Rilmn mΦ lΘkjΘkn m16 = Rilmn kΦ lΘkmΘnj m25 = kΦ i mΘnk nΘmj
m8 = Rilmn mΦ nΘkjΘkl m17 = Rilmn mΦ nΘlkΘkj
m9 = Rilmn mΦ lΘknΘkj m18 = Rklmn kΦ lΘmiΘnj (B.46)
– 25 –
JHEP03(2020)126
B.1 Additional identities
Contracting (A.6) with Θ and one covariant derivative, or the derivative of the dilaton, in
all possible ways gives the identities
0 = g12 − 4g13 − 2g14 + g15 + g19 , (B.47)
0 = 2g1 + g3 − 2g4 − 4g7 + 2g8 + 2g10 + g16 + 2g17 − 4g18 , (B.48)
0 = 2g1 + 2g3 − 4g5 + 2g6 + g8 − g16 + 2g17 − 4g18 , (B.49)
0 = ˆf1 + ˆf4 + 2 ˆf5 − 4 ˆf6 + iRklmnΘmnΘkl , (B.50)
0 = 2m4 + 2m12 − m13 + 4m16 + 4m18 + 2m19 , (B.51)
0 = 2m3 + m4 − 2m9 + 2m12 + m13 + 2m15 + 2m17 + 4m18 , (B.52)
0 = 2f14 + 2f18 + f19 − 4f20 − f22 + 2f23 + 4f24 − 2f26 , (B.53)
0 = f14 + 4f16 + 2f17 + 2f19 + f22 + 2f23 + 4f24 − 2f25 . (B.54)
The last two imply, using the previous ones, that m19 = m2 and
0 = g2 + g21 + g22 + h2 − 2h3 (B.55)
In addition we can derive the following identity
2h5 = 2RiklpRjkmnΘpm
Θnl = −2 l kki × n kkjΘnl
= −2 l(Rjknm kki × kmΘnl) + Rlnkm kki × mkjΘnl + Rlnjm kki × kkmΘnl
= l(Rjknm kΘmiΘnl) + l(Rjknm mΘkiΘnl) + l(Rjknm iΘkmΘnl)
−
1
2
RklmnΘkl i jΘmn − RlnkmRmijpΘkpΘnl −
1
2
RklpiRjpmnΘklΘmn
= −
1
2
· f1 − · f5 − · f6 −
1
2
g20 +
1
2
h1 + h9 . (B.56)
Open Access. This article is distributed under the terms of the Creative Commons
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any medium, provided the original author(s) and source are credited.
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Marginal deformations of WZW models and the
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Riccardo Borsato et al
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1
Journal of Physics A: Mathematical and Theoretical
Marginal deformations of WZW models
and the classical Yang–Baxter equation
Riccardo Borsato1,2 and Linus Wulff3
1
  Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de
Compostela, Santiago de Compostela, Spain
2
  Nordita, Stockholm University and KTH Royal Institute of Technology, Stockholm,
Sweden
3
  Department of Theoretical Physics and Astrophysics, Masaryk University,
611 37 Brno, Czech Republic
E-mail: riccardo.borsato@usc.es and wulff@physics.muni.cz
Received 15 January 2019, revised 12 April 2019
Accepted for publication 23 April 2019
Published 7 May 2019
Abstract
We show how so-called Yang–Baxter (YB) deformations of sigma models,
based on an R-matrix solving the classical Yang–Baxter equation (CYBE),
give rise to marginal current–current deformations when applied to the Wess–
Zumino–Witten (WZW) model. For non-compact groups these marginal
deformations are more general than the ones usually considered, since they
can involve a non-Abelian current subalgebra. We classify such deformations
of the AdS3 × S3
string.
Keywords: Wess–Zumino–Witten model, marginal deformations, string on
AdS(3), classical Yang–Baxter equation
1. Introduction
Conformal field theories (CFTs) in two dimensions are of interest for various areas of physics,
from condensed matter physics to string theory. In string theory they naturally arise on the
worldsheet of the string. In the context of holographic duality, certain two-dimensional CFTs
are also known to be dual to string theories on three-dimensional anti de Sitter spacetimes
[1–3]. An important instance of the AdS3/CFT2 duality is obtained by studying string theory
on AdS3 × S3
× T4
. In the case of pure NSNS flux the string is described by a Wess–Zumino–
Witten (WZW) model on the group F = SL(2, R) × SU(2), see [4–6] and references there. We
will be interested mainly in this setup. The CFT description of the worldsheet theory allows to
make precise statements about the AdS/CFT duality in this case. A recent example is the duality
between the symmetric product orbifold CFT and the string WZW model at level k  =  1 [7].
Generally speaking it is interesting to understand the conformal manifold of a CFT, i.e. the
space of marginal deformations generated by adding a local perturbation to the Lagrangian.
1751-8121/19/225401+30$33.00  © 2019 IOP Publishing Ltd  Printed in the UK
J. Phys. A: Math. Theor. 52 (2019) 225401 (30pp) https://doi.org/10.1088/1751-8121/ab1b9c
2
When applied to the WZW model under study, the marginal deformations correspond to
deformations of the supergravity background. They give us (at least in principle) a way to go
beyond the usual AdS3/CFT2 duality and extend it to cases in which e.g. the supersymmetry
or the conformal symmetry of the dual CFT2 are broken. Local marginal deformations of
WZW models were studied by Chaudhuri and Schwartz (CS) [8]. They found that a necessary
condition for local operators constructed out of the chiral and antichiral currents as
O(σ, ¯σ) = cab
Ja(σ)¯Jb(¯σ),(1.1)
with cab
some constant coefficients, to give a marginal deformation is that cab
satisfy
C · C + ¯C · ¯C = 0,(1.2)
where we have defined Cabc
≡ cda
ceb
f c
de , and ¯Cabc
≡ cad
cbe
f c
de , and the product is obtained
using the Killing metric Kab, e.g. C · C ≡ Cabc
Cdef
KadKbeKcf . Equation (1.2) is quartic in cab
,
and it involves also the structure constants of the algebra of F, the Lie group of the WZW
model. We will call it the weak CS condition. CS were interested in the case of CFTs where
the group F is compact. In that case (1.2) reduces to
Cabc
= 0, and ¯Cabc
= 0,(1.3)
which is an equation quadratic in cab
that we will call the strong CS condition. It imposes, in
fact, a stronger constraint, since it is only in the case of compact groups that it is equivalent to
(1.2). CS also showed that solutions of the strong condition correspond to Abelian subalgebras
of Lie(F), since when (1.3) holds it is always possible to identify linear combinations of Ja, ¯Ja
such that their OPEs do not have the term involving the structure constants—the so-called ‘no
simple-pole condition’, see (2.16). In that case the correlation functions of O are the same as
for an O constructed out of free bosons, which in turn implies that the deformation can be
completed to all orders in conformal perturbation theory in the deformation parameter. For
deformations satisfying only the weak CS condition on the other hand there is no guarantee
that they remain marginal beyond lowest order in the deformation parameter. In the literature
on marginal deformations of WZW models, see e.g. [9–18], we did not find examples that
satisfy the weak CS condition but not the strong one. Here we will construct such examples by
involving sufficient components of the chiral and antichiral currents of SL(2, R). In this sense
our results identify new directions to explore the conformal manifold.
In [10, 11, 19] it was argued that O(d, d) transformations provides the correct language to
obtain the exact (in the deformation parameter) version of the CFTs deformed by the Abelian
current–current operators. Indeed, such transformations do not break the isometries involved
in the deformation and one can show that the derivative of the action with respect to the deformation
parameter is given by
dS
dη
= −
T
2
d2
σ Jη¯Jη
,(1.4)
where Jη
and ¯Jη
are (anti)chiral currents of the deformed theory corresponding to the isometries
involved in the deformation. This makes it clear that the infinitesimal deformation can be
integrated to a finite one. The relevant so-called β-shifts of O(d, d), corresponding to a simple
shift of the B-field of the dual model obtained by performing T-duality on two U(1) isometries,
are also known as TsT (T-duality, shift, T-duality) transformations [20–23]. A TsT transformation
exploits an Abelian U(1)2
global symmetry of the sigma model to construct a deformation
parameterised by a continuous deformation parameter. Since the deformation is constructed
by exploiting T-duality, on-shell the deformed model is equivalent to the undeformed one, and
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
3
the deformation can be equivalently understood as a twist in the boundary conditions in the
compact direction of the worldsheet.
TsT transformations are known to belong to a larger class of deformations of sigma models,
usually calledYang–Baxter (YB) deformations. They first appeared in the context of integrable
models, since the deformations do not break the classical integrability of the original
model [24–26]. YB deformations are particularly interesting in the context of the AdS/CFT
correspondence, since they can be used to generate string backgrounds deforming the standard
ones appearing in the AdSd+1/CFTd dualities. Particularly important cases are those for
which integrability techniques may be applied. In the case of AdS5/CFT4 it was proposed
that the deformations of the AdS5 × S5
background should correspond to non-commutative
deformations of N = 4 superYang–Mills [27–30]. The nameYB comes from the fact that the
deformation is controlled by an object R which is an element of g ∧ g (where g is the algebra
of isometries of the starting background) and solves the classical Yang–Baxter equation4
(CYBE) on g. The simplest solutions to the CYBE are the so-called Abelian R-matrices, e.g.
R = T1 ∧ T2 with Ti ∈ g and [T1, T2] = 0. In this case the CYBE is trivially satisfied because
the relevant structure constants vanish. Abelian YB deformations were shown to be equivalent
to TsT transformations in [31]. On compact algebras, the CYBE only admits Abelian
solutions. On non-compact algebras, instead, more interesting non-Abelian solutions (i.e.
R-matrices constructed out of generators of a non-Abelian subalgebra) are possible.
Originally, in the construction of theYB-deformed sigma models, the CYBE was necessary
in order to preserve the classical integrability. Later it was understood that YB deformations
may be obtained from non-Abelian T-duality (NATD) [32, 33]. That interpretation revealed a
consistent generalisation of what is known about TsT transformations, since it became clear
that YB deformations correspond to a shift of the B-field of the dual (undeformed) sigma
model; the deformed model is then obtained by applying NATD on the subalgebra of g where
R is non-degenerate. After restricting the domain, R may be inverted and its inverse R−1
is a
Lie algebra 2-cocycle. The shift of the dual B-field is given by this 2-cocycle. In other words,
it is possible to go beyond the construction related to integrable models, and understand the
CYBE as being a constraint necessary to shift the dual B-field without modifying its field
strength H = dB. Consistently with this interpretation, in [34, 35] it was proposed to identify
YB deformations with the β-shifts of a larger group extending the known O(d, d) group
of Abelian T-duality, that in [34] was dubbed ‘non-Abelian T-duality group’. The logic of
NATD/β-shifts may be used to construct YB deformations of generic sigma models [34–36],
beyond those for which YB deformations were first introduced, the Principal Chiral Model
and (super)cosets5
. Here we will use the transformation rules of [36], which were derived
from the NATD construction and have the advantage of being applicable to a generic background
with isometries (even when the initial G  −  B is not invertible, as is the case for the
background metric and Kalb–Ramond field that we have to consider in this paper).
Because of their realization via NATD, YB models will be Weyl invariant at least to one
loop in σ–model perturbation theory (and exactly in the deformation parameter), provided
that the Lie algebra on which the R-matrix is non-trivial is unimodular (i.e. the trace of its
structure constants vanish). In this case the deformed background solves the standard supergravity
equations of motion. When the algebra is not unimodular there is a potential Weyl
anomaly [38, 39]. In that case the resulting background solves instead a generalization of the
standard supergravity equations [40, 41] controlled by a Killing vector field K. Even in these
4
There is also a version of these models where R solves instead the modified CYBE [24, 25]. The story in that case
is quite different and we will not consider it here.
5
Bakhmatov et al [37] put forward a first proposal for a set of transformation rules to go beyond these cases.
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
4
non-unimodular cases, it can happen that there is actually no Weyl anomaly. This is reflected
in the fact that the generalized supergravity equations can have ‘trivial’ solutions, i.e. solutions
with K = 0 but where nevertheless the other fields solve the standard supergravity equa-
tions [42]. In [42] it was shown that this can happen if K is null. In appendix E we show that
this condition can be weakened and K does not have to be null if the one-form X appearing in
the generalized supergravity equations takes the form X = dφ + ˜K with φ the dilaton and ˜K
another Killing vector. We will see that YB deformations of the AdS3 × S3
WZW model give
rise also to ‘trivial’ solutions, both ones with K2
  =  0 and with K2
= 0. We will also find some
examples with a genuine anomaly, corresponding to K not being null (and ˜K not defining a
Killing vector).
As we will discuss in more detail in section 3, at leading order in the deformation parameter
YB deformations correspond to current–current deformations. We are therefore led to study
YB deformations of strings on backgrounds containing an AdS3 subspace, expecting to find
marginal deformations of the corresponding WZW model. Particularly interesting for us are
the deformations that do not solve the strong version of the CS condition, but only the weak
one. We will construct explicitly such examples. Such possibilities are allowed because we
exploit also the non-compact part of the current algebra to generate the deformations.
The paper is organised as follows. In section  2 we review some aspects of the
SL(2, R) × SU(2) WZW model and of marginal current–current deformations that are important
for our discussion. In section 3 we review the transformation rules of YB deformations
and explain in which cases we can understand them as compositions of simpler YB transformations.
We will also explain the connection to the marginal current–current deformations.
Using the classification of R-matrices in appendix C, we later study deformations of AdS3 and
AdS3 × S3
. We give our conclusions in section 4. Appendix A collects some details on the
field redefinition used in section 3, and appendix B discusses the on-shell equivalence of the
YB models to the undeformed ones. In appendix D we consider the case of the sl3 algebra,
which is separate from the rest of the paper. In appendix E we extend the triviality condition
of [42].
2. Wess–Zumino–Witten model and marginal current–current deformations
In this section we review certain aspects of WZW models and their marginal current–current
deformations. Although the discussion can be made general, for concreteness we will take the
example of the SL(2, R) × SU(2) WZW model, since it is important for string theory applications
and it already contains all the salient features.
2.1. The AdS3 × S3
sigma model
We start with a sigma model describing the propagation of a string in AdS3 × S3
, that can be
viewed (after adding four free bosons) as the bosonic sector of the superstring . The sigma
model action is6
S =
k
2π
d2
σ
−∂x− ¯∂x+
+ ∂z¯∂z
z2
+
1
4
∂φi
¯∂φi +
1
2
∂φ2
¯∂φ1 sin φ3 .(2.1)
6
We work with a Lorentzian worldsheet and we introduce worldsheet coordinates σ±
= σ0
± σ1, so that
η+−
= η−+
= −2, +−
= − −+
= −2 and d2
σ = 1
2 dσ+
dσ−
. We also use the standard notation σ, ¯σ in place of
σ+
, σ−
, as well as ∂ = ∂σ, ¯∂ = ∂¯σ.
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
5
Here we are considering the pure NSNS background, and k will be the level of the WZW
model. The string tension is T = k/π, and the metric and B-field appearing in the sigma model
action follow from S = T d2
σ L = T
2 d2
σ ∂xm
(Gmn − Bmn)¯∂xn
and are7
ds2
= ds2
AdS3
+ ds2
S3 =
−dx+
dx−
+ dz2
z2
+
1
4
dφ2
3 + cos2
φ3dφ2
1 + (dφ2 + sin φ3dφ1)2
,
B =
dx+
∧ dx−
2z2
−
1
4
sin φ3dφ1 ∧ dφ2.
(2.2)
AdS3 is parameterised by the boundary coordinates x±
and the radial coordinate z, while the
angles φi parameterise the sphere. The AdS3 metric admits the following Killing vectors
km
0 ∂m = x+
∂x+ +
1
2
z∂z, km
+∂m = ∂x+ , km
−∂m = −(x+
)2
∂x+ − z2
∂x− − x+
z∂z,
¯km
0 ∂m = −x−
∂x− −
1
2
z∂z, ¯km
−∂m = ∂x− , ¯km
+∂m = −(x−
)2
∂x− − z2
∂x+ − x−
z∂z.
(2.3)
They satisfy [km
a ∂m, kn
b∂n] = −f c
ab k p
c ∂p (and similarly for ¯ka), where f c
ab are the structure constants
of the algebra of SL(2, R)
[S0, S±] = ±S±, [S+, S−] = 2S0.(2.4)
In these formulas and in the following we use a bar to distinguish the right copy of the algebra
from the left copy8
. For the sphere we have two copies of SU(2), whose algebra is generated
by Ta (a = 1, 2, 3) with commutation relations[Ta, Tb] = − abcTc. We will not write explicitly
all Killing vectors of S3
since we will not need them. For our purposes it will be enough to use
the two commuting Killing vectors
k1 = −∂φ1
and ¯k2 = ∂φ2
.(2.5)
The sigma-model action is invariant under the transformations generated by the above
Killing vectors, although in certain cases the B-field is not invariant but changes by a total
derivative. Therefore in general the corresponding Noether currents are given by
JA,± = km
A (Gmn ± Bmn)∂±xn
+ jA,±,(2.6)
where jA,± is defined by looking at the variation of the Lagrangian δAL = ε∂iji
A under the
infinitesimal global transformation. Because of our choice of gauge, in the AdS3 part only
ji
−
and ¯ji
+ are non-zero, and we also have ji
1 = ¯ji
2 = 0. In the following we will ignore the
transformations generated by S−, ¯S+, since for our discussion it will be enough to focus on the
(maximal solvable) subalgebra generated by
S0, S+, ¯S0, ¯S−, T1, ¯T2.(2.7)
All Noether currents that we will need to consider will therefore have jA,± = 0. Let us anticipate
that these Noether currents are not always equal to the chiral (resp. antichiral) currents
of the WZW description, which we shall denote by J (resp. ¯J) and write explicitly in the next
subsection. They agree up to ‘improvement terms’ that do not spoil the current conservation,
of the type ij
∂jc for some c. Restricting to the generators in (2.7), for AdS3 we have
7
In our conventions B = 1
2 Bnmdxm
∧ dxn
.
8
We will interchangeably place the bar on an object or on its index, in other words ¯ka or k¯a have the same meaning.
For readability sometimes we will prefer the former.
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
6
J0,+ = J0 −
1
2
∂ log z, J0,− = +
1
2
¯∂ log z, J+,+ = J+, J+,− = 0,
¯J0,− = ¯J0 +
1
2
¯∂ log z, ¯J0,+ = −
1
2
∂ log z, ¯J−,− = ¯J−, ¯J−,+ = 0,
(2.8)
while for S3
J1,+ = J1 +
1
4
∂φ1, J1,− = −
1
4
¯∂φ1,
¯J2,− = ¯J2 −
1
4
¯∂φ2, ¯J2,+ =
1
4
∂φ2.

(2.9)
This fact will later play an important role in our discussion.
2.2. The SL(2, R) × SU(2) WZW model
The action of the WZW model is SWZW = S1 + kΓ where k is the level and
S1[g] =
k
4π ∂B
d2
σTr(∂i
g−1
∂ig), Γ[g] = −
1
6π B
d3
σ ijk
Tr(g−1
∂igg−1
∂jgg−1
∂kg).(2.10)
Here g is an element of a group G, depending on coordinates on B, whose boundary ∂B is the
worldsheet of the string. In the following we will take the action9
SWZW[ga] − SWZW[gs] with
ga = ex+
S+
z2S0
e−x−
S−
∈ SL(2, R), gs = eφ1T1
eφ3T3
eφ2T2
∈ SU(2).(2.11)
We realise the generators of the algebra of SL(2, R)in terms of the Pauli matrices as S0 = σ3/2,
S+ = (σ1 + iσ2)/2,S− = (σ1 − iσ2)/2,andsimilarlyforSU(2)wetakeTa = i
2 σa.TheKilling
form is related to the trace in this representation as Kab = f d
ac f c
bd = 4Tr(SaSb), and similarly
for Ta. We will use the bilinear form induced by the trace, rather than the Killing form, to raise
and lower algebra indices.
The equations of motion for the action SWZW[g] imply chirality for the current J = ∂gg−1
,
and equivalently antichirality for the current ¯J = −g−1 ¯∂g, i.e. ¯∂J = 0, ∂¯J = 0. We decompose
the currents as J = JaSa
for AdS3 and J = JaTa
for S3
, and similarly for ¯J, where
S0
= 2S0, S±
= S and Ta
= −2Ta. Thanks to these definitions the component Ja of the chiral
current corresponds to the action of the generator Sa (or Ta in the case of the sphere) from
the left, while the component ¯Ja of the antichiral current corresponds to the action of the
same generator from the right. The same holds for the corresponding Killing vectors ka and
¯ka. In particular we have km
a ∂mga = +Saga, ¯km
a ∂mga = −gaSa for AdS and km
1 ∂mgs = −T1gs,
¯km
2 ∂mgs = +gsT2 for the sphere10
. In our parameterisation the components of the SL(2) currents
read
J0 =
z∂z − x+
∂x−
z2
, J− =
x+
(x+
∂x−
− 2z∂z)
z2
+ ∂x+
, J+ = −
∂x−
z2
,
¯J0 = −
z¯∂z − x− ¯∂x+
z2
, ¯J+ =
x−
(x− ¯∂x+
− 2z¯∂z)
z2
+ ¯∂x−
, ¯J− = −
¯∂x+
z2
,

(2.12)
9
The relative minus sign is needed to get the correct sign in front of the S3
metric. In the supersymmetric case it is
naturally accounted for by the supertrace.
10
The relative minus sign between AdS3 and S3
is again related to the fact that we are not using the supertrace in the
action and in order to define the components of the currents.
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
7
while for the SU(2) currents we have
J1 =
1
2
(−∂φ1 − s3∂φ2), J2 =
1
2
(−c1c3∂φ2 − s1∂φ3), J3 =
1
2
(−c1∂φ3 + c3s1∂φ2),
¯J2 =
1
2
(+¯∂φ2 + s3
¯∂φ1), ¯J1 =
1
2
(+c2c3
¯∂φ1 + s2
¯∂φ3), ¯J3 =
1
2
(+c2
¯∂φ3 − c3s2
¯∂φ1),
(2.13)
where we use the shorthand notation si = sin φi, ci = cos φi. The (anti)chiral currents appear
also when computing the Noether currents from the action SWZW = S1 + kΓ. In fact, invariance
of the WZW action under left transformations g → (1 + εL + . . .)g implies the conservation
of the Noether current J i
= 1
4 (ηij
− ij
)∂jgg−1
which is related to the chiral current as
J = (J+, J−) = (J, 0).(2.14)
Similarly, from the right transformations g → g(1 + εR + . . .) one finds the Noether current
¯J i
= −1
4 (ηij
+ ij
)g−1
∂jg related to the antichiral current as
¯J = ( ¯J+, ¯J−) = (0, ¯J).(2.15)
Conservation of the Noether current ∂iJ i
= 0 (respectively ∂i
¯J i
= 0) implies chirality of
J (respectively antichirality of ¯J). As we have already pointed out, in general these Noether
currents are not the same as those of the sigma model description, which we denoted by J .
2.3.  Marginal deformations
In [8] Chaudhuri and Schwartz considered two-dimensional CFTs with Ja, ¯Ja satisfying current
algebra relations11
Ja(σ)Jb
(σ ) ∼
i δb
a
2k(σ − σ )2
+
if b
ac Jc
2k(σ − σ )
,
¯Ja(¯σ)¯Jb
(¯σ ) ∼
i δb
a
2k(¯σ − ¯σ )2
+
if b
ac
¯Jc
2k(¯σ − ¯σ )
,

(2.16)
where we use  ∼  since we are omitting regular terms and f c
ab are structure constants of a Lie
algebra f. The authors of [8] were interested in exploring the space of marginal deformations
induced by dimension (1, 1) operators of the type
gO(σ, ¯σ) = gcab
Ja(σ)¯Jb(¯σ),(2.17)
where cab
are constant coefficients. The above operator is ‘integrably’ or exactly marginal (i.e.
can be completed to all orders in conformal perturbation theory in g) if it has no anomalous
dimension, and they found that a necessary condition for this to hold is that
Cabc
Cdef
KadKbeKcf + ¯Cabc ¯Cdef
KadKbeKcf = 0,(2.18)
where Kab is the Killing form and we have defined
Cabc
≡ cda
ceb
f c
de, ¯Cabc
≡ cad
cbe
f c
de.(2.19)
We will call (2.18) the weak Chaudhuri–Schwartz (CS) condition. Chaudhuri–Schwartz [8]
considered only the case of compact algebras, meaning that the Killing form Kab is negative
11
Since we are not normalising the currents with an explicit k and we raise/lower indices with the bilinear form
induced by the trace, as opposed to the Killing form, certain factors differ from [8].
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
8
definite and can be taken to be diagonal. In this case (2.18) becomes a sum of squares of Cabc
and ¯Cabc
, and it holds if and only if
Cabc
= 0, and ¯Cabc
= 0.(2.20)
We will call (2.20) the strong CS condition, because it is a stronger constraint in the case of
non-compact algebras. In [8] it was also shown that the strong condition is equivalent to being
able to rewrite
O(σ, ¯σ) = ˜cab˜Ja(σ)˜¯Jb(¯σ),(2.21)
where ˜Ja(σ), ˜¯Jb(¯σ) are linear combinations of the original Ja(σ), ¯Jb(¯σ) such that
˜Ja(σ)˜Jb
(σ ) ∼
i δb
a
2k(σ − σ )2
, ˜¯Ja(¯σ)˜¯Jb
(¯σ ) ∼
i δb
a
2k(¯σ − ¯σ )2
,(2.22)
i.e. the structure constants for this particular set of currents vanish. The absence of a simple
pole in these OPEs means that they are the same as similar ones for free bosons, which in turn
means that the β-function for the deformation parameter g vanishes and the deformation is
exactly marginal. In other words, deformations corresponding to Abelian subalgebras, which
is the only possibility in the compact case, are exactly marginal. When the Lie algebra f is
non-compact it is possible to find deformations that satisfy only the weak CS condition, as we
will see. A priori they are not guaranteed to be marginal beyond lowest order, and indeed we
will find both examples which are and those which are not.
In fact a sufficient condition on cab
such that the weak CS (2.18) holds is that the coefficients
cab
identify two solvable subalgebras of f (one corresponding to Ja and one to ¯Jb). This
follows directly from Cartan’s criterion for a solvable Lie algebra h
h solvable ⇐⇒ Tr(ab) = 0, ∀a ∈ h, b ∈ [h, h].(2.23)
If we are in such a situation then the two terms in (2.18) separately vanish because
Cabc
Cdef
Kcf = 0, ¯Cabc ¯Cdef
Kcf = 0.(2.24)
In the case of the SL(2, R) × SU(2) WZW model, we may for example identify the two solvable
subalgebras generated by {S0, S+, T1} and {¯S0, ¯S−, ¯T2}. Then if we call Ya ≡ {J0, J+, J1}
and ¯Ya = {¯J0, ¯J−, ¯J2} the list of the corresponding (anti)chiral currents, an operator
O(σ, ¯σ) = cab
Ya(σ)¯Yb(¯σ),(2.25)
will be marginal to lowest order for generic coefficients cab
. Notice that generically cab
will
not solve the strong CS condition (2.20).
All the solutions to the weak CS condition that we will generate from the CYBE on
g = fL ⊕ fR = sl(2, R)L ⊕ su(2)L ⊕ sl(2, R)R ⊕ su(2)R will be of this type. Indeed to solve
the CYBE it is enough to look at the subalgebra generated by {S0, S+, T1, ¯S0, ¯S−, ¯T2}. When
they come from the YB construction, the coefficients cab
will obviously not be generic, and
we will relate them to certain components of the R-matrix, see the discussion at the end of
section 3.2. The YB construction has the advantage of giving a way to go beyond the infinitesimal
deformation driven by O(σ, ¯σ), and gives a sigma-model action that is exact in the
deformation parameter.
As we have argued, we expect the CYBE to give solutions to the weak CS condition in
more generic situations. In appendix D we discuss a solution of the CYBE that provides coefficients
cab
that solve the weak CS condition without identifying solvable subalgebras.
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
9
3. Yang–Baxter and current–current deformations
3.1. Yang–Baxter deformations
We now review the transformation rules for the target space fields forYB deformations derived
in [36]. Given an initial sigma model with metric and Kalb–Ramond fields Gmn, Bmn, the background
of the YB deformed model is given by
˜G − ˜B = (G − B)[1 + ηΘ(G − B)]−1
,(3.1)
where for simplicity we are suppressing all spacetime indices. Here Θmn
= km
A RAB
kn
B is a tensor
constructed out of the Killing vectors km
A and of RAB
, which is a solution to the CYBE on
the Lie algebra g
RD[A
R|E|B
f
C]
DE = 0.(3.2)
In our case g = fL ⊕ fR is the sum of a left and a right copy of f = sl(2) ⊕ su(2), and G, B
were given in section 2.1. The derivation of [36] assumes that the B-field is invariant under the
isometries used in the deformation, i.e. the ones appearing in Θ. This is ensured by picking
the form of B in section 2.1 and using only the isometries generated by (2.7), which is enough
to generate any Yang–Baxter deformation (see appendix C).
The deformation produces also a shift of the dilaton calculated from the determinant12
e−2˜Φ
= e−2Φ
det[1 + ηΘ(G − B)],(3.3)
where Φ is the dilaton of the original background (in our case Φ = 0). In general YB backgrounds
are solutions to the equations of generalised supergravity [40, 41], so that in addition
to the usual fields one may have also a vector K computed as13
Km
= −
η
2
RAB
f C
ABkm
C ,(3.4)
which is a Killing vector of theYB background (mKn) = 0. For such generalised supergravity
solutions the role of (the derivative of) the dilaton is replaced by the vector14
Xm = ∂m
˜Φ − BmnKn
.(3.5)
When Km
vanishes one goes back to a standard supergravity solution. From (3.4) the relation
to the unimodularity condition of [43] is manifest. There exist also so-called ‘trivial solutions’
of generalised supergravity [42], i.e. when K does not vanish but it decouples from the
equations. A trivial solution is therefore both a solution of the generalised and the standard
supergravity equations. Later we will encounter examples of this type.
Let us comment on the fact that the YB transformations constructed in [36] were derived
by assuming a group of left isometries for the sigma model. This is necessary in order to apply
the NATD construction and twist the model with the corresponding Killing vectors kA. The
12
In the supersymmetric case the determinant is replaced by the superdeterminant.
13
Equation (3.4) may be obtained from the formula derived in [36] (i.e. Km
= ηΘmn
nn = ηkm
A RAB
nB with nA = f B
AB)
after using the identity RAB
f C
AB = −2f A
ABRBC
, which is a consequence of the CYBE. It is also easy to check that
(3.4) agrees with Km
= η
(0)
n Θmn
proposed in [30], where
(0)
n is the covariant derivative of the original undeformed
background. Indeed, using first the Killing equation for km
A and then the anti-symmetry of R, we have
(0)
m Θmn
= km
A RAB (0)
m kn
B = RAB
km
A ∂mkn
B. Knowing that Killing vectors satisfy [km
A ∂m, kn
B∂n] = −f C
AB k p
C∂p we obtain
again (3.4).
14
This expression applies in a gauge where B is invariant under the isometry generated by K, LKB = 0. Here we
stick with the original notation of [41]. In [36] and [42] Xm was used instead, but there is a risk of confusing it with
Xm = Xm + Km of [41].
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
10
isometries that we will exploit here to deform AdS3 × S3
, corresponding to the generators in
(2.7), belong both to the left and to the right copy of the symmetry group of the WZW model.
The reason why we can apply the above rules ofYB transformations is that the corresponding
sigma model may be constructed as a coset on SO(2, 2)/SO(1, 2) × SO(4)/SO(3) with a WZ
term. For example, focusing on AdS3, we may relate the generators of sl(2, R) to those of the
conformal algebra as
S0 = +
1
2
(D − J01), S+ = p+, S− = k−,
¯S0 = −
1
2
(D + J01), ¯S+ = k+, ¯S− = p−,

(3.6)
where e.g. p± = 1
2 ( p0 ± p1). Then we obtain the wanted sigma model action from
S = k
2π d2
σTr[g−1
∂g(P + b)g−1 ¯∂g] where g = exp(x+
p+ + x−
p−) exp(D log z), P projects
on the generators of the coset pi − ki and D, and finally b( p± − k±) = ±( p± − k±)
produces the B-field. In this formulation the isometries that we want to exploit, generated by
S0, S+, ¯S0, ¯S−, act from the left as g → hg and leave also the B-field invariant.
For later convenience, let us say at this point that it is easy to check that when an R-matrix
is given by the sum of two R-matrices, the corresponding background can be understood as
the composition of two successiveYB transformations. This is easily seen using the following
identity valid when Θ = Θ1 + Θ2
[1 + ηΘ1(G − B)]−1
1 + ηΘ2(G − B)[1 + ηΘ1(G − B)]−1 −1
= [1 + ηΘ(G − B)]−1
,(3.7)
which holds without assuming any property15
for Θi, neither antisymmetry nor CYBE. Thanks
to this formula it is straightforward to argue that the background metric and B-field of a YB
deformation generated by Θ = Θ1 + Θ2 are equivalent to those coming from the composition
of two successive deformations, e.g. first one generated by Θ1 and then one generated by Θ2
(or vice versa). The same holds for the transformation rule of the dilaton, and of the vector
K, which is linear in Θ (or equivalently R). Obviously, the interpretation as YB deformations
in the intermediate steps will be possible only if Θ1, Θ2 separately solve the CYBE, and if
the isometries needed to implement the second deformation are not broken by the first one.
In this case we will say that Θ is ‘decomposable’. Apart from these subtleties, it will often
prove useful to interpret a deformation generated by Θ = Θ1 + Θ2 as a composition of two
transformations.
Later we will encounter examples in which Θ1 generates the undeformed background
up to a (η-dependent) field redefinition. In this case one can say that Θ = Θ1 + Θ2 is
equivalent to the YB deformation generated by Θ2 alone only if the field redefinition
xm
→ xm
(x ) = x m
+ ηfm
(x ) needed to trivialise Θ1 is compatible with Θ2. It is easy to convince
oneself that the necessary compatibility condition is
A−1
Θ2(x m
+ ηfm
(x ))A−T
= Θ2(x m
),(3.8)
where Am
n = ∂xm
∂x n , and we are writing the explicit dependence of Θ2 on the coordinates.
3.2.  Relation to marginal current–current deformations
Before discussing YB deformations of AdS3 × S3
, let us make a simple observation: at leading
order in the deformation parameter, the YB deformation is of the form J J , where J are
15
Obviously we need to assume invertibility of the above operators.
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
11
the Noether currents of the sigma model. This is straightforwardly checked by expanding
the sigma model action S = T
2 d2
σ ∂xm
(˜Gmn − ˜Bmn)¯∂xn
to lowest order in the deformation
parameter16
S = S0 − η
T
2
d2
σ RAB
JA+JB− + O(η2
).(3.9)
While this is true for a generic sigma model, this observation is particularly interesting when
the original sigma model is related to a WZW model. If the Noether currents JAi coincided
with the chiral Noether currents of the WZW model JAi = {Jai, ¯Jai}, then we would automatically
obtain a current–current deformation of the type J¯J. In fact, from (2.14) and (2.15)
one immediately finds that17
d2
σ ij
RBA
JAiJBj = 4 d2
σ Ra¯b
Ja
¯J¯b. As we have seen,
though, in general J i
A = J i
A + ij
∂jcA, and the discussion is more subtle because (3.9) will
contain additional terms together with the wanted J¯J ones. We are about to show that for
YB deformations of the AdS3 × S3
sigma model these additional terms can be removed by
proper field redefinitions. We can therefore relate YB deformations to the deformations of the
type cab
Ja
¯Jb considered by CS. From this discussion it is also clear that we should identify
the coefficients cab
of CS with an ‘off-diagonal’ block of the R-matrix. More explicitly, since
R ∈ g ∧ g we are solving the CYBE on an algebra which is the sum of a left and a right copy
g = fL ⊕ fR, the R-matrix can be decomposed as
R =
RLL RLR
RRL RRR
(3.10)
with RT
LL = −RLL, RT
RR = −RRR and RT
LR = −RRL. The relation to the coefficients of CS is
therefore cab
= Rab
LR.We can therefore generate solutions to the (weak) CS condition from solutions
of the CYBE, and we will find several non-trivial examples in the following. Obviously
Abelian R-matrices (R = a ∧ b, [a, b] = 0) will give solutions of the strong CS condition.
When dealing with non-compact algebras the CYBE allows also for solutions that are not
of the Abelian type. Some of them will give coefficients cab
that do not solve the strong CS
condition. They all solve the weak CS condition as already explained at the end of section 2.3.
It would be very interesting to understand more deeply the relation between the space of
solutions of the CYBE (3.2) and the weak CS condition (2.18) in the case of a generic algebra
g = fL ⊕ fR. It is interesting to notice that in order to solve the CYBE one may need also
components of the diagonal blocks RLL and RRR, while in the weak CS condition these will
not enter. In fact, given Ra¯b
LR = ca¯b
and taking the CYBE on mixed left/right indices (where
we use an explicit bar for indices of the right copy of the algebra), one gets for example
cd¯a
Reb
f c
de + cb¯d
cc¯e
f ¯a
¯d¯e
+ Rdc
ce¯a
fb
de = 0. Depending on the coefficients ca¯b
one may also need
non-vanishing left-left Rab
components in order to solve this equation18
, but the CS condition
is not sensitive to them.
Let us also comment that, differently from what was claimed in [44], the strong CS condition
(2.20) is not the CYBE, not even when one further imposes the unimodularity condition
(and in fact our R9 in table 3 is a counter example to that claim).
16
Recall that we are restricting to the case when the isometries used to construct the deformation leave not just the
action but also the Lagrangian invariant, so that j Ai  =  0. This was the assumption also in the derivation done in [36].
17
Notice that the R-matrix may have also non-vanishing components with both indices in the left (or both in the
right) copy of the algebra, but these will not contribute in the final expression, and the contributions coupling currents
with the same chirality cancel out.
18
Such an example is given by the fourth R-matrix in table 1.
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
12
3.3.  Field redefinition
In order to display the current–current structure of the deformed model it is convenient to
write the Lagrangian in the form
L = L0 −
1
2
ηJA+[(1 + ηRM)−1
R]AB
JB−,
(3.11)
where L0 is the undeformed Lagrangian and MAB = km
A kn
B(G − B)mn. This follows directly
from the form of ˜G − ˜B in (3.1) and the definition of the Noether currents in (2.6) upon recalling
that we will pick R so that the last term in J does not contribute. To compare this to the
discussion of current–current deformations of the WZW model we need to perform a field
redefinition that replaces the Noether currents in the above expression with the chiral cur­
rents19
. In appendix A we find such a field redefinition for a general deformation specifying to
AdS3 × S3
for concreteness. Here we will just say that for all deformations that we consider
we find that the Lagrangian can be written in the form
L = L0 −
1
2
ηˆJa[(1 + ηRM )−1
R]a¯aˆ¯J¯a,(3.12)
where M is a shorthand for M after the field redefinition. ˆJa, ˆ¯J¯a are modifications of the chiral
currents of the undeformed WZW model. Their explicit form is given in (A.25) for deformations
of AdS3, and in section 3.5 for deformations of AdS3 × S3
.
At leading order in η the above Lagrangian becomes
L = L0 −
1
2
ηJaRa¯a¯J¯a + O(η2
),(3.13)
so that the comparison to the current–current deformations considered by CS is now manifest.
3.4. YB deformations of AdS3
Let us start by looking at YB deformations that deform only AdS3. We will start with the
simplest ones which are TsT-transformations. They come from the Abelian R-matrices of
sl(2, )L ⊕ sl(2, )R which (up to automorphisms) are20,21
S+ ∧ ¯S−, S0 ∧ ¯S0, S+ ∧ ¯S0.(3.14)
For the first one we obtain, from (3.1) and (3.3), the supergravity background of a deformation
of AdS3 × S3
× T4
ds2
= −
dx+
dx−
z2 − η
+
dz2
z2
+ ds2
S3 + ds2
T4 ,
B =
dx+
∧ dx−
2(z2 − η)
−
1
4
sin φ3dφ1 ∧ dφ2, e−2˜Φ
= 1 −
η
z2
.

(3.15)
In this case the isometries involved in the deformation procedure correspond to Noether currents
that agree with the (anti)chiral currents, see (2.8). We therefore automatically get that to
19
It is actually enough to look at the lowest order in η if we are considering infinitesimal deformations. The action
written in terms of the Noether currents as in (3.11) was given also in [44] but the rewriting in terms of the chiral
currents is missing there.
20
From now on we will use the components RAB
to construct R = RAB
TA ∧ TB ∈ g ∧ g.
21
We could consider also R = S0 ∧ ¯S−, but it is related to R = S+ ∧ ¯S0 if we also exchange the left and right copy
of the algebra.
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
13
leading order the deformation of the Lagrangian is given by the marginal operator ηJ+
¯J−. In
[45] the above background was argued to be the dual of the ‘single-trace’ T¯T deformation of
the symmetric product orbifold CFT2. This is also is accordance with the fact this particular
YB deformation is just a TsT transformation involving the two boundary coordinates22
. At
finite order in the deformation parameter the Lagrangian is given by (3.12), (A.25) and (A.1)
L = L0 −
1
2
η
1 − η
z2
J+
¯J−.(3.16)
The derivative of the action with respect to the deformation parameter is given by
dS
dη
= −
T
2
d2
σ Jη
+
¯Jη
−,(3.17)
where Jη
+ = (1 − ηz−2
)−1
J+ and ¯Jη
− = (1 − ηz−2
)−1¯J− are (anti)chiral currents of the
deformed model. This background was analysed in [46]. Let us also note that in this case the
deformation parameter can be absorbed by a rescaling of the coordinates. There are therefore
only three cases: η > 0, η = 0 and η < 0. The first of these is not globally well behaved since
the dilaton becomes imaginary when crossing z =
√
η. For η < 0 the solution interpolates
between two CFTs: the SL(2, ) WZW model and a linear dilaton background (plus two
decoupled bosons). It would be interesting to study further the implications of the existence
of this interpolating solution but we will not do so here. See also [45] for a discussion on the
different interpretations depending on the sign of the deformation parameter.
For the remaining two Abelian R-matrices we obtain by a similar calculation the
Lagrangians23
L = L0 −
1
2
η
(4 + η)z2
4z2 + η(4 + η)x+x−
J0
¯J0,
L = L0 −
1
2
η
z2
z2 + ηx−
J+
¯J0.

(3.18)
For all the Abelian examples one finds, as already mentioned, that
dS
dη
= −
T
2
d2
σ Ra¯a
Jη
a
¯Jη
¯a ,(3.19)
Table 1. Non-Abelian R-matrices of sl(2)L ⊕ sl(2)R up to SL(2, R)L × SL(2, R)R
inner automorphisms and swaps of L ↔ R. For convenience we also write the marginal
operators that they give rise to, and whether they satisfy the strong CS condition.
R = RAB
TA ∧ TB Deformation? cab
Ja
¯Jb Strong CS?
R1 = S0 ∧ S+ Trivial deformation 0 Yes
R2 = (S0
¯S−) ∧ S+ TsT ±J+
¯J− Yes
R3 = (S0 − a¯S0) ∧ S+ TsT aJ+
¯J0 Yes
R4 = (S0 − ¯S0) ∧ (S+ ± ¯S−) Not SUGRA J+
¯J0 ± J0
¯J− No
R5 = S0 ∧ S+ ± ¯S0 ∧ ¯S− + λS+ ∧ ¯S− TsT λJ+
¯J− Yes
22
We can have a TsT interpretation because we can implement the sequence T-duality, shift, T-duality in terms of
the coordinates x0
, x1
, instead of the null coordinates x±
.
23
The J0
¯J0-deformation was considered also in [47].
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
14
where Jη
a , ¯Jη
¯a are (anti)chiral currents of the deformed model. For the last two Abelian examples
they are
R = S0 ∧ ¯S0 : Jη
0 = αJ0, ¯Jη
0 = α¯J0, α ≡
4z2
1 + η
2
4z2 + η(4 + η)x+x−
,
R = S+ ∧ ¯S0 : Jη
+ = αJ+, ¯Jη
0 = α¯J0, α ≡
z2
z2 + ηx−
.

