Introduction to AdS/CFT correspondence
1 Introduction
The goal of these lectures is to give a brief introduction to the AdS/CFT
correspondence. We will mainly follow famous review paper that was published
in 1999 soon after the formulation of given correspondence. Then the
subject has developed in many directions and it is not possible to give an
overview of all these directions.
As is well known string theories are well defined in strictly given number
of dimensions. In fact, 10 dimensional string theory is described by a string
with fermionic excitations on the world-sheet and the massless spectrum
of given theories gives supergravity field theories. It is important that all
closed string theories contain the graviton as the massless excitation and
hence it is believed that string theory could be a candidate for a quantum
theory of gravity. Since we know from every day experience that our world
is three dimensional we should find mechanism how string theory describes
phenomena in three dimensions. Usually we consider string theory on R4
×
M6, where M6 is some six dimensional compact manifold. Then, the low
energy interactions and physical content are determined by the geometry of
M6.
We also know that the strong interactions are described by Quantum
Chromodynamics (QCD), that is the gauge theory based on the group SU(3).
QCD possesses an asymptotic freedom which means that the effective coupling
constant decreases with increasing the energy scale. At low energies QCD is
strongly coupled which means that standard perturbative treatment is not
well suited. To overcome this difficulty it is possible to implement the numerical
simulations when we defince QCD on the lattice. It was further suggested
by ’tHooft [2] that the theory could simplify when the number of colors N is
large, where N is the rank of the gauge group SU(N). It was believed that
we could solve exactly the theory with N = ∞ and then we could extrapolate
these results to the case 1/N = 1/3 in some way. It is also very interesting
that the diagrammatic expansion of the field theory resembles string theory
in the sense that the large N theory is a free string theory with the string
coupling constant 1/N. In this way the large N limit gives a connection
between gauge theories and string theories.
It is very important that the string theory is not consistent in four flat
1
dimensions. More precisely, it is possible to quantize string theory in four
dimensions at least in principle, but the prise is that the anomaly appears
that leads to the emergence of Liouville field. It is natural to interpret this
field on the world-sheet as an additional coordinate so that it is natural to
interpret string as moving in five dimensions.
It is also important to stress that the arguments that the gauge theories
in the large N limit are related to string theory are very general and are valid
for any gauge theory. Let us consider the gauge theory where the coupling
does not run with the energy scale. Given theory is conformally invariant.
A simple example is supersymmetric SU(N) or U(N) gauge theory in four
dimensions with four spinor supercharges N = 4. This theory contains gauge
fields (gluons), four fermions and six scalar fields in the adjoint representation
of the gauge field. The Lagrangian of such theories is completely determined
by the supersymmetry. Further, this theory has a global SU(4) R-symmetry
that rotates the six scalar fields and the four fermions. The conformal group
in four dimensions is SO(4, 2), that includes the Poincare transformations
as well as scale transformations and special conformal transformations. By
principle of duality we should expect that these symmetries should be symmetries
of dual string theory. Natural idea is to find the five dimensional
geometry that has these symmetries. It turns out that there is only one space
with this isometry, five dimensional Anti-de-Sitter space (AdS5), which is the
maximally symmetric solution of Einstein’s equations with a negative cosmological
constant. Clearly we have to add five more dimensions to get ten
dimensional space-time in order to have string theory defined in ten dimensions.
As we said above the gauge theory has SU(4) ≃ SO(6) global symmetry
we search for five dimensional space that possesses this symmetry. It turns
out that SO(6) global symmetry is the symmetry of the five sphere S5
. In
other words we should expect the duality between N = 4 U(N) Yang-Mills
theory and ten dimensional superstring theory on AdS5 × S5
. Of course, we
should be more precise with this statement.
The idea of the duality between these two theories was based on the
analysis of the extended objects in string theories, which are D-branes and
black holes. As we known D-branes are extended objects that are presented in
all string theories. We use notation Dp-brane which means that this brane is
extended in p− spatial dimensions. Tensions of these objects are proportional
to 1/gs, where gs is a string coupling constant and we see that in case when
the string coupling constant is small these objects are much heavier than
ordinary strings. It is very important that unlike of other extended objects
2
in string theories D-branes have very simple definitions: They are defined
in string perturbation theory as surfaces where open strings can end [3].
Clearly these strings have infinite number of excitations as ordinary string
theory but we are mainly interested in the massless modes that propagate on
the world-volume of Dp-branes. They are modes that describe the transverse
fluctutations of branes, gauge fields living on the world-volume of Dp-brane
and their fermionic partners so that resulting theory is supersymmetric with
16 supercharges. If we have N coincident Dp-branes, now open strings can
start and end on different branes so that it is natural that they carry indices
that run from 1 to N. As a result we find that the low energy dynamics
is described by U(N) gauge theory. Further, Dp-branes are charged under
p+1-form gauge potencials known as Ramond-Ramond fields. These massless
modes are parts of closed string modes in corresponding closed string theory.
If we now add D-branes together then due to the fact that they couple to
Ramond-Ramond fields we find that they generate the flux of corresponding
field strength, that in the end contribute to the stress energy tensor and
consequently has an impact on the gravity so that the space is curved. More
precisely, we can find solutions of the equations of motions where Dp-branes
are sources and and they presence lead to the emergence of non-trivial gauge
fields and curved spacetime, for some review, see [4].
As we show below when we consider set of N coincident D3-branes and
study the near horizon geometry we find that it has the form of AdS5×S5
. On
the other hand we know that the low energy dynamics on their world-volume
is governed by N = 4 U(N) SYM theory. These two descriptions are valid for
different regimes in space of the coupling constants. Explicitly, perturbative
field theory is valid when gsN is small while the low energy description is
valid when the radius of curvature is much larger than the string scale that
implies that gsN should be large. However then we can conjecture that SYM
at strong coupling is described by the near horizon region of the brane whose
geometry is AdS5 × S5
. The subject of this lecture is to give review of this
conjecture.
1.1 Large N Gauge Theories as String Theories
Let us consider U(N) Yang-Mills (YM) theory in four dimensions. The action
of given theory has the form
S = −
1
g2
Y M
d4
xTrFµνFµν
, Fµν = ∂µAν − ∂νAµ − i[Aµ, Aν] , (1)
3
where Aµ is hermitean field taking its value in the adjoin representation of
U(N). Explicitly, we have
A†
µ = Aµ , F†
µν = Fµν . (2)
We know that Aµ has mass dimension [Aµ] = M, where M is some mass scale.
Further, [∂µ] = M so that [Fµν] = M2
. As a result we find that [g2
Y M ] = M0
so that the coupling constant is dimensionless. On the other hand we know
that beta function for pure SU(N) YM theory is
µ
dgY M
dµ
= −
11
3
N
g3
Y M
16π2
+ O(g5
Y M ) (3)
Solving this equation we obtain
g2
Y M =
24
11π2
1
ln µ
Λ
, (4)
where Λ is some mass scale. and we see that the coupling constant decreases
with mass scale which is the property known as asymptotic freedom. On
the other hand we see that when the energy scale is decreases the coupling
constant increases and the perturbative calculations will not be valid. ΛQCD
is the scale where the coupling constant formally diverges.
Our goal is to understand the limit N → ∞ and its impact on the scaling
of the coupling gY M . Let us consider the beta function for pure SU(N) YM
given above, when we introduce the parameter
λ ≡ g2
Y M N (5)
so that
µ
dgY M
dµ
= −
11
3
λgY M
16π2
(6)
and we see that the left and right side are of the same order if we take the
large N limit while λ is kept fixed. This limit is known as ’tHooft limit. This
limit is also valid when we include matter fields in the adjoint representation
on condition that given theory is still asymptotic free. In case of the conformal
theory as for example N = 4 SYM theory we find that there is also possibility
to consider the limit λ → ∞.
Let us consider general theory with fields Φa
I , where a is index in the
adjoint representation of SU(N) and I is some label of field. We further
4
presume that the 3−point vertices of all these fields ar proportional gY M , the
4−point functions to g2
Y M . In other words we presume that the Lagrangian
density has the form
L ∼ Tr(dΦIdΦI) + gY M cIJK
Tr(ΦIΦJ ΦK) + g2
Y M dIJKL
Tr(ΦIΦJ ΦKΦL) , (7)
where cIJK
, dIJKL
are constants. If we rescale the fields ar ˜ΦI ≡ gY M ΦI we
find that the Lagrangian density takes the form
L =
1
g2
Y M
Tr(d˜ΦId˜ΦI) + cIJK
Tr(˜ΦI
˜ΦJ
˜ΦK) + dIJKL
Tr(˜ΦI
˜ΦJ
˜ΦK
˜ΦL) (8)
We would like to see what happens to correlation functions in the limit of
large N when λ is constant. Clearly we could thought that it corresponds to
the classical limit since by definition g2
Y M = λ
N
→ 0. On the other hand the
number of components of the fields goes to infinity as well so that we should
be more careful. It is isefull to write Feynman diagrams of the theory (8) in
the double line notation where the adjoint field Φa
is represented as a direct
product of a fundamental and an anti-fundamental field Φi
j so that we have
for the propagator for SU(N) theory (we omit space-time dependence)
Φi
jΦk
l ∝ (δi
l δk
j −
1
N
δi
jδk
l ) (9)
We see that there is small mixing term proportional to 1
N
due to the fact that
for SU(N) theory we have the trace of the field to be equal to zero Φi
i = 0.
Then on the left side vanishes identically while the right side is equal to
Φi
iΦk
l ∝ (δi
l δk
i −
1
N
δi
iδk
l ) = δk
l − δk
l = 0 . (10)
However this term is proportional to 1
N
so that it gives subleading contribution
and can be ignored. Then we can see that any Feynman diagram of
ajdoint fields has the form of the network of double lines. For example, in
case of the vacuum diagram these double lines form the edges in triangulation
of a surface. This surface is oriented since the lines have an orientation as
follows from the fact that fundamental index points in one direction while
anti-fundamental index in opposite direction. Further, when we compactify
space by adding a point at infinity each diagram corresponds to a compact,
closed, oriented surface.
5
The important question is the powers of N and λ associated with such
a diagram. From (8) we see that the vertex has the factor 1
g2
Y M
= N
λ
, while
propagators are proportional to λ
N
as follows from the kinetic term in (8).
There is also additional powers of N from the sum over indices in the loops
which gives a factor of N for each loop in the diagram. In other words we
find that a diagram with V vertices, E propagators corresponding E edges
in simpicial decomposition and F-loops , corresponding faces in simplicial
decomposition comes with following coefficient
NV −E+F
λE−V
= Nχ
λE−V
, (11)
where χ = V − E + F is the Euler character of the surface corresponding to
given diagram. On the other hand we know that for closed oriented surfaces
χ = 2 − 2g where g is the genus of the surface. Finally we find that the
perturbative expansion of any diagram in the field theory may be written as
a double expansion
∞
g=0
N2−2g
∞
i=0
cg,iλi
=
∞
g=0
N2−2g
fg(λ) , (12)
where fg(λ) is some polynomial in λ. We see that in the large N limit the
dominant contribution will come from surfaces of maximal χ that corresponds
surface with minimal genus, which are surfaces with topology of
sphere (plane) that will give contribution of N2
, while all other diagrams
will come with powers of 1/N2
. These leading order diagrams are known as
planar diagrams and come with factor N2
.
It is remarkable that form of the expansion (12) has the form of the
expansion as in case of closed oriented strings when we identify 1/N as the
string coupling constant. This analogy is one of the strongest motivations for
the belief that the string theories and field theories are related and it also
suggests that this relation will be mostly easily seen in the large N limit
where the dual string theory is weakly coupled. It is also important to stress
that this analogy is based on the perturbative theory which generally does
not converge so that we have to be careful with definite conclusions. We
should rather consider this relation as the indication of this correspondence.
Further, we know that field theory has effects as instantons that are nonperturbative
in 1/N expansion. Then the exact matching will require that
there should be found corresponding string theory interpretation.
6
It is also important to stress that this analogy can be extended in the direction
where we correspondence between additional insertion on the field
theory side and the vertex operators in the string theory interpretation.
Further, given theory could be easily generalized when we incorporate matter
in the fundamental representations. Matter in the fundamental representation
has a single-line propagators in the diagrams which can be interpreted
as the boundary of the corresponding surface. In other words now we have
to sum over surfaces with boundaries as in case of open string theories.
The previous discussion suggests that gauge theories are dual to string
theories with a coupling constant proportional to 1/N in the’t Hooft limit.
On the other there is no indication which string theory is dual to a particular
gauge theory. It seems to be rather difficult to find corresponding string
theory directly due to the fact that it is very difficult to formulate string theory
in four dimensions or generally below the critical dimension. Explicitly,
we have to add additional field on the string world-sheet in order to ensure
consistency of given theory. This field is known as Liouville fied and can be
interpreted as a fifth space-time dimensions. The idea that five dimensional
string theory could be related to four dimensional gauge theories was firstly
proposed by Polyakov in [5, 6], where the coupling of Liouville field to the
other fields has some specific form. The AdS/CFT correspondence provides
concrete realization of this idea where five additional dimensions, together
with radial coordinate on AdS, leads to a standard ten dimensional string
theory.
1.2 Black p−Branes
The first hint of the relation between large N field theories and string theory
came from the study of p−branes in string theory. Originally the p−branes
were found as classical solutions in supergravity that, as is well known, is
the low energy limit of string theory. In 1995 J. Polchinski discovered that
D-branes have full string theory description [3]. Then by comparing these
two descriptions led to the discovery of AdS/CFT correspondence.
As we know, string theory has variety of classical solutions. We restrict
ourselves to the examples of parallel Dp-branes. Let us consider type II string
theory in ten dimensions. We search the solutions that carry electric charge
with respect to Ramond-Ramond (R-R) (p + 1) form Ap+1, where p is even
in IIA theory while p is odd in IIB theory. It can be also shown that these
theories contain also magnetically charged (6−p)-branes that are electrically
7
charged under the dual dA7−p = ∗dAp+1 potential, where ∗ is Hodge dual.