(3.20)
Following [46], the above result is another way to see that the YB model provides a deformation
that is marginal exactly in the deformation parameter.
In order to find deformations that at least potentially are not TsT, one should look at the class
of non-Abelian R-matrices. The full list of non-Abelian R-matrices of sl(2, R)L ⊕ sl(2, R)R
(up to SL(2, R)L × SL(2, R)R automorphisms) is given in table 1. They are special cases of the
R-matrices for sl(2, )L ⊕ sl(2, )R ⊕ su(2)L ⊕ su(2)R classified in appendix C. Analysing
the first example R1 = S0 ∧ S+ one finds that, after the field redefinition x+
→ x+
− η
2 log z,
not only the leading order in η vanishes in the action, but also all the higher orders. This is
obvious from equation (3.12) and the fact that R with an anti-chiral index vanishes. In other
words the deformation is trivial, since its effect is to give the undeformed AdS3 background in
new (η-dependent) coordinates. To be more precise, from R1 we get back undeformed AdS3 up
to a non-vanishing K = −η∂x+, from (3.4). This is of course a ‘trivial solution’ of generalised
supergravity (i.e. one that solves also standard supergravity upon dropping K, notice that K is
null)24
.
The above result for R1 = S0 ∧ S+ turns out to be useful to analyse some of the following
examples in the table. It is easy to see that R2 and R3 in table 1 are of the form R = S0 ∧ S+ + R
and that in both cases R is compatible with the field redefinition that trivialises the effect of
the piece S0 ∧ S+, see (3.8). Therefore these two R-matrices are equivalent to two of the TsT
transformations already discussed25
, generated by S+ ∧ ¯S− and S+ ∧ ¯S0. Alternatively it is
easy to see this directly from the Lagrangian in (3.12) and (A.25).
The last two R-matrices in table 1, instead, give backgrounds that are not of this type. From
(3.4) we find that they have K = −η(∂x+ ± ∂x− ) and K = −η(∂x+ ∂x− ) respectively, neither
of which is null. The only way they can give solutions of standard supergravity is then,
as shown in appendix E, if X = dφ + ˜K with φ the dilaton and ˜K an independent Killing
vector field. We can extract ˜K from the equation dK + i˜KH = 0 in (E.1) and for R4 one finds
˜K = η(∂x+ ∂x− ) ± η2
z−1
∂z + O(η3
) which, at lowest order, differs from K only in the sign
of the ∂x+-term. However from the lowest order correction to the action J+
¯J0 ± J0
¯J− we read
off the deformed metric
z−2
(−dx−
dx+
+ dzdz) − ηz−4
(zdz(dx−
dx+
) + (±x+
− x−
)dx+
dx−
) + O(η2
),(3.21)
which is only invariant under δx+
= η , δx−
= ±η which is generated by K, and not under
δx+
= η , δx−
= η , δz = ±η2
z−1
which is generated by ˜K. Therefore ˜K is not Killing
and the R4-background is not a solution to standard supergravity. Note that it only fails to be a
solution at order η2
which is consistent with the fact that the leading-order deformation satisfies
the weak CS condition. We will not consider this background further here since our interest
is mainly in string theory applications. For R5, instead, one can show that ˜K is a Killing
vector of the deformed metric and it satisfies (E.1), meaning that we get a ‘trivial solution’ of
24
Obviously a similar discussion holds also for R = ¯S0 ∧ ¯S−.
25
Also in this case the equivalence holds up to a non-vanishing K = −η∂x+ that is null and decouples from the
equations of generalised supergravity.
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
15
generalised supergravity for generic λ. Actually, using (3.12) and (A.25), for R5 we find the
Lagrangian26
L = L0 +
ηz2
(η − 4λ)
2 (η(η − 4λ) + 4z2)
J+
¯J−,(3.22)
which is exactly that of the TsT example in (3.16) with η → ηλ − η2
/4. This background
therefore provides an example of the more general kind of trivial solution of the generalized
supergravity equations discussed in appendix E for which K2
= 0.
This example also shows how the identification of the deformation parameters can be nontrivial.
In fact, although at leading order the marginal deformation is given only by ηλJ+
¯J−, the
deformation exact in η shows that the deformation parameter of the TsT is instead ηλ − η2
/4,
and in particular it does not vanish even when λ = 0.
It is interesting to look at the marginal operators of the type cab
Ja
¯Jb that are generated by
each R-matrix. We write them for convenience in table 1. While they all solve the weak CS
condition, they all solve also the strong CS condition (which guarantees that they are exactly
marginal) except for the fourth one (which we have seen fails to be marginal beyond lowest
order).
3.5. YB deformations of AdS3 × S3
Looking at deformations of AdS3 × S3
gives a richer set of possibilities. In this case we want
to solve the CYBE on the algebra sl(2)L ⊕ sl(2)R ⊕ su(2)L ⊕ su(2)R.
The simplest example to start with is R = S+ ∧ ¯T2. This is an Abelian R-matrix and therefore
corresponds to a TsT mixing AdS3 and S3
. In [48, 49] it was argued that this background
is dual to the single-trace T¯J deformation of the CFT2. From the YB procedure one explicitly
finds
ds2
= ds2
AdS3
+ ds2
S3 +
η
4z2
dx−
(dφ2 + 2 sin φ3dφ1),
B = B0 +
ηdx−
∧ (dφ2 + 2 sin φ3dφ1)
8z2
, e−2Φ
= 1.

(3.23)
Table 2. Rank-2 non-Abelian R-matrices of sl(2)L ⊕ sl(2)R ⊕ su(2)L ⊕ su(2)R up
to SL(2, R)L × SL(2, R)R × SU(2)L × SU(2)R inner automorphisms and swaps of
L ↔ R. With T we denote a generic linear combination of T1, ¯T2.
R = RAB
TA ∧ TB Deformation?
R1 = (S0 − ¯S0 + T) ∧ (S+ ± ¯S−) R4  +  TsT=⇒ not SUGRA
R2 = (S0 − a¯S0 + T) ∧ S+ R3  +  TsT=⇒ TsT
R3 = (S0
¯S− + T) ∧ S+ R2  +  TsT=⇒ TsT
R cab
Ja
¯Jb Strong CS?
R1 ±(J0 + aJ1)¯J− + J+(¯J0 + b¯J2) No
R2 J+(a¯J0 + b¯J2) Yes
R3 J+(±¯J− + b¯J2) Yes
26
For concreteness we take R5 = S0 ∧ S+ + ¯S0 ∧ ¯S− + λS+ ∧ ¯S−, but similar results apply also for R5 = S0 ∧ S+−
¯S0 ∧ ¯S− + λS+ ∧ ¯S−.
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
16
From (2.9) we see that the Noether current ¯J2 differs from the antichiral current ¯J2, and therefore
field redefinitions are needed in order to put the action in the form that makes the chirality
structure manifest. After the field redefinition x+
→ x+
− η
4 φ2, or equivalently by looking
directly at (A.22), (A.16) and (A.23) the Lagrangian becomes
L = L0 −
1
2
ηJ+
¯J2.(3.24)
This deformation is special since the leading linear order is exact. Obviously
dS/dη = −T
2 J+
¯J2.
We will now focus on non-Abelian R-matrices. These are classified for
sl(2)L ⊕ sl(2)R ⊕ su(2)L ⊕ su(2)R in appendix C. Since the CYBE on the two copies of
su(2) implies that the subset of generators from this part of the algebra should be Abelian,
our classification is useful also to study deformations of the full AdS3 × S3
× T4
or the more
generic AdS3 × S3
× S3
× S1
. Every time an su(2) generator appears, it may as well be
replaced by another compact generator, as long as all compact generators involved form an
Abelian subalgebra. For concreteness we will only look at deformations of AdS3 × S3
. The
rank-2 non-Abelian R-matrices are collected in table 2. Since they are special cases of the
rank-4 R-matrices, listed in table 3, we will not consider them separately.
To compute the Killing vector K that appears in the generalized supergravity equations we
note that equation (3.4) implies that only the non-Abelian generators matter and we can set
T = T = T = 0 (where these are generic linear combinations of T1, ¯T2). We find therefore
the same answer as for the SL(2, ) case in the previous section and comparing to that analysis
we see that only R4 (and therefore R1) does not give a supergravity solution. Since we are
interested in string theory applications we will focus on the analysis of R5 through R10. Recall
that in this way we automatically consider also the rank-2 R-matrices R2 and R3. From (A.22),
(A.16) and (A.23) it follows that (after the field redefinitions) the Lagrangian takes the form
L = L0 −
1
2
ηˆJa[(1 + ηRM )−1
R]a¯aˆ¯J¯a,(3.25)
where for R5–R8
ˆJa = Ja + ηδ0
ayiYi
ARA+
J+, ˆ¯J¯a = ¯J¯a − ηδ
¯0
¯aYi
ARA ¯−
yi
¯J−,(3.26)
while for R9
ˆJa = eδa
Ja, ˆ¯J¯a = e
¯δ¯a ¯J¯a, δ+ = −¯δ¯− = −ηYi
ARA0
yi.(3.27)
It is not hard to show that R5, R7 and R8 are in fact equivalent to TsT transformations. Consider
the first one. The deformed Lagrangian is
L = L0 −
1
2
η[Ja + ηδ0
ayiYi
ARA+
J+][(1 + ηRM )−1
R]a¯a¯J¯a.(3.28)
However, due to the form of the R-matrix and the fact that M+A = 0 this is equal to
L = L0 −
1
2
ηJa[(1 + ηRM )−1
R]a¯a¯J¯a,(3.29)
where furthermore we can replace R by the Abelian R-matrix obtained by dropping the
S0 ∧ S+-term in R5. This deformation therefore reduces to a sequence of commuting TsT
transformations. The same conclusion applies to R8 as is easily seen from the form of the R
matrix. For R7 we have
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
17
L = L0 −
1
2
η[Ja + ηδ0
ayiYi
ARA+
J+][(1 + ηRM )−1
R]a¯a
[¯J¯a − ηδ
¯0
¯aYi
ARA ¯−
yi
¯J−],

(3.30)
but again the terms with a  =  0 and ¯a = ¯0 drop out due to the form of the R-matrix and the fact
that M+A  =  0 and this reduces to
L = L0 −
1
2
ηJa[(1 + ηRM )−1
R]a¯a¯J¯a.(3.31)
It is also not hard to see, using again the form of the matrices M and R, that one can again
replace R by the Abelian R-matrix obtained by dropping the S0 ∧ S+-term in R7. The resulting
background is therefore also a TsT. The fact that R5, R7 and R8 are equivalent to TsT
backgrounds may be argued also from the fact that the R-matrices are Abelian up to the
S0 ∧ S+-term, and that (3.8) holds.
For R6 the deformed Lagrangian is
L = L0 −
1
2
η[Ja + ηδ0
ayiYi
ARA+
J+][(1 + ηRM )−1
R]a¯a¯J¯a.(3.32)
Again, using the form of R and the fact that M+A  =  0, this simplifies to
L = L0 −
1
2
ηJa[(1 + ηRM )−1
R]a¯a¯J¯a.(3.33)
Table 3. Rank-4 non-Abelian R-matrices of sl(2)L ⊕ sl(2)R ⊕ su(2)L ⊕ su(2)R up
to SL(2, R)L × SL(2, R)R × SU(2)L × SU(2)R inner automorphisms and swaps of
L ↔ R. With T, T , T we denote generic linear combinations of T1, ¯T2.
R = RAB
TA ∧ TB Deformation?
R4 = R1 + aT1 ∧ ¯T2 R1+TsT=⇒not SUGRA
R5 = R2 + bT1 ∧ ¯T2 R2+TsT=⇒TsT
R6 = (S0 + T) ∧ S+ + T ∧ (¯S0 + T ) SUGRA
R7 = (S0 + T) ∧ S+ + T ∧ (¯S− + T ) TsT
R8 = R3 + aT1 ∧ ¯T2 R3+TsT=⇒TsT
R9 = (S0 + ¯S0 + T) ∧ T + S+ ∧ ¯S− SUGRA, not TsT
R10 = (S0 + T) ∧ S+ ± (¯S0 + T ) ∧ ¯S− + λS+ ∧ ¯S− TsT
R cab
Ja
¯Jb Strong CS?
R4 (J0 + aJ1)¯J− + J+(¯J0 + b¯J2) + aJ1
¯J2 No
R5 J+(a¯J0 + b¯J2) + bJ1
¯J2 Yes
R6 aJ+
¯J2 + bJ1
¯J0 + cJ1
¯J2 Yes
R7 aJ+
¯J2 + bJ1
¯J− + cJ1
¯J2 Yes
R8 J+(¯J− + b¯J2) + aJ1
¯J2 Yes
R9 aJ0
¯J2 + bJ1
¯J0 + cJ1
¯J2 + J+
¯J− No (a = 0 or b = 0)
R10 cJ+
¯J2 + dJ1
¯J− + λJ+
¯J− Yes
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
18
However, in this case we cannot get an equivalent background simply by dropping the
S0 ∧ S+-term in R6
27
. Let us work out the background for the simplest case, T = T = 0 and
T = aT1 + b¯T2. One finds
L = L0 − ηf−1
2bηJ+
¯J2 + 2abη(1 − η
x−
z2
)J1
¯J2 −
1
2
η2
(a2
+ b2
− 2ab sin φ3)J+
¯J0 + 8aJ1
¯J0
(3.34)
where f = 16 + η2
(a2
+ b2
− 2ab sin φ3)[1 − ηx−
/z2
]. One can show that in fact the (anti)
chiral currents entering this action J+ , J1, ¯J0 and ¯J2 extend to chiral currents to all orders in η,
i.e. the corresponding isometries are not broken by the deformation. Since this is a characteristic
feature of TsT backgrounds it is natural to guess that this background can be generated
in that way. It is not hard to show this explicitly in the special cases a  =  1, b  =  0 and a  =  0,
b  =  1 in which cases it is equivalent to the backgrounds generated by
R = T1 ∧ ¯S0 −
η2
16
S+ ∧ ¯S0 and R =
4η
16 + η2
S+ ∧ ¯T2 −
η2
16 + η2
S+ ∧ ¯S0,

(3.35)
respectively.
R9 is the only unimodular example, i.e. the only one with K  =  0. The deformed Lagrangian
is
L = L0 −
1
2
ηˆJa[(1 + ηRM )−1
R]a¯aˆ¯J¯a,(3.36)
with
ˆJa = eδa
Ja, ˆ¯J¯a = e
¯δ¯a ¯J¯a, δ+ = −¯δ¯− = −ηYi
ARA0
yi.(3.37)
For simplicity we will set T  =  0 and T = T1. We can argue that this example is not equivalent
to a TsT as follows. To order η2
the Lagrangian is
L = L0 −
1
2
η(1 +
η
z2
)J+
¯J− +
1
2
ηJ1
¯J0 −
1
8
η2
J0
¯J0 −
1
2
η2 x−
z2
J1
¯J− + O(η3
).

(3.38)
The action is clearly not invariant under the isometry corresponding to constant shifts of x−
and therefore the corresponding chiral current J+ does not extend to the deformed theory.
Instead the equations of motion lead to chiral currents
Jη
+ = (1 +
η
z2
)J+ − η
x−
z2
J1 + O(η2
), Jη
1 = J1 + O(η2
),(3.39)
while the remaining equations of motion read
∂[(1 +
η
z2
)¯J−] − ηJ1
¯J− = O(η2
), ∂[¯J0 + η
x−
z2
¯J−] − ηJ+
¯J− = O(η2
).
(3.40)
At the same time we have
dS
dη
= −
T
2
Jη
+(1 +
η
z2
)¯J− − J1
¯J0 + 3η
x−
z2
J1
¯J− +
1
2
ηJ0
¯J0 + O(η2
) ,
(3.41)
27
One way to see this is that R6 − R1 is not compatible with the coordinate redefinition needed to undo the transformation
with R1 = S0 ∧ S+ (see equation (3.8)), therefore one does not expect to be able to undo the effect of the R1
piece of the R-matrix.
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
19
which clearly cannot be written as a bilinear in deformed chiral currents. Explicitly, we obtain
the background28
ds2
= −
ηdφ1(x+
dx−
+ x−
dx+
) + 4dx−
dx+
+ η − z2
dφ2
1
η(ηx−x+ − 4) + 4z2
+
dz2
z2
+
dφ2
2 + dφ2
3
4
−
2dφ2 sin φ3 ηx−
dx+
+ η − z2
dφ1
η(ηx−x+ − 4) + 4z2
,
B =
−4dx−
∧ dx+
− 2 sin φ3 dφ2 ∧ (ηx−
dx+
+ η − z2
dφ1) + ηdφ1 ∧ (x+
dx−
− x−
dx+
)
2η(ηx−x+ − 4) + 8z2
,
e−2Φ
=1 +
η(−4 + ηx+
x−
)
4z2
.

(3.42)
Finally, R10 = R5 + T ∧ S+ + T ∧ ¯S−. Since the sl(2, R) R-matrix R5 does not break the
isometries generated by S+, ¯S−, we conclude that the additional terms in R10 have the only
effect of adding further TsT transformations on top of the background generated by R5, which
is itself a TsT background.
Interestingly, the matrices R4 (which may be decomposed in terms of R1) and R9 give rise
(at leading order in the deformation parameter) to marginal deformations of the WZW model
that obviously satisfy the weak CS condition, but not the strong one.
4. Discussion
InthispaperwehaveconstructedYBdeformationsofstringsonthepureNSNSAdS3 × S3
× T4
background. Together with Abelian YB deformations, which are known to reproduce TsT
transformations, we also constructed non-AbelianYB deformations. While some non-Abelian
R-matrices give rise to backgrounds that cannot be obtained simply from TsT transformations,
we found that others generate again TsT backgrounds, or even no deformation at all29
. We
expect this to be related to the fact that the initial G  −  B is degenerate.
For example, the Jordanian R-matrix R1 = S0 ∧ S+ gives back the undeformed AdS3
background up to an η-dependent field redefinition (and up to a non-vanishing K = −η∂x+).
Recalling that theYB deformation is equivalent to a shift of the B-field plus NATD, this observation
suggests that AdS3 with NSNS flux has a certain property of self T-duality, when we
dualise the non-Abelian algebra of isometries generated by S0, S+ and we also regularise the
action by performing a B-field gauge transformation.
Although we used our classification of R-matrices to deform, for concreteness, only the
AdS3 × S3
part of the background, our results may also be used to obtain deformations involving
the T4
of AdS3 × S3
× T4
, or even deformations of the more general AdS3 × S3
× S3
× S1
background. Indeed, in all our expressions of the R-matrices the generators T1, ¯T2 may be
substituted with any other two commuting generators of the compact part of the isometry
algebra30
. Let us also mention that the string on these AdS3 backgrounds is integrable [50, 51]
and that our deformations preserve the classical integrability.
To leading order in the deformation parameter, all our YB deformations reduce to the marginal
current–current deformations of the type considered by Chaudhuri and Schwartz in [8].
While they are all marginal to lowest order, since they satisfy what we called the ‘weak CS
28
Here we are writing the background that is obtained before doing the field redefinition.
29
Except for the introduction of a (decoupled) non-vanishing vector K in the equations of generalised supergravity.
30
That is because the CYBE implies that the restriction of an R-matrix to a compact algebra must be Abelian. We
are therefore free to choose which Abelian subalgebra we wish to consider.
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
20
condition’, some of them do not satisfy the ‘strong CS condition’ and the celebrated ‘no
simple-pole condition’ which guarantee exact marginality. Indeed we found examples which
failed to be marginal beyond lowest order (and at one-loop in 1/k), all involving the R-matrix
R4 in table 1, and one example, R9 in table 3, which remains marginal to all orders in η at
least up to one loop in 1/k. The relation between the space of solutions of the CYBE and that
of the weak CS condition is an interesting question. The former is a quadratic equation for R,
while the latter is a quartic equation for the coefficients cab
of the current–current deformation,
related to the left-right block of the R-matrix simply as cab
= Rab
LR. While we expect all
solutions of the CYBE to generate solutions of the weak CS condition (including the trivial
ones) it seems hard to prove this statement for a generic Lie algebra. In appendix D we took a
digression from the setup of the paper and we considered the CYBE on the sl3 algebra, finding
again that it generates non-trivial solutions to the weak CS condition (that do not solve
the strong one). We do not rule out the possibility of having solutions to the weak CS condition
that cannot be ‘completed’ to a solution of the CYBE equation. In section 3 we actually
discussed a more generic criterion (related to the solvability of the subalgebras involved) not
requiring CYBE, to construct solutions of the weak CS condition.
In [52] a marginal deformation constructed out of Abelian currents (a TsT transformation)
was interpreted in terms of spectral flow. It would be interesting to understand if this can be
generalised to the non-Abelian set-up.
We would like to stress that we worked out the YB deformations in the sigma model
description. It would be very interesting to understand how to formulate the YB deformation
directly at the level of the WZW action. Such a construction was performed in [53–55] for R
a solution of the modified CYBE31
. The formulation of the deformation of the WZW action
may be obtained from the construction in terms of NATD, in the spirit of [32] and [33]. An
alternative may be to use the language of E models [56–58], see [59] for a recent application
in similar contexts.
One motivation to carry out this work came from the recent developments on the T¯T deformation
[60, 61] and its generalisations. The components T, ¯T of the stress-energy tensor of a
(quite generic) two-dimensional relativistic field theory may be used to construct a ‘doubletrace’
operator generating an irrelevant perturbation of the theory. The deformation is solvable
in the sense that the spectrum of the deformed theory may be computed exactly in the deformation
parameter as a function of the spectrum of the original undeformed theory. In [45] a
‘single-trace’ version of the T¯T deformation of the symmetric product orbifold CFT was considered.
It was argued that the irrelevant deformation of the ‘spacetime’ CFT governed by32
O(x) ∝
N
i=1 Ti(x)¯Ti(¯x), where i labels each copy in the symmetric product, corresponds to a
marginal deformation of the dual WZW model that infinitesimally is just the current–current
deformation J+(σ)¯J−(¯σ), where J+, ¯J− are the left and right SL(2, R) currents generating
shifts of the boundary coordinates x+
, x−
. Another deformation, similar in spirit to the above
one, was studied in [48] and [49] after replacing the ¯T with an antichiral U(1) current of the
compact factor33
. It was argued in [48, 49] that the deformation of the dual WZW model is
governed again by a marginal deformation bilinear in the currents (where now the antichiral
current belongs to the compact part of the algebra). Such marginal deformations of the WZW
31
There the special propriety R3
  =  −R was used, so that we do not expect their results to be immediately applicable
to the case of the homogeneous CYBE. Moreover, here we want R to be a solution of the CYBE on fL ⊕ fR that also
couples the left and right copy of the algebra.
32
We refer to [45] for the connection to Little String Theories. The above operator should be compared to
the original double-trace version studied in [60, 61] and given by T(x)¯T(¯x), where T(x) =
N
i=1 Ti(x) and
¯T(¯x) =
N
i=1
¯Ti(¯x).
33
This deformation is in fact the single-trace version of the one first constructed in [62].
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
21
model may be completed to finite values of the deformation parameter in terms of TsT (or
equivalently certain O(d, d)) transformations. Since TsT deformations are a subclass of YB
ones, it would be interesting to understand if it is possible to provide a holographic interpretation
also for the YB deformations of AdS3 × S3
× T4
considered here. (The connection to
YB models was also pointed out the recent paper [44].) We expect our marginal deformations
of the WZW model to correspond to deformations of the dual CFT2 which generalise the
(single trace version of the) T¯T construction. It would be very interesting to understand for
example the case of the non-Abelian R-matrix R9, which gives rise to the marginal deformation
aJ0
¯J2 + bJ1
¯J0 + cJ1
¯J2 + J+
¯J−. The non-Abelianity of the generators involved forbids
the usual iteration of the infinitesimal deformation in order to obtain the exact one. The YB
deformation, despite the non-Abelianity, provides the realisation of the finite deformation on
the worldsheet of the string.
Acknowledgments
We thank R Conti, B Hoare and D Thompson for useful discussions. LW thanks the participants
of the workshop ‘A fresh look at AdS3/CFT2’ in Villa Garbald, Castasegna, for stimulating
discussions. The work of RB is supported by the Maria de Maeztu Unit of Excellence
MDM-2016-0692, by FPA2017-84436-P, by Xunta de Galicia (ED431C 2017/07), and by
FEDER. His work was supported by the ERC advanced grant No 341222 while at Nordita.
Appendix A.  Details on the field redefinition
The matrix M in (3.11) is a direct sum of the AdS and sphere part, M = Ma ⊕ Ms. They take
the form (we restrict to A, B = {0, +, ¯−, ¯0} and A, B = {1, ¯2} respectively, which is all we
need since M always comes multiplied with R on both sides)
Ma =





1
4 0 0 −1
4
0 0 0 0
−x+
z2 − 1
z2 0 0
−1
4 + x+
x−
z2
x−
z2 0 1
4





, Ms =
1
4 0
−1
2 sin φ3
1
4
.(A.1)
We will make an ansatz for the field redefinition based on the isometry transformations
used to construct the model, but with the transformation parameters depending linearly on
y = (ln z, φ1, φ2) (since we want to cancel terms involving ∂y and ¯∂y). We therefore consider
x±
→ x ±
= eηbi
±yi
[x±
+ ηai
±yi], z → z = eηbi
yi
z, φ1,2 → φ1,2 = φ1,2 + ηai
1,2yi,(A.2)
where i = 1, 2, 3 and bi
= (bi
+ + bi
−)/2. For the right-moving Noether currents JA+ we get
J0+ = J0 + ηai
+yiJ+ −
1
2
∂ ln z + ∂yi[ηbi
+M00 − ηbi
−M¯00 + ηai
−eηbi
−yi
M¯−0],
(A.3)
J++ = e−ηbi
+yi
J+ + ∂yi[ηai
−eηbi
−yi
M¯−+ − ηbi
−M¯0+],(A.4)
J¯0+ = −
1
2
∂ ln z −
1
2
ηbi
∂yi,(A.5)
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
22
J1+ = J1 +
1
4
∂φ1 + ∂yi[ηai
2M¯21 − ηai
1M11],(A.6)
J¯2+ =
1
4
∂φ2 +
1
4
ηai
2∂yi.(A.7)
These expressions take a more natural form if we further assume that
ai
1 = −Yi
ARA1
, ai
2 = Yi
ARA¯2
, ai
+ = Yi
ARA+
, ai
− = Yi
ARA ¯−
, bi
+ = Yi
ARA0
, bi
− = −Yi
ARA¯0
,
(A.8)
for some constants Yi
A to be determined. Then we have
JA+ = eδA
JA + ∂yiYi
B[ηRM ]B
A + ∆A(A.9)
with δ+ = −ηYi
ARA0
yi and
∆0 = −
1
2
∂ ln z + ηyiYi
ARA+
J+ + [eηbi
−yi
− 1]∂yiYi
AηRA ¯−
M¯−0,(A.10)
∆+ = [eηbi
−yi
− 1]∂yiYi
AηRA ¯−
M¯−+,(A.11)
∆¯0 = −
1
2
∂ ln z, ∆1 =
1
4
∂φ1, ∆¯2 =
1
4
∂φ2,(A.12)
the other components vanishing. Using these expressions the transformed Lagrangian becomes
L = L0 −
1
2
ηeδa
Ja[(1 + ηRM )−1
R]aB
JB− +
1
2
η(∂yiYi
A − ∆A)[(1 + ηRM )−1
R]AB
JB− −
1
2
η∂yiYi
ARAB
JB−.
(A.13)
Picking Yi
A such that ∂yiYi
A − ∆A vanishes to lowest order in η we get that its non-zero components
should be
Y1
0 = Y1
¯0 = −
1
2
, Y2
1 = Y3
¯2 =
1
4
(A.14)
and the Lagrangian becomes
L = L0 −
1
2
η∂yiYi
ARAB
JB− −
1
2
ηˆJa[(1 + ηRM )−1
R]aB
JB−(A.15)
where we have defined
ˆJa = eδa
Ja + ηδ0
ayiYi
ARA+
J+ + η[e−¯δ−
− 1]∂yiYi
ARA ¯−
M¯−a,(A.16)
with ¯δ− = ηYi
ARA¯0
yi. A short calculation shows that the transformed undeformed Lagrangian
is (up to total derivatives)
L0 = L0 +
1
2
ηYi
ARAa
Ja + ηδ0
ayjY j
BRB+
J+
¯∂yi +
1
2
η∂yiYi
ARA¯a¯J¯a
+
1
2
η[e−¯δ−
− 1]Yi
ARA ¯−
∂yi
¯J− −
1
2
η∂yiYi
ARAB
Y j
B
¯∂yj

(A.17)
and using this together with the fact that JB− = ¯JB − Yi
B
¯∂yi we get
L = L0 +
1
2
ηYi
ARAa
(Ja + ηδ0
ayjY j
BRB+
J+)¯∂yi −
1
2
ηˆJa[(1 + ηRM )−1
R]aB
JB−
+
1
2
η[eηbi
−yi
− 1]∂yiYi
ARA ¯−
J¯−−.

(A.18)
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
23
Finally we use the fact that
JB− = e
¯δB ¯JB − [1 + ηM R]B
A
Yi
A
¯∂yi + ¯∆B(A.19)
with ¯δ¯− = ¯δ−
was defined above and
¯∆¯− = −η[e−δ+
− 1]M¯−+R+A
Yi
A
¯∂yi,(A.20)
¯∆¯0 = −ηYi
ARA ¯−
yi
¯J− − η[e−δ+
− 1]Yi
AM¯0+R+A ¯∂yi,(A.21)
and the remaining components vanishing. Finally the Lagrangian becomes
L = L0 −
1
2
ηˆJa[(1 + ηRM )−1
R]a¯aˆ¯J¯a +
1
2
η[eδ+
− 1]J+R+A
Yi
A
¯∂yi −
1
2
η[e
¯δ−
− 1]∂yiYi
ARA ¯−¯J−
−
1
2
η2
[e−¯δ−
− 1]∂yiYi
ARA ¯−
M¯−+[e−δ+
− 1]Y j
BR+B ¯∂yj,

(A.22)
with (recall that δ+ = −ηYi
ARA0
yi and ¯δ¯− = ηYi
ARA¯0
yi)
ˆ¯J¯a = e
¯δ¯a ¯J¯a − ηδ
¯0
¯aYi
ARA ¯−
yi
¯J− − ηM¯a+[e−δ+
− 1]R+A
Yi
A
¯∂yi.(A.23)
The terms involving eδ+
− 1 and e
¯δ−
− 1 are not expressed in terms of the chiral currents
but they only appear at order η2
, so they do not interfere with the comparison to infinitesimal
current–current deformations. In fact they vanish for most of the deformations of AdS3 × S3
,
e.g. for AdS3 deformations we have R0¯0
RA+
= R0¯0
RA ¯−
= 0 and using this we find
L = L0 −
1
2
ηˆJa[(1 + ηRM )−1
R]a¯aˆ¯J¯a,(A.24)
with
ˆJa = Ja −
1
2
ηδ0
a(R0+
+ R
¯0+
) ln zJ+, ˆ¯J¯a = ¯J¯a +
1
2
ηδ
¯0
¯a(R0 ¯−
+ R
¯0 ¯−
) ln z¯J−.
(A.25)
Appendix B.  On-shell equivalence
Here we will demonstrate the on-shell equivalence of the YB deformed sigma models to the
original ones by deriving the explicit (non-local) field redefinition that relates them. We will
do that by following the NATD transformation and following field redefinition that are used
to get the action of theYB model as in [36]. In the notation of [36], let us start from the action
S =
T
2 Σ
AI
∧ (GIJ ∗ −BIJ)AJ
+ 2dzm
∧ (GmI ∗ −BmI)AI
+ dzm
∧ (Gmn ∗ −Bmn)dzn
,

(B.1)
where A  =  g−1
dg and we have set fermions to zero for simplicity. When going to the NATD
model one relates the original degrees of freedom to the new ones encoded in the Lagrange
multiplier ν as
(1 ± ∗)AI
= −(1 ± ∗) (dνJ + dzm
[ G − B]mJ) NJI
, NIJ
± = ±GIJ − BIJ − νKfK
IJ
−1
.(B.2)
The YB model appears after the redefinition34
34
It is assumed that the initial B-field is actually shifted as BIJ → BIJ − η−1
R−1
IJ .
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
24
νI = η−1
tr TI
1 − Ad−1
˜g
log Ad˜g
R−1
log ˜g ,(B.3)
which implies
dνI = η−1
R−1
˜g (˜g−1
d˜g)
I
, N = ηR˜g (1 + η(G − B)R˜g)
−1
= η (1 + ηR˜g(G − B))
−1
R˜g,
(B.4)
where N = N+ = −NT
−
. Combining all redefinitions we find that the original A  =  g−1
dg
(which will depend on some coordinates xi
) is related to the new degrees of freedom of theYB
model (i.e. the coordinates ˜xi
parameterising ˜g, together with the coordinates zm
that remain
spectators) as
(1 ± ∗)AI
= (1 ± ∗)[(1 ± ηR˜g(G B))−1
]I
J (˜g−1
d˜g)J
ηRJK
˜g (G B)Kmdzm
.
(B.5)
Using formula (4.21) of [36] we find that it can be written as
(1 ± ∗)A = (1 ± ∗)(1 ± ηR˜g(G B))−1 ˜V, ˜VM
≡ δM
I (˜g−1
d˜g)I
+ δM
m dzm
.
(B.6)
This can almost be written as a relation involving only the derivatives of the redefined
coordinates
(1 ± ∗)W = (1 ± ∗)(1 ± ηΘ(G B))−1
d˜X, d˜XM
≡ δM
i d˜xi
+ δM
m dzm
,
(B.7)
where
WM
= δM
i
i
I
˜I
j dxj
+ δM
m dzm
, g−1
dg = dxi I
i TI, ˜g−1
d˜g = d˜xi ˜I
i TI(B.8)
and where all indices that have been omitted above are curved indices M = {i, m}. In general
i
I
˜I
j = δi
j and G, B, Θ may depend on ˜x. Thanks to the above field redefinition we can argue
that solutions to the classical equations of the YB model can be mapped to solutions of the
undeformed model, and vice versa.
Appendix C.  All non-Abelian R-matrices of sl(2, )2
⊕ su(2)2
We will classify non-Abelian R-matrices solving the classical Yang–Baxter equation relevant
to deformations of AdS3 × S3
. The R-matrix is an anti-symmetric matrix with indices in the
isometry algebra, in our case
sl(2, )L ⊕ sl(2, )R ⊕ su(2)L ⊕ su(2)R.(C.1)
The classical Yang–Baxter equation (CYBE)
R[A|B|
RC|D|
f
E]
BD = 0,(C.2)
implies that RAB
is non-degenerate on a subalgebra and zero elsewhere. Calling the inverse
ωAB the CYBE is equivalent to ωA[BfA
CD] = 0, i.e. ω is a Lie algebra 2-cocycle on the (dual of
the) subalgebra where R defined. Since it is also invertible this subalgebra is a quasi-Frobenius
(sometimes also called symplectic) subalgebra35
. Therefore R-matrices solving the CYBE on
35
If ω is exact, i.e. ωAB = fC
ABXC for some XC, the algebra is Frobenius.
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
25
some Lie algebra are in one-to-one correspondence with quasi-Frobenius subalgebras of this
Lie algebra [63].
Semi-simple Lie algebras cannot be quasi-Frobenius and therefore such algebras of dimension
4 (or 2) must be solvable [64]. In our case things are particularly simple since we have a
sum of four 3 dimensional Lie algebras. When can therefore restrict our attention to subalgebras
of the maximal solvable subalgebra of the isometry algebra which we take to be
s = span{S0, S+, ¯S0, ¯S−, T1, ¯T2}.(C.3)
The only non-Abelian solvable Lie algebra of dimension 2 is r2 with Lie bracket
[e1, e2] = e2. The possible R-matrices of rank 2 are given by embeddings of r2 into s and up to
SL2
(2) × SU2
(2) automorphisms they are
R1 = (S0 − ¯S0 + T) ∧ (S+ ± ¯S−)
R2 = (S0 − a¯S0 + T) ∧ S+
R3 = (S0
¯S− + T) ∧ S+
(C.4)
where T is any linear combination of T1 and ¯T2.
For the rank 4 case we need to find 4-dimensional solvable subalgebras of s, which are
furthermore quasi-Frobenius (symplectic). The complete list of 4-dimensional quasi-Frobenius
algebras can be found in [65]. Taking into account the fact that [s, s] = span{S+, ¯S−} is
2-dimensional we can rule out any algebra with more than two independent linear combinations
of generators arising from commutators. It is also trivial to see that the Heisenberg algebra
with non-trivial Lie bracket[e1, e2] = e3 is not a subalgebra, h3 ⊂ s. Using these two facts
the list of 4-dimensional solvable subalgebras is reduced to (note that r4,−1,0 = ⊕ r3,−1)
⊕ r3, ⊕ r3,λ, ⊕ r3,γ, r2 ⊕ r2, r2, d4,1.(C.5)
The same paper lists the symplectic ones which are
⊕ r3,0 [e1, e2] = e2
⊕ r3,−1 [e1, e2] = e2, [e1, e3] = −e3
⊕ r3,0 [e1, e2] = −e3, [e1, e3] = e2
r2 ⊕ r2 [e1, e2] = e2, [e3, e4] = e4
r2 [e1, e3] = e3, [e1, e4] = e4, [e2, e3] = e4, [e2, e4] = −e3
d4,1 [e1, e2] = e3, [e4, e3] = e3, [e4, e1] = e1
.
(C.6)
It is not hard to show, using the fact that the elements arising from commutators are of the
form aS+ + b¯S− for some a, b, that r3,0 is not a subalgebra of s and therefore neither is r2. It
is also not hard to show that neither is d4,1. For the remaining ones we find the embeddings
(again up to automorphism)
⊕ r3,0 e1 = S0 − b¯S0 + e1, e2 = S+ + a¯S− (a = 0 or a = b = 1)
⊕ r3,−1 e1 = S0 + ¯S0 + e1, e2 = S+, e3 = ¯S−
r2 ⊕ r2 e1 = S0 + e1, e2 = S+, e3 = −¯S0 + e3, e4 = ¯S−
.

(C.7)
Primed generators denote any linear combination that commutes with the remaining generator.
Only the second algebra is unimodular (i.e. the trace of its structure constants vanish) and
is contained in the classification of unimodular R-matrices in [43].
The rank 4 R-matrices can then be read off from the classification in [65]. Up to inner automorphisms
and exchanging the left and right copy of the algebra they are
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
26
R4 = (S0 − ¯S0 + T) ∧ (S+ ± ¯S−) + aT1 ∧ ¯T2
R5 = (S0 − a¯S0 + T) ∧ S+ + bT1 ∧ ¯T2
R6 = (S0 + T) ∧ S+ + T ∧ (¯S0 + T )
R7 = (S0 + T) ∧ S+ + T ∧ (¯S− + T )
R8 = (S0
¯S− + T) ∧ S+ + aT1 ∧ ¯T2
R9 = (S0 + ¯S0 + T) ∧ T + S+ ∧ ¯S−
R10 = (S0 + T) ∧ S+ ± (¯S0 + T ) ∧ ¯S− + λS+ ∧ ¯S−

(C.8)
where T, T , T are linear combinations of T1 and ¯T2. The rank 2 R-matrices are contained as
special cases of R4, R5, R8 when the last term vanishes.
A rank 6 R-matrix is only possible if s is itself quasi-Frobenius. It is in this case but the
resulting R-matrix is just R10 + aT1 ∧ ¯T2 and therefore leads just to a TsT transformation of
the R10 deformation. Therefore we will not consider it further here.
Appendix D.  R-matrix on parabolic subalgebra of sl3
Let f denote the 6-dimensional parabolic Lie subalgebra of sl3 [66] with basis (see example
3.6 of [67])
(e1, . . . , e6) = (E12, E13, E21, E23, E11 − E22, E22 − E33).(D.1)
The Lie brackets are given by
[e1, e3] = e5, [e1, e4] = e2, [e1, e5] = −2e1, [e1, e6] = e1, [e2, e3] = −e4,
[e2, e5] = −e2, [e2, e6] = −e2, [e3, e5] = 2e3, [e3, e6] = −e3, [e4, e5] = e4, [e4, e6] = −2e4.
(D.2)
Thisalgebraisnotunimodularsince fi6
i
= −3 = 0 andalsonotsolvablesincetr([e1, e3]e5) = 0
but it is quasi-Frobenius with 2-cocycle
ω =a13e1
∧ e3
+ a14e1
∧ e4
− 2a16e1
∧ e5
+ a16e1
∧ e6
+ a23e2
∧ e3
− a14e2
∧ e5
− a14e2
∧ e6
− 2a36e3
∧ e5
+ a36e3
∧ e6
− a23e4
∧ e5
+ 2a23e4
∧ e6
.
(D.3)
One may take e.g.
ω = e1
∧ (2e5
− e6
) + e2
∧ e3
+ e4
∧ (2e6
− e5
)(D.4)
the inverse being
R = −
1
3
e1 ∧ (2e5 + e6) − e2 ∧ e3 −
1
3
e4 ∧ (2e6 + e5).(D.5)
For the SL(3) WZW model we can get a deformation by embedding this algebra in the diagonal
SL(3) ⊂ SL(3) × SL(3). After identifying the coefficients of CS with the off-diagonal
components of the R-matrix and using the definitions in (2.19) we find
C12¯3
= −1, C13¯2
= −1, C15¯1
= −
2
3
, C16¯1
= −
1
3
, C23¯4
= 1, C25¯5
=
2
3
,
C26¯5
=
1
3
, C34¯2
= 1, C45¯4
= −
1
3
, C46¯4
= −
2
3
, C56¯2
=
1
3
.

(D.6)
Once again we are using a bar for indices in the right copy of the algebra. Using the fact that
the non-zero components of the Killing metric are
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
27
K13 = K31 = 1, K55 = K66 = 2, K56 = K65 = −1,(D.7)
one finds that Cab¯c
K¯c¯dCef¯d
= 0 but the weak CS condition is satisfied since C2
  =  0.
Appendix E.  More general ‘trivial’ solutions of generalized supergravity
In [42] it was shown that the generalized supergravity equations can have ‘trivial’ solutions,
i.e. ones that are also solutions of standard supergravity even though the Killing vector K is
non-zero. Writing36
X = dφ + iKB it was shown that for φ to have a gauge invariant meaning
K must be null. While this is a natural condition it is not strictly necessary. It is possible to
have a situation where K is not null that still leads to a standard supergravity solution, as we
will show here. However in that case one must pick the correct gauge for the B-field to read off
the dilaton from the expression X = dφ + iKB, as in a different gauge one may find a different
φ which will not be the correct dilaton of a standard supergravity solution. To avoid having to
deal with gauge-transformations of the B-field we will write X = dφ + ˜K where φ should be
identified with the dilaton. Below we derive the conditions on ˜K for a trivial solution of the
supergravity equations. Note that if we pick a gauge so that B is invariant under the isometry
generated by K we have ˜K = iKB + dφ for some φ .
We will ignore the RR fields in our discussion. Looking at the generalized supergravity
equations in [42] it follows from the generalized Einstein equation that for (G, H, φ) to solve
the standard supergravity ˜K must be a Killing vector (of the metric, other fields do not a
priori have to be invariant). The remaining equations lead to the following conditions to have
a trivial solution
d˜K + iKH = 0, LKφ + K · ˜K = 0,
dK + i˜KH = 0, L˜Kφ + K2
+ ˜K2
= 0.
(E.1)
If ˜K is proportional to K we get precisely the solutions considered in [42], in particular K
is null. But it can also happen that ˜K and K are linearly independent, as the example in (3.22)
shows. Such solutions are clearly much less generic than the solutions considered in [42] since
they require at least two Killing vectors. They are also harder to identify since they require first
extracting the correct dilaton and ˜K and then verifying the equations above.
In [42] it was argued that the analysis based on the generalized supergravity equations agrees
with what one gets by looking at the non-local terms in the sigma model action induced by
non-Abelian T-duality on a non-unimodular group [38, 39]. The analysis was done for standard
YB sigma models where the more general possibility of an independent Killing vector ˜K
does not arise. However, the general form of the non-local terms proposed there, namely
Lσ = α dσ ∧ K − α dσ ∧ ∗X + O(α 2
),(E.2)
is consistent with the general analysis above since, up to total derivatives, this is equal to
α σ(dK + i˜KH) − α σ(d ∗ ˜K + i˜KH)+α φR(2)
(E.3)
and the first term vanishes by (E.1) and the second is proportional to the equations of motion
projected along the Killing vector ˜K. The order α 2
terms were also considered in [42] and,
given the present analysis, these will be modified so that they now involve a combination of
the second and last equation in (E.1) rather than just K2
.
36
We recall that this was denoted just by X in [42], but we prefer to avoid confusion with the vector X = X + K of
[41].
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
28
ORCID iDs
Riccardo Borsato https://orcid.org/0000-0002-0156-441X
Linus Wulff https://orcid.org/0000-0002-9168-3470
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467 183–201
R Borsato and L WulffJ. Phys. A: Math. Theor. 52 (2019) 225401
JHEP08(2018)027
Published for SISSA by Springer
Received: June 26, 2018
Accepted: August 6, 2018
Published: August 8, 2018
Non-abelian T-duality and Yang-Baxter deformations
of Green-Schwarz strings
Riccardo Borsatoa and Linus Wulffb
a
Nordita, Stockholm University and KTH Royal Institute of Technology,
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
b
Department of Theoretical Physics and Astrophysics, Masaryk University,
611 37 Brno, Czech Republic
E-mail: riccardo.borsato@su.se, wulff@physics.muni.cz
Abstract: We perform non-abelian T-duality for a generic Green-Schwarz string with
respect to an isometry (super)group G, and we derive the transformation rules for the
supergravity background fields. Specializing to G bosonic, or G fermionic but abelian, our
results reproduce those available in the literature. We discuss also continuous deformations
of the T-dual models, obtained by adding a closed B-field before the dualization. This
idea can also be used to generate deformations of the original (un-dualized) model, when
the 2-cocycle identified from the closed B is invertible. The latter construction is the
natural generalization of the so-called Yang-Baxter deformations, based on solutions of the
classical Yang-Baxter equation on the Lie algebra of G and originally constructed for group
manifolds and (super)coset sigma models. We find that the deformed metric and B-field
are obtained through a generalization of the map between open and closed strings that
was used also in the discussion by Seiberg and Witten of non-commutative field theories.
When applied to integrable sigma models these deformations preserve the integrability.
Keywords: Sigma Models, String Duality, Superstrings and Heterotic Strings, Integrable
Field Theories
ArXiv ePrint: 1806.04083
Open Access, c The Authors.
Article funded by SCOAP3
.
https://doi.org/10.1007/JHEP08(2018)027
JHEP08(2018)027
Contents
1 Introduction 1
2 Summary of the transformation rules 3
2.1 Rules of (bosonic) NATD 3
2.2 Rules of YB deformations 5
3 NATD of Green-Schwarz strings 6
3.1 The case with local symmetry 8
3.2 Extracting X and K from anomaly terms 9
3.3 Extracting the RR fields 10
4 Deformations 12
4.1 Deformed T-dual models 12
4.2 Yang-Baxter deformations 13
4.2.1 A convenient rewriting 16
4.3 Two examples of YB deformations 17
4.3.1 YB deformation of the D3-brane background 17
4.3.2 YB deformation of AdS3 × S3 × T4 with H-flux 20
5 Conclusions 21
A Conventions 22
B An example with local symmetry 22
1 Introduction
While ordinary abelian T-duality is an exact symmetry of string perturbation theory, its
non-abelian generalization [1] is not [2, 3]. It should be rather viewed as a solutiongenerating
technique in supergravity, since it (typically) maps one string background to
another, inequivalent one. Starting with the work of [4], which gave a prescription for
the transformation of the RR fields, it has been successfully applied to construct several
interesting supergravity solutions, e.g. [5–10].
Like its abelian version, non-abelian T-duality (NATD) can be understood as a canonical
transformation [11–13], so that the dualization preserves the (classical) integrability of
the sigma model (when present). To be more precise, starting from a sigma model whose
equations of motion are equivalent to the flatness of a Lax connection, one obtains a dual
model whose equations of motion can also be put into Lax form. Here we want to exploit
this property in order to generate integrable deformations of sigma models, following the
– 1 –
JHEP08(2018)027
ideas of [14–16].1 The deformations are interesting also because they (partially) break the
initial isometries. We remark that integrability is not essential for the construction, and the
deformations can be carried out also for non-integrable models. Some of the deformations
constructed here may be viewed as continuous interpolations between the “original” model
and the “dual” one obtained after applying NATD.
Starting from a generic type II Green-Schwarz superstring whose isometries contain
a (super)group G, we work out the transformation rules for the supergravity background
fields under NATD with respect to G. The derivation is performed in section 3, where
all orders in fermions are taken into account by working in superspace. When choosing a
bosonic G and focusing on the bosonic supergravity fields, the transformation rules reproduce
those of [4, 20], including the Ramond-Ramond (RR) fields whose transformations
were conjectured by analogy with the abelian case [21]. Moreover, when the Lie algebra
of G consists of only (anti)commuting fermionic generators, we also reproduce the rules
for fermionic T-duality derived in [22] from the pure spinor string. As expected, we show
that after NATD one still obtains a kappa symmetric Green-Schwarz superstring. It follows
from [23] that the target space is therefore a solution of the generalized supergravity
equations of [23, 24]. When G is unimodular (i.e. the structure constants of its Lie algebra
satisfy fJ
IJ = 0) the background fields satisfy the standard type II supergravity equations,
and the (dualized) sigma model is Weyl invariant. When G is not unimodular there is
typically an anomaly which breaks Weyl invariance and obstructs the interpretation of the
dual model as a string [25, 26]. We will also discuss exceptions to this, given by the “trivial
solutions” of [27].
Deformations of the non-abelian T-dual backgrounds may be generated by adding
a closed B-field before dualizing. The deformation will be controlled by one or more
continuous parameters that enter the definition of this B. From the point of view of
the original model, adding a B-field with dB = 0 does not affect the local physics, since
this term does not change the equations of motion. We will nevertheless obtain a nontrivial
deformation and a dependence on B in the equations of motion after applying
NATD, since this transformation involves a non-local field redefinition.2 Writing B =
1
2(g−1dg)J ∧ (g−1dg)I ωIJ with g ∈ G, the condition dB = 0 is equivalent to ω being a
2-cocycle on the Lie algebra of G. The resulting models were dubbed deformed T-dual
(DTD) models in [15], and we refer to section 4.1 for more details.
In [15, 16] it was proved that a DTD model constructed from a principal chiral model
(PCM) or supercoset sigma model with ω invertible is actually equivalent (thanks to a
local field redefinition) to the so-called Yang-Baxter (YB) sigma models [28–33] based
on an R-matrix solving the classical Yang-Baxter equation.3 The R-matrix is related to
1
Another class of integrable deformations related to NATD are the so-called λ-deformations of [17–19].
2
If B is not just closed but also exact, it contributes to the action of the original model as a total derivative
and it can be dropped. Even if kept, the dependence on this B can be removed by a (local) field redefinition
even after applying NATD. Therefore an exact B generates a trivial deformation of the dual model.
3
These are sometimes called “homogeneous” YB models. In the “inhomogeneous” YB models R solves
the modified classical Yang-Baxter equation. They were first introduced in [28, 29] and later generalized to
the supercoset case in [34], where the so-called η-deformation of AdS5 ×S5
was constructed. The inhomogeneous
YB models are not related in such a simple way to NATD and we will not consider them further here.
– 2 –
JHEP08(2018)027
the 2-cocycle simply as R = ω−1. The equivalence was first proposed and checked on
various examples in [14].4 When the R-matrix acts only on an abelian subalgebra YB
deformations are simply TsT (T-duality -shift- T-duality) transformations [36], so that we
can think of YB deformations as the “non-abelian” generalization of TsT transformations.
Here we propose to use the connection to NATD in order to extend the applicability of
YB deformations, from just PCM and supercoset models to a generic sigma model with
isometries. We do this in section 4.2 by carrying out the field redefinition which leads from
the DTD model to the YB model in the case of invertible ω. Although the construction
comes from a deformation of the dual model, when sending the continuous deformation
parameter of the YB model to zero we recover the original model. These deformations
may be particularly interesting for the AdS/CFT correspondence, and in section 4.3.1 we
use our results to “uplift” a YB deformation of AdS5 ×S5 — that cannot be interpreted as
(a sequence of) TsT transformations — to a deformation of the full D3-brane background,
of which AdS5 × S5 is the near-horizon limit.
For YB deformations of the PCM or (super)coset models, it is easy to see that the
background metric and B-field are related to the metric of the original model by a map
that coincides with the open/closed string map used also by Seiberg and Witten in [37].
For YB deformations the open string non-commutativity parameter is identified with the
R-matrix itself [38]. Based on this observation it was suggested in [39] that this map
could be used to generate solutions to (generalized) supergravity.5 Our results, based
on the construction of [15], generalize this to cases with a non-vanishing B-field in the
original model. Our derivation also ensures that the YB backgrounds are automatically
solutions of the (generalized) supergravity equations. Yet another approach to such general
(homogeneous) YB deformations was proposed in the context of doubled field theory, since
known YB deformations were shown to be equivalent to so-called β-shifts [41–43]. In
section 4.3.2 we check in an example that a recent solution generated in [43] coincides with
the one obtained from our method based on NATD.
In the next section we collect the transformation rules for the background fields under
NATD and under a generic YB deformation.
2 Summary of the transformation rules
In this section we wish to present and summarize in a self-contained way the transformation
rules derived in the paper, so that the reader may consult them without the need of going
through the whole derivation.
2.1 Rules of (bosonic) NATD
Here we summarize the NATD transformation rules for the bosonic supergravity fields only,
when we take G to be an ordinary (i.e. non-super) Lie group. The general transformations
4
An equivalent construction, applying NATD on a centrally extended algebra, was used there. See
also [35].
5
In [40] it was shown that the map generates solutions of the generalized supergravity equations if the
non-commutativity parameter satisfies the classical Yang-Baxter equation.
– 3 –
JHEP08(2018)027
can be found in section 3 (for the case of G a supergroup see footnote 11). It is convenient
to rewrite the background fields in a way that makes the G isometry manifest. The metric,
for example, will be written in the following block form
Gµν =
Gmn Gmj
Gin Gij
, Gin = ℓI
i GIn, Gij = ℓI
i ℓJ
j GIJ . (2.1)
We have chosen coordinates such that we can split indices into (i, m), where i takes dim G
values and m labels the remaining spectator fields which do not transform under G. We
have collected our conventions in appendix A. It is also convenient to rewrite certain blocks
by extracting ℓI
i , defined by g−1∂ig = ℓI
i TI, where g ∈ G and I = 1, . . . , dim G is an index
in g (the Lie algebra of G) so that [TI, TJ ] = fK
IJ TK. The dependence on the coordinates xi
(i.e. the coordinates to be dualized) is all in ℓI
i , so that GIJ , GIm, Gmn only depend on the
spectators xm. The transformation rules will be presented in terms of these objects, and
we will continue to call them “metric” and “B-field” also when writing them with indices
(m, I) instead of (m, i). In order to have a uniform derivation and presentation, we do
not restrict further the range of the index I even when a local symmetry is present.6 We
refer to section 3 for more details. Setting fermions to zero the transformation rules for
the metric and B-field in (3.6)–(3.8) read7
˜Gmn = Gmn − (G − B)N(G − B) (mn)
, (2.2)
˜GmI =
1
2
(G − B)N mI
−
1
2
N(G − B) Im
, ˜GIJ = N(IJ) ,
˜Bmn = Bmn + (G − B)N(G − B) [mn]
, (2.3)
˜BmI = −
1
2
(G − B)N mI
−
1
2
N(G − B) Im
, ˜BIJ = −N[IJ] ,
where NIJ = δIKNKLδLJ etc. and
NIJ
= GIJ − BIJ − νKfK
IJ
−1
. (2.4)
The transformation of the RR fields, encoded in the bispinor (for more on the conventions
see [23, 44])
S12
=



F(0)
−
1
2
F
(2)
ab Γab
+
1
4!
F
(4)
abcdΓabcd
IIA
− F(1)
a Γa
−
1
3!
F
(3)
abcΓabc
−
1
2 · 5!
F
(5)
abcdeΓabcde
IIB
, (2.5)
given in (3.30) is given by the action of a Lorentz transformation Λ ∈ O(1, 9) as
˜S12
= ˆΛS12
, Λab
= ηab
− 2EI
a
NIJ
EJ
b
, (2.6)
6
Therefore the range of (m, I) can exceed ten. Both the original and the final action are still written
only in terms of ten physical coordinates thanks to the local symmetry that survives NATD and removes
the additional degrees of freedom, see the discussion in section 3.1.
7
The coordinates νI that result from the dualization naturally have lower indices, since they parameterize
the dual space. To have the standard upper placement of indices also in the dualized model we declare that
those indices are raised with the Kronecker delta νI
= δIJ
νJ , and the total set of coordinates is (xm
, νI
).
– 4 –
JHEP08(2018)027
where GIJ = EI
aEJ
bηab, and we denote by ˆΛ the Lorentz transformation acting on spinor
indices that multiplies S12, defined such that Λa
bΓb = ˆΛT Γa ˆΛ. Finally the generalized
supergravity fields K and X given in (3.17)–(3.18) become8
Km
= 0 , KI
= nI
, Xm = ∂m φ +
1
2
ln det N − ˜BmInI
, XI = − ˜BIJ nJ
. (2.7)
They involve the trace of the structure constants9 of g, nI = δIJ nJ with nI = fJ
IJ . As
already mentioned, in the generic case we must write the results in terms of the fields K
and X. Indeed when nI = 0 the background solves the generalized type II supergravity
equations [23, 24] but not the standard ones, and the sigma model is scale but not Weyl
invariant at one loop. When g is unimodular, nI = 0 and we get a solution of standard
type II supergravity consistent with the results of [25, 26]. In that case, since X is a total
derivative we can write X = d˜φ in terms of a dual dilaton
˜φ = φ +
1
2
ln det N . (2.8)
It was shown in [27] that there exist special “trivial” solutions of the generalized supergravity
equations which solve the standard supergravity equations although K is not zero.
For this to happen K must be null and, in addition to a condition involving the RR fields
which we ignore here, it should satisfy dK = iKH. Using the rules of NATD presented
here the latter condition can be written as
n( ˜G − ˜B) = 0 . (2.9)
Since ( ˜G− ˜B)IJ = NIJ is invertible by assumption, it has no zero-eigenvector and therefore
it would seem that no trivial solution can be generated by NATD. However, the condition
written above is not invariant with respect to B-field gauge transformations, so that the
conclusion can change. This will actually play a role in the discussion of the closely related
YB models.
2.2 Rules of YB deformations
For YB deformations the rules are a bit simpler in the sense that we do not have to write
the background fields in the block-form as previously. The result can be phrased in different
ways, see section 4.2. Here we will describe the results in terms of Killing vectors of the
original background. The final result of our derivation is that in order to apply a YB
deformation one should first construct
Θµν
= kµ
I RIJ
kν
J , (2.10)
where RIJ solves the classical Yang-Baxter equation (4.6) and kµ
I are a collection of Killing
vectors labeled by I that are properly normalized so that they satisfy (4.24). Then the
8
Here we drop the tilde since these fields are not present before dualization. Also note that we have
raised the index on K with ˜G−1
in order to get a simpler expression. We assume the original dilaton φ to
respect the G isometry, so that is depends only on the spectators, but this assumption can be relaxed.
9
The identification of K with the trace of the structure constants was suggested earlier in [45].
– 5 –
JHEP08(2018)027
background metric and B-field of the YB model are simply obtained by the following
generalization of the open/closed string map
˜G − ˜B = (G − B)[1 + ηΘ(G − B)]−1
, (2.11)
where we have omitted indices µ, ν. The RR bispinor transforms as
˜S12
= ˆΛS12
, Λab
= ηab
− 2ηEµ
a ˆNµ
νΘνρ
Eρ
b
, (2.12)
where ˆNν
µ = δµ
ν + ηΘµρ(Gρν − Bρν)
−1
. We further have
Kµ
= ηΘµν
nν , Xµ = ∂µ φ −
1
2
ln det[1 + ηΘ(G − B)] − η ˜BµνΘνρ
nρ , (2.13)
and, when the Killing vectors used to construct Θ define a unimodular algebra fJ
IJ = 0,
we find the deformed dilaton
˜φ = φ −
1
2
ln det[1 + ηΘ(G − B)] . (2.14)
We refer to section 4.2 for the derivation and a discussion of trivial solutions for YB
deformations.
3 NATD of Green-Schwarz strings
In this section we apply NATD to a generic Green-Schwarz string with isometries. To
perform NATD we assume that we can bring the supervielbein to the form
EA
= (g−1
dg)I
EI
A
(z) + dzM
EM
A
(z) , (A = (a, α) , a = 0, . . . , 9 , α = 1, . . . , 32) ,
(3.1)
with g ∈ G encoding the coordinates we want to dualize and zM = (xm, θα) denoting
the remaining (spectator) coordinates. The isometry (sub)group G to be dualized acts
as g → ug, z → z for a constant element u ∈ G. To avoid extra awkward signs, we
will present the derivation when G is an ordinary Lie group, but we will write the end
result for the dualized geometry such that it applies also to the case when G is a super
Lie group. The index I takes dim G values and since we want to include the case in which
a local symmetry of the sigma model (which we do not fix) is a subgroup of G, we allow
the possibility that the total range of indices (m, I) is greater than ten. In that case the
local symmetry can be used at the end to remove the spurious coordinates and leave the
ten physical ones. In that case EI
a also involves a projection matrix [20], the simplest
example being a supercoset geometry where EI
a is proportional to the projector on the
coset directions (usually denoted by P(2)).
The (classical) Green-Schwarz string action is
S = T
Σ
1
2
Ea
∧ ∗Eb
ηab + B , (3.2)
where we are using worldsheet form notation and the supervielbein Ea and NSNS twoform
potential B are understood to be pulled back to the worldsheet Σ. To perform NATD
– 6 –
JHEP08(2018)027
we write this action in first order form using (3.1), replacing g−1dg → A and adding a
Lagrange multiplier term to enforce the flatness of A
S′
=
T
2 Σ
AI
∧ (GIJ ∗ −BIJ )AJ
+ 2dzM
∧ (GMI ∗ −BMI)AI
+ (−1)degN
dzM
∧ (GMN ∗ −BMN )dzN
+ νI(2dAI
− fI
JKAJ
∧ AK
) . (3.3)
The components of the (super) metric are GIJ = EI
aEJ
bηab, GIM = GMI = EI
aEM
bηab
and GMN = EM
aEN
bηab. Integrating out A gives10
(1±∗)AI
= −(1±∗) dνJ + dzM
[∓G − B]MJ NJI
∓ , NIJ
± = ±GIJ − BIJ − νKfK
IJ
−1
(3.4)
and the dual action
˜S =
T
4 Σ
dνI + dzM
[G − B]MI NIJ
+ ∧ (1 + ∗) dνJ − dzM
[G + B]MJ
+ dνI − dzM
[G + B]MI NIJ
− ∧ (1 − ∗) dνJ + dzM
[G − B]MJ
+ 2(−1)N
dzM
∧ (GMN ∗ −BMN )dzN
= T
Σ
1
2
˜Ea
± ∧ ∗ ˜Eb
±ηab + ˜B . (3.5)
In the last step we have written the dualized action in Green-Schwarz form by defining two
possible sets of dual supervielbeins11
˜EA
± = dzM
EM
A
− dνI + dzM
[∓G − B]MI NIJ
∓ EJ
A
. (3.6)
The dual B-field can also be written in two equivalent ways
˜B =
1
2
dzN
∧ dzM
BMN +
1
2
dνI + dzM
[±G − B]MI ∧ NIJ
± dνJ − dzM
[±G + B]MJ .
(3.7)
We choose ˜Ea
+ to be the dual bosonic supervielbein, while ˜Ea
− is related to it by a Lorentz
transformation as follows
˜Ea
= ˜Ea
+ , ˜E′a
= ˜Ea
− = ˜Eb
Λb
a
, Λb
a
= δa
b − 2EIbNIJ
+ EJ
a
. (3.8)
This is easily seen to follow from the useful identity
dνI + dzM
[G − B]MI NIJ
+ = dνI − dzM
[G + B]MI NIJ
− + 2 ˜Ea
EIaNIJ
+ . (3.9)
10
These solutions and the following action are written so that they hold also when G is a supergroup.
11
When G is a supergroup the correct expressions are obtained by writing things in a form which is
symmetric between N+ and N− (and where contracted indices are adjacent), e.g.
˜Ea
± = dzM
EM
a
−
1
2
dνI + dzM
[∓G − B]MI NIJ
∓ EJ
a
+
1
2
EI
a
NIJ
± dνJ + dzM
[∓G − B]MJ .
This will be true also for the expressions for Λ, K, X and ˜S to be derived below.
– 7 –
JHEP08(2018)027
It is interesting to compute the determinant of the Lorentz transformation Λ. We have
(suppressing the indices)
det Λ = exp(tr ln Λ) = exp −tr
∞
n=1
(2EN+E)n
n
= exp −tr
∞
n=1
(2GN+)n
n
= exp[tr ln(1 − 2GN+)] = exp[tr ln(1 − (N−1
+ + N−T
+ )N+)] = det(−N−T
+ N+)
= (−1)dim G
. (3.10)
This shows that this Lorentz transformation is an element of SO(1, 9) only when dim G is
even. When dim G is odd, i.e. one dualizes on an odd number of directions, the Lorentz
transformation involves a reflection. In the latter case its action on spinors contains an
odd number of gamma matrices, which means that one goes from type IIA to type IIB or
vice versa, cf. (3.28).
3.1 The case with local symmetry
Here we wish to give more details on the case when the original sigma model has a local
symmetry that is a subgroup of G. We will explain how the results of the previous section
apply also in that case. We will assume that the action (3.2) is invariant under a local
group H ⊂ G that acts on g from the right as12 g → gh, h ∈ H. Our goal will be to show
that if the local H invariance is not fixed before the dualization, NATD can still be applied
in the usual way and the dual action naturally inherits the local H symmetry. Therefore
this ensures that the additional degrees of freedom can be removed also in the dual model,
and that we are left only with physical ones of the correct number.
The action (3.2) is invariant under g → gh if the couplings are H invariant and project
out h, the Lie algebra of H
(Ad−1
h )K
I(GKL∗−BKL)(Ad−1
h )L
J = GIJ ∗−BIJ , yI
(GIJ ∗−BIJ ) = 0 = (GIJ ∗−BIJ )yJ
,
(GMJ ∗−BMJ )(Ad−1
h )J
I = GMI ∗−BMI, (GMI ∗−BMI)yI
= 0. (3.11)
Here y ∈ h. This local symmetry may be used to remove dim H degrees of freedom from
the parametrization of g, so that the total number of physical bosonic fields (including
spectators) is ten. We do not fix this local invariance yet, since this allows us to gauge the
whole G isometry and fix the gauge g = 1 to arrive at the action (3.3). This first order
action is still invariant under a local H which is now implemented as
A → h−1
Ah + h−1
dh, ν → h−1
νh . (3.12)
Here ν = νITI is taken to be an element of g∗, the dual of the Lie algebra of G. We
refer to section 4.1 for our conventions regarding g∗. At the moment of integrating out AI
from (3.3) one may worry about the invertibility of the relevant linear operators, given that
the couplings project out the components in h as assumed above. We consider cases when
the operators ±GIJ − BIJ − νKfK
IJ are invertible on the whole algebra g, so that also the
12
One may equivalently discuss this local invariance by introducing a vector valued in the Lie algebra of
H, so that integrating out such vector the original action is obtained.
– 8 –
JHEP08(2018)027
components of A in h can be integrated out. Obviously, since ±GIJ − BIJ are degenerate,
the invertibility of the operators must be ensured by the term νKfK
IJ . We recall that ν has
not been gauged-fixed yet, and that we have a total of dim G such fields. In general the
invertibility will hold only locally, meaning that there may be values of νI such that the
operators NIJ
± become singular. Those loci will correspond to singularities in target space
that we cannot remove. It is easy to check that the dual action (3.5) is still invariant under
the local H symmetry, which is now simply implemented by ν → h−1νh. We can then fix
the local symmetry at the level of the dual action, at the same time making sure that we
have the correct number of degrees of freedom and that the gauge fixing is done correctly.
Our reasoning is completely analogous to that of [20, 46]. There the degenerate matrices
±GIJ − BIJ are regulated by taking ±GIJ − BIJ + λ (Idh)IJ , where Idh is the identity
on h. The parameter λ is kept during the dualization and sent to zero only at the end. It
is clear that the λ → 0 limit is non-singular only if the degeneracies of ±GIJ − BIJ are
lifted by the additional term νKfK
IJ . Therefore the way coset models are treated in [20, 46]
is analogous to ours. For concreteness we work out an explicit example in appendix B.
3.2 Extracting X and K from anomaly terms
The easiest way to extract the generalized supergravity fields X and K is to look at
the terms in the action induced at the quantum level by the NATD change of variables
g−1dg → A in the path integral measure [26].13 It was shown in [27] that these non-local
terms take the form
˜Sσ =
1
2π Σ
dσ ∧ K − dσ ∧ ∗X −
1
2
α′
dσ ∧ ∗dσ |K|2
, (3.13)
where σ = ∂−2√
gR(2) is the conformal factor. From the first two terms it is easy to read off
X and K. To compute ˜Sσ we include the σ-dependent terms in the first order action (3.3).
They are [26]
Sσ =
1
2π Σ
σnId ∗ AI
− Φd ∗ dσ , (3.14)
where nI = fJ
IJ , the trace of the structure constants, d ∗ dσ = d2ξ
√
gR(2) and Φ is the
dilaton superfield of the original background. Integrating out A as before but now including
these terms, and keeping track of the det N from the measure, we obtain
(1 ± ∗)AI
= −(1 ± ∗) dνJ + dzM
[∓G − B]MJ ∓ α′
nJ dσ NJI
∓ (3.15)
and
˜Sσ =
1
4π Σ
nIdσNIJ
+ ∧(1+∗)(dνJ −dzM
[G+B]MJ ) (3.16)
−nIdσNIJ
− ∧(1−∗)(dνJ +dzM
[G−B]MJ )−2dσ∧∗d Φ+
1
2
lndetN+ +O(α′
).
13
A more direct, but lengthier, approach uses the superspace constraints as we do below to extract the
RR fields, see for example [47].
– 9 –
JHEP08(2018)027
Comparing to (3.13) we find
K =
1
2
(dνJ +dzM
[G−B]MJ )NJI
+ −(dνJ −dzM
[G+B]MJ )NJI
− nI , (3.17)
X = d Φ+
1
2
lndetN+ +
1
2
(dνJ +dzM
[G−B]MJ )NJI
+ +(dνJ −dzM
[G+B]MJ )NJI
− nI .
(3.18)
These expressions simplify when written in terms of ˜G and ˜B as in (2.7).
3.3 Extracting the RR fields
The simplest way to find the RR fields is to compute the superspace torsion TA = dEA +
EB ∧ ΩB
A and compare to the superspace torsion constraints of [23, 44], see e.g. [47]. In
particular the Ea ∧ Eα1-term in Tα2 takes the form14
T2
= −
1
8
Ea
(E1
ΓaS12
) + . . . (3.19)
from which we can read off the RR bispinor S. Here Ea is the bosonic supervielbein and
Eα1, Eα2 with α = 1, . . . , 16 are the two fermionic supervielbeins, corresponding to the
two Majorana-Weyl spinors of type II supergravity. For convenience of the presentation
we will use type IIA notation so that E1 = 1
2 (1 + Γ11)E1 and E2 = 1
2 (1 − Γ11)E2 but the
type IIB expressions are essentially identical.
To compute ˜T2 and then extract the RR fields of the dualized model, we must first find
the form of the fermionic supervielbeins ˜E1, ˜E2. We therefore start with the constraint on
the bosonic torsion
Ta
= −
i
2
EΓa
E = −
i
2
E1
Γa
E1
−
i
2
E2
Γa
E2
, (3.20)
and we can compute ˜Ta from ˜Ea.15 By assumption the constraint on Ta holds in the
original model before dualization. In our adapted coordinates (3.1) it takes the form16
2∂[M EN]
a
+ 2Ω[M|b|
a
EN]
b
= (−1)N
iEM Γa
EN , (3.21)
∂M EI
a
+ ΩMb
a
EI
b
− ΩIb
a
EM
b
= iEM Γa
EI , (3.22)
fK
IJ EK
a
+ 2Ω[J|b|
a
EI]
b
= iEJ Γa
EI . (3.23)
We will also need the constraints on H = dB which are
H = −
i
2
Ea
EΓaΓ11E +
1
6
Ec
Eb
Ea
Habc = −
i
2
Ea
E1
ΓaE1
+
i
2
Ea
E2
ΓaE2
+
1
6
Ec
Eb
Ea
Habc .
(3.24)
In our adapted coordinates we have
3∂[M BNP] = HMNP , 2∂[M BN]I = HMNI ,
∂M BIJ + fK
IJ BMK = HMIJ , 3fL
[IJ BK]L = HIJK , (3.25)
14
To improve the readability we suppress the spinor index α and drop the explicit ∧’s from now on.
15
It might appear that one needs to know the spin connection to do this but this is not the case. Instead
the fermionic vielbeins and spin connection can be read off by computing d ˜Ea
as we will see.
16
The anti-symmetrization is graded, e.g. Y[M ZN] = 1
2
(YM ZN − (−1)MN
YN ZM ).
– 10 –
JHEP08(2018)027
where HIJK = EK
CEJ
BEI
AHABC etc. Using these relations we can compute the exterior
derivative of ˜Ea
± in (3.6) and we find
d ˜Ea
± = −
i
2
˜E±Γa ˜E± +
i
2
˜E±Γb(1 ± Γ11) ˜E±(±EI
a
NIJ
± EJ
b
) − ˜Eb
±
˜EC
±ΩCb
a
± i ˜Eb
±
˜E±Γb(1 ∓ Γ11)EINIJ
∓ EJ
a
± ˜Ec
±
˜Eb
± ΩIbc ±
1
2
EI
d
Hbcd NIJ
∓ EJ
a
. (3.26)
Using our definition of the dualized bosonic supervielbein, ˜Ea = ˜Ea
+, this can be recast,
using the definition (3.6), as17
˜Ta
= d ˜Ea
+ ˜Eb ˜Ωb
a
= −
i
2
Λa
b
˜E1
+Γb ˜E1
+ −
i
2
˜E2
−Γa ˜E2
− . (3.27)
Comparing to the standard form (3.20) we can read off the fermionic supervielbeins of the
dualized model18
˜E1
= ˆΛ ˜E1
+ , ˜E2
= ˜E2
− , (3.28)
where the action of the Lorentz transformation on spinors is defined by Λa
bΓb = ˆΛT Γa ˆΛ.
We are now ready to compute the fermionic torsion and extract the dualized RR fields by
comparing to (3.19). Following the same lines as above we find
d ˜E2
− =
1
4
(Γab
˜E2
−) ˜EC
−ΩC
ab
+
1
2
˜EB
−
˜EA
−T2
AB − i ˜E1
−Γa
˜E1
−(E2
I NIJ
− EJ
a
)
− 2i ˜Ea
−
˜E1
−ΓaE1
I (NIJ
+ E2
J ) − ˜Eb
−
˜Ea
− ΩIab −
1
2
HabcEI
c
NIJ
+ E2
J . (3.29)
Extracting the ˜Ea ˜E1-terms we can read of the RR bispinor which takes the form
˜S12
= ˆΛS12
+ 16iˆΛE1
I NIJ
+ E2
J . (3.30)
The first term is a Lorentz transformation acting on one side of the original bispinor in
agreement with the NATD transformation rules first proposed in [4], by analogy with the
abelian case. The second term starts at quadratic order in fermions if one dualizes on a
bosonic algebra. However, in cases involving fermionic T-dualities the bosonic background
is affected by the second term. In the case of a single fermionic T-duality it reproduces the
transformation rule derived in [22].19
17
Note that (3.9) implies ˜E− = ˜E+ − 2 ˜Ea
EIaNIJ
+ EJ .
18
We also find the spin connection of the dualized background
˜Ωab
= ˜EC
+ ΩC
ab
−4i ˜E2
+Γ[a
E2
I NIJ
− EJ
b]
−2 ˜Ec
ΩIc
[a
−
1
2
EI
d
Hcd
[a
NIJ
− EJ
b]
+ ˜Ec
ΩI
ab
+
1
2
EI
d
Hd
ab
NIJ
− EJc
−4i ˜Ec
EI
[a
NIJ
+ E2
J Γb]
E2
K NKL
− ELc −2i ˜Ec
EI
a
NIJ
+ E2
J ΓcE2
K NKL
− EL
b
.
19
In the pure spinor formalism used there one does not directly see the Lorentz transformation acting on
half of the fermionic directions since the pure spinor description has a larger symmetry with independent
Lorentz transformations for bosons and the two fermionic directions. However, setting the fermions to zero
Λ becomes trivial and all transformations, including those of the RR fields, match.
– 11 –
JHEP08(2018)027
To be sure that the sigma model after NATD still has kappa symmetry, or equivalently
that the background solves the generalized supergravity equations [23], one must also verify
that ˜H = d ˜B satisfies the correct constraints (3.24) (up to dimension zero). A direct
calculation using (3.7) and (3.9) shows that ˜H is indeed of the right form (3.24).20 This
proves that the dual model is indeed a Green-Schwarz string invariant under the standard
kappa symmetry transformations, and it completes the derivation of the dualized target
space fields which therefore solve the equations of (generalized) supergravity [23].
4 Deformations
NATD may be viewed as a solution-generating technique for supergravity backgrounds.
Here we slightly modify the procedure to generate continuous deformations of the dual
model, which will be called deformed T-dual (DTD) models. Later we will show that a
subclass of DTD models may be recast in the form of a deformation that reduces to the
original sigma model when sending the deformation parameter to zero. This subclass will
be identified with a generalization of YB deformations.
4.1 Deformed T-dual models
In order to define DTD models, we start from the original sigma-model, before applying
NATD, and we shift the B-field as
BIJ → BIJ − ζ ωIJ . (4.1)
Here ωIJ is constant and anti-symmetric in its indices. We use ζ as a parameter to keep
track of the shift, or in other words the deformation. The shift affects only the components
of the B-field along g, and it does not spoil the global G isometry. We demand that the
new term appearing in the action (i.e. ζ(g−1dg)I ∧ ωIJ (g−1dg)J ) should not modify the
theory on-shell, in other words that it should be a closed B-field. It is easy to see that this
happens if and only if ωIJ satisfies the 2-cocycle condition
ωI[J fI
KL] = 0 , (4.2)
where the antisymmetrization involves all three indices J, K, L. We further demand that
the B-field ζ(g−1dg)I ∧ ωIJ (g−1dg)J is closed but not exact, i.e. the shift should not be
a gauge transformation. Thanks to this additional condition, after applying NATD the
resulting deformation is non-trivial, i.e. the ζ-dependence cannot be removed by a field
redefinition. The non-exactness of B is equivalent to ωIJ not being a coboundary, i.e.
ωIJ = cKfK
IJ for any constant vector cK. Non-trivial deformations are therefore classified
by elements of the second Lie algebra cohomology group H2(g).
20
One also finds
˜Habc = −
1
2
Habc +
3
2
Λ[a
d
Hbc]d − 6EI[aNIJ
+ Ω|J|bc] − 12i(EI[aNIJ
+ E2
|J|)Γb(E|K|c]NKL
+ E2
L) .
– 12 –
JHEP08(2018)027
We can view the 2-cocycle as an element of g∗ ⊗ g∗ by writing ω = ωIJ TI ∧ TJ .
Alternatively we may view it as a map from g to the dual vector space (we continue to call
this ω without fear of creating confusion) ω : g → g∗, whose action is given by
ω(TK) = ωIJ TI
tr(TJ
TK) = ωIKTI
. (4.3)
To proceed further we will endow the dual vector space g∗ with a Lie algebra structure
with structure constants ˜fIJ
K so that g has a bialgebra structure. Therefore g⊕g∗ becomes
a Lie algebra with Drinfel’d double commutation relations21
[TI, TJ ] = fK
IJ TK , [TI
, TJ
] = ˜fIJ
K TK
, [TI, TJ
] = fJ
KITK
+ ˜fJK
I TK . (4.4)
This is always possible since we can always take g∗ to be abelian with ˜fIJ
K = 0. In general
this construction is far from unique and there exist many possible choices of Lie algebra
structure on g∗, however this choice will have no effect in what follows. The 2-cocycle
condition (4.2) can now be written
ω[TI, TJ ] = PT
([ωTI, TJ ] + [TI, ωTJ ]) , (4.5)
where PT projects on g∗. Note that if we take g∗ to be abelian we can drop the projector
and this equation just says that ω is a derivation on the Lie algebra g ⊕ g∗. This is the
choice that is most useful for the general discussion here.22
Apart from the shift BIJ → BIJ − ζωIJ , nothing changes in the derivation of the
transformation of the action and of the background fields under NATD. Therefore, the
transformation rules derived in section 3 and presented in section 2.1 are valid also for
DTD if we shift BIJ → BIJ − ζωIJ . The resulting DTD background is a deformation of
the NATD background, and it reduces to it when ζ = 0. We refer to [15, 16] for some
explicit examples of DTD models obtained from PCM or from the superstring on AdS5×S5.
4.2 Yang-Baxter deformations
We will now construct deformations of the original background, rather than its NATD.
We introduce a deformation parameter η such that η = 0 gives back the original sigma
model. These deformations will be obtained from the DTD construction, where we identify
η = ζ−1. We identify them with Yang-Baxter deformations, since they are generated by
solutions of the classical Yang-Baxter equation and they generalize the original construction
for PCM and (super)cosets to generic (Green-Schwarz) sigma models.
The construction is possible when ωIJ is invertible. Writing R = ω−1 : g∗ → g it is easy
to verify that the 2-cocycle condition for ω implies that R solves the classical Yang-Baxter
equation
[Rx, Ry] − R([Rx, y] + [x, Ry]) = 0 , ∀x, y ∈ g∗
, ⇐⇒ RL[I
R|M|J
f
K]
LM = 0 , (4.6)
21
This is very similar to how one realizes NATD as a special case of Poisson-Lie T-duality [48] and it
would be interesting to consider the extension of our construction to the Poisson-Lie case.
22
In the PCM case considered in [15] or the supercoset model case considered in [16] there is a natural
Lie algebra structure on g∗
, inherited from the full isometry group. This is the structure that was chosen
in [15, 16]. Nevertheless, as already mentioned this choice has no consequence in our construction, and a
more natural choice may be for example to take g∗
abelian.
– 13 –
JHEP08(2018)027
where the action of the operator is again defined by R(TI) = TKRKI. The above is
equivalent to the more familiar form of the classical Yang-Baxter equation
[r12, r13] + [r12, r23] + [r13, r23] = 0 , (4.7)
written in terms of r = RIJ TI ∧TJ ∈ g⊗g, where the subscripts of rij denote the spaces in
g⊗g⊗g where it acts. To recast the DTD model as a deformation of the original model we
need to replace the coordinates νI, which parametrize the dual space, by a group element
g ∈ G. The invertible map ω : g → g∗ allows us to do this by writing [15]
νI = ζtr TI
1 − Ad−1
g
log Adg
ω log g . (4.8)
Using the 2-cocycle condition it can be shown that this implies23
dνI = η−1
R−1
g (g−1
dg) I
, νKfK
IJ = η−1
R−1
IJ − η−1
(R−1
g )IJ , (4.9)
where Rg = Ad−1
g RAdg. Using this in the definition of NIJ in (2.4) we get
N = ηRg (1 + η(G − B)Rg)−1
= η (1 + ηRg(G − B))−1
Rg . (4.10)
With these substitution rules it is easy to check that the DTD action is recast into the
following form24
S =
T
2 Σ
(g−1
dg)I
∧ ( ˜GIJ ∗ − ˜BIJ )(g−1
dg)J
+ 2dzM
∧ ( ˜GMI ∗ − ˜BMI)(g−1
dg)J
+ (−1)N
dzM
∧ ( ˜GMN ∗ − ˜BMN )dzN
− η−1
(dgg−1
)I
∧ ωIJ (dgg−1
)J
, (4.11)
where we isolated the last term which does not behave well in the η → 0 limit. This term
is again a closed B-field thanks to the 2-cocycle condition satisfied by ω, and therefore it
does not contribute to the equations of motion. We define the action of the YB model as
the above one where the closed B = η−1(dgg−1)I ∧ ωIJ (dgg−1)J is removed. Dropping it
we do not modify the on-shell theory, so that if the original model is classically integrable
this property is inherited also by the YB deformation. In this way we can also achieve
a non-singular η → 0 limit, which yields the original undeformed model as is clear from
the expressions given below. This also implies that YB deformations may be viewed as
interpolations between the original model (obtained just by sending η → 0) and the dual
one (which is recovered in the equivalent DTD formulation after sending ζ → 0, which
is η → ∞).
23
The easiest way to show this is to extend ω to act as a derivation on the universal enveloping algebra
of g. With this definition we can write ην = g−1
ω(g) ∈ g∗
. We can now compute dν and the two equivalent
expressions ω(dg) = ω(gg−1
dg) = ηgνg−1
dg + gω(g−1
dg) and ω(dg) = ω(dgg−1
g) = ω(dgg−1
)g + ηdgν.
This gives us the two equations.
24
We still use tilde to denote transformed metric and B-field, but now they differ from the ones of NATD.
The transformations rules are given below.
– 14 –
JHEP08(2018)027
Setting fermions to zero and assuming a bosonic group G, we then read off
˜Gmn = Gmn − η (G − B) ˆNRg(G − B) (mn)
, (4.12)
˜GmI =
1
2
(G − B) ˆN mI
+
1
2
ˇN(G − B) Im
, ˜GIJ = (G − B) ˆN (IJ)
,
˜Bmn = Bmn + η (G − B) ˆNRg(G − B) [mn]
, (4.13)
˜BmI = −
1
2
(G − B) ˆN mI
+
1
2
ˇN(G − B) Im
, ˜BIJ = − (G − B) ˆN [IJ]
,
while the RR bispinor is again transformed by a Lorentz transformation ˆΛ acting on spinor
indices from the left25
˜S12
= ˆΛS12
, Λab
= ηab
− 2ηEI
a ˆNI
J (Rg)JK
EK
b
. (4.15)
In the above we have also defined
ˆNJ
I = δI
J + η(Rg)IK
(GKJ − BKJ )
−1
,
ˇNI
J
= δJ
I
+ η(GJK − BJK)(Rg)KI −1
= R−1
g
ˆNRg I
J
.
(4.16)
Using (4.9) in (3.17) and (3.18) we find26
Km
= 0, KI
= η[Rgn]I
, Xm = ∂m φ+
1
2
lndet ˆN −η ˜BmI[Rgn]I
, XI = −η ˜BIJ [Rgn]J
.
(4.17)
At this point we wish to comment on the possibility of having “trivial” solutions of the
generalized supergravity equations, namely ones that solve the more restricting standard
supergravity equations while K does not vanish. This is possible if [27]
0 = KI
( ˜G− ˜B)IJ = −η[nRg(G− B) ˆN]J = [n( ˆN − 1)]J ⇐⇒ KI
(G− B)IJ = 0 , (4.18)
i.e. the original G − B must be degenerate. Such trivial solutions are possible for YB
deformations since we do not need to assume that G − B is non-degenerate. They are, at
least naively, not possible for NATD since there they would imply that the dual ˜G − ˜B
is degenerate, which is not allowed by assumption, see section 3. This discrepancy has to
do with the fact that when going from DTD to YB we did not just change coordinates,
we also shifted B by dropping the extra closed term in (4.11). Explicit trivial solutions
were found in [50], and more recently in [43] by double field theory β-shifts starting from
AdS3 × S3 × T4 with non-zero B-field. It is clear from the present discussion that these
solutions can be equivalently generated from the construction of YB deformations provided
here. An example is provided in section 4.3.2.
25
For YB deformations Λ ∈ SO(1, 9) and it is therefore useful to parametrize it in terms of an antisymmetric
matrix Aab
as Λ = (1 + A)−1
(1 − A) which implies A = (1 − Λ)(1 + Λ)−1
, where we lowered one
index with ηab to obtain e.g. Λa
b. Then the Lorentz transformation on spinor indices Λa
b
Γb = ˆΛT
Γa
ˆΛ can
be written as a finite sum [49]
ˆΛ = [det(η+A)]−1/2
Æ −
1
2
AabΓab
, Æ
1
2
AabΓab
≡ 1+
n=5
n=1
1
n!2n
Aa1b1 ···Aanbn Γa1b1···anbn
. (4.14)
26
In the expression for X we have used the fact that d(ln det[ηRg]) = tr(R−1
g dRg) = 2fI
JI [g−1
dg]J
=
2(g−1
dg)I
nI .
– 15 –
JHEP08(2018)027
4.2.1 A convenient rewriting
As remarked in the introduction the deformed metric and B-field can be obtained from the
original G and B by the following generalization of the open/closed string map used by
Seiberg and Witten
˜G − ˜B = (G − B)[1 + ηRg(G − B)]−1
. (4.19)
This is readily seen after noticing that, since Rg has only IJ indices, the following operator
is of block form
1 + ηRg(G − B) =
δm
n 0
η[Rg(G − B)]I
n δI
J + η[Rg(G − B)]I
J
, (4.20)
and it is straightforward to invert it giving
[1 + ηRg(G − B)]−1
=
δm
n 0
−η[ ˆNRg(G − B)]I
n
ˆNI
J
, (4.21)
where we used ˆNJ
I = [δI
J + η(Rg)IK(GKJ − BKJ )]−1. It is easy to check that (4.19)
indeed reproduces the formulas (4.12)–(4.13) for the transformed metric and B-field.
So far we have worked with explicit group elements and algebra indices. It is sometimes
convenient to translate the results so that the information on the initial isometries of the
model is encoded in a set of Killing vectors. Thanks to this rewriting the YB deformation
may be applied without the need of introducing an explicit parametrization of the group
G. Isometries of the metric and B-field are translated into equations for a family of Killing
vectors kµ
I , where I = 1, . . . , dim(G) is the index to enumerate them. In particular, the
metric possesses an isometry when shifting infinitesimally the coordinates Xµ → Xµ +
ǫIkµ
I + O(ǫ2), if kµ
I satisfy the Killing vector equation
∇µkI ν + ∇νkI µ = 0 . (4.22)
In order to make a connection with the formulation in terms of the group element g,
it is enough to notice that its variation δg under an infinitesimal transformation can be
understood in two ways, either as δxi∂ig, or as ǫITIg, the latter being the infinitesimal
version of the global transformation g → exp(ǫITI)g. We recall that indices i, j are used
to label coordinates xi on the group G. This leads to the identification
kJ
I ≡ kµ
I ℓJ
µ = tr(TJ
Ad−1
g TI) = (Ad−1
g )J
I , where g−1
dg = ℓI
TI . (4.23)
Obviously, ℓI
µ and kµ
I are non-zero only for µ = i. The structure constants of the Lie
algebra may be recovered by computing
LkI
kJµ − LkJ
kIµ = −fK
IJ kKµ , (4.24)
where L is the Lie derivative. Now let us notice that we can rewrite
ΘIJ
≡ (Rg)IJ
= kI
KRKL
kJ
L , (4.25)
– 16 –
JHEP08(2018)027
and that, before fixing any local symmetry (if present), the matrix ℓI
i is invertible. Let us
denote the inverse by ℓi
I so that ℓI
i ℓi
J = δI
J . This allows us to convert all algebra indices I, J
in (4.19) into curved indices i, j. Therefore the YB deformation of the metric and B-field
may also be written as
˜G − ˜B = (G − B)[1 + ηΘ(G − B)]−1
. (4.26)
This formula is then equally valid both when we use indices {m, I} or {m, i}. When a
local symmetry is present we arrive at the same result since the local invariance can be left
unfixed until the end. With a similar reasoning we may rewrite also the transformation
rule for the dilaton when nI = fJ
IJ = 0. In fact, when computing the determinant of ˆNI
J
we may as well extend it to all µ, ν indices. Since the (inverse of the) operator is in the
block-form (4.20), it is clear that det( ˆNµ
ν) = det( ˆNI
J ). This also means that we can
obtain the deformed dilaton simply by calculating
˜φ = φ −
1
2
ln det[1 + ηΘ(G − B)] . (4.27)
More generally, when nI = 0 we may write
Kµ
= ηΘµν
nν , Xµ = ∂µ
˜φ − η ˜BµνΘνρ
nρ . (4.28)
4.3 Two examples of YB deformations
We wish to work out two examples of YB deformations that do not fall under the (super)coset
construction. In addition to the intrinsic interest of the following (deformed)
backgrounds, the calculations also illustrate the applicability of our method.
4.3.1 YB deformation of the D3-brane background
Our first motivation is to understand a YB deformation of AdS5 × S5 generated by an
R-matrix that cannot be interpreted as a sequence of TsT transformations. In particular,
we want to use the formula (4.26) to “uplift” the YB deformation from the AdS5 × S5
background to the full D3-brane background, before taking the near-horizon limit. This
is in the spirit of [51, 52], where the uplift to the brane background was done for YB
deformations that are (sequences of) TsT transformations. For the sake of the discussion
we focus on the NS-NS sector, where the dilaton is constant (we set it to zero for simplicity),
B = 0 and the metric is
ds2
= H−1/2
dxidxi
+ H1/2
(dr2
+ r2
ds2
S5 ) , H = 1 +
(α′)2L4
r4
, (4.29)
where i = 0, . . . , 3 and ηij = diag(−1, 1, 1, 1). The above metric has an ISO(1, 3) Poincar´e
isometry acting on the xi coordinates, and an SO(6) isometry acting on the five-dimensional
sphere S5. We will now deform the background by exploiting the Poincar´e part of the
isometries. The Killing vectors in this case may be written as
Translations: kµ
[pi] = δµ
i , Lorentz: kµ
[Jij] = −δµ
i xj + δµ
j xi , i, j = 0, . . . , 3. (4.30)
– 17 –
JHEP08(2018)027
We wish to “uplift” the YB deformation of AdS5 × S5 worked out in section 6.4 of [47],
where the R-matrix was chosen to be
R = p1 ∧ p3 + (p0 + p1) ∧ (J03 + J13) . (4.31)
That is possible since this R-matrix is constructed out of generators that are isometries also
of the D3-brane background before the near-horizon limit. Following (4.25) we therefore
construct
Θµν
= 2 kµ
[p1]kν
[p3] + (kµ
[p0] + kµ
[p1])(kν
[J03] + kν
[J13]) − µ ↔ ν . (4.32)
More explicitly, in the block with µ, ν = 0, . . . , 3 it is
Θµν
= 2