Then we find that R−R charges have to be quantized according to the Dirac
procedure.
In order to find explicit solution, let us consider low energy effective action
in the string frame
S =
1
(2π)7l8
s
d10
x
√
−g e−2ϕ
(R + 4gµν
∇µϕ∇νϕ) −
2
(8 − p)!
F2
p+2 , (13)
where ls is the string length, Fp+2 is the field strength of the (p + 1)−form
potential defined as Fp+2 = dAp+1. Note that for p = 3 we have ∗F =
F and given case is known as self-dual. This case is also problematic due
to the lacking of the Lagrangian formulation as follows from the fact that
schematically the kinetic term for the gauge field has the form F ∧ ∗F
which in case F = ∗F gives F ∧ F so that the equations of motion that
follow from this action are d ∗ F = dF = d2
A4 = 0.
Then we are trying to find solution corresponding to the electric source
of charge N for Ap+1. From the point of view of the transverse space we have
point object so that it is natural to require that the metric is spherically
symmetric in (9 − p) dimensions. Further, since Dp-brane is extended in p
dimensions we require SO(p) symmetry in these directions. In other words
we presume that the metric has the form
ds2
= ds2
10−p + eα
p
i=1
dxi
dxi , (14)
where ds2
10−p is a Lorentzian-signature metric in (10−p) dimensions. Further,
since this Dp-brane is source of the R-R field we have the condition
S8−p
∗Fp+2 = N , (15)
where S8−p
is (8 − p)−sphere surrounding the source. The resulting metric
in the string frame has the form
ds2
= −
f+(ρ)
f−(ρ)
dt2
+ f−(ρ)
p
i=1
dxi
dxi+
f−(ρ)−1
2
−5−p
7−p
f+(ρ)
dρ2
+r2
f−(ρ)
1
2
−5−p
7−p dΩ2
8−p ,
(16)
8
with non-trivial profile of the dilaton field
e−2ϕ
= g−2
s f−(ρ)−p−3
2 , (17)
where the functions f±(ρ) have the form
f±(ρ) = 1 −
r±
ρ
7−p
(18)
where gs is the asymptotic string coupling constant. The parameters r+ and
r− are expressed through the mass M (per unit volume) and the RR charge
N of the solution by
M =
1
(7 − p)(2π)7dpl8
P
((8 − p)r7−p
+ − r7−p
− ) , N =
1
dpgpl7−p
s
(r+r−)
7−p
2 , (19)
where lP = g
1/4
s ls is the 10− dimensional Planck length and dp is a numerical
factor
dp = 25−p
π
5−p
2 Γ
7 − p
2
. (20)
The metric in the Einstein frame (gE)µν is defined by multiplying the string
frame metric gµν by the factor gse−ϕ so that the action takes the standard
Einstein-Hilbert form with the canonical kinetic term
S =
1
(2π)7l8
P
d10
x
√
−gE RE −
1
2
gµν
E ∇µϕ∇νϕ + . . . . (21)
This solution has the horizon at ρ = r+ and curvature singularity at ρ = r−,
when we also presume that r+ > r−. On the other hand there is special
solution when r+ = r− and we find
M =
1
(2π)7gslp+1
s
N (22)
that is extremal p-brane Generally for r+ > r− we find
M ≥
1
(2π)7gslp+1
s
(23)
that is known as non-extremal black p-brane. We call this brane black since
there is an event horizon for r+ > r− whose area goes to zero in the extremal
9
limit r+ = r−. It is important to stress that the extremal solution with p ̸= 3
has a singularity we find that the supergravity description breaks down near
ρ = r+ and hence the full description of given p-brane can be done in terms
of superstring theory.
Let us now consider more explicitly the extremal limit r+ = r− where the
line element has the form
ds2
= f+(ρ) −dt2
+
p
i=1
dxi
dxi +f+(ρ)−3
2
−5−p
7−p dρ2
+ρ2
f+(ρ)
1
2
−5−p
7−p dΩ2
8−p .
(24)
We see that the original Euclidean group of black p-brane that was SO(p)
was extended to the Poincare group SO(p, 1). In order to investigate the
geometry of the extremal solution near the horizon we define new coordinate
r as
r7−p
= ρ7−p
− r7−p
+ (25)
and introduce the isotropic coordinates
ra
= rθa
(a = 1, . . . , 9 − p),
a
(θa
)2
= 1 . (26)
Then the metric has the form
ds2
=
1
H(r)
−dt2
+
p
i=1
dxi
dxi + H(r)
9−p
a=1
dra
dra
, (27)
where
eϕ
= gsH(r)
3 − p
4
, (28)
and where
H(r) ≡
1
f+(ρ)
= 1 +
r7−p
+
r7−p
, r7−p
+ = dpgsNl7−p
s . (29)
It is important to stress that previous solution is the solution of the supergravity
equations of motion for any function H(r) which is a harmonic function
in the (9 − p) dimensions that are transverse to the p−brane. As a result we
can consider more general solution
H(r) = 1 +
k
i=1
r7−p
(i)+
|r − ri|7−p
, r7−p
(i)+
, (30)
10
where
r7−p
(i)+ = dpgsNil7−p
s . (31)
Physically this solution corresponds to the confinguration of parallel extremal
p-branes located at k different positions r = ri, where each carries Ni units of
R−R charge. It is very important to stress that these extremal D-branes are
extremal. As a consequence of this fact it can be shown that when we have
two separated D-branes that the gravity attraction is exactly canceled with
the force of the gauge field since two D-branes repel. This is in sharp contrast
with the configuration D-brane-D-anti-brane which is famous example of the
unstable configuration. Returning back to D-branes the fact that there is
no force between them leads to the possibility that these D-branes can be
localized at different points ri.
Of course, the previous description of p-branes was given in terms of
supergravity approximation. Clearly this description is suitable on condition
when the curvature of p-brane geometry is small compared to the string scale
so that we can neglect the string theory corrections. In other words, we have
the condition r+ ≫ ls. Further, in order to supress string loop corrections we
have to demand that the effective string coupling eϕ
should be small. In case
of interest p = 3 the dilaton is constant and clearly this coupling constant
could be small everywhere in the 3−brane geometry when we demand that
gs < 1 and hence lP < ls. In other words the supergravity approximation is
valid when
lP < ls ≪ r+ . (32)
Since we know that
r7−p
+ = dpgsNl7−p
s (33)
we find that this expression is equivalent to
g1/4
s ls < ls ≪ d1/(7−p)
p g
1
7−p
s N
1
7−p ls
g1/4
s < 1 ≪ g
1
7−p
s N
1
7−p ⇒
g
7−p
4
s < 1 ≪ gsN
(34)
Clearly 1 > g
7−p
4
s and clearly N > gsN due to the fact that gs < 1 and hence
we can rewrite the upper non-equivality in the form
1 ≪ gsN < N . (35)
11
1.3 D-Branes
As we said previously extremal Dp-brane can be also defined as a D-brane
which is (p+1)−dimensional hyperplane in space-time where the open strings
can end. Clearly open string theory contains the loop diagram with topology
of cilinder that, from the closed string point of view corresponds to the propagation
of the closed string. In other words loop diagram for the open string
that ends on different D-branes corresponds to the propagation of the closed
string between these D-branes. Using this fact we obtain that D-brane
carry RR charge. Further, D-brane is invariant under half of the space-time
supersymmetry and it can be shown that in type IIA (IIB) string theory one
half of supersymmetry is preserved when p is even (odd). This fact is also
consistent with the R-R fields that appear in given theory. Say differently,
Dp-brane is BPS object that carries R-R charge.