0 0 0 −x−
0 0 0 −x− + 1
0 0 0 0
x− x− − 1 0 0





, (4.33)
where we introduced the standard light-cone coordinates x± = x0 ± x1. Now, using (4.26)
and (4.27) we obtain the following deformed metric, B-field and dilaton
d˜s2
= −
ˆη2ξ2
−H−1/2dξ2
−
4 (H − 4ˆη2ξ−)
−
H−1/2 H − 2ˆη2ξ− dξ−dx+
2 (H − 4ˆη2ξ−)
−
ˆη2H−1/2(dx+)2
(H − 4ˆη2ξ−)
+ H−1/2
dx2
2 +
H1/2dx2
3
H − 4ˆη2ξ−
+ H1/2
(dr2
+ r2
ds2
S5 ) ,
˜B =
ˆη
2
dx3 ∧ (2dx+ + ξ−dξ−)
H − 4ˆη2ξ−
, exp (−2˜φ) = 1 −
4ˆη2ξ−
H
.
(4.34)
We chose ˆη as deformation parameter and to simplify expressions we redefined ξ− = 2x−−1.
We now want to check that the near-horizon geometry of this YB deformation of the D3brane
background indeed yields the YB deformation of AdS5 × S5 of [47]. In the nearhorizon
limit one sends r → 0 and α′ → 0 while keeping the ratio r/α′ fixed. We achieve
this by rewriting r = α′L2/z and ˆη = ηL−2/α′, and then sending α′ → 0. We obtain
lim
α′→0
ds2
α′L2
= z−6
1−
4η2ξ−
z4
−1
z4
dx3
2
−η2
(dx+
)2
−
1
4
dξ− η2
ξ2
−dξ−+2dx+
z4
−2η2
ξ−
+
dx2
2+dz2
z2
+ds2
S5
lim
α′→0
B
α′L2
=
η
2
dx3∧(2dx++ξ−dξ−)
z4−4η2ξ−
, lim
α′→0
e−2φ
= 1−
4η2ξ−
z4
,
(4.35)
which indeed reproduces27 (the NS-NS sector of) the deformation of AdS5×S5 appearing in
section 6.4 of [47]. Uplifting the YB deformation to the D3-brane background is particularly
interesting since it also allows us to go far from the brane and understand how the flat space
27
In this paper we have a different convention for the sign of the B-field.
– 18 –
JHEP08(2018)027
in which it is embedded has been deformed. In the limit r → ∞ we have simply H → 1
ds2
= −
ˆη2ξ2
−dξ2
−
4 (1 − 4ˆη2ξ−)
−
1 − 2ˆη2ξ− dξ−dx+
2 (1 − 4ˆη2ξ−)
−
ˆη2(dx+)2
(1 − 4ˆη2ξ−)
+ dx2
2 +
dx2
3
1 − 4ˆη2ξ−
+ ds2
R6
B =
ˆη
2
dx3 ∧ (2dx+ + ξ−dξ−)
1 − 4ˆη2ξ−
, e−2φ
= 1 − 4ˆη2
ξ− .
(4.36)
Obviously, the above background may be also obtained directly as a YB deformation of flat
space with Θ given by (4.32). In the AdS/CFT correspondence one looks at open strings
stretching between D3-branes in flat space, whose low-energy limit produces N = 4 super
Yang-Mills. In the presence of a B-field as in the case considered here, open strings feel an
effective metric gµν and a non-commutativity parameter θµν that are related to the metric
and B-field Gµν, Bµν of the closed string by28 [37]
gµν
+
θµν
2πα′
= (Gµν − Bµν)−1
, (4.37)
where gµν is obviously obtained by taking the symmetric part of the right-hand-side, while
θµν the antisymmetric part. In general, if we apply the open/closed string map to a background
obtained by a YB deformation we get
g−1
+
θ
2πα′
= ( ˜G − ˜B)−1
= [(G − B)−1
+ ηΘ] ,
=⇒ g−1
= (G − B)−1
s , θ = 2πα′
[(G − B)−1
a + ηΘ] ,
(4.38)
where we directly relate the open-string quantities to the metric and B-field G, B of the
original model before the YB deformation, and subscripts s and a indicate the symmetric
and antisymmetric parts. In our specific example, before deforming, the brane system is
in a flat spacetime with vanishing B-field, meaning that the effective open-string metric
will coincide with the flat one, and the non-commutativity parameter will be essentially
defined by the YB R-matrix
gµν = Gµν , θµν
= 2πα′
ˆη Θµν
. (4.39)
This discussion is obviously generic and is not confined to the current example. Apart from
uncovering the non-commutativity structure, at this point one should also take the lowenergy
limit of open strings in the non-commutative spacetime. Here we are considering
a case with an electric B-field, and these instances are known to produce problems when
trying to take the low-energy limit [53]. It is therefore not clear whether the low-energy
limit yields a non-commutative gauge theory with θ as non-commutativity parameter. The
relation between gravity duals of non-commutative gauge theories and YB deformations
was first pointed out in [54].
28
As it is written, this open/closed string map assumes the invertibility of (G − B). The generalization
(of the inverse transformation) to the case of degenerate (G − B) is in fact given by our (4.26).
– 19 –
JHEP08(2018)027
Certain YB deformations of AdS5 × S5 are constructed out of generators that are not
isometries of the brane background and that become isometries only after taking the nearhorizon
limit. For these examples it is not clear how to uplift the YB deformation to the
brane background. It would be interesting to see if YB deformations can be extended also
to cases without isometries by using Poisson-Lie T-duality.
4.3.2 YB deformation of AdS3 × S3 × T 4 with H-flux
We now want to apply the YB deformation to a background with degenerate G−B, and we
will compare our results to those of [43]. There it was indeed shown that YB deformations
of AdS5 × S5 are equivalent to local β-transformations of the double theory, and it was
proposed that local β-shifts should be the natural way to generalize YB deformations to
generic backgrounds, including cases with degenerate G − B. The example we consider is
that of AdS3 × S3 × T4 with non-vanishing H-flux
ds2
=
dxidxi + dz2
z2
+ ds2
S3 + ds2
T4 , ds2
S3 =
1
4
dθ2
+ sin2
θdϕ2
+ (dψ + cos θdϕ)2
B =
dx0 ∧ dx1
z2
+
1
4
cos θdϕ ∧ dψ .
(4.40)
G − B is degenerate because of the rows (or columns) i = 0, 1. The dilaton is constant
and for simplicity we set it to zero. To generate a YB deformation we will make use of the
Killing vectors of the Poincar´e isometry
Translations: kµ
[pi] = δµ
i , Lorentz: kµ
[Jij] = −δµ
i xj + δµ
j xi , i, j = 0, 1 . (4.41)
In order to compare to the results of section 4.2.2 of [43] we take R = cipi ∧ J01 or
Θµν
= (ci
kµ
[pi])kν
[J01] − µ ↔ ν , (4.42)
where we sum over i = 0, 1. The classical YB equation is satisfied only when the parameters
satisfy c0 = ±c1. Now using (4.26) and (4.27) we obtain the YB deformed background
ds2
=
dxidxi
z2 − 2ηcjxj
+
dz2
z2
+ ds2
S3 + ds2
T4 ,
B =
dx0 ∧ dx1
z2 − 2ηcjxj
+
1
4
cos θdϕ ∧ dψ , e−2φ
= 1 −
2ηcixi
z2
,
(4.43)
which agrees with the background obtained in section 4.2.2 of [43]. This confirms in
a specific example the expected equivalence of YB deformations and local β-shifts even
beyond the standard (H = 0) supercoset case. As already noticed in [43] the above
background is actually a trivial solution since the vector K decouples from the generalized
supergravity equations.
– 20 –
JHEP08(2018)027
5 Conclusions
We have derived the transformation rules for the supergravity fields under NATD by carrying
out the dualization in the general case for the Green-Schwarz string. This generalizes
the derivation performed for the case of the supercoset in [16]. If the dualized group G
is not unimodular there is in general an anomaly, which is reflected in the fact that the
resulting background solves the generalized supergravity equations of [23, 24] rather than
the standard ones. We have also discussed a generalization where one adds a closed B-field
to the action prior to performing the duality transformation. This leads to so-called DTD
models and, in special cases, a generalization of Yang-Baxter models [28, 29]. We have also
seen that this gives us an interesting way to find examples that avoid the anomaly from
non-unimodularity of G along the lines discussed in [27].
Non-abelian T-duality can be embedded in the even more general framework of PoissonLie
T-duality [48]. Also this case can be formulated at the path integral level and an
anomaly arises in a similar way [55] (see also [56]). It would be interesting to extend our
analysis to this case which would also make further contact with [42]. It would also allow
us to extend DTD and YB deformations to cases without isometries, and perhaps help
to uplift all YB deformations of AdS5 × S5 to deformations of the brane background. It
would also be interesting to consider the case of open strings along the lines of the recent
paper [57].
We have found that a natural way to rephrase YB deformations is in terms of a
generalization of (the inverse of) the open/closed string map of Seiberg and Witten, thus
extending what was observed in the case of both homogeneous and inhomogeneous YB
deformations of PCM or (super)cosets. Since the inhomogeneous case cannot be formulated
in terms of our construction we have only considered the homogeneous one here, but it
would be interesting to see what happens if we take R in (4.19) to solve the modified classical
YB equation on the Lie algebra of G. The lessons learned from the supercoset case [24, 47,
58, 59] suggest that the resulting sigma model will possibly be kappa-symmetric, but that
the background fields will probably only solve the equations of generalized supergravity
rather than the standard ones.
When applied to classically integrable sigma models, the deformations studied here
preserve the integrability. It would be interesting to extend the integrability methods
developed in the context of the AdS/CFT correspondence [60, 61] also beyond the “abelian”
YB deformations considered so far, namely the “diagonal abelian” deformations (considered
e.g. in [62] and with an exact spectrum encoded in the equations of [63]), and the “offdiagonal
abelian” deformations (addressed e.g. at one loop in [64]).
Acknowledgments
We thank B. Hoare and S. van Tongeren for interesting and valuable discussions. RB also
thanks the Department of Theoretical Physics and Astrophysics of Masaryk University for
hospitality during part of this work. The work of R.B. was supported by the ERC advanced
grant No 341222.
– 21 –
JHEP08(2018)027
A Conventions
Let us summarize our index conventions in the following table
µ, ν, . . . : labels of all bosonic coordinates
I, J, . . . : indices of g (the Lie algebra of G) and of the dual g∗
i, j, . . . : labels of coordinates parameterizing the group G
M, N, . . . : labels of spectator coordinates, of which
m, n, . . . : labels of bosonic spectator coordinates
α, β, . . . : labels of fermionic spectator coordinates
A, B, . . . : indices of tangent space, of which
a, b, . . . : indices of bosonic tangent space
α, β, . . . : indices of fermionic tangent space
(A.1)
When working with (super)forms we define the components as An = 1
n!dzMn ∧ dzMn−1
. . . ∧ dzM1 AM1M2···Mn and we take the exterior derivative to act from the right, so that
d(An ∧Am) = An ∧dAm +(−1)mdAn ∧Am. The (graded) anti-symmetrization of n indices
is denoted by [· · · ] and it includes a factor 1/n!.
B An example with local symmetry
To make the discussion in section 3.1 more concrete we will here apply the rules of NATD
to an explicit example with local symmetry (a case also referred to “with isotropy”). We
will follow the discussion in section 3.1 and show that we reproduce an example worked
out in section 4.1 of [20]. The starting point is the AdS3 × S3 × T4 background with pure
RR flux, and the goal is to apply NATD on the SO(4) global isometry of S3, which has
obviously also a local SO(3) symmetry. The metric and the flux are given by
ds2
= ds2
AdS3
+ ds2
S3 + ds2
T4 , F3 = 2 vol(AdS3) + vol(S3
) . (B.1)
We describe S3 in terms of the coset SO(4)/SO(3), where the generators of so(4) satisfy
[Jab, Jcd] = δbcJad−δacJbd−δbdJac+δadJbc and admit the matrix realisation Jab = Eab−Eba,
in terms of the matrices (Eab)cd = δacδbd. Following [20] we enumerate the generators of
the coset part as TI = J1,I+1 where I = 1, 2, 3, and the generators of the subalgebra so(3)
as T4 = J23, T5 = J24, T6 = J34. The metric of the original S3 comes from the piece of
the action T
2 AI ∧ GIJ ∗ AJ , where A = g−1dg, g ∈ SO(4) and GIJ = diag(1, 1, 1, 0, 0, 0)
projects on the coset part of the algebra. We do not need to look at AdS3 and T4, since the
off-diagonal blocks GmI are 0 and therefore the AdS3 and T4 spaces are not affected by the
NATD transformations, see (2.2). It is easy to construct GIJ − νKfK
IJ that in this case is29









1 ν4 ν5 −ν2 −ν3 0
−ν4 1 ν6 ν1 0 −ν3
−ν5 −ν6 1 0 ν1 ν2
ν2 −ν1 0 0 ν6 −ν5
ν3 0 −ν1 −ν6 0 ν4
0 ν3 −ν2 ν5 −ν4 0