We believe that the extremal p-brane in supergravity and Dp-brane are
different descriptions of the same object. More precisely, there are descriptions
that can be used in some specific situations. Explicitly, D-brane is defined
using string world-sheet which is a well defined object in the string perturbation
theory. When we have N D-branes at the same position that the
effective loop expansion parameter for the open string is gsN rather than gs
since each open string boundary curve that ends on D-branes has Chan-Paton
factor N and the string coupling gs. Then the perturbative treatment is good
on condition when gsN ≪ 1. We immediately see that this is complementary
to the regime where the supergavity description can be used.
When we restrict to the massless modes in the spectrum of the states of
the open string ending on Dp-brane we find that it forms U(N) gauge theory
in (p + 1)−dimensions with 16− supercharges. There are (9 − p) scalar fields
Φi
in the adjoint representation of U(N) that has following physical meaning.
Let us presume that the vacuum expectation value < Φi
> has k different
eigenvalues with N1 identical values ϕi
1, N2 identical values with ϕ2. Say
differently, in matrix notation
< Φi
>=




ϕi
1IN1×N1 0 0 0
0 ϕi
2IN2×N2 0 0
. . . . . . . . . . . .
0 . . . . . . ϕi
kINk×Nk



 . (36)
This vacuum expectation value breaks the original symmetry of the theory
that is U(N) to the group that preserves (36) that is U(N1)×· · ·×U(Nk). Phy-
12
sically this situation describes configuration when N1 Dp-branes are localized
at xi
1 = ϕi
1l2
s, N2 Dp-branes localized at xi
2 = ϕi
2l2
s and so on. Then careful
analysis of the fluctuations around this vacuum state shows that there are
massive W-bosons in bi-fundamental representation of U(Ni) × U(Nj) that
correspond to the open strings stretched between Dp-branes localized at ⃗ri
and ⃗rj. In fact, analysis of the open string with the Dirrichlet boundary conditions
at ⃗X(σ = 0) = ⃗xi , ⃗X(σ = π) = ⃗xj leads to the spectrum where the
mass of the lightest states is proportional to the Euclidean distance |⃗ri −⃗rj|.
1.4 N = 4 SYM theory
First of all we have to explain what means theory with N supersymetries. N
is the number of irreducible spinors that are conserved charges that generate
supersymmetry transformations. It turns out that N = 4 is the maximal
number that it is possible for interacting four dimensional field theory without
gravity. In four dimension general spinor has four complex componets.
Then the Weyl projection eliminates two of them so that Weyl spinor has
two complex Weyl spinors. Since N = 4 we have 4 × 2 × 2 = 16 real superchages,
where the first 2 means number of complex components and the
second one means that one complex component has two real components.
Further, we consider the gauge group U(N) which means that all fields are
N × N hermitean matrices.
The simplest way how to derive N = 4 four dimensional gauge theory is to
consider N = 1 SYM theory in ten dimensions. (Note that in ten dimensions
the dimension of spinor space is 2
10
2 = 32 complex components, however now
we can impose Majorana and Weyl projection so that irreducible spinor has
16 real components which is exactly equal to number of supersymmetries of
N = 4 SYM theory in four dimensions.) This theory contains a ten-vector
field AM , a Majorana-Weyl spinor Ψ that both are in adjoint representation
of the gauge group. Then we derive four dimensional theory by dimensional
reduction which means that we presume that all fileds do not depend on six
of the (Euclidean) spatial dimensions. As a result we obtain four dimensional
theory with four-vector Aµ, six scalars ϕI
and four Weyl spinors ψA
. This four
dimensional SYM theory has a global SO(6) ∼ SU(4) global symmetry that
rotates six scalars. Due to this symmetry it also rotates four supercharges
QA
and it is known as R−symmetry.
13
Note About Kaluza-Klein compactification on a circle
Our goal is to compactify one space dimension on a circle S1
R with radius
R. It is convenient to decompose the coordinates xM
of RD
into a coordinate
y on the circle and the remaining coordinates xµ
. For simplicity we consider
the flat space-time metric so that the wave operator takes the form
1
√
−g
∂M [gMN
∂N ] = ∂µ[ηµν
∂ν] +
∂2
∂y2
. (37)
We are interested how various fields behave in the limit R → 0.
We begin with a scalar field ϕ(XM
) that obeys periodic boundary conditions
on S1
R that has following Fourier decomposition
ϕ(xµ
, y) =
1
√
2πR n∈Z
ϕ(x)einy/R
. (38)
Note that this field obeys the periodic boundary condition
ϕ(x, y + 2π) =
n∈Z
ϕ(x)e2πiny/R
e2πin
= ϕ(x) . (39)
Then D−dimensional kinetic term of a scalar field with mass m then decomposes
as follows
1
2
dD
x(−gµν
∂µϕ∗
∂νϕ + m2
ϕϕ∗
) =
=
1
2πR
dD−1
x
2πR
0
dy[−
n,m
ei(−n+m)y/R
(∂µϕnηµν
∂νϕm +
+
2π
R
2
mnϕnϕm + m2
ϕnϕm)] =
=
n
dD−1
x[−∂µϕnηµν
∂νϕn +
2πn
R
2
ϕ2
n + m2
ϕ2
n]
(40)
using the fact that
2πR
0
dyei(n+m)/Ry
= 0 , n ̸= −m ,
2πR
0
dyei(−n+m)/Ry
= 2πR , n = m .
(41)
14
In other words in the process of compactification we obtain infinite number
of scalar fields with masses m2
n = m2
+ 2πn
R
2
. We also see that in the limit
R → 0 the massive modes decouple and hence the dynamical modes are the
zero modes only. Zero modes are modes ϕ0 that do not carry momentum in
y direction.
Situation is more involved in case of gauge theory. Let us consider the
action
S = − dD
x
2πR
0
dyFMN FMN
= − dD
x
2πR
0
dy[FµνFµν
+ 2FyµFyµ
] .
(42)
Let us expand the gauge field as
Aµ =
1
√
2πR m
am
µ einy/R
, Ay =
1
√
2πR m
ϕmeiny/R
. (43)
Then we have
Fµν =
1
√
2πR n
[∂µan
ν − ∂νan
µ]einy/R
≡
n
fn
µνeiny/R
,
Fµνηµρ
ηνσ
Fρσ =
1
2πR m,n
fn
µνηµρ
ηνσ
fm
ρσei(n−m)y/R
,
Fµy =
1
√
2πR n
(∂µϕn −
in
R
an
µ)einy/R
,
Fµyηµρ
Fρy =
1
2πR m,n
(∂µϕn −
in
R
an
µ)ηµρ
(∂ρϕm +
im
R
am
ρ )ei(m−n)y/R
.
(44)
However we should remember that gauge theory possesses U(1) symmetry.
Let us fix part of this symmetry by fixing
ϕn = 0 , n ̸= 0 . (45)
This gauge fixing can be achieved and fixed by the Fourier modes θk(x), k ̸= 0
when we perform the Fourier decomposition of the gauge function θ(x) =
1√
2πR k eiky/L
θk(x) where the gauge function θ(x) appears in the gauge
transformations A′
µ = Aµ + ∂µθ. Using this fixing of the gauge symmetry
we obtain the action in the form
S = − dD
x
n
[fn
µνηµρ
ηνσ
fn
ρσ + 2∂µϕηµν
∂νϕ + 2
n
R
2
an
µηµν
an
ν ] (46)
15
We see that there are infinite number of massive vector fields that again
decouple in the limit R → 0. Note that the zero mode of the gauge field aµ
possesses the gauge symmetry under a′
µ = aµ + ∂µθ0, since the Fourier mode
θ0(x) remained unfixed in the gauge fixing process given above.
Let us again return to the analysis of N = 4 SYM theory. N = 4 SYM
theories are conformally invariant which implies that they are ultraviolet
finite. If we combine space-time symmetries which are Lorentz invariance,
translation invariance, supersymmetry, scale invariance, conformal invariance,
R−symmetry gives the superconformal supergroup called PSU(2, 2|4),
where supergroup has both bosonic and fermionic generators.
The parameters of N = 4 with the gauge group U(N) are Y M coupling
constant g, vacuum angle θ and the rank of the gauge group N. As we
argued above for large N the natural parameter is ’tHooft parameter λ =
g2
N. Then for fixed λ the theory has 1/N expanssion for N ≫ 1. There
is remarkable symmetry of the leading term in the expanssion (the planar
approximation) that is known as a dual conformal invariance. It turns out
that this amount of symmetry is sufficient to make the theory completely
integrable. Unfortunately it is not known whether this is property of the
planar approximation or whether it can be extended to the complete theory.
If we further introduce the complex parameter
τ =
θ
2π
+ i
4π
g2
, (47)
we find that U(N) theory has an SL(2, Z) duality symmetry where τ transforms
as
τ →
aτ + b
cτ + d
, (48)
For a = 0, b = 1, c = −1, d = 0 we find τ → −1
τ
which is known as S−duality.
For θ = 0 this relation gives g → 4π
g
which implies the relation between the
strong coupling theory and weak coupling. It can be shown that S-duality is
an exact non-abelian electric-magnetic equivalence.
N SYM theory has closed relation with the Dp-branes, for very nice review
of this correspondence, see [7]. As we argued above a stack of coincident
flat D3-branes has a world-volume theory which is N = 4 SYM U(N) theory.
As we also shown above the configuration with D-branes at different
transverse positions is given by the vacuum expectation value (36) which
leads to the emergence of massive bosons with the mass
mij = |⃗ri − ⃗rj|T (49)
16
where T is fundamental string tension. We see that this is the similar situation
as in case of Higgs mechanism. On the other hand there is a difference since
the gauge fields become massive when the scalar fields in the adjoint representation
of the original gauge group are eaten while in case of the Standard
model the SU(2) Higgs doublet is in the fundamental representation. This
difference between these two descriptions is emphasized in the names of these
two phenomena, Coulomb branch in case of our case while the second case
is known as Higgs branch.
Let us consider N = 4, d = 4 SYM with the gauge group U(N). Usually
we say that free U(1) multiplet decouples leading to interacting SU(N)
theory. In D3-brane intepretation this decoupling multiplet describes the dynamics
of the center of mass of the collection of D3-branes.
1.4.1 The Probe Approximation
Let us consider a D3-brane embedded in ten dimensional target space-time.
The probe approximation means that we neglect the back reaction of the
brane on the geometry and the other background fields. Since the brane is
source for one unit of flux we have to require that the background flux N is
large so that N + 1 ≈ N. As we know the world-volume of a single D-brane
contains U(1) gauge field strength Fαβ = ∂αAβ − ∂βAα. Since we presume
that the action does not contain derivatives of F we can say that these fields
vary sufficiently slowly so that their derivatives can be neglected. Similar
restriction we also apply for other world-volume fields.
In more details, the world-volume effective action for D3-brane is famous
Dirac-Born-Infeld action
S = −TD3 d4
σe−Φ
− det(Gαβ + 2πα′Fαβ) + SCHS , (50)
where
Gαβ = gMN ∂αxM
∂βxN
(51)
where Chern-Simons term has the form
SCHS = µ3
Σ
P[ C(n)
eB
]e2πα′F
, (52)
where C(n)
is sum over Ramond-Ramond fields where µ3 is RamondRamond
charge of given brane. Further P[. . . ] means pullback of given field
17
to the world-volume of D3-brane. Note that there are integrations of forms
over world-volume of D3-brane where the wedge products between forms is
implicitly presumed.
Now we would like to see how from DBI action the YM action can be
derived. Let us presume that D3-brane is embedded in the flat target spacetime
gMN = ηMN = diag(−1, 1, . . . , 1). Further we presume static gauge
where
xµ
= σµ
, µ = 0, 1, 2, 3 . (53)
Then we obtain
gαβ = gµν∂αxµ
∂βxν
+ gIJ ∂αxI
∂βxJ
= ηαβ + ∂αxI
∂βxI (54)
Then we obtain
− det(gαβ + 2πα′Fαβ) = − det η det(det(δµ
ν + ∂µϕI∂νϕI + 2πα′Fµ
ν) =
= − det η(1 +
1
2
∂µ
ϕI
∂µϕI −
(2πα′
)2
4
FµνFµν
) ,
(55)
using the fact that
det(I + M) = exp Tr ln(I + M) ≈ exp Tr(M −
1
2
M2
≈ 1 + TrM −
1
2
TrM2
.
(56)
In summary we find that the DBI action reduces to the YM action in the
limit of slowly varying fields. The generalization to non-abelian configuration
is given by non-abelian form of DBI action.
2 Conformal Field Theories and AdS Spaces
2.1 Conformal Field Theories
As we know symmetry plays fundamental role in the particle physics, where
Lorentz and Poincare symmetries are basic principles for the constructions of
corresponding actions. It is natural to ask the question whether there exists
non-trivial generalization of these symmetries. An interesting generalization
of Poincare invariance is the addition of a scale invariance symmetry that
implies that the physics on the different scales are the same. On the other
18
hand this symmetry is non-consistent wit the existence of S-matrix since the
S-matrix is based on the definition of the asymptotic states that are defined at
far infinity. Clearly in theory, that is invariant under scaling transformation,
the meaning of asymptotic distance does not make sense. Many interesting
field theories, as for example Yang-Mills theory in four dimensions, are scale
invariant, at least classically. In fact, generally scale invariance is broken by
quantum theory, since it requires cutoff that breaks scale invariance explicitly.