, (B.2)
29
This is the transpose of M of [20].
– 22 –
JHEP08(2018)027
and invert it to obtain NIJ . Notice that GIJ is not invertible, but we can invert GIJ −
νKfK
IJ . For special values of the coordinates νK also GIJ −νKfK
IJ becomes degenerate. After
fixing the gauge, some of these degeneracies will produce singularities in target space.
Taking the symmetric and antisymmetric parts of NIJ we can compute the deformed metric
and B-field. In the action the contributions are respectively T
2 dνIN(IJ) ∗ dνJ and
−T
2 dνIN[IJ]dνJ . These are still written in terms of all six dual coordinates νK, meaning
that we should fix the gauge. We fix it as in [20] setting ν1 = ν2 = ν6 = 0, and we also
rename ν3 = x1, ν4 = x2, ν5 = x3. In agreement with [20] we find that the B-field vanishes
and that the metric of the dualised sphere and the dilaton are
ds2
˜S3 =
dx2
2 x2
1 − x2
2
2
+ x2
2x2
3 + x2
2
x2
1x2
3
+
x2
2 + x2
3 + 1 dx2
3
x2
1
+
2x2dx2dx3 −x2
1 + x2
2 + x2
3 + 1
x2
1x3
+
2dx1
x1
(x2dx2 + x3dx3) + dx2
1,
e−2φ
= x2
1x2
3.
(B.3)
In order to compute the transformation of the RR fields we first need to compute the
Lorentz transformation Λ. Suppose we use labels in tangent space a = 0, . . . , 9 so that
a = 3, 4, 5 are the labels for the tangent space of the sphere. Then we can take EI
a to
be E1
3 = E2
4 = E3
5 = 1, and 0 otherwise. Calculating Λab = ηab − 2EI
aNIJ EJ
b in the
above gauge for νI we easily find (for the block with a, b = 3, 4, 5) Λ = diag(1, −1, −1).
As expected the Lorentz transformation is an element of SO(1, 9), since we have dualized
an even-dimensional group. In this case it is a simple reflection along a = 4 and a = 5.
Therefore on spinor indices it is realised just as the product of the two corresponding tendimensional
gamma matrices. The transformed RR fluxes obtained from ˜S = ˆΛS then
agree with the ones of [20].
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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JHEP10(2017)024
Published for SISSA by Springer
Received: August 23, 2017
Accepted: September 25, 2017
Published: October 4, 2017
On non-abelian T-duality and deformations of
supercoset string sigma-models
Riccardo Borsatoa and Linus Wulffb
a
Nordita, Stockholm University and KTH Royal Institute of Technology,
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
b
Department of Theoretical Physics and Astrophysics, Masaryk University,
611 37 Brno, Czech Republic
E-mail: riccardo.borsato@su.se, linus.wulff@gmail.com
Abstract: We elaborate on the class of deformed T-dual (DTD) models obtained by first
adding a topological term to the action of a supercoset sigma model and then performing
(non-abelian) T-duality on a subalgebra ˜g of the superisometry algebra. These models
inherit the classical integrability of the parent one, and they include as special cases the socalled
homogeneous Yang-Baxter sigma models as well as their non-abelian T-duals. Many
properties of DTD models have simple algebraic interpretations. For example we show that
their (non-abelian) T-duals — including certain deformations — are again in the same class,
where ˜g gets enlarged or shrinks by adding or removing generators corresponding to the
dualised isometries. Moreover, we show that Weyl invariance of these models is equivalent
to ˜g being unimodular; when this property is not satisfied one can always remove one
generator to obtain a unimodular ˜g, which is equivalent to (formal) T-duality. We also
work out the target space superfields and, as a by-product, we prove the conjectured
transformation law for Ramond-Ramond (RR) fields under bosonic non-abelian T-duality
of supercosets, generalising it to cases involving also fermionic T-dualities.
Keywords: Integrable Field Theories, String Duality
ArXiv ePrint: 1706.10169
Open Access, c The Authors.
Article funded by SCOAP3
.
https://doi.org/10.1007/JHEP10(2017)024
JHEP10(2017)024
Contents
1 Introduction 1
2 The deformed T-dual models 4
2.1 Relation to Yang-Baxter sigma models 6
3 Global symmetries 7
3.1 DTD of DTD models 8
3.2 DTD models not related to YB models by NATD 12
4 Kappa symmetry and Green-Schwarz form 13
5 Target space superfields 15
5.1 Supergravity condition and dilaton 18
6 Some explicit examples 19
6.1 A TsT example 20
6.2 A new example 22
7 Conclusions 22
A Useful identities 23
B Derivation of the action 24
C Classical integrability 25
1 Introduction
In this paper we investigate further the deformed T-dual (DTD) supercoset sigma models
introduced in [1], and we find results that are of interest also when considering the
undeformed case, i.e. when applying just non-abelian T-duality (NATD).
The construction of DTD models is equivalent to applying NATD on a centrally extended
subalgebra as first suggested in [2].1 The models are constructed by picking a
subalgebra of the (super)isometry algebra ˜g ⊂ g — the canonical example is the AdS5 ×S5
superstring where g = psu(2, 2|4) — and a 2-cocycle, i.e. an anti-symmetric linear map
ω : ˜g ⊗ ˜g → R satisfying
ω(X, [Y, Z]) + ω(Z, [X, Y ]) + ω(Y, [Z, X]) = 0 , ∀X, Y, Z ∈ ˜g . (1.1)
1
The first hint of the relation of YB models to NATD appeared in [3] for the case of Jordanian defor-
mations.
– 1 –
JHEP10(2017)024
Together with an element of the corresponding group ˜g ∈ ˜G, the 2-cocycle defines a 2-form
B = ω(˜g−1d˜g, ˜g−1d˜g) which is closed, i.e. dB = 0, thanks to the 2-cocycle condition. The
idea behind the construction is to add this topological term to the supercoset sigma model
Lagrangian and then perform NATD on ˜G. If ζB is added to the Lagrangian, with ζ
a parameter, the resulting model can be thought of as a deformation of the non-abelian
T-dual of the original model with deformation parameter ζ. The classical integrability of
the original sigma model is preserved by the deformation, since both adding a topological
term and performing NATD preserve integrability. We refer to [1] for more details on how
this procedure relates to the construction of [2]. Let us remark that DTD models may
be constructed starting from a generic σ-model, for example the principal chiral model as
in [1], and the starting model does not have to be (classically) integrable. In this paper we
will only consider the supercoset case.
It was proven in [1] that the so-called Yang-Baxter (YB) sigma models [4–7], defined
by an R-matrix solving the classical Yang-Baxter equation (CYBE), are equivalent to DTD
models with invertible ω. This relation was first conjectured and checked for many examples
— in the language of T-duality on a centrally extended subalgebra — in [2]. See also [8] for
a more detailed discussion of some of the examples. In [1] we used the fact that when ω is
invertible its inverse R = ω−1 solves the CYBE, and therefore defines a corresponding YB
model; by means of a field redefinition and relating the deformation parameters as η = ζ−1
we could prove the equivalence of the two sigma model actions [1].
Note that simply by setting the deformation parameter to zero, DTD models include
all non-abelian and abelian T-duals of the original supercoset model, including fermionic
T-dualities. Therefore all the statements we prove for DTD models apply also to (nonabelian)
T-duals of supercoset models. They are also easily seen to describe all so-called
TsT-transformations of the underlying supercoset model. In fact we will argue here that the
class of DTD models is closed under the action of NATD, as well as certain deformations,
meaning that applying these operations yields a new DTD model. They therefore represent
a very broad class of integrable string sigma models.
It was shown in [1] that these models are invariant under kappa symmetry, which
is needed to interpret them as Green-Schwarz superstrings. From the results of [9] it
follows that their target spaces must solve the generalised supergravity equations of [9, 10]
that ensure the one-loop scale invariance of the string sigma model. To have a fully
consistent superstring, however, we must require the stronger condition of Weyl invariance,
which implies that the target space should be a solution of the more stringent standard
supergravity equations. Here we show that Weyl invariance of the DTD model is equivalent
to the Lie algebra ˜g being unimodular, i.e. its structure constants should satisfy fj
ij = 0. In
fact, this condition is precisely the one found in [11, 12] when analysing the Weyl invariance
of bosonic sigma models under NATD by path integral considerations. The presence of ω
and the deformation does not modify the supergravity condition. When ω is invertible the
condition is also equivalent to unimodularity of the R-matrix R = ω−1, as defined in [13],
which was shown there to be the condition for Weyl invariance of YB models. The fact
that these conditions are the same was in fact an important hint that the latter should
have an interpretation involving NATD [2].
– 2 –
JHEP10(2017)024
Here we give the detailed proof of kappa symmetry for DTD models and extract the
target space superfields from components of the torsion as was done for η (i.e. YB) and
λ models in [13]. In particular, the RR fields and dilaton are difficult to extract by other
means but we find that they are given by the simple expressions
e−2φ
= sdet O , Sα1β2
= −8i[Adh(1 + 4Ad−1
f O−T
Adf )]α1
γ1Kγ1β2
, (1.2)
with O defined in (2.4) and S defined in (5.2) — for definitions of the remaining quantities
see sections 2 and 5. A by-product of these expressions is a formula for the transformation
of RR fields under NATD for the case of supercosets. As we show in section 5 it agrees,
for bosonic T-dualities, with the formula conjectured in [14], see also [15], but our formula
is valid also when doing fermionic T-dualities.
An advantage of the formulation of DTD models is that many statements about the
sigma model boil down to simple algebraic statements about the Lie algebra ˜g. One
example is the Weyl invariance condition already mentioned, while another concerns their
transformation under NATD — possibly including additional deformation. The advantages
are clear also when discussing the isometries of these models. We show that they fall into
two classes; in fact, besides the standard ones, i.e. the unbroken part of the G isometries,
there are also certain (abelian) shift isometries. We prove that T-dualising on either type of
isometry we get back a DTD model; in particular, T-dualising on the first type of isometries
is equivalent to the simple operation of enlarging ˜g by the corresponding generators, while
T-dualising on the shift isometries removes generators from ˜g. The latter operation can
be used to prove, in this context, that solutions of the generalised supergravity equations
are (formally) T-dual to solutions of the standard supergravity equations [10]. For more
general NATD, where one applies T-duality on both types of isometries at the same time, we
propose that the resulting model is still obtained in a similar way, namely simply by adding
to ˜g the isometry generators that lie outside of it and removing from it the generators that
are inside. We show that this conjecture is indeed consistent, i.e. the resulting model is a
well-defined DTD model, which turns out to be quite non-trivial. As already mentioned
this suggests that the class of DTD models is closed under (bosonic and fermionic) NATD,
including also the deformations considered here.
It was suggested in [1] that it might be possible to think of all DTD models as nonabelian
T-duals of YB models. Here we show that this is in fact not true by providing an
example of a DTD model which cannot be obtained from a YB model by NATD.
The outline of the paper is as follows. In section 2 we introduce the DTD models based
on supercosets, discuss their gauge invariances and the equivalence to YB models when ω
is invertible. Section 3 describes the two classes of global symmetries, or isometries, of
these models. We also address the question of what happens if one performs NATD and
deformation of a DTD model and argue that this gives a new DTD model, proving this in
simpler cases. Models which cannot be obtained by NATD of YB models are also discussed.
In section 4 we demonstrate the kappa symmetry of DTD models and write the DTD model
as a Green-Schwarz superstring. Given these results it is then straightforward to derive the
target space fields of the DTD model from components of the superspace torsion, which we
– 3 –
JHEP10(2017)024
do in section 5. This includes a derivation of the Weyl-invariance condition for these models.
In section 6 we work out the supergravity background for two examples of DTD models.
The first is equivalent to a well known TsT-background but is useful to demonstrate the
procedure. The second example is one of the new examples which cannot be obtained
from a YB model by NATD. We finish with some conclusions and open problems. Three
appendices contain some useful algebraic identities, a derivation of the DTD model action
and a proof of integrability.
2 The deformed T-dual models
As described in the introduction the deformed T-dual (DTD) models are constructed as
follows. We start with a supercoset sigma model, e.g. the AdS5 × S5 superstring [16] or
one of the other examples in [17, 18]. We single out a subalgebra ˜g ⊂ g of the (Z4-graded)
superisometry algebra and write the group element as g = ˜gf with ˜g ∈ ˜G and f ∈ G. This
parametrization is of course redundant and introduces a corresponding ˜G gauge symmetry
˜g → ˜g˜h−1 and f → ˜hf on which we will comment below. The second ingredient, which is
responsible for the deformation, is a Lie algebra 2-cocycle ω on ˜g satisfying (1.1). We add
to the original supercoset sigma model action the term
Sω =
T
4 Σ
ζω(˜g−1
d˜g, ˜g−1
d˜g) , (2.1)
where ζ is a parameter introduced to keep track of the deformation — if there exist many 2cocycles
we could introduce a parameter for each.2 As explained already, this is equivalent
to adding a B-field to the action, which is closed by virtue of the 2-cocycle condition.
This term is therefore topological and has no effect on local properties of the theory —
issues with boundary conditions are more subtle and will not be considered here. The final
step is to perform NATD on ˜g. This is done in the usual way by gauging the global ˜g
symmetry and integrating out the gauge field. This procedure guarantees that properties
like integrability are preserved, see appendix C for an explicit proof. However, since Tduality
is a non-local transformation of the fields of the sigma model, ω will now affect
local properties of the deformed model.
If ω is a coboundary, meaning that ω(X, Y ) = f([X, Y ]) for some function f : ˜g → R,
the B-field is exact; this is equivalent to no deformation at all since B is pure gauge —
alternatively a field redefinition can remove the ζ dependent contributions in the deformed
model. Therefore non-trivial deformations are classified by the second (Lie algebra) cohomology
group H2(˜g). The same group also classifies non-trivial central extensions of ˜g,
consistent with the interpretation of these models as arising from NATD on a centrally
extended subalgebra of the isometry algebra [2].
Performing the above procedure one obtains the DTD supercoset model action
S = −
T
2
d2
σ
γij − ij
2
Str Ji
ˆdf Jj + (∂iν − ˆdT
f Ji)O−1
(∂jν + ˆdf Jj) , γij
=
√
−hhij
,
(2.2)
2
If ω has mixed Grassmann even-odd components the corresponding deformation parameter ζ would be
fermionic. Since the interpretation of such a fermionic deformation is not so clear we will generally assume
that ω has only even-even and odd-odd components and that ζ is real.
– 4 –
JHEP10(2017)024
and we refer to appendix B for the details of its derivation. Here J = dff−1 encodes the
degrees of freedom in f, while ν ∈ ˜g∗ denotes the dualised degrees of freedom coming from
˜g. We have further defined
ˆdf = Adf
ˆdAd−1
f , ˆd = P(1)
+ 2P(2)
− P(3)
, ˆdT
= −P(1)
+ 2P(2)
+ P(3)
, (2.3)
where P(i) project onto the corresponding Z4-graded component of g = 3
i=0 g(i) and O−1
is the inverse3 of the linear operator O : ˜g → ˜g∗
O = ˜PT
( ˆdf − adν − ζω) ˜P . (2.4)
Given a basis {Ti} of ˜g and using the fact that g has a non-degenerate metric given by the
supertrace, we define the Lie algebra ˜g∗ ⊂ g dual to ˜g by taking as dual basis {Ti}, where
Str(TjTi) = δj
i . Then we have ˜P and ˜PT which are projectors onto ˜g and ˜g∗ respectively.
At the same time we are thinking of the 2-cocycle ω as a map ω : ˜g → ˜g∗ so that the
cocycle condition takes the form
ω[x, y] = ˜PT
([ωx, y] + [x, ωy]) , ∀x, y ∈ ˜g . (2.5)
Therefore, modulo the projector on the right-hand-side, ω acts as a derivation with respect
to the Lie bracket, similarly to adν which is a derivation thanks to the Jacobi identity.
In general one needs to make sure that the inverse O−1 exists in order to be able to
define the model, and this puts some restrictions on the subalgebra ˜g. By expanding in the
parameter ζ we can think of the DTD model as a deformation of the non-abelian T-dual
of the original model, since taking ζ = 0 reduces to ordinary NATD. Therefore, at least
for a small deformation parameter the invertibility is guaranteed if one can apply NATD
with respect to ˜g. There may also be cases in which NATD cannot be implemented but the
operator is invertible for finite values of ζ, i.e. the cocycle removes the 0-eigenvalues of O.
We now want to turn to the discussion of the gauge invariances of the action (2.2) of
DTD models. Besides the fermionic kappa symmetry, which will be discussed separately
in section 4, the action has two types of gauge invariances:
1. Local Lorentz invariance:
f → fh , h ∈ H = G(0)
. (2.6)
2. Local ˜G invariance:
f → ˜hf , ν → ˜PT
Ad˜hν + ζ
1 − eadx
adx
ωx , ˜h = ex
∈ ˜G ⊂ G . (2.7)
The former is obvious and, as in the case of supercosets, it boils down to the fact that
P(0) is missing in ˆd. As mentioned at the beginning of this section, the latter comes about
from the decomposition of the original group element as g = ˜gf where multiplication of ˜g
from the right by an element of ˜G can be compensated for by multiplying f on the left by
3
Notice that OO−1
= ˜PT
and O−1
O = ˜P rather than 1.
– 5 –
JHEP10(2017)024
the inverse group element. To verify that the action is indeed invariant under the second
type of symmetry we use the identities (A.7) and (A.8) that say how the transformations
of O and dν can be rewritten. Then the difference of the actions after and before the
transformation (2.7) is proportional to
d2
σ ij
Str 2∂iν˜h−1
∂j
˜h + ˜h−1
∂i
˜h(adν + ζω)(˜h−1
∂j
˜h) . (2.8)
The terms involving ν combine to a total derivative, and the one with ω is closed as already
remarked, meaning that it is also a total derivative at least locally. This establishes the
invariance of the action under the local transformation (2.7). This gauge invariance is
obviously present also in the case of NATD, where the shift of ν is absent since ζ = 0.
The classical integrability of DTD models may be argued by the fact that they are
obtained by adding a closed B-field and then applying NATD to the action of a supercoset,
since neither of these operations breaks classical integrability, see e.g. [19] for the argument
in the case of NATD. In appendix C we give a direct proof of the classical integrability of
these models by showing that, similarly to what was shown in the case of DTD of PCM
in [1], the on-shell equations can be recast into the flatness condition
ij
(∂iLj + LiLj) = 0 , (2.9)
for the Lax connection
Li = A
(0)
i + zA
(1)
i +
1
2
z2
+ z−2
A
(2)
i +
1
2
γij
jk
z−2
− z2
A
(2)
i + z−1
A
(3)
i , (2.10)
where z is the spectral parameter, Ai = Ai
+ +Ai
− and Ai
± ≡ Ad−1
f ( ˜Ai
± +Ji
±), with ˜Ai
± given
in (B.5). See appendix B for our notation. Notice that the presence of the Lax connection
still implies that we have conserved charges corresponding to the full original g symmetry.
However, in contrast to the case of supercosets, for DTD models one cannot argue any
more that they are all local, see appendix C.
2.1 Relation to Yang-Baxter sigma models
Given a DTD model with a cocycle ω which is non-degenerate on ˜g, we can show that the
action can be recast into the one of a YB model via a field redefinition. This result was
first presented in [1] and we collect here more details of the proof.
Given a non-degenerate ω we denote its inverse by R = ω−1. From the cocycle condition
for ω it follows that R solves the CYBE on ˜g∗. Conversely any solution of the CYBE
on g defines an invertible 2-cocycle on a subalgebra4 ˜g, which demonstrates the one-to-one
correspondence between DTD models with invertible ω and YB sigma models based on an
R-matrix solving the CYBE. The field redefinition that relates the two models is
ν = ζ ˜PT 1 − Ad¯g
adRx
ωRx , ¯g = eRx
∈ ˜G , (2.11)
4
This follows from the fact that the subspace on which R is invertible must be a subalgebra due to the
CYBE [20]. Since ω = R−1
is a 2-cocycle on this subalgebra the subalgebra is quasi-Frobenius. Note that
these results are true also for non-semisimple algebras and superalgebras.
– 6 –
JHEP10(2017)024
with x ∈ ˜g∗ so that Rx ∈ ˜g. In fact, using the identities in (A.5) and (A.4) we find
dν = ˜PT
(adν + ζω)(¯g−1
d¯g) , ˜PT
adν
˜P = ζ ˜PT
Ad−1
¯g ωAd¯g
˜P − ζω , (2.12)
and the action (2.2) becomes, after a bit of algebra,
S = −
T
2
d2
σ
γij − ij
2
Str g−1
∂ig ˆd 1 −
Rg
ˆd
Rg
ˆd − ζ
g−1
∂jg+ ¯g−1
∂i¯g(adν + ζω)¯g−1
∂j ¯g ,
(2.13)
where we have defined g = ¯gf and Rg = Ad−1
g RAdg. The last term vanishes up to a total
derivative and we are left precisely with the action of the YB sigma model [6, 7]
S = −
T
2
d2
σ
γij − ij
2
Str g−1
∂ig ˆd (1 − ηRg
ˆd)−1
(g−1
∂jg) , (2.14)
with deformation parameter η = ζ−1. In the special case when ˜g is abelian the DTD model
is equivalent to a TsT transformation of the original supercoset sigma model, in agreement
with the YB side for abelian R [2, 21].
Let us mention that one can also construct a YB model for an R-matrix solving the
modified CYBE, whose action takes essentially the same form as the above one [6]; however,
in that case it is not clear how to define the operator corresponding to ω, and the relation
to DTD models remains unclear. This case should be related by Poisson-Lie T-duality to
the λ-model of [19, 22].
We will argue in the next section that all (bosonic and fermionic) non-abelian T-duals
of YB sigma models can be described as DTD models with certain degenerate ω. The
converse is not true, in fact it is possible to identify DTD models which are not related to
YB models by NATD; we refer to section 3.2 for an example and a discussion on this.
3 Global symmetries
We will now describe the global symmetries, i.e. superisometries, of DTD models. Setting
ζ = 0 and ignoring the presence of ω this discussion reduces to what one would have in
the case of NATD. In order to identify the global symmetries of these models we study the
global transformations that leave the action invariant, modulo gauge transformations with
a global parameter, since the latter would not produce any Noether charge. We find two
types of global symmetries:5
1. Unbroken global G-transformations:
f → g0f , ν → ˜PT
Adg0 ν , g0 ∈ G and g0 /∈ ˜G,
such that (1 − ˜P)Adg0
˜P = 0 , ˜PT
Ad−1
g0
ωAdg0
˜P = ω .
(3.1)
The requirement g0 /∈ ˜G comes from the fact that for g0 ∈ ˜G a combination of this
isometry and the shift isometries described below is equivalent to a global ˜G gauge
transformation.
5
The two sets of transformations do not commute and their commutator is a transformation of the
second type.
– 7 –
JHEP10(2017)024
2. Global shifts of ν:
ν → ν + λ , λ ∈ ˜g∗
such that ˜PT
adλ
˜P = 0 . (3.2)
Note that the set of such λ’s will in general not close into a subalgebra, although the
corresponding isometry transformations of course commute since they are just shifts
of ν.
In the case when ω is invertible, which is equivalent to a YB sigma model with R = ω−1,
it is not hard to show that these isometries coincide with the ones of the YB model which
are normally written as t ∈ g such that Radt = adtR.
Having global symmetries at our disposal means that we can gauge them and implement
further NATD. Before discussing the details of this in the next subsection, we
would like to exploit this possibility to make a comment regarding Weyl invariance of DTD
models. As we prove in section 5, the target spaces of DTD models solve the standard
supergravity equations if and only if the Lie algebra ˜g is unimodular, i.e. fab
b = 0. The
standard supergravity equations are equivalent to the Weyl invariance at one-loop for the
sigma-model, as opposed to just the scale invariance implied by the generalised supergravity
equations [9, 10]. In the non-unimodular case fab
b = 0, and this defines a distinguished
element of ˜g; we can rotate the basis so that this element is T1, i.e. f1b
b = 0 and fab
b = 0
for a = 1. The important observation is that the dual of the generator T1 corresponds to
an isometry. In fact, taking the trace of the Jacobi identity we find fab
1 = 0 and therefore
Str(TbadT1 Ta) = fab
1
= 0 , (3.3)
where Ta ∈ ˜g∗. This confirms that T1 satisfies (3.2) and can be used to generate a shift
isometry. Using the results of the next subsection, applying T-duality along the isometry
direction T1 one obtains a DTD model where T1 is removed from ˜g, so that the subalgebra
that is left is now unimodular. Therefore, to each DTD model which is not Weyl invariant
we can associate a Weyl invariant one obtained by (formal6) T-duality along a particular
isometry direction. Obviously this possibility fails if there are obstructions to carrying out
the T-duality, e.g. if the isometry in question is a null isometry. More generally, solutions
of the generalised supergravity equations are formally T-dual to solutions of the standard
supergravity equations [9, 10], and the above argument shows this relation in the specific
context of DTD models.
3.1 DTD of DTD models
It is interesting to start from a DTD model as in (2.2) and further perform NATD, possibly
including a deformation by a cocycle. We do this on the one hand to show that the
application of these transformations on the sigma model does not require to start from
a supercoset formulation, on the other hand to show that after these transformations we
6
Our discussion of isometries is at the level of the classical sigma model action, where the dilaton only
appears in the combination F = eφ
F — together with RR fields — and in derivatives ∂φ. When performing
the T-duality we ignore the Fradkin-Tseytlin term, which will break the isometry referred to here.
– 8 –
JHEP10(2017)024
obtain a new DTD model. We will also use these results to argue that the example of the
next subsection is not related to a YB model by NATD.
We can apply NATD by gauging the global isometries discussed above and dualising
the corresponding directions. Obviously, the choice of the type of isometries that we want
to dualise will produce qualitative differences. In fact, if we consider isometries of the first
type (3.1) and dualise a subalgebra ˆg, we essentially enlarge the subalgebra ˜g. If instead we
consider isometries of the shift type (3.2) and dualise a subspace ¯V ∗ ⊂ ˜g∗, then we remove
generators from the subalgebra ˜g. The combination of isometry transformations that we
consider here is therefore
f = ˆgf , ν = ˜PT
(Adˆgν + ¯λ) , with ˆg ∈ ˆG , ¯λ ∈ ¯V ∗
. (3.4)
After gauging them in the usual way we obtain a sigma model action which is just the one
in (2.2), where we replace7
f → f , J → J + ˆA , dν → dˇν + ˇPT
[ ˆA, ˇν] + ¯a , (3.5)
where ˆA ∈ ˆg is the non-abelian gauge field corresponding to the ˆG isometries and ¯a ∈ ¯V ∗
is the abelian gauge field corresponding to the shift isometries. We add to the action
the terms8
− T d2
σ Str(ˆν ˆF+− + ¯ρ ¯f+− − ˆζ ˆA+ ˆω ˆA−) , (3.6)
where ˆF+− = ∂+
ˆA− − ∂−
ˆA+ + [ ˆA+, ˆA−] and ¯f+− = ∂+¯a− − ∂−¯a+, ˆν and ¯ρ are two new
Lagrange multipliers, and ˆω is a cocycle on ˆg. Integrating out ˆν and ¯ρ one obtains the
action from which we started; to apply NATD we integrate out ˆA and ¯a instead.
We will now describe what happens when we dualise either ˆg or ¯V ∗, and then use it
to argue what should happen in the most general case where one dualises on both at the
same time.9
Dualising type 1 isometries. Consider first isometries of type 1 above, where we have
ˆP + ˇP = ˜P and ˆP ˇP = 0. After a bit of algebra and dropping primes, we find that the new
action takes the form S = −T d2σStr(J+
ˆdf J− + (∂+ν − ˆdT
f J+)Q(∂−ν + ˆdf J−)) where
ν = ˇν + ˆν and Q is an operator acting on ˜g = ˇg ⊕ ˆg which can be written in a 2 × 2 block
form as
Q =
ˇO−1 + ˇO−1( ˆdf − adˇν)U−1( ˆdf − adˇν) ˇO−1 − ˇO−1( ˆdf − adˇν)U−1
−U−1( ˆdf − adˇν) ˇO−1 U−1
, (3.7)
7
We will now use the notation ˇν ∈ ˇg for the field and the subalgebra of the DTD model from which
we start. Similarly, we will denote the corresponding operators as ˇP, ˇO, etc. We do this because we want
to reserve the usual notation for the DTD model that is obtained at the end, after applying the further
deformation of NATD.
8
For the sake of the discussion here we fix conformal gauge γ+−
= γ−+
= −+
= − +−
= 2 where
σ±
= τ ± σ. In principle it is also possible to add a deformation for the second type of isometry by adding
a term ¯a¯ω ¯a, but we will not consider this possibility further here.
9
In the rest of this section we absorb the parameter ζ into ω to simplify the expressions.
– 9 –
JHEP10(2017)024
where10 U = ˆO − ˆPT ( ˆdf − adˇν) ˇO−1( ˆdf − adˇν) ˆP. It is straightforward to check that if we
take ω = ˇω + ˆω and define O as in (2.4), then its decomposition in block form is
O =
ˇO ˇPT ( ˆdf − adˇν) ˆP
ˆPT ( ˆdf − adˇν) ˇP ˆO
, (3.8)
and that Q = O−1. Therefore performing DTD by exploiting the unbroken isometries of
the first type is equivalent to the simple operation of enlarging the dualised subalgebra
as ˜g = ˇg ⊕ ˆg, which is a Lie algebra due to the isometry condition [ˆg, ˇg] ⊂ ˇg. As for the
deformation, we are just adding new contributions, and ω = ˇω + ˆω is a 2-cocycle on ˜g due
to the isometry conditions in (3.1).
Dualising type 2 isometries. For isometries of type 2 we have ¯PT that projects on
the space ¯V ∗, so that ¯P ˇP = ˇP ¯P = ¯P and ˜P = ˇP − ¯P. When integrating out ¯a± we get
equations where ¯P ˇO−1 appears, so that it is convenient to use the block decomposition on
the space ˜g ⊕ ¯V
ˇO−1
≡
O ˜PT ( ˆdf − ad˜ν − ˇω) ¯P
¯PT ( ˆdf − ad˜ν − ˇω) ˜P ¯PT ( ˆdf − ad˜ν − ˇω) ¯P
−1
(3.9)
=
O−1 + O−1( ˆdf − ad˜ν − ˇω)U−1( ˆdf − ad˜ν − ˇω)O−1 −O−1( ˆdf − ad˜ν − ˇω)U−1
−U−1( ˆdf − ad˜ν − ˇω)O−1 U−1
,
where U = ¯PT ( ˆdf − ad˜ν − ˇω) ¯P − ¯PT ( ˆdf − ad˜ν − ˇω)O−1( ˆdf − ad˜ν − ˇω) ¯P.
Note that ˜g = {x ∈ ˇg | Str(xλ) = 0 , ∀λ ∈ ¯V ∗} is indeed a subalgebra since for x, y ∈ ˜g
we have Str([x, y]λ) = −Str(xadλy) = 0 as a consequence of (3.2). In fact for x, y ∈ ˇg we
have in the same way [x, y] ∈ ˜g. This means in particular that if ¯V closes into a subalgebra
it must be abelian. Clearly ˇω reduces to a 2-cocycle ˜ω = ˜PT ˇω ˜P on ˜g.
After some algebra and dropping a total derivative dνd¯ρ-term, the dualised action
becomes
−T d2
σStr (J+ + ∂+ ¯ρ) ˆdf (J− + ∂− ¯ρ) + (∂+˜ν − ˆdT
f J+)O−1
(∂−˜ν + ˆdf J−)
+ (∂+˜ν − ˆdT
f J+)O−1
( ˆdf − ad˜ν − ˇω)∂− ¯ρ − ∂+ ¯ρ( ˆdf − ad˜ν − ˇω)O−1
(∂−˜ν + ˆdf J−)
− ∂+ ¯ρ( ˆdf − ad˜ν − ˇω)O−1
( ˆdf − ad˜ν − ˇω)∂− ¯ρ − ∂+ ¯ρ(ad˜ν + ˇω)∂− ¯ρ . (3.10)
As expected ¯ν = ˇν−˜ν has dropped out, since we have dualised the corresponding directions.
Finally ¯ρ can be removed by the field redefinition
f → ¯hf , ˜ν → ˜PT
Ad¯hν +
1 − Ad¯h
ad¯ρ
ˇω¯ρ , ¯h = e−¯ρ
, (3.11)
which resembles a ˜G gauge transformation except for the fact that ¯h /∈ ˜G. To check that
we match with the DTD action in (2.2) we use the fact that under the above redefinition
10
The operators ˇO, ˆO are obtained from O by dressing ν, ω and the projectors with checks or hats.
– 10 –
JHEP10(2017)024
O → ˇPT Ad¯hOAd−1
¯h
ˇP which follows from11
ˇPT
ad˜ν
ˇP → ˇPT
Ad¯h
ˇPT
adν
ˇPAd−1
¯h
ˇP + ˇPT
Ad¯h ˇωAd−1
¯h
ˇP − ˇω ,
d˜ν → ˇPT
Ad¯h(dν − adν(¯h−1
d¯h) − ˇω(¯h−1
d¯h)) .
(3.12)
The calculations are simple when ¯V is a (abelian) subalgebra since in that case ¯h−1d¯h =
−Ad−1
¯h
d¯ρ and the last d¯ρd¯ρ term vanishes up to a total derivative. When ¯V is not a
subalgebra it is clear that it must still work since these are abelian isometries and we can
just T-dualise one at a time. It is nevertheless instructive to show this explicitly. To do
this we use the fact that ¯h−1d¯h + Ad−1
¯h
d¯ρ is in ˜g since it involves commutators of elements
from ¯V . This simplifies the left-over terms to dσ2 ijStr(¯h−1∂i
¯h ˇω(¯h−1∂j
¯h)) which indeed
is a total derivative term and can be dropped. As anticipated, we get that T-dualising on
the shift isometries is equivalent to shrinking ˜g by removing the generators in ¯V .
Dualising type 1 and 2 isometries. We have seen that dualising on the isometries
outside of ˜g has the effect of adding the corresponding generators to ˜g. Similarly dualising
on isometries inside ˜g effectively removes the corresponding generators. The natural
conjecture is then that dualising on both types of isometries at the same time again just
adds/removes the generators outside/inside ˜g to give the ˜g of the resulting model.
To be more specific, start from a DTD model with a cocycle on the subalgebra12 ˇg and
imagine the most general NATD of this DTD model where we dualise isometries ti /∈ ˇg of
type 1 as in (3.1) and λI ∈ ˇg∗ of type 2 as in (3.2). Our conjecture is that this results in a
new DTD model where now
˜g = {x = ˇy + aiti , ˇy ∈ ˇg | Str(λI ˇy) = 0 , ∀λI such that Str(λI[ti, tj]) = 0 , ∀ti, tj} . (3.13)
In other words, ˜g is obtained by adding to ˇg all generators ti and by removing all elements
which are dual to λI, except when these are generated in commutators [ti, tj]. In fact,
we want the last condition on λI because the commutator of two isometries of type 1 can
generate an isometry of type 2, and if we are adding the ti we want to make sure that
they close into an algebra. Here we will not work out explicitly the transformation of the
action under this NATD since this is quite involved, we will rather just check that this
expectation makes sense and such a DTD model is well-defined.
To start, we must assume that the isometries on which we dualise form a subalgebra
of the isometry algebra. This implies the conditions
[ti, tj] = cij
k
tk + ˇcij
K ˇtK , ˇω(ˇtI ) = δI
I λI , ˇPT
adti λI = ciI
J
λJ , (3.14)
with some coefficients cij
k, ˇcij
k and ciI
J . The generators ˇtK ∈ ˇg appear because, as
already mentioned, the commutators of two ti can generate an element in ˇg. These must
still satisfy the second condition in (3.1) which translates to the second condition above.
11
These are proved using (A.4), (A.5) and ˜PAd¯h
ˇP = Ad¯h
˜P, the last being a consequence of [x, y] ∈ ˜g for
any x, y ∈ ˇg.
12
Also here we prefer to change notation and call ˇg the original subalgebra, so that ˜g will be used for the
algebra obtained after applying NATD.
– 11 –
JHEP10(2017)024
The first consistency check is to show that ˜g defined above indeed forms a subalgebra of
g so that the corresponding DTD model can be defined. Commuting two elements of ˜g
we get
[ˇy + aiti, ˇz + bjtj] = [ˇy, ˇz] − biadti ˇy + aiadti ˇz + aibj[ti, tj] . (3.15)
The isometry conditions in (3.1) indeed imply that the second and third term are in ˇg.
Taking the supertrace with λI satisfying Str(λI[ti, tj]) = 0 we get
Str([ˇy, ˇz]λI) + biciI
J
Str(ˇyλJ ) − aiciI
J
Str(ˇzλJ ) = −Str(ˇyadλI
ˇz) = 0 , (3.16)
where we used the conditions (3.14) and the fact that ˇy, ˇz ∈ ˜g and, in the last step, the
isometry condition (3.2) for λI. This proves that indeed ˜g in (3.13) defines a subalgebra
of g. To define a 2-cocycle on ˜g we take ω = ˜PT ˇω ˜P — we could also add an additional
deformation in the ti directions but we will not do so here— and we find
ω[ˇy + aiti, ˇz + bjtj] = ˜PT
[ˇωˇy, ˇz + biti] + [ˇy + aiti, ˇωˇz] + aibj ˇω[ti, tj]
= ˜PT
[ωˇy, ˇz + biti] + ˜PT
[ˇy + aiti, ωˇz] + aibj
˜PT
ˇω[ti, tj] , (3.17)
where we used the cocycle condition for ˇω, the fact that adti commutes with ˇω (3.1), and
in the last step we used (A.1). The first two terms are precisely what we want, it remains
to show that the last one vanishes. By the conditions (3.14) this term is proportional to
a combination of λI and therefore the ˜PT projection means that this term vanishes unless
Str([tk, tl]ˇω[ti, tj]) = 0 for some k, l. However
Str([tk, tl]ˇω[ti, tj]) =
1
2
Str(ˇω[[ti, tj], [tk, tl]])
=
1
2
Str( ˇPT
[ˇω[ti, tj], [tk, tl]]) +
1
2
Str( ˇPT
[[ti, tj], ˇω[tk, tl]])
=
1
2
ˇcij
I
Str( ˇPT
adλI
[tk, tl]) −
1
2
ˇckl
I
Str( ˇPT
adλI
[ti, tj]) = 0 , (3.18)
where we used the cocycle condition and the isometry condition in (3.2). Therefore ω is
indeed a 2-cocycle on ˜g and the corresponding DTD model is well-defined.
3.2 DTD models not related to YB models by NATD
Here we want to present an example of a DTD model which is not related to a YB model
by NATD.13 To argue that this is the case we use two important facts concerning the
dualisation of the two types of isometries discussed above. First, when dualising isometries
of type 1, thanks to the condition (3.1) the original ˇg will become an ideal of the larger
algebra ˜g that is obtained by adding the generators ti, i.e. by applying NATD. That means
that starting from a YB model — or, rather, its corresponding DTD model with nondegenerate
ω — NATD on isometries of type 1 will produce a DTD model with a cocycle
non-degenerate on an ideal of ˜g. When we include also isometries of type 2 it remains true
13
Let us mention that it is possible to find examples where ω — as well as any 2-cocycle in its equivalence
class — is non-degenerate on a space which does not close into an algebra. This corrects a statement in the
first version of [1].
– 12 –
JHEP10(2017)024
that what is left of ˇg forms a proper ideal inside ˜g, on which, however, ω does not have to
be non-degenerate. We also remark that, since they are realised as linear shifts, isometries
of type 2 are commuting and are therefore still present even after applying abelian Tduality
along them. After the dualisation the corresponding symmetry will be realised as
an isometry of type 1.
Consider the following algebra and corresponding 2-cocycle
˜g = span{p1, p2, p3, J12} , ω = k3 ∧ J12 , (3.19)
where we refer to [13] for our definitions and conventions on the generators of the conformal
algebra so(2, 4). The above 2-cocycle is defined on a space which is not an ideal of ˜g, and
it is clear that adding an exact term to ω cannot change this, since the only terms that
we could add are k1 ∧ J12 and k2 ∧ J12. According to the above discussion, this rules out
the possibility of this example coming from dualising isometries of type 1 of a YB model.
In fact, since there is no proper ideal in ˜g that contains the subspace {p3, J12} where ω
is defined, a combination of isometries of type 1 and type 2 is also ruled out. This leaves
only the possibility that this example is generated by T-dualising isometries of type 2 only.
If it were true that it comes from a YB model by dualising isometries of type 2, these
should be realised here as isometries of type 1 and we would be able to dualise them back
to find a YB model (in DTD form). However, in this example the only isometry of type 1
corresponds to p0, and adding p0 to ˜g does not help in making the cocycle non-degenerate
on the dualised algebra. We therefore conclude that the above example is not related to
a YB model by NATD,14 and we refer to section 6.2 for the corresponding supergravity
background.
The above example may be obtained by dropping one of the two terms in R11 in table
2 of [13], and similar examples coming from dropping a term in other rank 4 R-matrices
of [13] are e.g.
˜g = span{p1, p2, p3, p0 + J12} , ω = k3 ∧ (k0+ J12) , from R10 .
˜g = span{p0, p1, p2, J12} , ω = k0 ∧ J12 , from R13 .
˜g = span{p1, p2, J12, J03} , ω = J12 ∧ J03 , from R14 .
(3.20)
In each case it is easy to see that ω cannot be defined on an ideal in ˜g even if we add
exact terms — in the first case the only terms that we could add are k1 ∧ (k0 + J12) and
k2 ∧ (k0 + J12), in the second and third case they are k1 ∧ J12 and k2 ∧ J12. In the first case
the only isometry of type 1 corresponds to p0, while in the second and third there is no
isometry of type 1. Note that the second case can be embedded into so(2, 3) and therefore
gives a deformation also of AdS4.
4 Kappa symmetry and Green-Schwarz form
As we will show in a moment the action of DTD models is invariant under kappa symmetry
variations, and this will allow us to put it into the Green-Schwarz form. To show invariance
14
It would be interesting to understand whether this or similar examples are related to YB models in
other ways, e.g. contractions.
– 13 –
JHEP10(2017)024
under kappa symmetry we need to consider the variation of the action under the fields ν
and f, as well as the worldsheet metric γij. The variation of the action with respect to the
fields is computed in (C.1). To define a kappa symmetry variation we should also say how
δf and δν are expressed in terms of the kappa symmetry parameters ˜κ
(j)
i , each of them
being a local Grassmann parameter of grading j. We define Ai
± ≡ Ad−1
f ( ˜Ai
± + Ji
±), where
subscripts ± indicate that we act with the worldsheet projectors in (B.3) and ˜Ai
± is given
in (B.5); we take15
ˆdT
(f−1
δκf) = Ad−1
f δκν = −{i˜κ
(1)
i , A
(2)i
− } + {i˜κ
(3)
i , A
(2)i
+ } . (4.1)
This relation is fixed by noticing that after we impose it the total variation of the action
with respect to the fields simplifies considerably, and we find
(δf + δν)S = −
T
2
d2
σ 4 Str A
(2)i
− A
(2)j
− [A
(1)
+i , i˜κ
(1)
j ] + A
(2)i
+ A
(2)j
+ [A
(3)
−i , i˜κ
(3)
j ]
= −
T
2
d2
σ
1
2
Str A
(2)i
− A
(2)j
− Str W[A
(1)
+i , i˜κ
(1)
j ]
+ Str A
(2)i
+ A
(2)j
+ Str W[A
(3)
−i , i˜κ
(3)
j ] .
(4.2)
Here we used the property Ai
±Bj
± = Aj
±Bi
±, which follows from the identity Pij
± Pkl
± =
Pil
±Pkj
± , as well as the identity
A
(2)i
± A
(2)j
± =
1
8
W Str(A
(2)i
± A
(2)j
± ) + cij
18 , (4.3)
where cij is an expression which is not interesting for this calculation, and W =
diag(14, −14) is the hypercharge. The above variation does not vanish but it can be compensated
by the contribution coming from varying the worldsheet metric. In fact, we first
notice that the contribution of the terms involving the worldsheet metric to the action may
be written as
Sγ = −
T
2
d2
σγij
Str E
(2)
i E
(2)
j , (4.4)
where we have two possible choices for the bosonic vielbein which are related by a local
Lorentz transformation, either E(2) = A
(2)
+ or E(2) = A
(2)
− , where
A+ = Ad−1
f (J + O−T
(dν − ˆdT
f J)) , A− = Ad−1
f (J − O−1
(dν + ˆdf J)) . (4.5)
The subscript on A± is here used only to distinguish the two fields and should not be
confused with the ± used to denote the worldsheet projections; however, we choose this
notation since projecting on A± with Pij
± after reintroducing worldsheet indices we obtain
in fact the Ai
± used above.16 We declare the kappa symmetry variation of the worldsheet
metric to be
δκγij
= −
1
2
Str W[A
(1)i
+ , i˜κ
(1)j
+ ] + Str W[A
(3)i
− , i˜κ
(3)j
− ] , (4.6)
15
We write the kappa symmetry transformation in this way rather than the one in [1] because we want
P(0)
Ad−1
f δκν = 0.
16
A caveat is that the projections of A± in (4.5) with Pij
do not vanish, while Pij
A±j = 0. We trust
that this will not create confusion, since the notation has clear advantages and those projections will never
be needed.
– 14 –
JHEP10(2017)024
so that the total variation of the action under kappa symmetry transformations vanishes
(δf +δν +δγ)S = 0. The kappa symmetry transformations for the fields may be also recast
into the form
iδκzE(2)
= 0 , iδκzE(1)
= Pij
− {iκ
(1)
i , E
(2)
j } , iδκzE(3)
= Pij
+ {iκ
(3)
i , E
(2)
j } , (4.7)
where κ(1) = Adh˜κ(1) and κ(3) = ˜κ(3) and where we made a choice for the bosonic and
fermionic components of the supervielbeins
E(2)
= A
(2)
+ = AdhA
(2)
− , E(1)
= AdhA
(1)
+ , E(3)
= A
(3)
− . (4.8)
The above transformations are the standard ones for kappa symmetry, and the action also
takes the standard Green-Schwarz form
S = −
T
2
d2
σ γij
Str(E
(2)
i E
(2)
j ) − T B , (4.9)
where the B-field is
B =
1
4
Str(J ∧ ˆdf J + (dν − ˆdT
f J) ∧ O−1
(dν + ˆdf J)) . (4.10)
As already noticed, A
(2)
+ and A
(2)
− are related by a local Lorentz transformation, A
(2)
+ =
AdhA
(2)
− for some h ∈ G(0). For later convenience we can also relate other components of
A+ and A− as follows17
A− = MA+ , P(2)
M = Ad−1
h P(2)
, (4.11)
M = Ad−1
f [1 − ˜P − O−1
OT
− 4O−1
Adf P(2)
Ad−1
f (1 − ˜P)]Adf
= 1 − 4Ad−1
f O−1
Adf P(2)
,
while M−1 is given by the same expression as M but with O replaced by its transpose
OT = ˜PT ( ˆdT
f + adν + ζω) ˜P. From this we can derive the useful relation
M−1
− 1 = −(M − 1)Adh . (4.12)
5 Target space superfields
In this section we will derive the form of the target space supergravity superfields for the
DTD model. The calculations are very similar to the ones performed in [13] for the ηmodel
and λ-model. Once the action and kappa symmetry transformations are written in
Green-Schwarz form as in (4.9) and (4.7), the easiest way to extract the background fields
is by computing the torsion Ta = dEa + Eb ∧ Ωb
a and Tα = dEα − 1
4 (ΓabE)α ∧ Ωab where
17
As a consequence of this we have for example A
(3)
+ = E(3)
− P(3)
ME(2)
.
– 15 –
JHEP10(2017)024
Ωab is the spin connection superfield. It was shown in [9] that the constraints on the torsion
implied by kappa symmetry take the form18
Ta
= −
i
2
Eγa
E ,
TαI
=
1
2
EαI
Eχ +
1
2
(σ3
E)αI
Eσ3
χ −
1
4
EγaE (γa
χ)αI
−
1
4
Eγaσ3
E (γa
σ3
χ)αI
−
1
8
Ea
(Eσ3
γbc
)αI
Habc −
1
8
Ea
(EγaS)αI
+
1
2
Eb
Ea
ψαI
ab , (5.1)
for the type IIB case.19 The target space superfields contained here are the dilatino superfields
χαI, the gravitino field strengths ψαI
ab , where I = 1, 2 denotes the two Majorana-Weyl
spinors of type IIB, as well as the NSNS three-form field strength H = dB and “RR field
strengths” encoded in the anti-symmetric 32 × 32 bispinor
S = −iσ2
γa
Fa −
1
3!
σ1
γabc
Fabc −
1
2 · 5!
iσ2
γabcde
Fabcde . (5.2)
Kappa symmetry implies that the target space is generically only a solution of the generalised
type II supergravity equations defined in [9] and first written down, for the bosonic
sector, in [10]. However, when the (Killing) vector
Ka
= −
i
16
(γa
σ3
)αIβJ
αIχβJ (5.3)
vanishes one gets a solution of standard type II supergravity, and a one-loop Weyl invariant
string sigma model. In that case there exists a dilaton superfield φ such that χαI = αIφ
and the RR field strengths are defined in terms of potentials in the standard way F =
eφdC + · · · [23, 24].
Given that the supervielbeins for the DTD model are defined in terms of A± as in (4.8)
we need to compute the exterior derivative of A± defined in (4.5) to find the torsion. With
a bit of work one finds the deformed “Maurer-Cartan” equations20
dA+ =
1
2
{A+, A+} −
1
2
Ad−1
f O−T
Adf
ˆdT
{A+, A+} − 2{A+, ˆdT
A+} , (5.4)
dA− =
1
2
{A−, A−} −
1
2
Ad−1
f O−1
Adf
ˆd{A−, A−} − 2{A−, ˆdA−} , (5.5)
where we have used the identity (A.1) and the fact that, due to the Jacobi identity and the
2-cocycle condition (2.5), both adν and ω effectively act as derivations on the Lie bracket.
Projecting the first equation with P(2) and using (4.8) and (4.11) we get
dE(2)
= {A
(0)
+ , E(2)
} +
1
2
{E(1)
, E(1)
} +
1
2
{E(3)
, E(3)
} − {E(3)
, P(3)
ME(2)
}
− P(2)
MT
{E(2)
, E(3)
} +
1
2
{P(3)
ME(2)
, P(3)
ME(2)
}
+ P(2)
MT
{E(2)
, P(3)
ME(2)
} −
1
2
P(2)
MT
{E(2)
, E(2)
} . (5.6)
18
This is valid only for a suitable choice of the spin connection, which can however be extracted from the
same equations. We have dropped the ∧’s for readability.
19
Essentially identical expressions hold for type IIA, cf. [23].
20
We use anti-commutators rather than commutators because the objects that appear are one-forms, and
therefore naturally anti-commute.
– 16 –
JHEP10(2017)024
Using A
(0)
+ = 1
2 Aab
+ Jab, E(2) = EaPa etc. and the algebra in appendix A of [13] this gives the
form for the bosonic torsion Ta in (5.1) provided that we identify the spin connection with21
Ωab = (A+)ab + 2i(E2
γ[a)βMβ2
b] +
3i
2
Ec
Mα2
[a(γb)αβMβ2
c] +
1
2
Ec
(Mab,c − 2Mc[a,b]) . (5.7)
In a similar way, using (4.8) and (5.5) we find that
dE(3)
= {A
(0)
+ , E(3)
} + {P(0)
ME(2)
, E(3)
}
+ Ad−1
h {E(1)
+ P(1)
AdhME(2)
, E(2)
} +
1
2
P(3)
M{E(3)
, E(3)
}
+ 2P(3)
Ad−1
f O−1
Adf 2Ad−1
h {E(1)
+ P(1)
AdhME(2)
, E(2)
}+Ad−1
h {E(2)
, E(2)
} ,
(5.8)
which leads to the torsion Tα2 taking the form in (5.1) with the background fields given by22
Habc = 3M[ab,c] − 3iMα2
[a(γb)αβMβ2
c] ,
Sα1β2
= − 8i[Adh(1 + 4Ad−1
f O−T
Adf )]α1
γ1Kγ1β2
, (5.9)
χ2
α = −
i
2
γa
αβMβ2
a ,
ψα2
ab = 2[Ad−1
f O−1
Adf Ad−1
h ]α2
cdKab
cd
+
1
4
[AdhM]β1
[a(γb]S12
)β
α
.
Here KAB denotes the inverse of the metric defined by the supertrace Str(TATB) = KAB,
see appendix A of [13] for more details on our conventions.
Since the DTD model contains NATD as a special case we obtain as a by-product the
transformation rules for RR fields under NATD — starting from a supercoset model. As
a check we can compare this to the formula conjectured in [14] based on analogy to the
abelian case [25] — consistency of that formula was checked in some particular cases also
in [8]. Setting ζ = 0, which removes the deformation, and restricting to a bosonic ˜g, so
that ˜P = ˜P(P(0) + P(2)) = (P(0) + P(2)) ˜P, we find23
Sα1β2
= −8i[Adh|θ=0]α1
γ1Kγ1β2
+ fermions , (5.10)
which agrees with the transformations conjectured in [14]. Note that our result generalises
this to the case where also fermionic T-dualities are involved.
Finally we must compute Tα1 to extract the other dilatino superfield χ1. We find
dE(1)
= {AdhA
(0)
+ − dhh−1
, E(1)
} + Adh{E(2)
, E(3)
− P(3)
ME(2)
}
+
1
2
P(1)
AdhM−1
Ad−1
h {E(1)
, E(1)
}
+ 2P(1)
AdhAd−1
f O−T
Adf 2{E(2)
, E(3)
− P(3)
ME(2)
} + {E(2)
, E(2)
} . (5.11)
21
The components of M are defined as MTA = TBMB
A.
22
These expressions have obvious close analogies with the ones found for the η-model in [13].
23
Note that (P(0)
+ P(2)
)Adf P(1)
= 0+fermions.
– 17 –
JHEP10(2017)024
Taking the exterior derivative of the equation A
(2)
+ = AdhA
(2)
− , cf. (4.11), we find the
relation
[AdhA
(0)
+ − dhh−1
]ab = Ωab −
1
2
Ec
Habc + 2i(E1
γ[a)α[AdhM]α1
b], (5.12)
which can be used to show that the torsion again takes the form in (5.1), where the
remaining components of the background fields are24
χ1
α =
i
2
(γa
)αβ[AdhM]β1
a , ψα1
ab = 2[AdhAd−1
f O−T
Adf ]α1
cdKab
cd
−
1
4
(S12
γ[a)α
βMβ2
b] .
(5.13)
It remains only to analyse the question of when this is a solution to the standard or the
generalised type II supergravity equations, in other words to identify the conditions under
which Ka defined in (5.3) vanishes. We do this in the next subsection.
5.1 Supergravity condition and dilaton
By analogy with the calculations performed in [13] there is a natural candidate for the
dilaton superfield for the DTD model namely25
e−2φ
= sdet O . (5.14)
We will now show that this guess is indeed correct by verifying that its spinor derivatives
reproduces the dilatini found above. Using the formula for the supertrace StrM =
KABStr(TAMTB) we find
dφ = −
1
2
Str(dOO−1
) = −
1
2
KAB
Str ([J, ˆdT
f TA] − ˆdT
f [J, TA] + [dν, TA])O−1
TB
= −
1
2
KAB
Str [J, ˆdT
f TA] − ˆdT
f [J, TA] + [Adf
ˆdT
A+, TA]
+ [(adν + ζω)(Adf A+ − J), TA] O−1
TB
=
1
2
KAB
Str TA
ˆd[A+, Ad−1
f O−1
Adf TB]
+ [ ˆdT
A+, Ad−1
f O−1
Adf TB] − [A+, ˆdAd−1
f O−1
Adf TB]
+ KAB
Str [(Adf A+ − J), TA] ˜PTB . (5.15)
If the last term vanishes, then using (4.8), (5.13), (5.9) and (4.11) one may check that the
E(1,3)-terms are indeed equal to
Eα1
χ1
α + Eα2
χ2
α . (5.16)
Therefore χαI = αIφ which implies that Ka in (5.3) vanishes and we have a solution to
standard type II supergravity. Since (Adf A+ − J) ∈ ˜g can be regarded as an arbitrary
24
Just as in [13], one finds a superficially different expression for Habc namely
Habc = 3[AdhM][ab,c] + 3i[AdhM]α1
[a(γb)αβ[AdhM]β1
c] .
However consistency requires this to be the same as the expression in (5.9) and this can also be verified
explicitly similarly to [13].
25
The prime on the superdeterminant denotes the fact that we must restrict to the subspace where O is
defined, i.e. the subalgebra ˜g.
– 18 –
JHEP10(2017)024
element of the Lie algebra, the vanishing of the last term in (5.15) is equivalent to fAB
A = 0
for the structure constants of ˜g, i.e. ˜g must be unimodular. This condition is therefore
sufficient to get a standard supergravity solution. Following a calculation similar to the
one done in [13], computing Ka in (5.3) and requiring it to vanish one finds that this
condition is also necessary.26
Our results imply that the DTD model gives a one-loop Weyl invariant string sigma
model precisely27 when the subalgebra ˜g is unimodular. This is in fact the same condition
that was found long ago for NATD on bosonic sigma models by path integral considerations
[11, 12]. Since the DTD model includes NATD as a special case, the analysis here
coupled with the results of [9, 10], gives an alternative derivation of the Weyl anomaly for
NATD of supercosets.
A nice fact is that we do not have to impose extra conditions on the cocycle ω used
to construct the deformation. When ω is non-degenerate unimodularity of ˜g is equivalent
to unimodularity of R = ω−1 as defined in [13], see the discussion there; this is consistent
with the fact that the YB models are a special case of the DTD models.
6 Some explicit examples
Here we would like to collect some formulas that are useful when deriving the explicit
background for a given DTD model, and then work out two examples in detail. We denote
the generators of ˜g ⊂ g by Ti, i = 1, . . . , N = dim(˜g), and those of the dual ˜g∗ by Ti. They
satisfy Str(TiTj) = δi
j. The action of the projectors on a generic element x ∈ g may be
written as
˜P(x) = Str(Ti
x)Ti, ˜PT
(x) = Str(Tix)Ti
, (6.1)
where summation of repeated indices is assumed. Given a cocycle ω = 1
2 ωijTi ∧ Tj with
ωji = −ωij, its action on an element of the algebra is
ω(x) = ωijTi
Str(Tj
x), (6.2)
and it must satisfy the cocycle condition, which may be written as
Str Tk(ω[Ti, Tj] − [Ti, ωTj] + [Tj, ωTi]) = 0, ∀Ti, Tj, Tk ∈ ˜g. (6.3)
With the above definitions one may easily construct the operator O : ˜g → ˜g∗ defined
in (2.4), that can be encoded in an explicit N × N matrix
˜Oij = Str(O(Ti)Tj), (6.4)
26
In very special cases it is possible for Ka
to decouple from the remaining generalized supergravity
equations. One then obtains a background solving both the generalised and standard supergravity equations
depending on if Ka
is included or not. One such example is the pp-wave solution discussed in appendix B
of [26]. We thank B. Hoare and S. van Tongeren for pointing this out.
27
This is modulo possible subtleties with the special cases mentioned in the previous footnote. One should
also note that this condition is true provided one only allows a local (Fradkin-Tseytlin) counter-term. If
one relaxes this condition one can find a non-local counter-term also when Ka
is non-zero, since solutions of
the generalised supergravity equations are formally T-dual to solutions of the standard ones; see also [27].
This being said, cases where Ka
is null may be subtle and deserve further study.
– 19 –
JHEP10(2017)024
so that O(Ti) = ˜OijTj. The matrix ˜O can be inverted with standard methods and used
to construct the action of the inverse operator as O−1(x) = Str(xTi)( ˜O−1)ijTj, so that on
the basis generators O−1(Ti) = ( ˜O−1)ijTj. Obviously, when choosing a parametrisation
for the group element f, one should make sure that the corresponding degrees of freedom
cannot be gauged away by applying the local transformations discussed in section 2.
To obtain the background fields we use the results of section 5. The metric reads
as ds2 = ηabEaEb, where the components of the bosonic supervielbein are obtained by
Ea = Str(A+Pa), and the B-field is given by equation (4.10). From the superdeterminant
of the matrix ˜O it is also straightforward to compute the (exponential of the) dilaton
eφ = (sdet ˜O)− 1
2 . In order to determine the RR fields one first identifies the components of
the matrix Mab = Str((MPa)Pb) and then one constructs the local Lorentz transformation
on spinorial indices
(Adh)β
α = exp −
1
4
(log M)abΓab β
α , (6.5)
so that AdhΓaAd−1
h = M b
a Γb, where Γa are 32 × 32 Gamma-matrices.28 From (5.2)
and (5.9) one finds that the expression for RR fields is obtained by solving the equation
Γa
Fa +
1
3!
Γabc
Fabc +
1
2 · 5!
Γabcde
Fabcde Π = e−φ
[Adh(1 + 4Ad−1
f O−T
Adf )](4Γ01234)Π,
(6.6)
where Π = 1
2 (1 − Γ11) is a projector29 and (−4Γ01234)Π corresponds to the 5-form flux of
AdS5×S5. In order to find the component Fa1...a2m+1 it is then enough to multiply the above
equation by Γa1...a2m+1 and take the trace. As already explained, when the subalgebra ˜g
is bosonic the above result simplifies considerably, and only Adh remains inside square
brackets. After obtaining the components in tangent indices we translate them into form
language using F(2m+1) = 1
(2m+1)! Ea2m+1 ∧ . . . ∧ Ea1 Fa1...a2m+1 .
6.1 A TsT example
First we will work out a simple example where we dualise a two-dimensional abelian subalgebra
of the isometry of the sphere so(6), so that the deformation is equivalent to doing
a TsT there [28–30]. This example was worked out already in [2] for the NSNS sector, and
the RR fields were taken into account in [8] by following the T-duality rules of [14]. Here
we will use the matrix realisation of the psu(2, 2|4) superalgebra used in [13], see also [31].
We take ˜g to be the abelian algebra spanned by two Cartans of so(6), T1 ≡ J68, T2 ≡ J79,
and for the dual generators we may just take T1 = J68, T2 = J79. We parametrise the
bosonic fields as30
ν = ˜ϕiTi
, f = fa · exp(ϕP5) exp(−ξJ89) exp(− arcsin rP9), (6.7)
where fa is a coset group element parametrised by fields in AdS5. We take ω = T1 ∧ T2
which obviously satisfies the cocycle condition. The matrix corresponding to O is very
28
Alternatively one can use the 16 × 16 gamma matrices used in the previous section.
29
With these conventions the self-duality for the 5-form is F(5)
= ∗F(5)
.
30
The group elements parametrised by ϕ, ξ and r coincide with those in (A.1) of [32].
– 20 –
JHEP10(2017)024
simple
˜Oij =
2r2 sin2
ξ ζ
−ζ 2r2 cos2 ξ
, (6.8)
and it is easily inverted. Following the above discussion we immediately find the fields of
the NSNS sector
ds2
= ds2
a +
r2
ζ2 + r4 sin2
(2ξ)
(cos2
ξ d ˜ϕ2
1 + sin2
ξ d ˜ϕ2
2) + (1 − r2
)dϕ2
+ r2
dξ2
+
dr2
1 − r2
,
eφ
= (ζ2
+ r4
sin2
(2ξ))− 1
2 , B =
ζ
2
d ˜ϕ1 ∧ d ˜ϕ2
ζ2 + r4 sin2
(2ξ)
,
(6.9)
where ds2
a is the metric of AdS5. After computing the matrix Mab and the local Lorentz
transformation31 we get that only F(3) and F(5) are non-vanishing
F(3)
= 4r3
sin(2ξ)dϕ ∧ dξ ∧ dr,
F(5)
= − 2ζ(1 + ∗)
r3 sin(2ξ) d ˜ϕ1 ∧ d ˜ϕ2 ∧ dϕ ∧ dξ ∧ dr
ζ2 + r4 sin2
(2ξ)
.
(6.10)
Since ω is non-degenerate on ˜g we can relate the above background to a YB deformation of
AdS5×S5, see also section 2.1. In this particularly simple example the R-matrix of the YB
model is abelian, and therefore it corresponds just to a TsT transformation on the sphere,
see also [21]. In fact, consider the following TsT transformation on AdS5×S5
ϕ1 → T(ϕ1), ϕ2 → ϕ2 − 2ηT(ϕ1), T(ϕ1) → ϕ1, (6.11)
which produces the following background32
ds2
= ds2
a +
r2
1 + η2r4 sin2
(2ξ)
(cos2
ξ dϕ2
2 + sin2
ξ dϕ2
1) + (1 − r2
)dϕ2
+ r2
dξ2
+
dr2
1 − r2
,
eφ
= (1 + η2
r4
sin2
(2ξ))− 1
2 , B = −
ηr4 sin2
(2ξ)dϕ1 ∧ dϕ2
1 + η2r4 sin2
(2ξ)
,
(6.12)
for the NSNS sector and
F(3)
= 4ηr3
sin(2ξ)dϕ ∧ dξ ∧ dr,
F(5)
= − 2(1 + ∗)
r3 sin(2ξ) dϕ1 ∧ dϕ2 ∧ dϕ ∧ dξ ∧ dr
1 + η2r4 sin2
(2ξ)
,
(6.13)
for the RR sector. To match with the above TsT background we need to implement the
field redefinition (2.11) at the level of the DTD background, which in this case just reduces
to ˜ϕ1 = η−1ϕ2, ˜ϕ2 = −η−1ϕ1 since ˜g is abelian. We find agreement only if we also use the
gauge freedom for B to subtract the exact term 1
2η dϕ1 ∧ dϕ2; moreover we also need to
redefine the constant part of the dilaton to reabsorb a factor of η, which then appears in
front of the RR fields.
31
For 32 × 32 Gamma matrices we find convenient the basis used in [31].
32
As a starting point we take the undeformed AdS5×S5
background as written in [31].
– 21 –
JHEP10(2017)024
6.2 A new example
Let us now consider the example in (3.19)
˜g = span{p1, p2, p3, J12} , ˜g∗
= span −
1
2
k1, −
1
2
k2, −
1
2
k3, −J12 ω = k3 ∧ J12 .
(6.14)
In this case we have just one isometry of type 1 corresponding to p0, and the isometries
of type 2 are k3 and J12. Inspired by the parametrisation used in (6.19) of [13] we
parametrise33
ν = ˜ξ J12 + ˜r k1 + ˜x3
k3 , f = exp(x0
p0) exp(log zD) . (6.15)
The above is a good parametrisation because it is not possible to remove degrees of freedom
by applying gauge transformations. This will be confirmed e.g. by the fact that we get a
non-degenerate metric in target space. We find that the (matrix corresponding to the)
operator O is
˜Oij =