On the other hand N = 4 SYM theory in four dimension is well defined
quantum field theory where the scaling symmetry is not broken. We also
know that quantum field theories are characterized by renormalization group
flow from some scale invariant UV fixed point, that could be free theory, to
some scale invariant IR fixed point, that could be also free.
Very important question is whether unitary interacting scale-invariant
theories are also invariant under full conformal group. The definite answer to
this question exists in two dimensions where it was proved that scale invariant
theories are always conformaly invariant. In the case of four dimensions the
situation is more involved with recent extensive discussion of this point where
however no definitive conclusion has been reached yet. In any case we now
review the main properties of conformal group.
2.1.1 Conformal Group and Conformal Algebra
Conformal group is the group of transformations that preserve the form of
the metric up to scale factor
g′
µν(x′
) = Ω(x)gµν(x) . (57)
The set of conformal transformations forms a group where the Poincare group
is subgroup of given group since it corresponds to the case Λ(x) = 1. In order
to investigate conformal group in more details let us consider an infinitesimal
transformation
x′µ
= xµ
+ ϵ(x) . (58)
Since this can be considered as a diffeomorphism transformation we find that
the metric transforms as
g′
µν(x′
) = gρσ(x)
∂xρ
∂x′µ
∂xσ
∂x′ν
(59)
that for (58) gives
g′
µν(x + ϵ) = gµν + gµρ∂νϵρ
+ ∂µϵσ
gσν (60)
19
Now the requirement that the transformations are conformal implies
g′
µν = gµν + fgµν = gµν + ∂µϵρ
gρν + gµρ∂νϵρ
(61)
and hence
∂µϵρ
gρν + gµρ∂νϵρ
= fgµν (62)
where the factor f can be determined by the trace of given expression
f =
2
d
∂µϵµ
(63)
In what follows we presume that the metric gµν is flat gµν = ηµν. Then if we
apply additional derivative on (62) ∂ρ we find
∂ρ∂µϵν + ∂ρ∂νϵµ = ∂ρfηµν
(64)
At the same time taking partial derivative with ∂µ of equation (62) now with
indices ρ, ν we find
∂µ∂ρϵν + ∂µ∂νϵρ = ∂µfηρν (65)
If we combine these two equations we obtain
2∂ρ∂µϵν + ∂ν[∂ρϵµ + ∂µϵρ] = ∂ρfηµν + ∂µfηρν ⇒
2∂ρ∂µϵν = ∂ρfηµν + ∂µfηρν − ∂νfηµρ
(66)
where in the last step we again used e.q. (62) with µ, ρ indices. Now contracting
this expression with ηρµ
we obtain
2∂ν
∂νϵµ = (2 − d)∂µf . (67)
Then we apply ∂ν on this expression and ∂2
on (62) we obtain
2∂2
∂νϵµ = (2 − d)∂µ∂νf ⇒
2∂2
∂µϵν = (2 − d)∂µ∂νf ⇒
∂2
(∂µϵν + ∂νϵµ) = (2 − d)∂µ∂νf ⇒
(2 − d)∂µ∂νf = ηµν∂2
f ,
(68)
20
where in the first step we used ∂µ∂νf = ∂ν∂µf and than the first equation
again. Finally we contract this expression with ηµν
and we obtain
(d − 1)∂2
f = 0 . (69)
Now using these equations we derive the explicit form of the conformal trans-
formations.
We see that for d = 1 there are no constraints on function f and hence
any smooth transformation is conformal in one dimension. Of course, this is
trivial result since the notion of angle does not exist in one dimension. The
case d = 2 corresponds to the two dimensional conformal field theory which
is analyzed in case of string theory so that it is not interesting for us. For that
reason we restrict ourselves to the case d ≥ 3. In this case we find ∂2
f = 0
from (69) and then from (68) we find ∂µ∂νf = 0, which however implies that
f is at most linear in the coordinates
f(x) = A + Bµxµ
, A, Bµ = const . (70)
Now inserting this expression into (66) we find that ∂µ∂νϵρ is constant which
means that ϵµ is at most quadratic in coordinates, so that we can write
ϵµ = aµ + bµνxν
+ cµνρxν
xρ
, cµνρ = cµρν . (71)
Since the constraints equations hold for all x we can treat each power of
coordinate separately. First of all it is clear that the constant term aµ is not
restricted by these equations and it corresponds to the well known infinitesimal
translation. Focusing on the linear term we insert it into the equation
(62) and we obtain
bνρδρ
µ + bµρδρ
ν =
2
d
bλ
ληµν
(72)
Since bµν can be expressed as linear combination of the symmetric, antisymmetric
tensor and trace we find that the previous equation can be solved
with
bµν = αηµν + mµν , mµν = −mνµ (73)
since then bµνηνµ
= dα(mµνηνµ
= 0) so that we really find that (72) is
obeyed by (73). Now the pure trace corresponds to the infinitesimal scale
21
transformations while antisymmetric part is an infinitesimal rigid rotation.
Finally we consider the quadratic term. First of all we have
f =
2
d
∂µϵµ
=
4
d
cµ
νρδρ
µxρ
=
4
d
cµ
µρxρ
. (74)
Then we obtain
cµνρ = ηµρbν + ηµνbρ − ηνρbµ , (75)
where
bµ =
1
d
cσ
σµ . (76)
Corresponding special conformal transformation has the form
x′µ
= xµ
+ 2(xρ
bρ)xµ
− bµ
(xν
xν) . (77)
Now we would like to discuss how fields transform under these transformations.
Let us consider general transformation in the form
x′µ
= xµ
+ ωa
δxµ
δωa
, (78)
where ωa are parameters of these transformations. Physical fields transform
under these transformations as
Φ′
(x′
) = Φ(x) + ωa
δF
δωa
(x) (79)
where the functional F follows from the transformation properties of these
fields. Let us define generator Ga of symmetry transformation from following
relation
δωΦ(x) ≡ Φ′
(x) − Φ(x) ≡ −iωaGaΦ(x) . (80)
Now from previous definitions we have
Φ′
(x + ωa
δxµ
δωa
) = Φ(x) + ωa
δF
δωa
⇒
Φ′
(x) + ∂µΦωa
δxµ
δωa
= Φ(x) + ωa
δF
δωa
⇒
δωΦ(x) = Φ′
(x) − Φ(x) = −ωa
δxa
δωa
∂µΦ + ωa
δF
δωa
(81)
22
and hence we have
iGaΦ =
δxµ
δωa
∂µΦ −
δF
δωa
. (82)
For simplicity we consider the case where F(Φ) = Φ. Now we have ϵµ
= aµ
and the generator has the form
Pµ = −i∂µ , (translation) (83)
In case of dilation we have δxµ
δα
= xµ
and hence the generator has the form
D = −ixµ
∂µ . (84)
In case of special conformal transformation we have
δxµ
δbν
= 2xν
xµ
− δµ
ν (xρ
xρ) (85)
and hence the generator has the form
Kµ = −i(2xµxν
∂ν − (xρ
xρ)∂µ) (86)
In the same way we can determine the generator of Lorentz rotation
Lµν = i(xµ∂ν − xν∂µ) (87)
Now we can calculate the algebra of these generators in the standard way as
the commutators of these operators acting on the test function and we find
[D, Pµ] = iPµ ,
[D, Kµ] = −iKµ ,
[Kµ, Pν] = 2i(ηµνD − Lµν)
[Kρ, Lµν] = i(ηρµKν − ηρνPµ) ,
[Pρ, Lµν] = i(ηρµPν − ηρνPµ) ,
[Lµν, Lρσ] = i(ηνρLµσ + ηµσLνρ − ηµρLνσ − ηνσLµρ) .