2
z2 0 0 0
0 2
z2 0 2˜r
0 0 2
z2 2ζ
0 −2˜r −2ζ 0





, (6.16)
which is clearly invertible. We find the following NSNS sector fields
ds2
=
−(dx0)2 + dz2
z2
+ d˜r2
z2
+
d˜ξ2
4z2 (ζ2 + ˜r2)
+
˜r2z2(d˜x3)2
ζ2 + ˜r2
+ ds2
s ,
eφ
=
16 ζ2 + ˜r2
z4
− 1
2
, B = −
ζd˜ξ ∧ d˜x3
2 (ζ2 + ˜r2)
,
(6.17)
where ds2
s is the metric on S5. In the RR sector we have only three-form flux
F(3)
= −
8(dx0 ∧ d˜ξ ∧ dz)
z5
. (6.18)
According to the discussion in section 2.1 the above background is not related to a YB
model by NATD.
7 Conclusions
We have argued that DTD models based on supercosets represent a large class of integrable
string models which is closed under NATD as well as (certain) deformations. Besides
being a useful tool to generate new integrable supergravity backgrounds it would be very
interesting if these deformations could be understood on the dual field theory side. In
the case when the 2-cocycle is invertible these models are equivalent to YB sigma models,
which have been argued to correspond to non-commutative deformations, e.g. [33, 34],
33
Even if present, one could remove k2 in ν by means of a gauge transformation.
– 22 –
JHEP10(2017)024
of the field theory [35–37] (see also [38]). This interpretation is consistent with the fact
that TsT transformations are special cases of these models [21, 39] and this includes the
so-called β and γ-deformations which have a known interpretation in N = 4 super YangMills
[28, 29, 40, 41]. Recently a certain limit of the γ-deformation has been used to
construct a simplified integrable scalar field theory [42, 43] and it would be very interesting
to explore similar limits of the more general class of deformations considered here to see
whether one can learn more about the AdS/CFT duality for those cases.
Another important question is how the DTD model relates to the other known deformations
of the AdS5 × S5 string, i.e. the η-model with R-matrix solving the modified
CYBE [44] and the λ-model [22]. These two deformations are related by Poisson-Lie Tduality
and the fact that the latter is Weyl-invariant [13] while the former is not [10, 31]
is explained by the fact that the obstruction to the duality at the quantum level again
involves the trace of the structure constants [45].34 The fact that NATD is used also in the
construction of the λ-model suggests that there might be a bigger picture relating it to the
DTD construction considered here. In fact this seems to be part of an even bigger picture
of general integrable deformations of sigma models where T-duality and its generalizations
play a central role, see for example the recent paper [46].
Acknowledgments
We thank B. Hoare, S. van Tongeren and A. Torrielli for interesting and useful discussions,
and A. Tseytlin for illuminating discussions and comments on the manuscript. The
work of R.B. was supported by the ERC advanced grant No 341222. We also thank the
Galileo Galilei Institute for Theoretical Physics (GGI) for the hospitality and INFN for
partial support during the completion of this work at the program New Developments in
AdS3/CFT2 Holography.
A Useful identities
A useful identity is
˜P[ ˜PT
x, (1 − ˜P)y] = 0 , ∀x, y ∈ g (A.1)
which is easily proven by taking the supertrace with an element of g. We will also need
some relations related to the well-known formula for the derivative of the exponential map
dex
= ex 1 − e−adx
adx
dx . (A.2)
Let x ∈ ˜g and define a similar looking object µ = ˜PT e−xδex, where δ is the derivation
acting as δ(x) = ω(x) on x ∈ ˜g. Note that this derivation is compatible with the Lie
bracket due to the 2-cocycle condition (2.5), and following the same computations needed
to prove the identity above, one may show that
µ = ˜PT 1 − e−adx
adx
ωx. (A.3)
34
We thank A. Tseytlin for this comment.
– 23 –
JHEP10(2017)024
Taking y ∈ ˜g, from the definition of µ we find ˜PT adµy = ˜PT Ad−1
ex δ(Adex y) − δy which
implies
˜PT
adµ
˜P = ˜PT
e−adx
ωeadx ˜P − ω. (A.4)
Another useful identity valid for the derivative of µ is
dµ = µe−x
dex
+ δ(e−x
dex
) + ˜PT
de−x
ex
µ = ˜PT
(adµ + ω)(e−x
dex
) . (A.5)
Now, the identity (A.1) implies that
˜PT
ad ˜PT Ad˜hν
˜P = ˜PT
Ad˜hadνAd−1
˜h
˜P = ˜PT
Ad˜h
˜PT
adν
˜PAd−1
˜h
˜P (A.6)
and together with (A.4) it implies that if we redefine ν → ˜PT Ad˜hν + ζµ as in (2.7) then
the operator in (2.4) transforms as
O → ˜PT
Ad˜hOAd−1
˜h
˜P. (A.7)
Moreover, using (A.5) we also find
dν → ˜PT
Ad˜h(dν − (adν + ζω)(˜h−1
d˜h)). (A.8)
B Derivation of the action
To derive the action of DTD models we start from the action of a supercoset sigma model,
see e.g. [47], and we rewrite the group element as g = ˜gf, where ˜g ∈ ˜G ⊂ G. We then
gauge the ˜G symmetry and introduce the gauge fields ˜Ai. If we fix the gauge ˜g = 1 we
essentially achieve ˜g−1d˜g → ˜A when comparing to the initial supercoset action. At this
point we add a Lagrange multiplier to impose the flatness of ˜Ai, plus a ω-dependent term
which deforms the model
S = −
T
2
d2
σ
γij − ij
2
Str ( ˜Ai + Ji) ˆdf ( ˜Aj + Jj)
− ij
Str ν(∂i
˜Aj + ˜Ai
˜Aj) −
ζ
2
˜Aiω ˜Aj . (B.1)
Instead of integrating out ν we integrate out ˜A, so that we obtain the equations of motion
Pij
− O ˜Aj + ∂jν + ˆdf Jj + Pij
+ OT ˜Aj − ∂jν + ˆdT
f Jj = 0, (B.2)
where
Pij
± =
γij ± ij
2
, (B.3)
are projectors
Pij
+ + Pij
− = γij
, Pil
±P j
±l = Pij
± , Pil
±P j
l = 0. (B.4)
Here we used also γij = ikγkl
lj. We also define V i
± ≡ Pij
± Vj, and it is useful to remember
Pij
± AiBj = Ai γijBj
±. We then solve for ˜A±
˜Ai
− = O−1
−∂i
−ν − ˆdf Ji
− , ˜Ai
+ = O−T
+∂i
+ν − ˆdT
f Ji
+ . (B.5)
– 24 –
JHEP10(2017)024
The action on the solutions to the equations of motion is
S = −
T
2
d2
σ Str J+i
ˆdf J i
− + (∂+iν − ˆdT
f J+i)O−1
(∂i
−ν + ˆdf Ji
−)
= −
T
2
d2
σ
γij − ij
2
Str Ji
ˆdf Jj + (∂iν − ˆdT
f Ji)O−1
(∂jν + ˆdf Jj) .
(B.6)
C Classical integrability
Here we wish to be more explicit and show that the on-shell equations of DTD models
can be recast into the flatness condition for a Lax connection. The argument follows the
one presented in [1] in the case of DTD of Principal Chiral Models. First we compute the
equations of motion for f and ν, which are obtained by the straightforward variations δf S
and δνS of the action
δf S = +
T
2
d2
σ Str f−1
δf C ,
δνS = −
T
2
d2
σ Str δν F
˜A
= −
T
2
d2
σ Str (Ad−1
f δν)FA
,
(C.1)
where we defined
C ≡ ∂+i( ˆdAi
−) + ∂−i( ˆdT
Ai
+) + [A+i, ˆdAi
−] + [A−i, ˆdT
Ai
+],
FA
≡ ∂+iAi
− − ∂−iAi
+ + [A+i, Ai
−] = − ij
(∂iAj + AiAj),
(C.2)
and similarly for F
˜A. Notice that P(0)C = 0. For convenience we also introduced the
(projections of the) field Ai
± ≡ Ad−1
f ( ˜Ai
± + Ji
±), where ˜Ai
± is given in (B.5). On the one
hand, imposing the equations of motion δνS = 0 is enough to get FA = 0. Notice that this
equation is equivalent to imposing separately F
˜A = 0 and FJ ≡ ∂+iJi
−−∂−iJi
+−[J+i, Ji
−] =
0. On the other hand, the equations of motion δf S = 0 imply that C vanishes only on
a certain subspace of the superalgebra g. In fact, in the special case when the whole
superalgebra is dualised ˜g = g, there is no f for which we can compute the variation of
the action, and we should find an independent argument to claim that the equation C = 0
holds. We will now show that an appropriate (rotated) projection of C by ˜PT indeed
vanishes without appealing to the equations of motion for f. Consider the equations of
motion for ˜Ai
± in (B.2) and let us rewrite them as Ei
± − Mi⊥
± = 0 where
Ei
+ ≡ +(∂i
+ + ad ˜Ai
+
)ν − ˆdT
f (Ji
+ + ˜Ai
+) − ζω ˜Ai
+,
Ei
− ≡ −(∂i
− + ad ˜Ai
−
)ν − ˆdf (Ji
− + ˜Ai
−) + ζω ˜Ai
−.
(C.3)
Since we choose Mi⊥
± to take values only in the complement of ˜g∗, taking ˜PT Ei
± = 0 gives
indeed (B.2). Clearly (∂+i +ad ˜A+i
)(Ei
− −Mi⊥
− )+(∂−i +ad ˜A−i
)(Ei
+ −Mi⊥
+ ) = 0 is identically
true since it just follows from the above equations, and working out all the terms we find
Adf C = [ν, F
˜A
] + ζωF
˜A
− (∂−i + ad ˜A−i
)Mi⊥
+ − (∂+i + ad ˜A+i
)Mi⊥
− . (C.4)
– 25 –
JHEP10(2017)024
After projecting with ˜PT all terms with Mi⊥
± disappear. The remaining terms on the
right-hand-side of the above equation vanish thanks to the flatness of ˜A (F
˜A = 0) implied
by the equations of motion for ν. To conclude, we obtain ˜PT (Adf C) = 0 as wanted,
which together with the equations of motion for f is enough to claim C = 0 on the whole
superalgebra.
The on-shell equations FA = 0 and C = 0 formally take the same form as those for a
supercoset, where in that case A is the Maurer-Cartan form, see also [47, 48]. Therefore
one may follow the derivation done in the case of the supercoset, and find that they are
encoded in the flatness condition
ij
(∂iLj + LiLj) = 0, (C.5)
for the Lax connection
Li = A
(0)
i + zA
(1)
i +
1
2
z2
+ z−2
A
(2)
i +
1
2
γij
jk
z−2
− z2
A
(2)
i + z−1
A
(3)
i , (C.6)
where z is the spectral parameter. The existence of a Lax connection implies the presence
of a tower of conserved charges, see e.g. [49] for a review. However, differently from the
case of the supercoset, now fewer of them can be argued to be local. In fact, thanks to the
gauge transformation it is always possible to define
Li = hLih−1
− ∂ihh−1
, (C.7)
so that Li is also flat. In the case of the supercoset, after noticing that Li(z = 1) =
Ai = g−1∂ig, one may choose h = g so that the new Lax connection vanishes at z = 1
Li(z = 1) = 0. Expanding around that point one finds
Li(z = 1 + w) = w g A
(1)
i − 2γij
jk
A
(2)
k − A
(3)
i g−1
+ O(w2
), (C.8)
so that the flatness condition for Li at order w implies the conservation ∂iAi = 0 for the
current
Ai
= ij
g A
(1)
j − 2γjk
kl
A
(2)
l − A
(3)
j g−1
. (C.9)
This is how in the supercoset case one can argue from the Lax connection that the isometries
corresponding to the superalgebra g correspond to local charges. In the case of DTD models
A is not of the Maurer-Cartan form, and in general it is not possible to find a group element
h for which a gauge-equivalent Lax connection vanishes at z = 1. With the exception of
the isometries discussed in section 3, we therefore expect that the initial symmetries of the
undeformed model are traded for non-local charges.
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
– 26 –
JHEP10(2017)024
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Integrable Deformations of T-Dual σ Models
Riccardo Borsato*
and Linus Wulff†
The Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom
(Received 21 October 2016; published 16 December 2016)
We present a method to deform (generically non-Abelian) T duals of two-dimensional σ models, which
preserves classical integrability. The deformed models are identified by a linear operator ω on the dualized
subalgebra, which satisfies the 2-cocycle condition. We prove that the so-called homogeneous Yang-Baxter
deformations are equivalent, via a field redefinition, to our deformed models when ω is invertible. We
explain the details for deformations of T duals of principal chiral models, and present the corresponding
generalization to the case of supercoset models.
DOI: 10.1103/PhysRevLett.117.251602
Introduction.—Integrable models in two dimensions
have played a pivotal role in the understanding of (quantum)
field theory, have numerous applications in condensed
matter theory, and have recently attracted attention also in
the context of the AdS=CFT correspondence [1], which
relates certain string theories on (d þ 1)-dimensional anti–
de Sitter (AdS) backgrounds to conformal field theories in
d dimensions. The most studied example that exhibits
integrable structures is that of the superstring on AdS5 × S5
[2] and its dual N ¼ 4 super Yang-Mills theory in four
dimensions [3], see Refs. [4,5] for reviews. On the string
side the two-dimensional world sheet theory is classically
integrable; i.e., there is a Lax pair whose flatness condition
is equivalent to the equations of motion of the σ model. The
Lax pair depends on an auxiliary spectral parameter z, and
its expansion around a fixed z0 yields an infinite set of
conserved charges, see Ref. [6] for a review. Integrability
has provided the most stringent tests of AdS=CFT, culminating
with the possibility of computing the spectrum of the
quantum theory in the large N limit exactly [7–10].
Given this tremendous success it is natural to ask whether
other theories that are not maximally (super)symmetric are
still integrable. Integrability could then also be a guiding
principle to discover new models that are interesting in their
own right. The β deformation [11–13] or certain gravity
duals of noncommutative gauge theories [14,15] are examples
that are integrable but reduce to the maximally
symmetric case only when a deformation parameter is sent
to zero. These instances actually fall into a larger class that
goes under the name of Yang-Baxter (YB) models [16–19],
sometimes also called η deformations after the deformation
parameter. AYB model is identified by an R matrix solving
the classical Yang-Baxter equation (CYBE), which in
general has a rich set of solutions. Each R generates a
background that reduces to the undeformed model (e.g.,
AdS5 × S5) in the η → 0 limit. Here, we will not consider
the case of the “modified” CYBE.
In this Letter we explore another possibility; we deform
the original σ model by adding a topological term (a closed
B field) and then apply non-Abelian T duality (NATD) [20]
with respect to a subgroup ~G of the isometry group G. The
special case when ~G is Abelian gives so-called TsT
transformations [11–13]. We refer to the resulting actions
as deformed T dual (DTD) models, since sending the
deformation parameter ζ → 0 they reduce to NATD.
DTD models are in one-to-one correspondence with the
2-cocycles ω of the Lie algebra of ~G. The cocycle condition
(3) guarantees that integrability is preserved, and plays the
same role as the CYBE for YB models.
The analogy goes even further. When ω is invertible its
inverse R ¼ ω−1
solves the CYBE, and each solution of the
CYBE corresponds to an invertible 2-cocycle [21]. We use
this identification to show that the action of YB models can
be recast in the form of DTD models, where the two
deformation parameters are simply related by η ¼ ζ−1. As
explained later, this translates into our language a recent
conjecture by Hoare and Tseytlin [22]. We prove it by
providing the explicit field redefinition that relates YB to
DTD models. The field redefinition is local, albeit in
general nonlinear, and it allows us to interpolate between
a certain σ model (ζ → ∞) and its NATD (ζ → 0). In the
case when ω is degenerate, the DTD model is equivalent to
a combination of YB deformation and NATD.
We first construct the DTD of the principal chiral model
(PCM), since it provides a simpler setup where all the
essential features already appear. Later, we generalize it to
the case of supercosets, which is more relevant to the study
of deformations of superstrings. The supercoset case will be
described in more detail elsewhere [23].
DTD of the PCM.—We start from a PCM parametrized
by a group element g ∈ G, with the familiar action
S½gŠ ¼ − 1
2
R
Trðg−1
∂þgg−1
∂−gÞ. Since we want to dualize
a ~G subgroup of the left copy of G [24] we rewrite [25]
S½f; ~A; νŠ ¼ −
1
2
Z
Trðð ~Aþ þ JþÞð ~A− þ J−Þ þ ν ~Fþ−Þ: ð1Þ
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Here, J ¼ dff−1 is a right-invariant Maurer-Cartan form
for f ∈ G, depending on fields that remain spectators under
NATD. At the same time ~A ∈ ~g and ν ∈ ~gà identify each of
the two T-dual frames. If Ti are generators for ~g, a basis for
the dual algebra ~gÃ
is given by Ti
, where TrðTiTj
Þ ¼ δj
i .
The curvature of ~A is ~Fþ− ¼ ∂þ
~A− − ∂−
~Aþ þ ½ ~Aþ; ~A−Š.
The original PCM is recovered upon integrating out ν since
~Fþ− ¼ 0 implies that ~A is pure gauge, i.e., ~A ¼ ¯g−1
d¯g for a
¯g ∈ ~G, and we get the desired action with g ¼ ¯gf. The
NATD with respect to ~G, on the other hand, is obtained by
integrating out ~A.
We now add a deformation with parameter ζ given by
S0
½f; ~A; νŠ ¼ S½f; ~A; νŠ þ
ζ
2
Z
Trð ~Aþω ~A−Þ: ð2Þ
Here, ω: ~g → ~gÃ
is a linear antisymmetric [i.e., TrðxωyÞ ¼
−TrðωxyÞ] map satisfying the cocycle condition [26]
ωadxy ¼ adxωy − adyωx; ∀ x; y ∈ ~g: ð3Þ
This property is needed to have local ~G invariance also
for ζ ≠ 0, which ensures that # d:o:f: ¼ dimðGÞ [27].
Equations of motion for ~A give
R
Trðδ ~A∓EÆÞ ¼ 0, where
EÆ ≡ ð1 Æ adν Æ ζωÞ ~AÆ ∓ ∂Æν þ JÆ: ð4Þ
This implies ~PT
EÆ ¼ 0, where ~P projects onto ~g, ~PT
onto
~gÃ
. We solve these equations by defining the linear operator
~O ¼ ~PT
ð1 − adν − ζωÞ ~P, which is a map ~g → ~gÃ
~A− ¼ ~O−1
ð−∂−ν − J−Þ; ~Aþ ¼ ~O−T
ð∂þν − JþÞ ð5Þ
and ~O−T
is the inverse of its transpose. Note that
~O−1 ~O ¼ ~P as the lhs is defined only on ~g. Evaluating S0
on the solution we get the DTD action
S0
½f; νŠ ¼ −
1
2
Z
TrðJþJ− þ ð∂þν − JþÞ ~O−1
ð∂−ν þ J−ÞÞ:
ð6Þ
A second interpretation of DTD comes from integrating out
ν rather than ~A from Eq. (2), which gives again ~A ¼ ¯g−1
d¯g.
The resulting action is a topological deformation of the
PCM, since the cocycle condition implies that B ¼
ζωð¯g−1
d¯g; ¯g−1
d¯gÞ is closed. At the classical level adding
this term has no effect, and in fact this picture of a
deformation that is trivial in the dual frame is reminiscent
of YB models: in some cases they correspond to TsT
transformations [22,28–30], which are just field redefinitions
in a T-dual frame. Since DTD is a NATD of a
topological deformation of the PCM, it is classically
integrable, where NATD can be applied thanks to closure
of B. In fact, the equation ~A ¼ ¯g−1
d¯g with ~A given in
Eq. (5) allows us to relate the variables of the deformed
model to those of the original PCM. In the special case of
Abelian subalgebra ~g the relation simplifies and the
deformed model becomes equivalent to the PCM with
twisted boundary conditions, consistent with the TsT
interpretation [12].
A third interpretation of DTD comes from the possibility
of applying NATD to a centrally extended subalgebra. This
idea first appeared in Ref. [22] and was the original
motivation for considering the deformation (2). One can
indeed replace ~A in Eq. (1) with ~A0
∈ ~gc:e: ¼ ~g ⊕ c and c
central; similarly ν0
∈ ~gÃ
c:e:. We decompose ~A0
¼ ~A þ ~Ac
,
ν0
¼ ν þ νc
with obvious notation, and extend the definition
of the trace Trðc2Þ ¼ 1, TrðcgÞ ¼ 0. Equations for
~Ac
imply that νc
is constant, νc
¼ ζc. At this point
Trðν0 ~F0
þ−Þ ¼ Trðν ~Fþ−Þ þ ζfab
~Aa
þ
~Ab
−, where fab are the
structure constants introduced by the central extension
½Ta; TbŠ ¼ fc
abTc þ fabc. Introducing a map ω whose
components are ωab ¼ −fab we just notice that it is
antisymmetric and satisfies the cocycle condition, a consequence
of the Jacobi identity in ~gc:e: projected on c.
For some ω’s DTD reduces to just NATD; i.e.,
the deformation parameter can be removed by a field
redefinition. This happens when ω is a coboundary, i.e.,
ωðx; yÞ ¼ fð½x; yŠÞ for some function f. Therefore, nontrivial
deformations are in one-to-one correspondence with
2-cocycles modulo coboundaries, i.e., with elements of the
second cohomology group H2
ð~gÞ. The same holds also for
nontrivial central extensions. In particular, there are none
for semisimple ~g. Trivial deformations are equivalently
described as adding an exact B field to the PCM.
An example.—Before continuing our general discussion,
let us provide an explicit example: a PCM on Uð2Þ. We use
generators Tj ¼ iσj ∈ suð2Þ and T4 ¼ i1, with duals Tj
¼
−ði=2Þσj and T4 ¼ −ði=2Þ1. We parametrize the group
element by g¼expðiθ1Þexpðiϕþσ1ÞˇgðξÞexpðiϕ−σ2Þ, where
ϕÆ ¼ ðϕ1 Æ ϕ2Þ=2 and ˇgðξÞ ¼ diagði−1=2
eiξ
; i1=2
e−iξ
Þ. The
PCM action yields the metric of S3
× S1
ds2
¼ dξ2
þ sin2
ξdϕ2
1 þ cos2
ξdϕ2
2 þ dθ2
: ð7Þ
Suppose we want to dualize the coordinates ϕþ in S3
and θ
in S1
, corresponding to the Abelian subalgebra ~g ¼
spanfT1; T4g. We take f ¼ ˇgðξÞ expðiϕ−σ2Þ and ν ¼
2ð~ϕþT1
þ ~θT4
Þ, where ~ϕþ, ~θ are dual coordinates. We
deform the dual theory by taking ω ¼ 2T1
∧ T4
; namely,
ωT1 ¼ −2T4
, ωT4 ¼ 2T1
. From Eq. (6) we find the action
of DTD S0 ¼
R
∂þXiðGij − BijÞ∂−Xj, with the metric and
B field
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ds2 ¼ dξ2 þ ð1 þ ζ2Þ−1ðd ~ϕ2
þ þ ðζ2 þ sin22ξÞdϕ2
−
þ d~θ2
þ 2ζ cos 2ξd~θdϕ−Þ;
B ¼ ð1 þ ζ2
Þ−1
ðcos 2ξdϕ− − ζd~θÞ ∧ d ~ϕþ: ð8Þ
The ζ → 0 limit yields the T-dual model of S3
× S1
with respect to ~g. To relate this simple example to a YB
model it is enough to take ν ¼ η−1
RðϑT4
þ φþT1
Þ with
R ¼ 1
2 ðT4 ∧ T1Þ. However, when ~g is non-Abelian, the
field redefinition is more complicated, see Eq. (13).
Integrability.—Above we argued that DTD models must
be integrable; however, it is instructive to show this
explicitly to see how the cocycle condition enters and
write a Lax connection. We will show that the equations of
motion formally resemble those of the PCM, for which a
Lax pair is known. Suppose we consider a PCM with group
element g ¼ ¯gf, with ¯g ∈ ~G, f ∈ G. We prefer to rewrite its
on-shell equations in terms of the left and right currents
~A ¼ ¯g−1
d¯g and J ¼ dff−1
. To start, the flatness condition
for A ¼ g−1
dg is equivalent to FJ
¼ 0, F
~A
¼ 0
FJ
≡ ∂þJ− − ∂−Jþ − ½Jþ; J−Š;
F
~A
≡ ∂þ
~A− − ∂−
~Aþ þ ½ ~Aþ; ~A−Š: ð9Þ
Moreover, the equations of motion for the PCM, i.e.,
conservation of A, become C ¼ 0,
C ≡ ∂þðJ− þ ~A−Þ þ ∂−ðJþ þ ~AþÞ
þ ½ ~Aþ; J−Š þ ½ ~A−; JþŠ: ð10Þ
Let us now rederive the above equations for DTD models,
where now importantly ~A is identified as in Eq. (5). To start,
the flatness condition FJ
¼ 0 still follows from the
definition of J. Flatness for ~A, instead, now arises as the
equations of motion for ν, which are δνS0
½f; νŠ ¼
− 1
2
R
TrðδνF
~A
Þ ¼ 0. It is nice that the known mechanism
familiar from T duality of trading flatness for an equation of
motion still holds for DTD models.
The equations of motion for f are δfS0
½f; νŠ ¼
þ 1
2
R
Trðδff−1
CÞ ¼ 0, essentially as in the previous example
of the PCM. However, in that case it is only thanks to
the equations of motion for ¯g [i.e.,
R
Trð¯g−1δ¯gCÞ ¼ 0] that
one can claim C ¼ 0. In analogy to the PCM, it is then clear
that our task is to show that ~PT
C ¼ 0 also for DTD models.
We generalize the argument of Ref. [31] for NATD of the
PCM, and consider the equations EÆ ¼ M⊥
Æ, for some M⊥
Æ
for which ~PT
M⊥
Æ ¼ 0. They imply ~PT
EÆ ¼ 0; i.e., they are
equivalent to the solutions for ~A as in Eq. (5). They
obviously imply also the equation ð∂þ þad~Aþ
ÞðE− −M⊥
− Þþ
ð∂− þad~A−
ÞðEþ −M⊥
þÞ ¼ 0, which reads as
C ¼ ½∂− þ ad~A−
; ∂þ þ ad~Aþ
Šν
− ð∂− þ ad~A−
ÞM⊥
þ − ð∂þ þ ad~Aþ
ÞM⊥
−
þ ζ½ωð∂þ
~A− − ∂−
~AþÞ þ ad~Aþ
ω ~A− − ad~A−
ω ~AþŠ:
The first line on the right-hand side is rewritten as ½ν; ~Fþ−Š,
and hence vanishes thanks to the flatness of ~A. The second
line vanishes upon projecting with ~PT
[32]. Finally, the last
line vanishes thanks to the cocycle condition: using Eq. (3)
it is rewritten as −ζωð ~Fþ−Þ, which is again zero. Since also
~PT
C ¼ 0 holds, we conclude that the whole set of on-shell
equations for the DTD models is formally equivalent to
those of a PCM, provided the proper ~A is used. We can
furthermore write the Lax pair as
LÆ ¼
1
2
ð1 þ z∓2
ÞAd−1
f ð ~AÆ þ JÆÞ ð11Þ
with z a spectral parameter. In fact, the flatness condition
∂þL− − ∂−Lþ þ ½Lþ; L−Š ¼ 0 is equivalent to the on-shell
equations just derived.
Relation to Yang-Baxter models.—We now prove that
YB deformations for the PCM on the group G are
equivalent to DTD. This was checked for many particular
examples in Ref. [22]. YB models are identified by an R
matrix solving the CYBE on the Lie algebra g. If g ∈ G
SYB½gŠ ¼ −
1
2
Z
Tr