(88)
Now we would like to show that this algebra is isomorphic to the algebra of
SO(d, 2) and can have the standard form of SO(d, 2) alegra with signature
(−, +, +, . . . , +, −). We define these generators as
Jµν = Lµν , Jµd =
1
2
(Kµ − Pµ) , Jµ(d+1) =
1
2
(Kµ + Pµ) , J(d+1)d = D . (89)
23
These generators obey SO(d + 1, 1) commutation relations
[Jab, Jcd] = i(ηadJbc + ηbcJad − ηacJbd − ηbdJac) . (90)
where ηab = diag(−1, 1, . . . , 1, −1).
It is also very useful to consider conformal field theory in Euclidean space
where the conformal group is SO(d+1, 1). Now Rd
is conformally equivalent
to Sd
and hence the field theory on Rd
(with appropriate boundary conditions
at infinity) is isomorphic to the theory on Sd
.
2.1.2 Conformal Invariance in Classical Field Theory
We say that given field theory is conformal field theory at the classical level if
its action is invariant under conformal transformations. In order to formulate
such a theory we have to give a prescription of the conformal transformations
on the classical fields.
Let us now consider an infinitesimal conformal transformation parameterized
by ωg. We would like to find a matrix representation Tg such that a
multicomponent field Φ(x) transforms as
Φ′
(x′
) = (1 − iωgTg)Φ(x) , (91)
where the generator Tg has to be added to the space-time part that was
given above in order to find the full generators of symmetry. In order to find
such a generator we consider a subgroup of the Poincare group that leaves
the point x = 0 invariant which is the Lorentz group. Then we introduce
a matrix representation Sµν in order to define the action of infinitesimal
Lorentz transformations on the field Φ(0)
LµνΦ(0) = SµνΦ(0) , (92)
where Sµν is spin operator whose explicit form is given by nature of the field
Φ. Then in order to calculate the action of this operator on the field at the
space-time point x we have to perform the translation of given operator as
eixρPρ
Lµνe−ixρPρ
= Sµν − xµPν + xνPµ (93)
where we used the Hausdorff formula for two A and B operators
e−A
BeB
= B + [B, A] +
1
2!
[[B, A], A] +
1
3!
[[[B, A], A], A] + . . . (94)
24
so that we generally obtain
PµΦ(x) = −i∂µΦ(x) ,
LµνΦ(x) = i(xµ∂ν − xν∂µ)Φ(x) + SµνΦ(x) .
(95)
In the same way we proceed with the conformal group. The subgroup that
leaves the origin x = 0 invariant is generated by rotations, dilations and
special conformal transformations. If we look on the algebra of generators of
corresponding symmetry we obtain an algebra that has similar structure as
Poincare algebra with the dilatation, where now Kµ play the similar roles
as Pµ. We denote Sµν, ¯△ and κµ as the respective values of the generators
Lµν, D and Kµ at x = 0. Clearly they have to form a matrix representation
of the reduced algebra
¯△, Sµν = 0 ,
¯△, κµ = −iκµ ,
[κν, κµ] = 0 ,
[κρ, Sµν] = i(ηρµκν − ηρνκµ) ,
[Sµν, Sρσ] = i(ηνρSµσ + ηµσSνρ − ηµρSνσ − ηνσSµρ) .
(96)
Then we translate given operators, or more precisely, calculate the action of
these operators on the field at the point x when we calculate
eixµPµ
De−ixµPµ
= D + xν
Pν ,
eixρPρ
Kµe−ixρPρ
= Kµ + 2xµD − 2xν
Lµν + 2xµ(xν
Pν) − x2
Pµ ,
(97)
and hence we obtain the action of the operators on the physical fields
DΦ(x) = (−ixν
∂µ + ˆ△)Φ(x) ,
KΦ(x) = (κµ + 2xµ
ˆ△ − xν
Sµν − 2ixµxν
∂ν + ix2
∂µ)Φ(x) .
(98)
Now we demand that Φ(x) belong to an irreducible representation of the
Lorentz group and then by Schur’s lemma, any matrix that commutes with
25
all the generators Sµν must be a multiple of the identity. Consequently the
matrix ˆ△ is a multiple of the identity and from the algebra of the matrix we
find that κµ has to vanish. ˆ△ is then number equal to −i△ where △ is the
scaling dimension of the field Φ which is the scaling dimension of the field Φ
defined as
x′
= λx ,
Φ′
(λx) = λ−△
Φ(x) .
(99)
It is important to stress that previous form of the operators correspond to
the conformal algebra realized on the space-time functions as differential
operators. On the other hand in the quantum field theories these generators
should correspond to the operators in Heisenberg picture, where they act on
the local fields (again, local operators) in form of the commutators
[Pµ, Φ(x)] = i∂µΦ(x) ,
[Lµν, Φ(x)] = (i(xµ∂ν − xν∂µ) + Sµν)Φ(x) ,
[D, Φ(x)] = i(−△ + xµ
∂µ)Φ(x) ,
[Kµ, Φ(x)] = (i(x2
∂µ − 2xµxν
∂ν + 2xµ△) − 2xν
Sµν)Φ(x) ,
(100)
where of course Sµν are matrices of the finite dimensional representation of
the Lorentz group that act on the indices of the field Φ. Further, △ is the
scaling dimension matrix that acts on the collection of the local operators Φ
(with fixed spin) that may not be digagonalizable in general.
Further, the commutation relations given above imply that Pµ raises the
scaling dimension of the field while Kµ lowers it. In unitary field theories
there is a lower bound on the dimension of fields, for example in case of
the scalar field we have △ ≥ (d − 2)/2. Then clearly each representation of
the conformal group contains operators of lowest dimension that has to be
annihilated by Kµ at x = 0. These operators ar called as primary operators.
Then non-primary operators may be obtained by acting Pµ further.
We define the vacuum state as the state that obeys
Pµ |0⟩ = Jµν |0⟩ = D |0⟩ = Kµ |0⟩ = 0 . (101)
Space-time symmetry and energy-momentum tensor
26
We introduced the symmetry of given system as an algebra of conserved
charges that act on the Hilbert space. On the other hand in field theories
we usually postulate that these symmetries are given as integrals of local
conserved charges. In other words, if the current is conserved
∂µjµ
= 0 (102)
then we can construct the conserved charge
Q = dd−1
xj0 . (103)
Let us now presume that the action under infinitesimal translation x′µ
=
xµ
+ ϵµ
, ϵµ
= const. Then the corresponding conserved current has the form
Tµν
c = −ηµν
L +
δL
δ∂µΦ
∂ν
Φ , ∂µTµν
= 0 (104)
that by N¨oether assumption obeys the conserved law
∂µ
Tc
µν = 0 . (105)
Generally the energy-momentum tensor that is derived by N¨oether procedure
is not symmetric
Tc
µν ̸= Tc
νµ (106)
but it can be made symmetric by Belinfante prescription.