g−1∂þg
1
1 − ηRg
g−1∂−g

: ð12Þ
R is invertible on a certain subalgebra and its inverse is a
2-cocycle [21]. As anticipated, we identify R ¼ ω−1
, where
ω is the operator defining the DTD model. Then, R: ~gÃ
→ ~g.
The two deformation parameters will be related by η ¼ ζ−1
.
We first split the group element parametrizing the YB
model as g ¼ ~gf, where ~g ∈ ~G and f ∈ G. We identify f
with the homonym appearing on the DTD side. Our proof
of equivalence of the two actions will then consist in giving
the field redefinition relating ~g and ν. Since R is invertible,
we can always take ~g ¼ expðRXÞ for some X ∈ ~gÃ
. One
can check that taking X ¼ ην þ ðη2
=2Þ ~PT
½Rν; νŠ þ Oðη3
Þ
the two actions are equivalent up to terms that are at least
cubic in η. The generalization to all orders can be obtained
by requiring that the dfdf terms in the two actions match.
This leads to the condition ð1 − ηR~gÞ−1
¼ 1 − ~O−1
whose
solution can be shown to be
ν ¼
1
η
~PT 1 − e−adRX
adRX
X ¼
1
η
~PT
1 − Ad−1
~g
log Ad~g
ω log ~g: ð13Þ
It follows that dν ¼ ð ~PT
− ~OÞ~g−1
d~g or, equivalently,
AÆ ¼ Ad−1
f ðJÆ þ ~AÆÞ; ð14Þ
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where we defined AÆ ¼ ð1 Æ ηRgÞ−1ðg−1∂ÆgÞ on the YB
side. Using these relations it is not hard to check that the
two actions are the same up to the topological term
ζωð~g−1
d~g; ~g−1
d~gÞ, which has no effect in the classical
theory as remarked earlier.
We have proven the equivalence of the DTD and YB
models when ω is nondegenerate. In the case of a
degenerate ω it is always possible to choose it in such a
way that it is nondegenerate on a subalgebra ˆg ⊂ ~g [33] and
acts trivially on its complement ˇg in ~g, also an algebra
thanks to Eq. (3). We interpret it as NATD on ˇg of the YB
model corresponding to restricting ω to ˆg.
DTD of supercosets.—The construction of DTD models
for supercosets follows the steps explained in the simpler
case of the PCM. Here, we only present the main results,
whose derivation will be collected in Ref. [23].
We still denote by G the group of superisometries, e.g.,
PSUð2; 2j4Þ for superstrings on AdS5 × S5
, see Ref. [34]
for a review. Its Lie superalgebra g admits a Z4 decomposition,
and we denote by PðjÞ
the projectors onto the four
subspaces. They typically appear in the combination
ˆd ¼ Pð1Þ þ 2Pð2Þ − Pð3Þ or its transpose ˆdT
. The absence
of Pð0Þ
in ˆd is necessary for the local gð0Þ
invariance of the
action, i.e., local Lorentz transformations. The action for
DTD models of supercosets is [35]
S0
½f;νŠ
¼ −
T
2
Z
StrðJþ
ˆdfJ− þð∂þν− ˆdT
f JþÞ ~O−1
ð∂−νþ ˆdfJ−ÞÞ;
ð15Þ
where ˆdf ≡ Adf
ˆdAd−1
f . We keep the same definitions for J,
ν, which however now take values in superalgebras.
Moreover, now ~O ¼ ~PT
ðˆdf − adν − ζωÞ ~P.
The model is integrable since we can write down a Lax
pair. This is more conveniently expressed in terms of
A ¼ Ad−1
f ð ~A þ JÞ, where
~Aþ ¼ ~O−T
ðþ∂þν − ˆdT
f JþÞ;
~A− ¼ ~O−1
ð−∂−ν − ˆdfJ−Þ: ð16Þ
The flatness condition ∂þL− − ∂−Lþ þ ½Lþ; L−Š ¼ 0 for
LÆ ¼ A
ð0Þ
Æ þ zA
ð1Þ
Æ þ z∓2A
ð2Þ
Æ þ z−1A
ð3Þ
Æ ð17Þ
is equivalent to the on-shell equations of the DTD model.
DTD models of supercosets possess kappa symmetry,
and therefore correspond to solutions of the generalized
supergravity equations of Refs. [36,37]. Kappa symmetry
transformations are δff−1
¼ ˆdT
f ðδνÞ ¼ ρ1;− þ ρ3;þ, where
ρj;Æ ¼ fiAdfκðjÞ
; J
ð2Þ
Æ þ ~A
ð2Þ
Æ g ð18Þ
and κðjÞ
, j ¼ 1, 3 are two local parameters of grading j. The
action (15) is invariant under these transformations upon
using the Virasoro constraints. If we were not fixing
conformal gauge, the variation of the action would be
compensated by the variation of the world sheet metric.
From these kappa symmetry transformations it is possible
to extract the background fields of DTD models [23].
The equivalence to YB models for invertible ω’s
holds also in the case of DTD models of supercosets.
Remarkably, the field redefinition is still given by Eq. (13)
as for the PCM. We have further verified that kappa
symmetry transformations of YB models [18] take the
above form under this field redefinition, when we fix the ~G
gauge to get δff−1 ¼ ˆdT
f ðδνÞ.
Conclusions.—We provided a unified picture of (nonAbelian)
T duality and homogeneous YB deformations as
DTD of σ models. As pointed out in Ref. [22], an advantage
of this formulation is that it can be realized at the path
integral level, giving a better handle on the quantum theory.
In fact, it also explains why the condition for one-loop
Weyl invariance, i.e., unimodularity of ~g, is the same for
both the YB model and NATD [30,38,39].
Despite the close relation, it is still worth viewing the
DTD models as a distinct class of deformations. In fact, the
field redefinition that relates it to the YB model is singular
in the two undeformed limits; the YB model becomes
degenerate when taking the undeformed (i.e., ζ → 0) limit
of DTD models, and vice versa. Therefore, the interpretation
as deformation applies to just one of the two models in
the T-dual pair. It would be interesting to understand if
there is any connection to the λ model of Refs. [31,40,41],
which is also a deformation of NATD and is related to the
inhomogeneous YB deformation [16–18].
Although our motivation was integrability, such deformations
can be applied also to nonintegrable models, which
provides an interesting and potentially useful way to
generate new supergravity solutions.
We thank Ben Hoare, Stijn van Tongeren, and Arkady
Tseytlin for interesting discussions and comments on the
Letter. R. B. thanks also Bogdan Stefański for related
discussions. R. B. thanks also Wim Hennink and his group
in Utrecht for the kind hospitality during part of this
project. This work was supported by ERC Advanced
Grant No. 290456. The work of L. W. was also supported
by STFC Consolidated Grant No. ST/L00044X/1.
*
riccardo.borsato@su.se
†
l.wulff@imperial.ac.uk
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[40] T. J. Hollowood, J. L. Miramontes, and D. M. Schmidtt,
J. High Energy Phys. 11 (2014) 009.
[41] T. J. Hollowood, J. L. Miramontes, and D. M. Schmidtt,
J. Phys. A 47, 495402 (2014).
PRL 117, 251602 (2016) P H Y S I C A L R E V I E W L E T T E R S
week ending
16 DECEMBER 2016
251602-5
JHEP10(2016)045
Published for SISSA by Springer
Received: September 1, 2016
Accepted: October 7, 2016
Published: October 10, 2016
Target space supergeometry of η and λ-deformed
strings
Riccardo Borsato and Linus Wulff
Blackett Laboratory, Imperial College,
London SW7 2AZ, U.K.
E-mail: r.borsato@imperial.ac.uk, l.wulff@imperial.ac.uk
Abstract: We study the integrable η and λ-deformations of supercoset string sigma models,
the basic example being the deformation of the AdS5 × S5 superstring. We prove that
the kappa symmetry variations for these models are of the standard Green-Schwarz form,
and we determine the target space supergeometry by computing the superspace torsion.
We check that the λ-deformation gives rise to a standard (generically type II*) supergravity
background; for the η-model the requirement that the target space is a supergravity
solution translates into a simple condition on the R-matrix which enters the definition of
the deformation. We further construct all such non-abelian R-matrices of rank four which
solve the homogeneous classical Yang-Baxter equation for the algebra so(2, 4). We argue
that most of the corresponding backgrounds are equivalent to sequences of non-commuting
TsT-transformations, and verify this explicitly for some of the examples.
Keywords: Integrable Field Theories, Supergravity Models, Superstrings and Heterotic
Strings, Sigma Models
ArXiv ePrint: 1608.03570
Open Access, c The Authors.
Article funded by SCOAP3
.
doi:10.1007/JHEP10(2016)045
JHEP10(2016)045
Contents
1 Introduction and summary of results 1
2 η and λ-deformed string sigma models 6
2.1 Lagrangians of the deformed models 7
2.2 Kappa symmetry transformations in Green-Schwarz form 10
3 Target superspace for the λ-model 11
4 Target superspace for the η-model 14
4.1 Dilaton and supergravity condition 15
5 Non-abelian R-matrices and the unimodularity condition 16
6 Some examples of supergravity backgrounds 22
6.1 h3 ⊕ Ê 24
6.2 r′
3,0 ⊕ Ê 25
6.3 r3,−1 ⊕ Ê 26
6.4 n4 26
7 Conclusions 27
A 4-graded superisometry algebras 28
B Useful results for the deformed models 30
B.1 λ-model 31
B.2 η-model 33
1 Introduction and summary of results
A remarkable property of the AdS5 × S5 superstring sigma model is its classical integrability
[1], see [2] for a review. In fact, this property extends to several other symmetric
space string backgrounds [3, 4]. Recently two interesting deformations of the AdS5 × S5
superstring sigma model1 were proposed which preserve the integrability. The η-model [5]
and λ-model [6], named after the corresponding deformation parameters. The former is
based on the Yang-Baxter deformation of [7–9], it generalises the case of bosonic coset
models [10], and its essential ingredient is an R-matrix which satisfies the modified classical
Yang-Baxter equation (MCYBE). The λ-model was originally proposed by [11] and
1
These deformations extend to any 4-symmetric supercoset sigma model, i.e. symmetric space RR
string background preserving supersymmetry.
– 1 –
JHEP10(2016)045
it extends the case of bosonic cosets [12] (see also [13]). The construction is based on a
G/G gauged Wess-Zumino-Witten (WZW) model, and it is more naturally interpreted as
a deformation of the non-abelian T-dual of the original string. The two deformations are
closely related; in fact, in both cases the original symmetry algebra gets q-deformed [14, 15]
(with q real and root of unity respectively), and the two models are related, at least at the
classical level, by the Poisson-Lie T-duality of [16, 17], see [18–20].
The attempt of interpreting these deformations as string theories has raised interesting
questions. In fact, both models possess a local fermionic symmetry believed to be the standard
kappa symmetry — which was expected to guarantee a string theory interpretation.
However, the target space of the η-model derived in [21, 22]2 does not solve the type IIB
supergravity equations [22], but rather a generalisation of them as suggested in [25]. These
generalised equations ensure scale invariance for the sigma model, but are not enough to
have the full Weyl invariance, which is present only when the target space satisfies the
standard equations of supergravity. For the λ-model, on the other hand, it was shown that
the target space does solve the supergravity equations, at least in the case of λ-deformed
AdS2 × S2 × T6 [26] and AdS3 × S3 × T4 [27] string sigma models.3
A possible resolution for the puzzle posed by the η-model could have been that, after
all, the possessed local fermionic symmetry was not the standard kappa symmetry of GreenSchwarz.
However this state of affairs was clarified recently in [30] where it was shown that,
contrary to what was commonly believed, kappa symmetry of the type II Green-Schwarz
superstring does not imply the full equations of motion of type II supergravity.4 Rather it
implies a weaker (generalized) version of these equations, whose bosonic subsector coincides
with the equations written down in [25]. These generalized supergravity equations involve
a Killing vector field Ka, and reduce to the standard type II supergravity equations when
this vector field is set to zero. This fact implies that kappa-symmetric backgrounds whose
metric does not allow for isometries must in fact solve the standard type II equations.
The λ-model falls into this class, which is consistent with the fact that the corresponding
target spaces were found to be supergravity backgrounds.5 On the other hand, the ηmodel
typically leads to a target-space metric which possesses isometries, so that a priori
it is not possible to exclude the possibility that it solves only the generalized supergravity
equations. It should be mentioned that, given a solution of the generalized supergravity
equations and provided that Ka is space-like, it is possible to find a genuine supergravity
solution which is formally T-dual to it [25, 32] (i.e. only at the classical level of the sigma
model, ignoring the fact that the dilaton is linear in the coordinate along which T-duality
is implemented). We will not consider this possibility here.
2
See [23] for lower dimensional examples of bosonic truncations and [24] for a review.
3
These results differ from the ones proposed in [28]. The metric in target space of the λ-deformed
AdS5 × S5
was obtained in [29].
4
Earlier indications of this was seen in the pure spinor string in [31].
5
We will actually see that the kappa symmetry transformations of the λ-model take the standard form
only after inserting proper factors of i (see section 2.2). This leads to a target space geometry which is a
solution of type II* rather than type II supergravity. In the case of AdS2 × S2
× T6
[26] it was shown how
one can get a standard (and real) type IIB background by analytic continuation, or equivalently by picking
a different coordinate patch. The same should be true for the deformation of AdS3 × S3
× T4
[27], and
probably AdS5 × S5
.
– 2 –
JHEP10(2016)045
Target space supergeometry. The procedure for the η-deformation can be generalised6
also to the case when the R-matrix satisfies the classical Yang-Baxter equation (CYBE) [33–
35]. Therefore several solutions exist and the question is which choices lead to a string
theory, i.e. a target space that solves the standard type II supergravity equations. Here
we will answer this question and find a simple (necessary and sufficient) condition on R.
We will also determine the form of the target space (super) fields for both the η and the
λ-model in terms of the ingredients that define them (see section 2 for their definition);
we check that the models can be written in Green-Schwarz form and we work out the
superspace torsion. The target space fields can then be read off by comparing to the
expressions in [30, 36]. This gives a simple way of extracting the target space backgrounds,
much simpler than previous methods. The metric and B-field are easily read off directly
from the sigma model Lagrangian, see (2.7). The NSNS three-form and RR fluxes are
found to be given by the expressions7
Habc = 3M[ab,c] − 3i
ˆη2
−λ2
M ˆα2
[a(γb)ˆαˆβM
ˆβ2
c] , (1.1)
S ˆα1ˆβ2
= 8i
[Adh(1 + 2ˆη−2 − 4O−1
+ )]ˆα1
ˆγ1
iλ[Adh(1 + λ(1 − λ−4)O−1
+ )]ˆα1
ˆγ1
Kˆγ1ˆβ2
, (1.2)
where the upper (lower) expression in curly brackets refers to the η (λ) model and ˆη =
1 − cη2. The RR field strengths are encoded in the bispinor defined as [30, 36]
S = −iσ2
γa
Fa −
1
3!
σ1
γabc
Fabc −
1
2 · 5!
iσ2
γabcde
Fabcde , (1.3)
where for standard supergravity backgrounds F = eφF contains the exponential of the
dilaton. The remaining ingredients in these equations are defined in section 2, in particular
the operators O+, M and the group element h are defined in (2.5), (2.2), (B.2) and (2.12).
From our computation we obtain also the Killing vector of the generalised type II equations
Ka
= −
i
16
(γa
)ˆαˆβ
(∇ˆα1χˆβ1 − ∇ˆα2χˆβ2) , (1.4)
where χI (I = 1, 2) are the would be dilatino superfields
χ1
ˆα =
i
2
ˆη
−1
γb
ˆαˆβ
[AdhM]
ˆβ1
b , χ2
ˆα = −
i
2
ˆη
iλ
γa
ˆαˆβ
M
ˆβ2
a . (1.5)
When Ka vanishes we have a standard supergravity solution and the dilaton is given by8
e−2φ
= sdet(O+) . (1.6)
6
We will use the names “η-deformation” and “Yang-Baxter deformation” for both the homogeneous
(CYBE) and inhomogeneous (MCYBE) cases, as we can treat them both at the same time.
7
Note that here we write the λ-model as a solution of type IIB supergravity, and the corresponding RR
flux is imaginary. The background is real when written as a solution of type IIB*. The reason for this is a
non-standard sign in the kappa symmetry transformations of the lambda model, see sec 2.2.
8
For the λ-model this formula was argued in [6]. It is also consistent with the form of the bosonic dilaton
suggested in [37] for the η-model based on bosonic R-matrices.
– 3 –
JHEP10(2016)045
For the λ-model Ka automatically vanishes and the target space is always a supergravity
solution, consistently with the observation of [30] and the previous findings [26, 27].
The η-model as a string. For the η-model the situation is more subtle. Let us review
some details at this point and recall that the η-deformation is defined by an antisymmetric
R-matrix on the algebra R : g → g, RT = −R, satisfying the (M)CYBE
[R(x), R(y)] − R([R(x), y] + [x, R(y)]) = c[x, y] , ∀x, y ∈ g ,
c = 0 CYBE
c = ±1 MCYBE
. (1.7)
In section 4.1 we prove that the condition Ka = 0 for the η-model is equivalent to the
following algebraic condition on the R-matrix9
STr(Radx) = 0 , ∀x ∈ g (i.e. RB
AfA
BC = 0) . (1.8)
We will refer to R-matrices satisfying this condition10 as “unimodular”, for reasons that
will be clear in section 5. Therefore the η-model has an interpretation as a string sigma
model precisely for the unimodular R-matrices.
Let us consider the η-deformation based on an R-matrix which is a non-split11 (c =
1 in (1.7)) solution of the MCYBE for the supercoset on AdS5 × S5 with superalgebra
psu(2, 2|4), as in [5]. A standard choice is to take R that multiplies by −i (+i) positive
(negative) roots of the complexified algebra, and annihilates Cartan elements. Choices of
different real forms of the superalgebra correspond to inequivalent R-matrices, but one can
check that none of the examples considered so far [5, 14, 22, 39] are unimodular, which
is consistent with the findings of [22, 39]. We are not aware of a complete classification
of solutions of the MCYBE for psu(2, 2|4), which leaves open the possibility of having
unimodular non-split R-matrices that would lead to genuine string deformations. We will
not analyze this question further here.
As first pointed out in [33], there is a rich set of solutions to the CYBE (c = 0 in (1.7))
which can be used to define an η-deformation of the supercoset. These R-matrices can
be divided into two classes: abelian and non-abelian. Writing the R-matrix as (sums over
repeated indices are understood)
R =
1
2
rij
bi ∧ bj , (R(x) = rij
biStr(bjx), x ∈ g), (1.9)
abelian R-matrices are the ones for which [bi, bj] = 0 ∀i, j while non-abelian ones have
[bi, bj] = 0 for some i, j. The unimodularity condition (1.8) takes the form
rij
[bi, bj] = 0 . (1.10)
9
Essentially the same condition was argued to appear from the analysis of vertex operators of the βdeformed
AdS5 × S5
superstring in [38], see equation (87) there. That discussion would correspond to the
truncation of our deformed action at order O(η2
). We thank Arkady Tseytlin for pointing this reference
out to us.
10
It is easy to see that this condition is compatible with the (M)CYBE.
11
For the split case (c = −1) there exist no solution for the compact subalgebra su(4) ⊂ psu(2, 2|4). It
seems then not possible to have a split solution for the full superalgebra.
– 4 –
JHEP10(2016)045
This is trivially satisfied by any abelian R-matrix, which is consistent with observations in
the literature, see e.g [37, 40, 41]. This is also in line with the expectation that abelian Rmatrices
always have an interpretation in terms of (commuting) TsT-transformations12 [35].
For non-abelian R-matrices the unimodularity condition (1.10) is non-trivial, and it is
interesting to find all the compatible ones. In fact, as explained in section 5 it rules out
most of the R-matrices of the so-called Jordanian type, which is the only class considered
in the literature so far [33, 35, 37, 40, 41].
Here we will focus on the problem of classifying all R-matrices which satisfy the CYBE
on the bosonic subalgebra so(2, 4) ⊕ so(6) ⊂ psu(2, 2|4) and are unimodular. The question
is non-trivial only for non-abelian R-matrices, which we classify by the rank. From (1.10),
any unimodular R-matrix of rank two R = a ∧ b must be abelian, i.e. [a, b] = 0, so nonabelian
unimodular R-matrices have at least rank four. In tables 1 and 2 we write down all
non-abelian rank four R-matrices for so(2, 4) (the second table gives the inequvalent ones
from the point of view of the string sigma model), and in table 3 we provide the bosonic
isometries and the number of supersymmetries that they preserve. These R-matrices are
constructed in section 5, where we also show that the only other possibility is rank six.
The extension to so(2, 4) ⊕ so(6) is essentially trivial as it turns out that they must be
abelian.13 in so(6). Therefore there are no new marginal deformations of the dual CFT.14
Notice that R6, R13 and R15 can be embedded in so(2, 3) and can therefore be used to define
deformations of AdS4. To have non-abelian deformations of AdS3, instead, one must
involve also generators from the sphere.
Because abelian R-matrices seem to generate backgrounds which can be equivalently
obtained by doing (commuting) TsT-transformations on the undeformed model, one might
suspect that η-deformed strings always correspond to TsT-transformations. With the exception
of the last three R-matrices our results appear to be consistent with this expectation,
see section 5 for a discussion.
The outline of the rest of the paper is as follows. In section 2 we first review the
definitions of the deformed models, we introduce a notation that highlights their similarities,
and prove that the local fermionic symmetries of both deformed models are of the
standard Green-Schwarz form. In section 3 we derive the target space supergeometry for
the λ-model, and by comparing to the results of [30] we extract the corresponding background
fields. Section 4 achieves the same goal for the η-model. Here we also show how the
unimodularity condition for the R-matrix is derived. In section 5 we study this condition
in detail. We discuss its compatibility with Jordanian R-matrices, and derive all rank-four
non-abelian unimodular R-matrices for so(2, 4) which solve the CYBE. In section 6 we
consider the case of backgrounds generated by R-matrices which act only on the bosonic
12
TsT stands for T-duality — shift — T-duality [42–44]. Here we use it in the most general possible
sense, e.g. including non-compact and fermionic T-dualities.
13
This includes R-matrices mixing generators of AdS and S, e.g. as in the so-called dipole deformations
of [45].
14
This statement remains to be true also if we further allow the R-matrix to act non-trivially on supercharges:
after imposing unimodularity, preservation of the so(2, 4) isometry, reality and CYBE, we find
that the only possible R-matrices are abelian and they act just on so(6).
– 5 –
JHEP10(2016)045
R1 = p1 ∧ p2 + (p0 + p3) ∧ (J01 − J13)
R2 = p1 ∧ p2 + (p0 + p3) ∧ (p3 + J01 − J13)
R3 = p1 ∧ (J02 − J23) + (p0 + p3) ∧ (p2 + J01 − J13)
R4 = (p1 − J02 + J23) ∧ (k0 + k3 + 2p3 − 2J12) + 2(p0 + p3) ∧ (p2 + J01 − J13)
R5 = p1 ∧ (J02 − J23) + (p0 + p3) ∧ (D + J03)
R6 = p1 ∧ J03 + 2p0 ∧ p3
R7 = J03 ∧ J12 + 2p0 ∧ p3
R8 = p1 ∧ p2 + (p0 + p3) ∧ J12
R9 = p1 ∧ p2 + (p0 + p3) ∧ (p3 + J12)
R10 = p1 ∧ p2 + p3 ∧ (p0 + J12)
R11 = p1 ∧ p2 + p3 ∧ J12
R12 = p1 ∧ p2 + p0 ∧ (p3 + J12)
R13 = p1 ∧ p2 + p0 ∧ J12
R14 = p1 ∧ p2 + J12 ∧ J03
R15 = p1 ∧ p3 + (J01 − J13) ∧ (p0 + p3)
R16 = p1 ∧ p3 + (p2 + J01 − J13) ∧ (p0 + p3)
R17 = p1 ∧ (p3 + J02 − J23) + (p0 + p3) ∧ (p2 + J01 − J13)
Table 1. All non-abelian unimodular rank-four R-matrices (CYBE) of so(2, 4) up to automorphisms
of the corresponding subalgebras (see section 5).
subalgebra. We work out certain examples generated by the R-matrices previously derived,
and we check in some cases that they are equivalent to sequences of TsT transformations
on the original undeformed model.
2 η and λ-deformed string sigma models
The η and λ deformations are deformations of supercoset sigma models that preserve
the classical integrability of the original models. In the string theory context the most
studied example is the deformation of the AdS5 × S5 string15 described by a PSU(2,2|4)
SO(1,4)×SO(5)
supercoset sigma model [47]. However, there are many other backgrounds where at least
a subsector of the string worldsheet theory is described by a supercoset sigma model, e.g.
AdS4 × È3
[48–50], AdS3 × S3 × T4 [51], AdS2 × S2 × T6 [52] and several others [3].
We start by reviewing the definitions of the deformed models. The relevant superalgebra
conventions are collected in appendix A.
15
Another supercoset closely related to this is the pp-wave background of [46].
– 6 –
JHEP10(2016)045
R1 = (p1 + a(J01 − J13)) ∧ p2 + (p0 + p3) ∧ (J01 − J13)
R2 = (p1 + a(p3 + J01 − J13) + b(p0 + p3)) ∧ p2 + (p0 + p3) ∧ (p3 + J01 − J13)
R3 = (p1 + a(p2 + J01 − J13)) ∧ (p1 + J02 − J23) + (p0 + p3) ∧ (p2 + J01 − J13)
R4 = ((p1−J02+J23)+2a(p2+J01−J13)+2b(p0+p3)) ∧ (k0+k3+2p3−2J12+c(p0+p3))
+2d(p0 + p3) ∧ (p2 + J01 − J13)
R5 = p1 ∧ (J02 − J23) + a(p0 + p3) ∧ (D + J03)
R6 = p1 ∧ J03 + 2p0 ∧ p3
R7 = J03 ∧ J12 + 2p0 ∧ p3
R8 = p1 ∧ p2 + (p0 + p3) ∧ J12
R9 = p1 ∧ p2 + a (p0 + p3) ∧ (p3 + J12)
R10 = p1 ∧ p2 + a p3 ∧ (p0 + J12)
R11 = p1 ∧ p2 + p3 ∧ J12
R12 = p1 ∧ p2 + a p0 ∧ (p3 + J12)
R13 = p1 ∧ p2 + p0 ∧ J12
R14 = p1 ∧ p2 + J12 ∧ J03
R15 = (p1 + a(p0 + p3)) ∧ p3 + (J01 − J13) ∧ (p0 + p3)
R16 = (p1 + a(p0 + p3)) ∧ p3 + (p2 + J01 − J13) ∧ (p0 + p3)
R17 = (p1 + a(p0 + p3)) ∧ (p1 + p3 + J02 − J23) + (p0 + p3) ∧ (p2 + J01 − J13)
Table 2. All non-abelian unimodular rank four R-matrices (CYBE) of so(2, 4) up to inner auto-
morphisms.
2.1 Lagrangians of the deformed models
The η-model Lagrangian takes the form [5, 33]
L = −
(1 + cη2)2
4(1 − cη2)
(γij
− εij
)Str(g−1
∂ig ˆd O−1
− (g−1
∂jg)) , (2.1)
where g is a group element of G, i, j are worldsheet indices, γij is the (Weyl-invariant)
worldsheet metric and ε01 = +1. Here η is the deformation parameter, and setting η = 0
yields the Lagrangian of the undeformed supercoset sigma model. The deformation involves
the Lie algebra operators
O+ = 1 + ηRg
ˆdT
, O− = 1 − ηRg
ˆd , (2.2)
where Rg = Ad−1
g RAdg, RT = −R and R satisfies the (M)CYBE (1.7). Our derivation is
general and we will not need to pick a particular solution of (1.7): we only need to assume
the above properties for R, and we will treat the homogeneous (c = 0, CYBE) and the
inhomogeneous (c = 1, MCYBE) cases at the same time. In the Lagrangian the following
– 7 –
JHEP10(2016)045
supercharges bosonic isometries
R1 8 p0 + p3, p1, p2, p0 − p3 − 2(J02 − J23), (a = 0)
8 p0 + p3, p1 + a(J01 − J13), p2, (a = 0)
R2 8 p0 + p3, p1, p2, p0 − p3 − J01 − J02 + J13 + J23, (a = 0)
8 p0 + p3, p1 + a(J01 − J13), p2, (a = 0)
R3 8 p0 + p3, p1, J02 − J23, (a = 0)
8 p0 + p3, p1 + (J02 − J23), J02 − J23 − a(J01 − J13 + p2), (a = 0)
R4 0 −J02 + J23 + p1 + 2a(J01 − J13 + p2), p0 + p3, 2J12 − 2p3 − k0 − k3,
R5 8 D + J03, p0 + p3,
R6 0 J03, p1, p2,
R7 0 J03, J12,
R8 0 p0, p3, J12,
R9 0 p0, p3, J12,
R10 0 p0, p3, J12,
R11 0 p0, p3, J12,
R12 0 p0, p3, J12,
R13 0 p0, p3, J12,
R14 0 J03, J12,
R15 8 p0 + p3, p1, p2,
R16 8 p0 + p3, p1, p2,
R17 8 p0 + p3, p1, J02 − J23 − p2 + p3,
Table 3. For each R-matrix of table 2 we indicate the number of unbroken supercharges and we
list the unbroken bosonic isometries.
combinations of projection operators appear
ˆd = P(1)
+ 2ˆη−2
P(2)
− P(3)
, ˆη = 1 − cη2 .
ˆdT
= −P(1)
+ 2ˆη−2
P(2)
+ P(3)
, where ˆd + ˆdT
= 4ˆη−2
P(2)
.
(2.3)
The λ-model is defined as a deformation of the G/G gauged WZW model. To get a standard
string sigma model one integrates out the gauge-field which leads to a Lagrangian16
somewhat similar to that of the η-model, namely [6]
L = −
k
2π
(γij
− εij
)Str(g−1
∂ig(1 + B0 − 2O−1
− )(g−1
∂jg)) . (2.4)
16
This is the classical Lagrangian. At the quantum level there is also a Fradkin-Tseytlin term R(2)
φ
present, where φ is the dilaton superfield, generated by integrating out the gauge-field, whose form will be
discussed in section 3.
– 8 –
JHEP10(2016)045
Here k is the level of the WZW model,17 and the Lie algebra operators O± are now defined
as
O+ = Ad−1
g − ΩT
, O− = 1 − Ad−1
g Ω . (2.5)
In this case things are written in terms of the combinations of projectors
Ω = P(0)
+λ−1
P(1)
+λ−2
P(2)
+λP(3)
,
ΩT
= P(0)
+λP(1)
+λ−2
P(2)
+λ−1
P(3)
, 1 − ΩΩT
= 1 − ΩT
Ω = (1 − λ−4
)P(2)
.
(2.6)
Both the Lagrangian (2.1) of the η and (2.4) of the λ-model can be formally written
in the same way18
L = −
T
2
γij
Str(A
(2)
−i A
(2)
−j ) +
T
2
εij
Str(A−iBA−j) , (2.7)
in terms of the one-forms
A± = O−1
± (g−1
dg) , (2.8)
where the string tension T and the operator B (responsible for the B-field) in the two cases
are
η − model : T =
1 + cη2
1 − cη2
2
, B =
ˆη2
2
(P(1)
− P(3)
+ η ˆdT
Rg
ˆd) ,
λ − model : T =
k
π
(λ−4
− 1) , B = (λ−4
−1)−1
(OT
−B0O−+ΩT
Adg −Ad−1
g Ω) .
(2.9)
An important role is played by the operator
M = O−1
− O+ (2.10)
which relates A− to A+ as A− = MA+. Using the expressions in (B.2) it is not hard to
show that
MT
P(2)
M = P(2)
, (2.11)
which implies that the operator P(2)MP(2) implements a Lorentz transformation on the
subspace with grading-2 of the superisometry algebra. This implies that there exists an
element h ∈ H = G(0) ⊂ G such that
P(2)
MP(2)
= Ad−1
h P(2)
= P(2)
Ad−1
h . (2.12)
The fact that Adh is a Lorentz transformation implies the basic relation between the action
on vectors and spinors
[Adh]ˆγ
ˆαγa
ˆγˆδ
[Adh]
ˆδ
ˆβ = [Adh]a
bγb
ˆαˆβ
. (2.13)
We refer to appendix B for some useful identities satisfied by the operators entering the
deformed models.
17
B0 = −BT
0 is related to the original WZ-term, see section 3.
18
We have used (2.3), (2.6), AdT
g = Ad−1
g and RT
g = −Rg.
– 9 –
JHEP10(2016)045
2.2 Kappa symmetry transformations in Green-Schwarz form
Both the η and λ model have a local fermionic symmetry which removes 16 of the 32
fermions, and here we show that it takes the form of the standard kappa symmetry of the GS
superstring. The transformations for the local fermionic symmetry take the form [5, 6, 33]
O−1
+ (g−1
δκg) = Pij
− {i˜κ
(1)
i , A
(2)
−j } + ζs
Pij
+ {i˜κ
(3)
i , A
(2)
+j } , (2.14)
where we denote the parameter by ˜κ, which is related to the kappa symmetry parameter κ
of the GS string as explained below. The above transformations are accompanied by the
variation of the worldsheet metric
δκγij
=
ζ2
2
Str(W[(P+i˜κ(1)
)i
, (P+A
(1)
+ )
j
]) + Str(W[(P−i˜κ(3)
)i
, (P−A
(3)
− )
j
]) , (2.15)
where we have defined
Pij
± =
1
2
(γij
±εij
) , ζ =
ˆη
λ
, s =
0 η − model
1 λ − model
. (2.16)
Using the fact that A
(2)
− is related to A
(2)
+ by a gauge transformation, i.e.
A
(2)
− = P(2)
MA
(2)
+ = Ad−1
h A
(2)
+ , (2.17)
we can write the kappa transformations as19
iδκ E(2)
= 0 , iδκ E(1)
= Pij
− {iκ
(1)
i , E
(2)
j } , iδκ E(3)
= Pij
+ {iκ
(3)
i , E
(2)
j }
δκγij
=
1
2
Str(W[(P+iκ(1)
)i
, (P+E(1)
)
j
]) +
1
2
Str(W[(P−iκ(3)
)i
, (P−E(3)
)
j
]) ,
(2.18)
where κ(1) = ζAdh˜κ(1) and κ(3) = (−i)sζ˜κ(3). This shows that the kappa symmetry
variations have the standard GS form, and at the same time it allows us to identify the
supervielbeins with projections of A± as20
E(2)
≡ Ea
Pa = A
(2)
+ , E(1)
≡ Eˆα1
Q1
ˆα = ζAdhA
(1)
+ , E(3)
≡ Eˆα2
Q2
ˆα = is
ζA
(3)
− . (2.19)
In terms of these the Lagrangian (2.7) takes the standard form
L = −
T
2
γij
Str(E
(2)
i E
(2)
j ) +
T
2
εij
Bij , (2.20)
where the B-field can be read off from (2.9).
19
In writing the transformations in this form we used (B.4).
20
The explicit i in E(3)
and κ(3)
in the case of the λ-model is needed to put the transformations in the
standard type IIB form. The reason for having i can be traced to the relative sign between P(1)
and P(3)
in (2.6) compared to (2.3). Alternatively, insisting on manifest reality of the model, the kappa symmetry
transformations and superspace constraints become those of type IIB* rather than type IIB. This is rather
natural since the λ-model is a deformation of the non-abelian T-dual of the AdS5 ×S5
string, which involves
also a T-duality in the time direction.
– 10 –
JHEP10(2016)045
Since the action and kappa symmetry transformations take the standard GS form,
it follows from the analysis of [30] that the target superspace of these models solves the
generalized type II supergravity equations derived there. If the Killing vector Ka appearing
in these equations vanishes, they reduce to the standard supergravity equations. In the
next sections we will derive the form of the target space supergeometry for the η and λdeformed
strings. Having identified the supervielbeins of the background superspace we
can find the supergeometry by calculating the torsion21
Ta
= dEa
+ Eb
∧ Ωb
a
, T ˆαI
= dEˆαI
−
1
4
(γabEI
)ˆα
∧ Ωab
(I = 1, 2) , (2.21)
and reading off the background superfields by comparing to the general expressions derived
in [30]. These are valid for a generalized type II supergravity background and reduce to
those of a standard supergravity background (see e.g. [36]) only when Ka = 0. We will
see that the λ-model background is a solution to standard (type II*) supergravity. For the
η-model background we will derive the condition on the R-matrix of the η-model for it to
give rise to a standard type II background.
3 Target superspace for the λ-model
In this section we present the derivation for the λ-model. We refer to appendix B.1 for
more details. The supervielbeins are defined in terms of projections of A± by (2.19). To
calculate the torsion we therefore need to calculate the exterior derivative of A±. Using
A+ = O−1
+ (g−1dg) where O± are defined in (2.5) we find
dA+ = O−1
+ (dO+ ∧ A+) + O−1
+ (g−1
dg ∧ g−1
dg)
= −O−1
+ {g−1
dg, Ad−1
g A+} +
1
2
O−1
+ {g−1
dg, g−1
dg}
= −
1
2
O−1
+ {Ad−1
g A+, Ad−1
g A+} +
1
2
O−1
+ {ΩT
A+, ΩT
A+}
= −
1
2
{A+, A+} −
1
2
O−1
+ (ΩT
{A+, A+} − {ΩT
A+, ΩT
A+}) , (3.1)
where we used the fact that g−1dg = O+A+ = (Ad−1
g − ΩT )A+ to write everything in
terms of A+. An almost identical calculation gives
dA− =
1
2
{A−, A−} +
1
2
O−1
− Ad−1
g (Ω{A−, A−} − {ΩA−, ΩA−}) . (3.2)
In the above equations it is useful to expand out the expressions inside parenthesis,
see (B.5), (B.6). Projecting equation (B.5) with P(2) we find
dE(2)
=
1
2
{E(1)
, E(1)
} +
1
2
{E(3)
, E(3)
} − {A
(0)
+ , E(2)
} − iλ{E(3)
, P(3)
ME(2)
}
− iλP(2)
MT
{E(2)
, E(3)
} −
1
2
λ2
{P(3)
ME(2)
, P(3)
ME(2)
} −
1
2
P(2)
MT
{E(2)
, E(2)
}
− λ2
P(2)
MT
{E(2)
, P(3)
ME(2)
} . (3.3)
21
Our conventions are the same as those of [30]. In particular d acts from the right and components of
superforms are defined as ωn = 1
n!
EAn
∧ · · · ∧ EA1
ωA1···An .
– 11 –
JHEP10(2016)045
where the result has been rewritten in terms of the supervielbeins (2.19), and we have
used (B.4) and (2.12). Using the explicit form of the commutators in (A.1) and (A.2)
we find that the component Ta of the torsion takes the standard form (here and in the
following we drop the ∧’s for readability)
Ta
= dEa
+ Eb
Ωb
a
= −
i
2
E1
γa
E1
−
i
2
E2
γa
E2
, (3.4)
if we identify the spin connection as22
Ωab = −(A+)ab − 2λ(E2
γ[a)ˆαM ˆα2
b] −
3i
2
λ2
Ec
M ˆα2
[a(γb)ˆαˆβM
ˆβ2
c] +
1
2
Ec
(Mab,c − 2Mc[a,b]) .
(3.5)
To derive the other components of the torsion we first need to compute the exterior
derivative of the fermionic supervielbeins. Using (B.6) and (2.19) we find
dE(3)
=
i
2
λP(3)
M{E(3)
, E(3)
} − {A
(0)
+ , E(3)
} + {P(0)
ME(2)
, E(3)
}
− iλ 1 + λ(1 − λ−4
)P(3)
(OT
+)−1
Ad−1
h {E(2)
, E(1)
} − {E(2)
, AdhP(1)
ME(2)
}
+
i
2
λ(1 − λ−4
)P(3)
(OT
+)−1
Ad−1
h {E(2)
, E(2)
} . (3.6)
Since we have already identified the form of the spin connection (3.5) from the previous
computation, we can now find the corresponding component of the torsion (2.21) and
compare it to the standard form given in [30], i.e.
T ˆα2
= Eˆα2
E2
χ2
−
1
2
E2
γa
E2
(γaχ2
)ˆα
+
1
8
Ea
(E2
γbc
)ˆα
Habc −
1
8
Ea
(E1
γaS12
)ˆα
+
1
2
Eb
Ea
ψˆα2
ab ,
(3.7)
where H is the NSNS three-form, S the RR bispinor, χI
ˆα the dilatino and ψˆαI
ab the gravitino
field strength superfields. We find that T ˆα2 takes the above form if we identify
Habc = 3M[ab,c] + 3iλ2
M ˆα2
[a(γb)ˆαˆβM
ˆβ2
c] , (3.8)
S ˆα1ˆβ2
= −8λ Adh(1 + λ(1 − λ−4
)O−1
+ )
ˆα1
ˆγ1Kˆγ1ˆβ2
, (3.9)
χ2
ˆα =
1
2
λγa
ˆαˆβ
M
ˆβ2
a , (3.10)
ψˆα2
ab =
i
4
λ(1 − λ−4
)[(OT
+)−1
Ad−1
h ]ˆα2
cdKab
cd
−
1
4
[AdhM]
ˆβ1
[a(γb])ˆβˆγSˆγ1ˆα2
. (3.11)
As already remarked, the RR bispinor superfield is imaginary if we interpret the λ-model
target space as a solution of type II supergravity, as here, rather than type II* supergravity.23
This determines the bosonic target space fields, with the exception of the dilaton
which we will determine shortly. First, let us calculate also the remaining components of
the femionic superfields, which we will extract from the corresponding component of the
22
Here we rewrote A
(0)
± = 1
2
Aab
± Jab and used the relation between components of M and MT
in (A.11).
23
Let us recall that at least in some cases it is possible to define a real type II background, after analytic
continuation or proper choice of coordinate patch [26, 27].
– 12 –
JHEP10(2016)045
torsion, T ˆα1. From (B.5) and using (2.19) we find
dE(1)
= −{AdhA
(0)
+ + dhh−1
, E(1)
} +
1
2
λ(1 − λ−4
)P(1)
AdhO−1
+ Ad−1
h {E(1)
, E(1)
}
− iλAdh{E(2)
, E(3)
}−λ2
Adh{E(2)
, P(3)
ME(2)
}−iλ2
(1−λ−4
)P(1)
AdhO−1
+ {E(2)
, E(3)
}
−
1
2
λ(1 − λ−4
)P(1)
AdhO−1
+ {E(2)
, E(2)
} + 2λ2
{E(2)
, P(3)
ME(2)
} . (3.12)
Using this expression we find24
T ˆα1
= Eˆα1
E1
χ1
−
1
2
E1
γa
E1
(γaχ1
)ˆα
−
1
8
Ea
(E1
γbc
)ˆα
Habc −
1
8
Ea
(E2
γaS21
)ˆα
+
1
2
Eb
Ea
ψˆα1
ab ,
(3.13)
is again of the standard form given in [30], where S
ˆβ2ˆα1 = −S ˆα1ˆβ2 and
χ1
ˆα =−
i
2
γb
ˆαˆβ
[AdhM]
ˆβ1
b , ψˆα1
ab =−
1
2
λ(1−λ−4
)[AdhO−1
+ ]ˆα1
cdKab
cd
−
i
4
λ(S12
γ[a)ˆα
ˆβM
ˆβ2
b] .
(3.14)
We complete the set of background superfields for the λ-model by noting that the B-field
can be written in the two equivalent forms
B = (λ−4
− 1)−1
B0 + Str(g−1
dg ∧ A−) , dB0 =
1
3
Str(g−1
dg ∧ g−1
dg ∧ g−1
dg) ,
= (λ−4
− 1)−1
B0 − Str(g−1
dg ∧ ΩT
A+) ,
(3.15)
and that the dilaton is given by
e−2φ
= sdet(O+) = sdet(Adg − Ω) . (3.16)
This result for the dilaton arises from integrating out the gauge-fields in the deformed
gauged WZW model [6]. To verify that the λ-model gives rise to a standard supergravity
background25 it is enough to verify that the dilatino’s found in (3.10) and (3.14) are indeed
the spinor derivatives of φ
∇ˆα2φ =
i
2
λK
ˆβ1ˆγ2
STr(Q1
ˆβ
M[Q2
ˆα, Q2
ˆγ]) = χ2
ˆα ,
∇ˆα1φ =
1
2
(1 − λ−4
)[Ad−1
h ]
ˆβ
ˆαSTr(Pa
O−1
− [Q1
ˆβ
, Pa]) = χ1
ˆα .
(3.17)
24
To calculate this component of the torsion we must first find the Lorentz-transformed spin connection
AdhA
(0)
+ + dhh−1
appearing in the first term, see equation (B.9) and the corresponding derivation.
25
As pointed out in [30] this was clear from the fact that the metric of the λ-model does not admit any
isometries, so that the Killing vector Ka
of the generalized supergravity equations vanishes.
– 13 –
JHEP10(2016)045
4 Target superspace for the η-model
The calculations for the η-model proceed along the same lines as those for the λ-model
with only minor differences. We begin by calculating the derivative of A+
dA+ = O−1
+ (dO+ ∧ A+) + O−1
+ (g−1
dg ∧ g−1
dg)
= ηO−1
+ Rg{g−1
dg, ˆdT
A+} − ηO−1
+ {g−1
dg, Rg
ˆdT
A+} +
1
2
O−1
+ {g−1
dg, g−1
dg}
=
1
2
O−1
+ {A+, A+} + ηO−1
+ Rg{A+, ˆdT
A+} + η2
O−1
+ Rg{Rg
ˆdT
A+, ˆdT
A+}
−
1
2
η2
O−1
+ {Rg
ˆdT
A+, Rg
ˆdT
A+}
=
1
2
O−1
+ {A+, A+} −
1
2
cη2
O−1
+ { ˆdT
A+, ˆdT
A+} + ηO−1
+ Rg{A+, ˆdT
A+} , (4.1)
where we used the fact that g−1dg = O+A+ and in the last step we used the fact that R
(as well as Rg) satisfies the (M)CYBE equation, so that
{Rg
ˆdT
A+, Rg
ˆdT
A+} − 2Rg{Rg
ˆdT
A+, ˆdT
A+} − c{ ˆdT
A+, ˆdT
A+} = 0 . (4.2)
The result for dA− is simply obtained by changing the sign of η and replacing ˆdT → ˆd in
the above expression
dA− =
1
2
O−1
− {A−, A−} −
1
2
cη2
O−1
− { ˆdA−, ˆdA−} − ηO−1
− Rg{A−, ˆdA−} . (4.3)
After rewriting dA+ as in (B.13) and projecting with P(2) we find
dE(2)
= {A
(0)
+ , E(2)
} +
1
2
{E(1)
, E(1)
} +
1
2
{E(3)
, E(3)
} − 2ˆη{E(3)
, P(3)
O−1
− E(2)
}
+ 4ˆη−1
P(2)
O−1
+ {E(2)
, E(3)
} − 8P(2)
O−1
+ {E(2)
, P(3)
O−1
− E(2)
}
+ 2ˆη2
{P(3)
O−1
− E(2)
, P(3)
O−1
− E(2)
} + 2ηˆη−2
P(2)
O−1
+ Rg{E(2)
, E(2)
} , (4.4)
where we have used (2.19) to write the result in terms of the supervielbeins, together
with (B.4) and (2.12). We check again that the bosonic torsion Ta takes the standard
form (3.4), where we can now identify the spin connection for the η-model background as
Ωab = (A+)ab + 2iˆη(γ[aE2
)ˆαM ˆα2
b] +
3i
2
ˆη2
Ec
M ˆα2
[a(γb)ˆαˆβM
ˆβ2
c] −
1
2
Ec
(2Mc[a,b] − Mab,c) .
(4.5)
As before, we continue by computing the remaining components of the torsion. First,
from (B.14) we get
dE(3)
= {A
(0)
+ , E(3)
} + ˆηP(3)
O−1
− {E(3)
, E(3)
} + 2{P(0)
O−1
− E(2)
, E(3)
}
+ P(3)
(4O−1
− − 1 − 2ˆη−2
)Ad−1
h {E(2)
, E(1)
} − 2ηˆη−1
P(3)
O−1
− RgAd−1
h {E(2)
, E(2)
}
+ 2ˆηP(3)
(4O−1
− − 1 − 2ˆη−2
){Ad−1
h E(2)
, P1
O−1
− E(2)
} , (4.6)
– 14 –
JHEP10(2016)045
which we use to check that also T ˆα2 is of the standard form (3.7). To do this we make use
of the spin connection (4.5) and we identify the following superfields for the η-model
Habc = 3M[ab,c] − 3iˆη2
M ˆα2
[a(γb)ˆαˆβM
ˆβ2
c] , (4.7)
S ˆα1ˆβ2
= 8i[Adh(1 + 2ˆη−2
− 4O−1
+ )]ˆα1
ˆγ1Kˆγ1ˆβ2
, (4.8)
χ2
ˆα = −
i
2
ˆηγa
ˆαˆβ
M
ˆβ2
a , (4.9)
ψˆα2
ab = −2ηˆη−1
[O−1
− RgAd−1
h ]ˆα2
cdKab
cd
+
1
4
ˆη[AdhM]
ˆβ1
[a(γb]S12
)ˆβ
ˆα
. (4.10)
To identify the last component of the spinor superfields we must compute torsion T ˆα1.
Starting from (B.13) we find
dE(1)
= {AdhA
(0)
+ − dhh−1
, E(1)
} + ˆηP(1)
AdhO−1
+ Ad−1
h {E(1)
, E(1)
}
+ P(1)
Adh(4O−1
+ − 1 − 2ˆη−2
){E(2)
, E(3)
} + 2ηˆη−1
P(1)
AdhO−1
+ Rg{E(2)
, E(2)
}
− 2ˆηP(1)
Adh(4O−1
+ − 1 − 2ˆη−2
){E(2)
, P(3)
O−1
− E(2)
} . (4.11)
Using this expression we can check26 that T ˆα1 is standard, see (3.13), where S
ˆβ2ˆα1 =
−S ˆα1ˆβ2 and
χ1
ˆα =
i
2
ˆηγb
ˆαˆβ
[AdhM]
ˆβ1
b , ψˆα1
ab = 2ηˆη−1
[AdhO−1
+ Rg]ˆα1
cdKab
cd
−
1
4
ˆη(S12
γ[a)ˆα
ˆβM
ˆβ2
b] .
(4.12)
Let us also note that in the case of the η-model the B-field can be written in the two ways
B =
ˆη2
4
Str(g−1
dg ∧ ˆdT
A+) = −
ˆη2
4
Str(g−1
dg ∧ ˆdA−) , (4.13)
which are equivalent thanks to the properties of O± under transposition.
4.1 Dilaton and supergravity condition
Unlike in the case of the λ-model, the η-model does not come with a natural candidate
dilaton. Indeed, in general the target space geometry of the η-model is a solution of the
generalized type II supergravity equations of [25, 30] rather than the standard ones, and a
dilaton does not exist. One of our goals is to determine precisely when a dilaton exists for
the η-model. To do this, let us define a would-be dilaton in the same way as the dilaton is
defined in the λ-model
e−2φ
= sdet(O+) = sdet(1 + ηRg
ˆdT
) . (4.14)
For this to be the actual dilaton of the η-model its spinor derivatives must coincide with
the dilatinos in (4.9) and (4.12). In (B.18) we write down the result for dφ. In particular
26
As in the previous section, we need to first find an expression for AdhA
(0)
+ − dhh−1
, see (B.16).
– 15 –
JHEP10(2016)045
we find27
∇ˆα2φ = −2ˆη−1
STr(Pa
O−1
+ [Q2
ˆα, Pa]) −
η
2
ˆη−1
KAB
STr(TARg[TB, Q2
ˆα])
= χ2
ˆα −
η
2
ˆη−1
KAB
STr([TA, RTB]gQ2
ˆαg−1
) , (4.15)
∇ˆα1φ = −ˆη[Ad−1
h ]
ˆβ
ˆα(Kˆγ1ˆδ2
STr(Q2
ˆδ
O−1
+ [Q1
ˆβ
, Q1
ˆγ]) −
η
2
KAB
STr(TARg[TB, Q1
ˆβ
]))
= χ1
ˆα +
η
2
ˆη[Ad−1
h ]
ˆβ
ˆαKAB
STr([TA, RTB]gQ1
ˆβ
g−1
) . (4.16)
Therefore a sufficient condition for the η-model to lead to a standard supergravity background
is that
KAB
STr([TA, RTB]gQI
ˆαg−1
) = 0 , (4.17)
or, since g is an arbitrary group element (modulo gauge-transformations),
STr(Radx) = 0 , ∀x ∈ g (i.e. RB
AfA
BC = 0, or RBC
fA
BC = 0) . (4.18)
To see that this condition is also necessary we calculate the Killing vector superfield Ka
appearing in the generalized supergravity equations of [30], which in general is given by
Ka
= −
i
16
(γa
)ˆαˆβ
(∇ˆα1χˆβ1 − ∇ˆα2χˆβ2) , (4.19)
and whose result is collected in (B.19). The η-model has a standard type II supergravity
solution as target space if Ka = 0. In fact, it must be that it vanishes order by order in
the deformation parameter η. At linear order we find the equation
KAB
STr([TA, RTB]gPag−1
) = 0 , (4.20)
which, since g ∈ G is arbitrary implies (4.18). Therefore the condition (4.18) is both
necessary and sufficient, and also the higher order terms in η in (B.19) vanish when this
condition is fulfilled.
5 Non-abelian R-matrices and the unimodularity condition
In this section we study the unimodularity condition (1.8) for the R-matrix. First we
analyse its compatibility with a class of non-abelian R-matrices — the Jordanian ones —
and then we explain how to classify all unimodular R-matrices solving the CYBE on the
bosonic subalgebra of the superisometry algebra.
Following [53] we define an “extended Jordanian” R-matrix for a Lie superalgebra g as
follows: we fix a Cartan element h (deg(h) = 0) and a positive root e as well as a collection
of roots eγ±i with i ∈ {1, 2, . . . , N} such that deg(e) = deg(eγi ) + deg(eγ−i ) (mod 2) and
satisfying
[h, e] = e , [h, eγi ] = (1 − tγi )eγi , [h, eγ−i ] = tγi eγ−i , (tγi ∈ )
[eγ±i , e] = 0 , [eγk
, eγl
] = δk,−le , (k > l ∈ {±1, ±2, . . . , ±N}) . (5.1)
27
Here we used the fact that O−1
± P(0)
= P(0)
.
– 16 –
JHEP10(2016)045
The extended Jordanian R-matrix is then defined as
R = h ∧ e +
N
i=1
(−1)deg(eγi ) deg(eγ−i )
eγi ∧ eγ−i . (5.2)
It is now easy to see that for a bosonic deformation, i.e. deg(e) = 0, we have
rij
[bi, bj] = (N0 − N1 + 1)e , (5.3)
with N = N0 + N1, N0 (N1) being the number of bosonic (fermionic) roots eγi . For this
to vanish we need precisely one more fermionic eγi than bosonic. This is clearly a very
strong restriction on the allowed Jordanian R-matrices. Let us note that this result is
compatible with the findings of [37, 40, 41], where Jordanian R-matrices acting only on
bosonic generators were found to produce backgrounds which do not solve the standard
supergravity equations. We have considered certain examples of bosonic Jordanian Rmatrices
(namely R = J01 ∧ (P0 − P1), R = J03 ∧ (J01 − J13) and R = D ∧ pi, i = 0, . . . , 3)
and we have checked that it is not possible to find a positive and a negative fermionic root
satisfying (5.1) without spoiling the reality of the extended R-matrix. If possible, it would
be interesting to find extended Jordanian unimodular R-matrices for psu(2, 2|4), but we
will not analyze this question further here.
From now on we will restrict to the bosonic subalgebra so(2, 4) ⊕ so(6) ⊂ psu(2, 2|4).
Let us recall some known facts about solutions to the CYBE, (1.7) with c = 0, for ordinary
Lie algebras. The first important fact, due to Stolin [54, 55], is that there is a one-to-one
correspondence between constant solutions of the CYBE for a Lie algebra g and quasiFrobenius
(or symplectic) subalgebras f ⊂ g (see also [56]). Notice that we do not need to
assume anything about the Lie algebra g, in particular it does not need to be simple. A
Lie algebra is quasi-Frobenius if it has a non-degenerate 2-cocycle ω, i.e.
ω(x, y) = −ω(y, x) , ω([x, y], z) + ω([z, x], y) + ω([y, z], x) = 0 , ∀x, y, z ∈ f . (5.4)
It is Frobenius if ω is a coboundary, i.e. ω(x, y) = f([x, y]) for some linear function f. If R
is a solution to the CYBE for g, then there is a subalgebra f on which R is non-degenerate.
This subalgebra is necessarily quasi-Frobenius, and writing R in the form (1.9) the 2cocycle
is the inverse of the R-matrix, i.e. ω(bi, bj) = (r−1)ij. The converse is also true,
i.e. if f ⊂ g is quasi-Frobenius then the inverse of the 2-cocycle ω gives a solution to the
CYBE, as is easily verified. Therefore, finding solutions to the CYBE for a given g reduces
to finding all quasi-Frobenius subalgebras28 of g. A fact with important consequences for
our analysis is that if g is compact then f must be abelian [58]. This leads to the conclusion
that deformations involving only S5 (i.e. marginal deformations of the dual CFT) must
necessarily have abelian R-matrices.
We now show that the unimodularity condition (1.8) for the R-matrix adds a further
property to the quasi-Frobenius subalgebra f. If we write the structure constants as fi
jk
in some basis, the 2-cocycle condition is
(r−1
)i[jfi
kl] = 0 . (5.5)
28
This was done for sl(2) and sl(3) in [57].
– 17 –
JHEP10(2016)045
f Defining Lie brackets
Ê4 —
h3 ⊕ Ê [e1, e2] = e3
r3,−1 ⊕ Ê [e1, e2] = e2, [e1, e3] = −e3
r′
3,0 ⊕ Ê [e1, e2] = −e3, [e1, e3] = e2
n4 [e1, e2] = −e4, [e4, e2] = e3
Table 4. The four-dimensional real unimodular quasi-Frobenius Lie algebras. In all cases the
2-cocycle can be taken as ω = e1
∧ e4
+ e2
∧ e3
, where ei
denotes the dual basis of f∗
.
Contracting this equation with rjk we get (r−1)ilfi
jkrjk = −2fi
il, which together with the
unimodularity condition for the R-matrix written as (1.10), i.e. fi
jkrjk = 0, implies
fi
il = 0 ⇔ tr(adx) = 0 ∀x ∈ f . (5.6)
Therefore f is a unimodular Lie algebra. Clearly the converse is also true and we have
established that solutions of the CYBE for a Lie algebra g which satisfy the condition (1.8)
are in one-to-one correspondence with unimodular quasi-Frobenius subalgebras of g.
For this reason we refer also to the R-matrices which satisfy (1.8) as unimodular.
A quasi-Frobenius Lie algebra must clearly have even dimension, and if the dimension
is two the algebra must be abelian to respect unimodularity. To find a non-abelian Rmatrix
we must therefore consider at least the case of rank four. Luckily the real quasiFrobenius
Lie algebras of dimension four were classified in [59], and the five unimodular
ones (Corollary 2.5 in [59]) are listed in table 4. The task of finding all R-matrices of rank
four which solve the CYBE and lead to a deformation of the AdS5 ×S5 string with a proper
supergravity background is therefore reduced to finding all inequivalent embeddings of these
subalgebras in so(2, 4) ⊕ so(6). The most interesting problem is to find the embedding of
the non-abelian algebras29 in so(2, 4). This is still quite challenging, but it becomes simpler
by the following observation. A unimodular quasi-Frobenius Lie algebra is solvable [58],
and solvable subalgebras of so(2, 4) must be embeddable in one of the maximal solvable
subalgebras of so(2, 4), see [60] for a proof of this. Besides the Cartan subalgebra which
is not relevant for our purposes, Patera, Winternitz and Zassenhaus in [61] showed that
there are two maximal solvable subalgebras of so(2, 4), s1 and s2 of dimension 9 and 8
respectively. It is most convenient to write them using the conformal form of the so(2, 4)
algebra, with dilatation generator D, translations and special conformal generators pi, ki
(i = 0, . . . 3) and Lorentz transformations and rotations Jij. They are related to the form
of so(2, 4) in (A.1) with Kij
kl = −2δk
[iδl
j] by
pi = Pi + Ji4 , ki = −Pi + Ji4 , D = P4 , (5.7)
29
The extension to so(2, 4) ⊕ so(6) is essentially trivial and amounts to adding in commuting generators
from so(6) in such a way that the commutation relations of the algebra are preserved.
– 18 –
JHEP10(2016)045
and the non-vanishing commutators are
[D, pi] = pi , [D, ki] = −ki , [pi, kj] = −2ηijD + 2Jij , (5.8)
[Jij, pk] = 2ηk[ipj] , [Jij, kk] = 2ηk[ikj] , [Jij, Jkl] = ηikJjl − ηjkJil − ηilJjk + ηjlJik .
The metric on the Lie algebra is given by tr(DD) = 1, tr(pikj) = −2ηij, tr(JijJkl) =
−2ηi[kηl]j . The two non-abelian maximal solvable subalgebras of so(2, 4) then take the
form
s1 = span(pi, J01 − J13, J02 − J23, J03, J12, D) ,
s2 = span(p0 + p3, p1, p2, J01 − J13, J02 − J23, J12, J03 − D, k0 + k3 + 2p3) , (5.9)
up to automorphisms. Our task is reduced to finding all embeddings of the non-abelian
algebras in table 4 in s1 and s2. To simplify this problem further we will single out the
element e3 in this table30 and use automorphisms generated by elements of s1 (s2) to
simplify it as much as possible. Using this freedom we can bring e3 to one of the following
forms
s1 : (1) e3 = p1 , (2) e3 = J02 − J23 , (3) e3 = p1 + J02 − J23 , (4) e3 = p0 ,
(5) e3 = p3 , (6) e3 = p0 + p3 , (7) e3 = p0−p3+J01−J13 , (5.10)
s2 : (1) e3 = p1 , (2) e3 = p0 + p3 , (3) e3 = ap1+bp2+J01−J13 . (5.11)
The rest is a straightforward if slightly tedious calculation. The results are summarized
in tables 5–8. Note that in writing these embeddings we have used automorphisms of the
four-dimensional subalgebras which are not always inner automorphisms of so(2, 4). This
must be accounted for when constructing the list of inequivalent R-matrices. In table 1
in the introduction we write the corresponding R-matrices, R = e1 ∧ e4 + e2 ∧ e3 up to
automorphisms. In table 2 instead we list the inequivalent, modulo inner automorphisms
of so(2, 4), R-matrices. This is the result which is interesting from the string sigma model
perspective, since inner automorphisms correspond to field redefinitions in the sigma model,
i.e. coordinate transformations in target space. In table 3 we write down the bosonic
isometries and the number of supercharges that each R-matrix preserves. Given a generator
t of the superalgebra g, the condition that it is preserved by the R-matrix is given by
[t, R(x)] = R([t, x]) , ∀x ∈ g . (5.12)
Most of these R-matrices all have a form which suggests that they should correspond
to non-commuting TsT-transformations,31 in the sense that they involve sequences of Tdualities
along non-commuting directions. All but the last three R-matrices in table 1 have
the form
R = a ∧ b + c ∧ d , (5.13)
30
The reason for picking e3 is that it always arises as a commutator of two other elements. Since the last
three generators in s1 or s2 are never generated in commutators, they do not appear in e3.
31
Here we use TsT in a generalized sense, where we can involve also non-compact directions.
– 19 –
JHEP10(2016)045
where [a, b] = [c, d] = 0 and c, d generate isometries of the corresponding background.
It is natural to conjecture that such R-matrices correspond to two successive TsTtransformations,
the first using isometries a, b and the second using isometries c, d. Note
that unlike in standard applications of TsT-tranformations, e.g. [62], the pairs of isometries
a, b and c, d do not commute with each other. This means that after the first TsT is implemented,
it is necessary to make a change of coordinates in order to realize the isometries
of the second TsT transformation as shift isometries. We will confirm this in section 6,
when we will check in some examples that the deformed backgrounds are indeed equivalent
to such sequences of TsT-transformations. These considerations suggest a very simple
picture for how TsT-transformations are interpreted at the level of the R-matrix: the TsTtransformation
involving isometries a, b should be simply implemented by adding a term
a∧b to the R-matrix.32 Notice that the number of free parameters entering the definitions
of the R-matrices (plus the overall deformation parameter) does not need to be equal to
the number of TsT-transformations implemented. In fact, the number of parameters could
be reduced in some cases, if they can be reabsorbed by means of field redefinitions. In
other cases one might have more parameters than expected, which suggests the possibility
of applying TsT-transformations on linear combinations of the isometric coordinates.
The structure of the last three R-matrices in table 1 is different, and one observes
that now a, c generate isometries. However, one can check explicitly that the background
corresponding to R15, for example, is self-dual (up to field redefinitions) under a TsTtransformation
involving a, c.33 This example is particularly instructive because it can be
embedded in so(2, 3): in this algebra, the deformed background does not preserve other
bosonic isometries than a, c, which suggests that backgrounds corresponding to the algebra
n4 are not of TsT-type. Note that n4 is the only algebra considered which is not the direct
sum of a three-dimensional algebra and a commuting generator. One possibility is that
non-abelian T-duality of the corresponding subalgebra should instead play a role in the
interpretation of these backgrounds. A hint towards this direction comes from the results
of [63], where it was shown that a conformal anomaly is encountered when implementing
non-abelian T-duality on a subalgebra, unless all generators have vanishing trace.34 In the
case of the adjoint representation this condition is precisely that of unimodularity of the
corresponding subalgebra.
Let us now consider the case of higher ranks, which can only be six or eight. We
have not done a systematic study for the case of rank six R-matrices. One would first
need to identify all 6-dimensional subalgebras of s1 and s2, and check which of them are
unimodular and quasi-Frobenius. We have found that the subalgebra of s1 generated by
{pi, J03, J12} has both properties. It is straightforward to find the 2-form ω that solves the
cocycle condition (5.4), and invert it to find the corresponding R-matrix. For particular
choices of the free parameters this can be written e.g. as R = p0 ∧ p1 + p2 ∧ p3 + J01 ∧ J23.
32
It is easy to check that this is compatible with the CYBE, since a, b are isometries and satisfy (5.12).
33
Note that this is consistent with our above proposal on how to interpret the action of TsT at the level
of the R-matrix; in fact, in this case the addition of the term a ∧ c to R15 can be removed by an inner
automorphism of so(2, 4). Here a, c can be chosen to be p1, p0 + p3.
34
We thank Arkady Tseytlin for pointing this reference out to us.
– 20 –
JHEP10(2016)045
h3 ⊕ Ê e1 e2 e3 e4
1. p1 J01 − J13 p0 + p3 p2
2. p1 p3 + J01 − J13 p0 + p3 p2
3. p1 p2 + J01 − J13 p0 + p3 p1 + J02 − J23
4. 1
2 p1 − 1
2(J02 − J23) p2 + J01 − J13 p0 + p3 k0 + k3 + 2p3 − 2J12
Table 5. Embeddings of h3 ⊕ Ê in so(2, 4) up to automorphism.
r3,−1 ⊕ Ê e1 e2 e3 e4
1. −D − J03 J02 − J23 p1 p0 + p3
2. J03 p0 − p3 p0 + p3 p1
3. J03 p0 − p3 p0 + p3 J12
(4.) D + 2J03 p1 p0 + p3 −
Table 6. Embeddings of r3,−1 ⊕ Ê in so(2, 4) up to automorphism. The last case is an embedding
of r3,−1 which does not extend to an embedding of r3,−1 ⊕Ê. It is the only case where this happens
and included only since it is relevant for constructing all non-abelian R-matrices of so(2, 4) ⊕ so(6).
r′
3,0 ⊕ Ê e1 e2 e3 e4
1. J12 p2 p1 p0 + p3
2. p3 + J12 p2 p1 p0 + p3
3. p0 + J12 p2 p1 p3
4. J12 p2 p1 p3
5. p3 + J12 p2 p1 p0
6. J12 p2 p1 p0
7. J12 p2 p1 J03
Table 7. Embeddings of r′
3,0 ⊕ Ê in so(2, 4) up to automorphism.
n4 e1 e2 e3 e4
1. p3 J01 − J13 p0 + p3 p1
2. p3 p2 + J01 − J13 p0 + p3 p1
3. p1 + p3 + J02 − J23 p2 + J01 − J13 p0 + p3 p1
Table 8. Embeddings of n4 in so(2, 4) up to automorphism.
– 21 –
JHEP10(2016)045
We have also checked that there is no 8-dimensional subalgebra which is at the same
time unimodular and quasi-Frobenius. Therefore there is no rank eight R-matrix which
produces a background that solves the supergravity equations of motion. It is in fact
easy to check that s2 (which is 8-dimensional) is quasi-Frobenius but not unimodular.
To identify all 8-dimensional subalgebras of s1 (which is 9-dimensional), we first define
e = 9
j=1 λjej to be the generator which we want to remove, where ej are the generators
of s1. Then for a generic element X ∈ s1 we define its component perpendicular to e as
X⊥ = X − P(X), where P projects35 along e. Then the condition to have a subalgebra
is P[X⊥, Y ⊥] = 0, ∀X, Y ∈ s1. These equations give two possible solutions, depending on
some unconstrained parameters
(a) e = λ7J12 + λ8J03 + λ9D ,
(b) e = λ1(p0 − p3) + λ8(J03 − D) .
(5.14)
In the case (a) we find36 that the subalgebra is unimodular if λ7 = 0 and λ9 = 2λ8.
However, for this choice it is not quasi-Frobenius — the cocycle condition gives a 2-form
of rank six. In the case (b) the subalgebra is not unimodular for any choice of λ1, λ8.
6 Some examples of supergravity backgrounds
In this section we give a brief discussion on the η-model backgrounds generated by solutions
of the CYBE (c = 0), when we restrict R to act only on the bosonic subalgebra. In most
cases a convenient parameterisation of the group element g = ga · gs ∈ SO(2, 4) × SO(6) is
ga = exp xi
pi · exp (log z D) , (6.1)
where pi, D are the generators defined in (5.7). Here we will be interested only on deformations
of AdS, so we will not need to specify the parameterisation that we use for gs on
the sphere. In this coordinate system the undeformed metric takes the familiar form
ds2
η=0 =
ηijdxi dxj + dz2
z2
+ ds2
s . (6.2)
Because of our restriction on R, it is enough to look at the action of the operators O± on
the bosonic subalgebra. They take a block form
1 (O±)bc
a
0 (O±)b
a
, (6.3)
35
We define P(X) = e STr(Xe∗
), where e∗
is a dual to e, STr(ee∗
) = 1. We can take it as
e∗
= 9
j=1
λj
||λ||2 ej
, where ||λ||2
= 9
j=1 λ2
j and ej
are the duals of the generators in the basis such that
STr(eiej
) = δj
i .
36
In both cases (a) and (b) one needs to choose carefully a basis for the 8-dimensional subalgebra, in such
a way that the generators are linearly independent and non-degenerate for generic choices of the remaining
λj. A way to do it is to pick an orthogonal basis, and normalise the vectors such that they can be degenerate
only if λj = 0 ∀j.
– 22 –
JHEP10(2016)045
since37 O±P(0) = P(0). All the information about background fields of the deformed model
can be extracted by studying just the block (O+)b
a — or in other words P(2)O+P(2). Notice
that the results for (O−)b
a are simply obtained by changing the sign of the deformation
parameter η. The dilaton of the deformed model is easily obtained by computing the
determinant of (O+)b
a
eφ
= (det O+)−1/2
. (6.4)
The rest of the background fields are written in terms of (O−1
+ )b
a
— the inverse of the
block (O+)b
a. The vielbein components for the deformed model are
Ea
= (O−1
+ )a
b
eb
, (6.5)
where ea is the bosonic vielbein of the undeformed background, related to the MaurerCartan
form as
g−1
dg = ea
Pa +
1
2
ωab
Jab . (6.6)
The spacetime metric of the deformed background is then straightforwardly obtained, ds2 =
ηabEaEb. The B-field can be extracted immediately from the action of the bosonic σ-model,
and it reads as
B =
1
2
dXn
∧ dXm
Bmn =
1
2
(O−1
− )ab ea
∧ eb
, (6.7)
where it is assumed that indices are raised and lowered with ηab. To get the RamondRamond
fields we first need to consider the local Lorentz transformation given by M
in (2.10) and write its action on spinors
(Adh)
ˆβ
ˆα = exp −
1
4
(log M)abΓab ˆβ
ˆα , (6.8)
where here we have introduced a basis for 32 × 32 Gamma-matrices.38 The RR fields are
obtained by solving the equation (note that (1.2) simplifies considerably for R-matrices of
the bosonic subalgebra)
Γa
Fa +
1
3!
Γabc
Fabc +
1
2 · 5!
Γabcde
Fabcde Π = e−φ
Adh(−4Γ01234)Π (6.9)
where Π = 1
2 (1 − Γ11) is a projector and (−4Γ01234)Π encodes the 5-form flux of the undeformed
model. The various components of F’s are found by multiplying the above equation
by the relevant Gamma-matrix Γa1...a2m+1 and then taking the trace. This computation39
yields the F’s expressed with tangent indices, which are translated into form language by
F(2m+1) = 1
(2m+1)! Ea2m+1 ∧ . . . ∧ Ea1 Fa1...a2m+1 .
37
We recall that P(0)
and P(2)
are projectors on the subspaces spanned by the generators Jab and Pa
respectively. A useful matrix realisation of the algebra generators can be found in [22]. Here we identify
Pa = Pa, and Jab = −Jab, where Pa, Jab are the generators used in [22].
38
For a convenient basis see [22].
39
For F(5)
it is enough to look at half of the components, e.g. F0bcde, and construct the corresponding
form f(5)
. Then F(5)
= (1 + ∗)f(5)
, such that F(5)
= ∗F(5)
.
– 23 –
JHEP10(2016)045
In the rest of this section we present some backgrounds solving the standard supergravity
equations which we have derived by using the above procedure. We work out one
example for each of the 4-dimensional non-abelian subalgebras in table 4.
In section 5 we have argued that the R-matrices related to the subalgebras h3 ⊕ Ê,
r′
3,0 ⊕ Ê, r3,−1 ⊕ Ê should produce backgrounds which can be obtained by sequences of
TsT-transformations starting from AdS5 × S5. We check this explicitly for the backgrounds
that we have derived, where we follow the conventions of [32] for the T-duality
rules [64–66]. Because the isometries of the first TsT do not commute with those of the
second one, we will see that before doing the last step it is necessary to implement a
coordinate transformation, which realizes the second pair of isometries as shifts of the
corresponding coordinates. Let us mention that since we have chosen to have just one
overall deformation parameter η (i.e. we fix some free parameters in the definitions of the
possible R-matrices), the shifts of the two TsT-transformations are related to each other.
This does not need to be true for generic cases.
6.1 h3 ⊕ Ê
Let us choose the R-matrix (this corresponds to R1 in table 1 with x1 ↔ x3)
R = (J03 + J13) ∧ (p0 + p1) + p2 ∧ p3 , (6.10)
which preserves 4 bosonic isometries
p2 , p3 , p0 + p1 , p0 − p1 − 2(J02 + J12) , (6.11)
and 8 supercharges. Clearly, it is convenient to introduce lightcone coordinates x± = x0 ±
x1, since a shift of x+ will correspond to an isometry. The spacetime metric that we obtain is
ds2
= z−2
1 +
4η2
z4
−1
4η2
z−4
x−
dx−
(2dx2 − x−
dx−
) + dx2
2
+ dx3
2
+
−dx−dx+ + dz2
z2
+ ds2
s.
(6.12)
The dilaton depends only on the z-coordinate, while the B-field also on x−
eφ
= 1 +
4η2
z4
−1/2
, B =
2η(dx2 − x−dx−) ∧ dx3
(4η2 + z4)
. (6.13)
The RR-fluxes turn out to be quite simple
F(5)
= (1+∗)
2dx− ∧ dx+ ∧ dx2 ∧ dx3 ∧ dz
z(z4 + 4η2)
, F(3)
=
4η
z5
(2x−
dx2 −dx+
)∧dx−
∧dz. (6.14)
In order to show that this background can be obtained by a sequence of TsTtransformations,
we start from the deformed background and show that we can reach the
undeformed AdS5 × S5 by TsT-transformations. We will write T(xi) to indicate that we
apply T-duality along the isometric coordinate xi, and denote by ˜xi the dual coordinate.
In this case we need to do the sequence
T(x2), x3 → x3 − 2η˜x2, T(˜x2), T(ψ), w+
→ w+
− 2η ˜ψ, T( ˜ψ), (6.15)
– 24 –
JHEP10(2016)045
where we need to redefine the coordinates in the 013 space
x+
= 2(ψ2
w−
+ w+
), x−
= 2w−
, x3 = −2ψw−
, (6.16)
before applying the last TsT-transformation. Obviously, starting from AdS5×S5 and applying
these TsT-transformations backwards, we find the deformed background presented here.
6.2 r′
3,0 ⊕ Ê
In this case we can choose an R-matrix which involves generators along spacelike directions
(R11 in table 1)
R = J12 ∧ p3 + p2 ∧ p1 . (6.17)
It preserves 3 bosonic isometries
J12, p0, p3, (6.18)
and no supercharges. It is more convenient to use the parameterisation
ga = exp(ξJ12) · exp(rp1 + x0
p0 + x3
p3) · exp(log z D), (6.19)
since ξ will be isometric. In the undeformed case
ds2
η=0 =
−(dx0)2 + r2dξ2 + dr2 + dx3
2 + dz2
z2
+ ds2
s, (6.20)
so that (r, ξ) are a radial and an angular coordinate in the 1, 2 plane. Turning on the
deformation parameter we find
ds2
= z−6
1+
4η2 r2+1
z4
−1
dr2
4η2
r2
+z4
+r2
z4
dξ2
−8η2
r drdx3+dx3
2
4η2
+z4
+
dz2 − (dx0)2
z2
+ ds2
s (6.21)
The dilaton and the B-field now depend on r and z
eφ
= 1 +
4η2 r2 + 1
z4
−1/2
, B =
2η r dξ ∧ (dr + rdx3)
z4 + 4η2 (r2 + 1)
. (6.22)
For the RR-fluxes we find
F(5)
= (1 + ∗)
4r dx0 ∧ dr ∧ dξ ∧ dx3 ∧ dz
z (z4 + 4η2 (r2 + 1))
, F(3)
=
8η
z5
(dx3 − rdr) ∧ dx0
∧ dz. (6.23)
The sequence of TsT-transformations
T(x3), ξ → ξ + 2η˜x3, T(˜x3), T(x1), x2 → x2 − 2η˜x1, T(˜x1), (6.24)
(where r = x2
1 + x2
2, ξ = arctan(x1/x2)) yields undeformed AdS5 × S5.
– 25 –
JHEP10(2016)045
6.3 r3,−1 ⊕ Ê
The R-matrix (R6 in table 1 with x1 → x2, x3 → x1)
R = J01 ∧ p2 + 2p0 ∧ p1 , (6.25)
preserves 3 bosonic isometries
J01, p2, p3 , (6.26)
and no supercharges. As before, it is more convenient to parameterise the group element
in a different way
ga = exp(tJ01) · exp(ρp1 + x2
p2 + x3
p3) · exp(log z D), (6.27)
so that t is an isometry. In the undeformed case we have the spacetime metric
ds2
η=0 =
−ρ2dt2 + dρ2 + dx2
2 + dx3
2 + dz2
z2
+ ds2
s, (6.28)
while the defomation gives
ds2
=z−6
1−
4η2 ρ2+4
z4
−1
−ρ2
z4
dt2
−16η2
ρdρdx2+dx2
2
z4
−16η2
+dρ2
z4
−4η2
ρ2
+
dx3
2
z2
+
dz2
z2
+ ds2
s . (6.29)
The dilaton and the B-field depend on ρ and z
eφ
= 1 −
4η2(4 + ρ2)
z4
−1/2
, B =
2η ρ dt ∧ (2dρ − ρdx2)
z4 − 4η2 (4 + ρ2)
, (6.30)
and the RR-fluxes are
F(5)
= −(1 + ∗)
4ρ dt ∧ dρ ∧ dx2 ∧ dx3 ∧ dz
z (z4 − 4η2 (4 + ρ2))
, F(3)
=
8η(2dx2 + ρdρ) ∧ dx3 ∧ dz
z5
. (6.31)
We can get back the undeformed AdS5 × S5 background by applying the sequence of
TsT-transformations
T(x2), t → t + 2η˜x2, T(˜x2), T(x1), x0 → x0 − 4η˜x1, T(˜x1), (6.32)
where x1 = ρ cosh t, x0 = ρ sinh t.
6.4 n4
Let us consider the R-matrix (R15 in table 1 with x1 ↔ x3)
R = p1 ∧ p3 + (p0 + p1) ∧ (J03 + J13) (6.33)
which preserves the 3 bosonic isometries
p0 + p1 , p2 , p3 , (6.34)
– 26 –
JHEP10(2016)045
and 8 supercharges. The metric is given by
ds2
= z−6
1−
4η2ξ−
z4
−1
z4
dx3
2
−η2
(dx+
)2
−
1
4
dξ− η2
ξ2
−dξ−+2dx+
z4
−2η2
ξ−
+
dx2
2 + dz2
z2
+ ds2
s , (6.35)
where we preferred to redefine ξ− = 2x− − 1. The dilaton and the B-field depend on ξ−
and z
eφ
= 1 −
4η2ξ−
z4
−1/2
, B =
η(ξ−dξ− + 2dx+) ∧ dx3
2 (z4 − 4η2ξ−)
. (6.36)
The RR-fluxes are
F(5)
= (1+∗)
dξ− ∧ dx+ ∧ dx2 ∧ dx3 ∧ dz
z(z4 − 4η2ξ−)
, F(3)
=
2η
z5
ξ−dξ− − 2dx+
∧dx2 ∧dz. (6.37)
We have checked that this background is self-dual (after field redefinitions) under a TsTtransformation
involving p0 + p1 and p3. If we view it as a deformation of AdS4 there
are no other bosonic isometries at our disposal, so it appears that this background cannot
be generated by (bosonic) TsT-transformations. As remarked earlier, it would be very
interesting to understand if it can be generated by applying non-abelian T-duality.
7 Conclusions
We have derived the target space geometry of the η and λ-deformed type IIB supercoset
string sigma models. With this result we have checked that the λ-deformation leads to a
(type II*) supergravity background, while in general the η-deformation only to a “generalized”
one in the sense of [25, 30]. When this is the case, the sigma model is expected
to be scale invariant but not Weyl invariant, and therefore does not seem to define a consistent
string theory. We have identified the (necessary and sufficient) condition for the
η-model to have a standard supergravity background as target space. This is translated
into an algebraic condition on the R-matrix, which we refer to as the unimodularity condition.
It imposes strong restrictions on non-abelian R-matrices, and in fact all non-abelian
R-matrices considered in previous works do not lead to supergravity solutions.
We have also analyzed the problem of finding all unimodular R-matrices which solve the
CYBE for the bosonic subalgebra so(2, 4) ⊕ so(6) ⊂ psu(2, 2|4). The complete list of rank
four non-abelian R-matrices for so(2, 4) has been given and we have showed that the only
other non-abelian R-matrices in this case have rank six. We have argued that most of these
examples should correspond to a sequence of non-commuting TsT-transformations and have
verified this explicitly in some cases. It should be possible to understand these deformations
in terms of twisted boundary conditions for the string just as in the standard TsT case [44].
There are many similarities between the backgrounds we construct and that of HashimotoItzhaki/Maldacena-Russo
[67, 68] and the dual field theories are expected to be certain
non-commutative deformations of N = 4 super Yang-Mills, see [69] and in particular [70].
Many interesting open questions remain. It would be important to find all possible
unimodular R-matrices of psu(2, 2|4) to have a complete list of Yang-Baxter deformations
– 27 –
JHEP10(2016)045
of AdS5 × S5 with a string theory interpretation. A question is whether any of them are
of the Jordanian type. It is particularly interesting to investigate whether it is possible to
have unimodular R-matrices that solve the MCYBE rather than the CYBE, to solve one of
the puzzles of [22]. One could also try to give an interpretation to backgrounds generated
by non-unimodular R-matrices; in many cases one can associate to them a formally Tdual
model which does describe a string sigma model, so it is natural to wonder what
these backgrounds correspond to. See [41] for some investigations along these lines. It
would be also interesting to clarify if these deformed models have a connection to nonabelian
T-duality, in view of the similarities between our unimodularity condition and the
tracelessness condition of [63].
Our results are also useful to make further progress in the case of the λ-model. In
fact, we have written the NSNS and RR background fields in terms of the Lie algebra
operators which are used to define the deformation procedure, and after picking a certain
parameterisation for the group element this enables to obtain their explicit form. This
method is more efficient, albeit equivalent, to the ones used so far e.g. in [22, 26, 41]. One
could then check the proposal of [27] for the background of the λ-deformed AdS3 ×S3 ×T4
string, and finally derive the one for the AdS5 × S5 case. It would be interesting to
understand whether there is room to modify the definition of the λ-model, hence realising
other possible deformations of the string. In fact, in the current status the λ-model is
related through Poisson-Lie T-duality to the η-model based on the MCYBE, but there is
no known counterpart for deformations based on the CYBE.
Acknowledgments
We would like to thank Arkady Tseytlin for useful discussions and helpful comments on
a first draft of this manuscript. This work was supported by the ERC Advanced grant
No. 290456.
A 4-graded superisometry algebras
In this appendix we review some facts about the relevant superalgebras and explain our
notation and conventions. In [4] it was shown that for all cases of interest here40 the
superisometry algebra — which admits a 4-grading that extends the 2-grading of the
bosonic subalgebra — can be written in the same form. The bosonic subalgebra is of the
standard symmetric space form
[Jab, Pc] = 2ηc[aPb] , [Pa, Pb] =
1
2
Kab
cd
Jcd ,
[Jab, Jcd] = ηacJbd − ηbcJad − ηadJbc + ηbdJac . (A.1)
Here a, b, c = 0, . . . , 9 and Jab generate Lorentz-transformations and rotations while Pa
generate translations. Note that since the space is typically a product of factors Jab is
40
We restrict our attention to models with only RR flux since these have certain simplifying features like
4-symmetry.
– 28 –
JHEP10(2016)045
block-diagonal with components mixing different factors absent and this should be taken
into account in interpreting the last commutator above. In the case of RR backgrounds,
i.e. no NSNS three-form flux, the commutators involving the supercharges take the form
(here and in the rest of the paper we specialize to the type IIB case, but the type IIA case
works in the same way)
[Pa, QI
ˆα] = −i(QJ
KJI
γa)ˆα , [Jab, QI
ˆα] = −
1
2
(QI
γab)ˆα , (I, J = 1, 2)
{Q1
ˆα, Q1
ˆβ
} = {Q2
ˆα, Q2
ˆβ
} = iγa
ˆαˆβ
Pa , {Q1
ˆα, Q2
ˆβ
} = (γa
K12
γb
)ˆαˆβ Jab . (A.2)
Here ˆα = 1, . . . , N where 2N is the number of supersymmetries preserved by the background.
For AdS5 × S5 (psu(2, 2|4)) N = 16 and γa
ˆαˆβ
are the standard 16 × 16 symmetric
Weyl blocks or ‘chiral gamma-matrices’ (see for example the appendix of [36]). For
AdS3 × S3 × T4 (psu(1, 1|2)2) N = 8 and for AdS2 × S2 × T6 (psu(1, 1|2)) N = 4 and the
gamma-matrices γa
ˆαˆβ
involve an extra projector to make them 8 × 8 and 4 × 4 respectively.
The 4 automorphism acts as
Jab → Jab , Pa → −Pa , Q1
→ iQ1
, Q2
→ −iQ2
. (A.3)
We introduce projectors that split the generators TA = {Pa, Jab, QI
ˆα} according to their
4-grading as follows
P(0)
(TA) = Jab , P(1)
(TA) = Q1
ˆα , P(2)
(TA) = Pa , P(3)
(TA) = Q2
ˆα . (A.4)
Finally KAB appearing on the right-hand-side in (A.1) and (A.2) is the inverse of the Lie
algebra metric defined by the supertrace41
Str(TATB) = KAB , TA = {Pa, Jab, QI
ˆα} , (A.5)
e.g.
1
2
Kab
ef
Kef,cd = 2ηa[cηd]b . (A.6)
It can be expressed in terms of the geometry and fluxes of the corresponding symmetric
space supergravity background as
Kab
= ηab
, Kab
cd
= −Rab
cd
, KˆαI ˆβJ
=
i
8
S ˆαI ˆβJ
, (A.7)
where Rab
cd and SIJ
are the Riemann curvature and RR field strength bispinor respectively.42
Let us also note the relation
Kab
cd
(K12
γcd)ˆαˆβ = 8(γ[aK12
γb])ˆαˆβ . (A.8)
41
Note that our definition of K differs by a factor of i compared to the definition used in [4].
42
The curvature of AdS is Rab
cd
= 2δc
[aδd
b] while that of the sphere is Rab
cd
= −2δc
[aδd
b] in our conventions.
The RR flux takes the form
AdSn × Sn
× T10−2n
: S ˆαI ˆβJ
= −4i(σ2
)IJ
(Pγ01234
)ˆα ˆβ
,
where the projector P, with QI
= QI
P, is given by 1 for n = 5, 1
2
(1+γ6789
) for n = 3 and 1
2
(1+γ6789
)1
2
(1+
γ4568
) for n = 2.
– 29 –
JHEP10(2016)045
Finally for operators acting on the Lie algebra (i.e. endomorphisms) M : g → g we
define its components in the following way
M(TC) = TDMD
C . (A.9)
The transpose operator is defined with respect to the supertrace by
Str(TAM(TB)) = Str(MT
(TA)TB) , (A.10)
or
MAB = (−1)AB
(MT
)BA MAB = KACMC
B (A.11)
e.g.
(MT
)aˆβ1 = Kˆβ1ˆγ2Mˆγ2
a , (MT
)a,bc =
1
2
Kbc,deMde
a . (A.12)
The supertrace of the Lie algebra operator M is given by
Str(M) = (−1)A
MA
A = KAB
Str(TAMTB) . (A.13)
When we need to raise indices with KAB we use the convention
MA
= MBKBA
. (A.14)
To conclude, when writing generic commutation relations we write
[TA, TB] = fC
ABTC . (A.15)
B Useful results for the deformed models
In this appendix we collect some useful identities and expressions to obtain the results
presented in the main text. In the two deformed models, we can relate OT
± and O± by
λ − model : OT
− = Ad−1
g O+, η − model : OT
−
ˆdT
= ˆdT
O+ . (B.1)
Using the definitions of O±, we can express M defined in (2.10) in terms of O± and
projectors only
λ − model : M = −ΩT
+ (OT
+)−1
(1 − ΩΩT
) = −ΩT
+ (1 − λ−4
)(OT
+)−1
P(2)
,
η − model : M = O−1
− (O− + 2ηRg
ˆdP(2)
) = 1 − 2P(2)
+ 2O−1
− P(2)
,
(B.2)
which is useful to prove
λ − model : Ad−1
h P(2)
= O+(1 + Ω(OT
+)−1
)P(2)
= P(2)
(1 + (OT
+)−1
Ω)O+ ,
η − model : Ad−1
h P(2)
= O+(2P(2)
− 1)O−1
− P(2)
.
(B.3)
Note that using the expression for M we can express A− in terms of A+ as
A− = MA+ =
A+ + (M − 1)A
(2)
+
−ΩT A+ + (M + λ−2)A
(2)
+
. (B.4)
The rest of this appendix is devoted to the two deformed models separately.
– 30 –
JHEP10(2016)045
B.1 λ-model
The expressions for dA± in (3.1), (3.2) can be rewritten as
dA+ = −
1
2
{A+, A+}−
1
2
(1−λ−4
)O−1
+ ({A
(2)
+ , A
(2)
+ }−λ2
{A
(1)
+ , A
(1)
+ }+2λ{A
(2)
+ , A
(3)
+ }) , (B.5)
dA− =
1
2
{A−, A−}+
1
2
(1−λ−4
)(OT
+)−1
({A
(2)
− , A
(2)
− }−λ2
{A
(3)
− , A
(3)
− }+2λ{A
(2)
− , A
(1)
− }) , (B.6)
if we use
ΩT
{X, X} − {ΩT
X, ΩT
X} = (1 − λ−4
)({X(2)
, X(2)
} − λ2
{X(1)
, X(1)
} + 2λ{X(2)
, X(3)
}),
(B.7)
for X ∈ g, and the same for Ω but with X(1) and X(3) interchanged.
To calculate the component T ˆα1 of the torsion, we first need to compute the Lorentztransformed
spin-connection AdhA
(0)
+ +dhh−1. We do this by taking the exterior derivative
of both sides of the relation E(2) = AdhA
(2)
− , from which we find the equation
0 = {AdhA
(0)
+ + dhh−1
− A
(0)
+ , E(2)
} + λ(1 − λ−4
)P(2)
Adh(OT
+)−1
Ad−1
h {E(2)
, E(1)
}
+ {E(1)
, AdhP(1)
ME(2)
} − iλ{E(3)
, P(3)
ME(2)
} − iλP(2)
MT
{E(2)
, E(3)
}
−{AdhP(0)
ME(2)
, E(2)
}−
1
2
Adh{P(1)
ME(2)
, P(1)
ME(2)
}−
1
2
λ2
{P(3)
ME(2)
, P(3)
ME(2)
}
−
1
2
(1 − λ−4
)P(2)
Adh(OT
+)−1
Ad−1
h ({E(2)
, E(2)
} + 2λ{E(2)
, AdhP(1)
ME(2)
})
−
1
2
P(2)
MT
{E(2)
, E(2)
} − λ2
P(2)
MT
{E(2)
, P(3)
ME(2)
} , (B.8)
where we used (3.3) and (B.6). This equation determines AdhA
(0)
+ + dhh−1 completely:
this is obvious for the terms involving fermionic vielbeins, while for the terms involving Ea
it follows from symmetry in the same way that the condition Tab
c = 0 determines the spin
connection Ωab
c. Using the algebra (A.1), (A.2) as well as (B.3) the result is
[dhh−1
+ AdhA+]ab = −Ωab +
1
2
Ec
Habc + 2i(E1
γ[a)ˆα(AdhM)ˆα1
b] . (B.9)
Here we have used the fact, which will be proven below, that the expression that we find
Habc = 3[AdhM][ab,c] + 3iM ˆα1
[a[Adh]b|d|γd
ˆαˆβ
M
ˆβ1
c] , (B.10)
is equivalent to the one in (3.8). In fact, if we calculate H = dB using the first definition
– 31 –
JHEP10(2016)045
for B in (3.15) we find
H = dB =
1
3
(1 − λ−4
)−1
Str(ΩA− ∧ ΩA− ∧ ΩA−) − Str(A− ∧ A− ∧ A−)
−
1
2
(1 − λ−4
)−1
Str(A+ ∧ (Ω{A−, A−} − {ΩA−, ΩA−}))
= −Str((A
(0)
+ + A
(0)
− ) ∧ A
(2)
− ∧ A
(2)
− ) + λ2
Str(A
(2)
+ ∧ A
(3)
− ∧ A
(3)
− )
−
1
2
Str(A
(2)
− ∧ {A
(1)
− , A
(1)
− + 2λA
(1)
+ })
= Str(E(2)
∧ E(1)
∧ E(1)
)−Str(E(2)
∧ E(3)
∧ E(3)
)−Str(P(0)
AdhME(2)
∧ E(2)
∧ E(2)
)
− Str(E(2)
∧ P(1)
AdhME(2)
∧ P(1)
AdhME(2)
)
= −
i
2
Ea
E1
γaE1
+
i
2
Ea
E2
γaE2
+
1
3!
Ec
Eb
Ea
Habc , (B.11)
with Habc given by (B.10). On the other hand, if we start from B given in the second
line of (3.15), we find a result which is mapped to the previous one by the replacements
A− ↔ A+, Ω ↔ ΩT and A(3) ↔ A(1). This leads to the same form of H except now
with Habc given by (3.8), which proves the equivalence of the two expressions. Let us also
remark that this computation shows that the NSNS three-form superfield H = dB satisfies
the correct superspace constraints.
In order to check that the dilatinos in (3.10), (3.14) are in fact the spinor derivatives
of the dilaton φ, we start from (3.16) and compute
dφ = −
1
2
STr(O−1
− Ad−1
g dAdg) = −
1
2
KAB
STr(TAO−1
− [g−1
dg, TB])
= −
1
2
KAB
STr(TAO−1
− [O−A−, TB])
= −
1
2
KAB
STr(TAO−1
− [A−, TB]) +
1
2
KAB
STr(TAO−1
− Ad−1
g [ΩA−, AdgTB])
= −
1
2
KAB
STr(TAO−1
− [A−, TB]) +
1
2
KAB
STr(TAAdgO−1
− Ad−1
g [ΩA−, TB])
= −
1
2
KAB
STr(TAO−1
− [A−, TB]) +
1
2
KAB
STr(TAAdg(OT
+)−1
[ΩA−, TB])
= −
1
2
KAB
STr(TAO−1
− [A−, TB]) +
1
2
KAB
STr(TAΩO−1
− Ad−1
g [ΩA−, TB])
= −
1
2
KAB
STr(TAO−1
− [A−, TB]) +
1
2
KAB
STr(TAΩO−1
− ΩT
[ΩA−, TB])
+
1
2
(1 − λ−4
)λ2
KAB
STr(TAΩO−1
− P(2)
[ΩA−, TB]) , (B.12)
where we used (A.13) and in the last step we inserted 1 = 1 − ΩΩT + ΩΩT =
(1 − λ−4)P(2) + ΩΩT . It is easy to see that the A
(0)
− -terms cancel, as they must since they
transform as a connection.
– 32 –
JHEP10(2016)045
B.2 η-model
The expressions for dA± in (4.1), (4.3) can be rewritten as
dA+ =
1
2
{A+, A+} −
1
2
cη2
{ ˆdT
A+, ˆdT
A+} + (O−1
+ − 1) 4{A
(2)
+ , A
(3)
+ } + ˆη2
{A
(1)
+ , A
(1)
+ }
+ ηO−1
+ Rg{A
(2)
+ , ˆdT
A
(2)
+ } , (B.13)
dA− =
1
2
{A−, A−} −
1
2
cη2
{ ˆdA−, ˆdA−} + (O−1
− − 1) 4{A
(2)
− , A
(1)
− } + ˆη2
{A
(3)
− , A
(3)
− }
− ηO−1
− Rg{A
(2)
− , ˆdA
(2)
− } , (B.14)
where we have rewritten e.g. the last term in the expression for dA+ as
ηO−1
+ Rg{A
(2)
+ , ˆdT
A
(2)
+ }
+ (1 − O−1
+ )
1
2
{A+, A+}−
1
2
cη2
{ ˆdT
A+, ˆdT
A+}−4{A
(2)
+ , A
(3)
+ }−ˆη2
{A
(1)
+ , A
(1)
+ } .
(B.15)
As in the case of the λ-model, to calculate the component T ˆα1 of the torsion we must
first find the Lorentz-transformed spin connection AdhA
(0)
+ − dhh−1 (note the difference
in sign between the two models). We use the same method explained in the previous
subsection and we find
[AdhA
(0)
+ − dhh−1
]ab = Ωab −
1
2
Ec
Habc + 2iˆη(γ[aE1
)ˆα[AdhM]ˆα1
b] , (B.16)
where we write the components of Habc as
Habc = 3[AdhM][ab,c] − 3iˆη2
[Adh][a|d|M ˆα1
bγd
ˆαˆβ
M
ˆβ1
c] . (B.17)
This expression is equivalent to the one in (4.7), which is easy to verify by a calculation
similar to the one performed for the λ-model: the B-field written as in the first way of (4.13)
leads to Habc of the form (4.7), while the second way leads to the form in (B.17). The
same calculation also shows that the remaining components of the superform H satisfy the
standard supergravity constraints.
If we take (4.14) as the definition of the dilaton in the case of the η-model we find
dφ = −
1
2
ηKAB
STr(TA
ˆdT
O−1
+ Rg[g−1
dg, TB]) +
1
2
ηKAB
STr(TARg
ˆdT
O−1
+ [g−1
dg, TB])
= −
1
2
ηKAB
STr(TA
ˆdT
O−1
+ Rg[A+, TB]) −
1
2
KAB
STr(TAO−1
+ [A+, TB])
−
1
2
ηKAB
STr(TAO−1
+ [Rg
ˆdT
A+, TB]) −
1
2
η2
KAB
STr(TA
ˆdT
O−1
+ Rg[Rg
ˆdT
A+, TB])
= −
1
2
KAB
STr(TAO−1
+ [A+, TB]) +
1
2
cη2
KAB
STr(TA
ˆdT
O−1
+ [ ˆdT
A+, TB])
−
1
2
ηKAB
STr(TA
ˆdT
O−1
+ Rg[A+, TB]) −
1
2
ηKAB
STr(TAO−1
+ Rg[ ˆdT
A+, TB])
+
1
2
ηKAB
STr(TARg[ ˆdT
A+, TB]) , (B.18)
– 33 –
JHEP10(2016)045
where we used the (M)CYBE (1.7) in the last step. It is again easy to verify that the
A(0)-terms cancel, as they must.
Using (4.16) and (4.15) and (B.16), the explicit result for the vector Ka in (4.19) is
Ka
=
i
32
η(γa
)ˆαˆβ
KAB
STr
× [TA, RTB]Adg [(1 − ηRg)Ad−1
h Q1
ˆα, Ad−1
h Q1
ˆβ
] + ˆη−2
[(1 + ηRg)Q2
ˆα, Q2
ˆβ
]
+ fermions
= −
η
2
[ˆη−2
+ Adh]a
bKAB
STr([TA, RTB]gPbg−1
)
−
η2
32
ˆη−2
(γa
γc
)ˆα
ˆβ[Rg]
ˆβ2
ˆα2 − [Adh]a
b(γb
γc
)ˆα
ˆβ[Rg]
ˆβ1
ˆα1 KAB
STr([TA, RTB]gPcg−1
)
−
iη2
32
[Adh]a
b(γb
γc
K12
γd
)ˆα
ˆβ[Rg]
ˆβ2
ˆα1
− ˆη−2
(γa
γc
K21
γd
)ˆα
ˆβ[Rg]
ˆβ1
ˆα2 KAB
STr([TA, RTB]gJcdg−1
)
+ fermions . (B.19)
Open Access. This article is distributed under the terms of the Creative Commons
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any medium, provided the original author(s) and source are credited.
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