Let us further presume that the action is invariant under scale transfor-
mation
x′
= (1 + α)x , F(Φ) = (1 − α△)Φ . (107)
Note that for general variation we have following conserved current
jµ
a =
∂L
∂(∂µΦ)
∂νΦ − δµ
ν L
δxν
δωa
−
δL
δ∂µΦ
δF
δωa
, (108)
where
x′µ
= xµ
+ ωa
δxµ
δωa
,
Φ′
(x′
) = Φ(x) + ωa
δF
δωa
(x) .
(109)
27
Then in case of the scale invariance we find
jµ
D = −Lxµ
+
δL
δ(∂µΦ)
xν
∂νΦ +
δL
δ∂µΦ
△Φ =
= Tµ
cνxν
+
δL
δ∂µΦ
△Φ ,
(110)
Since by presumption the current jµ
D is conserved we find
∂µjµ
D = ∂µTµ
νxν
+ Tµ
cµ + △∂µ
δL
δ(∂µΦ)
= 0 .
(111)
Let us now define virial of the field Φ
V µ
=
δL
δ∂ρΦ
(ηµρ
△ + iSµρ
)Φ , (112)
where Sµν
is the spin operator of the field Φ. Let us also presume that virial
is the divergence of another tensor σνµ
V µ
= ∂σσσµ
, (113)
where the last condition is obeyed in a large class of physical theories. Then
we define
σµν
+ =
1
2
(σµν
+ σνµ
) ,
Xλρµν
=
2
d − 2
ηλρ
σµν
+ − ηλµ
σρν
+ + ηµν
σλρ
+ +
1
d − 1
(ηλρ
ηµν
− ηλµ
ηρν
)σα
+α ,
(114)
and we consider the following modified energy-momentum tensor
Tµν
= Tµν
c + ∂ρBρµν
+
1
2
∂λ∂ρXλρµν
, (115)
where the first two terms constitute the Belinfante tensor defined as
Tµν
B = Tµν
c + ∂ρBρµν
, Bρµν
= −Bµρν
. (116)
28
Note that the presence of the tensor Tρµν
does not affect the conserved law
∂µTµν
B = ∂µTµν
c + ∂µ∂ρBρµν
= ∂µTµν
. (117)
We search for Bρµν
such that the new tensor Tµν
B is symmetric. Then we can
certainly construct another conserved current
JJµν
ρ = xµ
T ν
Bρ − xν
T µ
Bρ (118)
so that
∂ρ
JJµν
ρ = ∂ρ
(xµ
Tν
Bρ − xν
Tµ
Bρ) =
= ηµρ
Tν
Bρ − ηρν
Tµ
Bρ + xµ
∂ρ
Tν
Bρ − xν
∂ρ
Tµ
Bρ =
= Tµν
B − Tνµ
B = 0
(119)
Returning to (115) we claim that the last term is needed in order to make
Tµν
traceless. Due to the symmetry properties of the tensor X we see that
it does not spoil the conserved property since
∂µ∂λ∂ρXλρµν
= 0 . (120)
It can be also shown that it does not spoil the property of the Belinfante
tensor since
Xλρµν
− Xλρνµ
=
2
(d − 2)(d − 1)
σα
+α(ηλµ
ηρν
− ηλν
ηρµ
) (121)
that vanishes when contracted with ∂λ∂ρ. Finally the trace of the new term
is
1
2
∂λ∂ρXλρµ
µ = ∂λ∂ρσλρ
+ = ∂µV µ
.
(122)
From definition of Bρµν
we find
∂ρBρµ
µ =
1
2
i∂ρ
δL
δ(∂µΦ)
Sµρ
Φ (123)
and then from the definition of the virial we obtain the final result
Tµ
µ = ∂µjµ
D , (124)
29
and therefore the scale invariance implies that the modified stress energy
tensor is traceless. Then we can construct the special conformal current
Kµ
(ρ) = [ρνx2
− 2xν(ρσxσ
)]Tνµ
(125)
for constant vector ρ. Note that this current obeys the equation
∂µKµ
(ρ) = [2ρνxµ − 2ηνµ(ρσxσ
) − 2xνρµ]Tνµ
+
+[ρνx2
− 2xν(ρσxσ
)]∂µTνµ
= 0
(126)
where the expression in the first bracket vanished due to the symmetry of
Tµν
and also from the fact that Tµν
is traceless and where the last term
vanished due to the conservation of Tµν
. In this way we are able to construct
the current corresponding to the special conformal transformations. Note
however that we have to made crucial presumption about the existence of
the virial current.
3 Supersymmetry and Gauge Theories
In order to understand the spectrum of operators in the QFT living on the
boundary we have to say few basic facts considering SUSY. We start with
Supersymmetry algebra in 3 + 1 dimensions.
3.0.1 Supersymmetry Algebra in 3 + 1 dimensions
We know that the Poincare symmetry of the flat space-time is generated
by Pµ and Lµν. Supersymmetry enlarges the Poincare algebra by including
spinor supercharges
Qa
α , α = 1, 2 left Weyl spinor ,
¯Q˙αa = (Qa
α)†
, right Weyl spinor ,
(127)
where N is the number of independent supersymmetries of the algebra. We
use two-component spinor notation α = 1, 2. The supercharges transform as
Weyl spinors of SO(1, 3) and commute with translations. The remaining susy
structure relations are
Qa
α, ¯Q ˙βb = 2σµ
α ˙β
Pµδa
b , Qa
α, Qb
β = 2ϵαβZab
(128)
30
By construction the generators Zab
are anti-symmetric in the indices a and b
and commute with all generators of the supersymmetry algebra. Sometimes
Zab
are called as central charges so that we have
Zab
= −Zba
, [Zab
, . . . ] = 0 . (129)
Clearly in case N = 1 we have Z = 0. The supersymmetry algebra is invariant
under a global phase rotation of all supercharges Qa
α which is a group
U(1)R. Further, in case N > 1, the different supercharges may be rotated into
one another under the unitarity group SU(N)R. Thes symmetrise of the supersymmetry
algebra are called R− symmetries. Generally these symmetries
could be broken by quantum mechanical effects.
We are mainly interested in N = 4 SYM theory where the Lagrangian
density has the form
L = Tr −
1
2g2
FµνFµν
+
a
i¯λa
¯σµ
Dµλa −
i
DµXi
Dµ
Xi
+
+
a,b,i
gCab
i λa[Xi
, λb] +
a,b,i
g ¯Ciab
¯λa
[Xi
, ¯λb
] +
g2
2 i,j
[Xi
, Xj
]2
,
(130)
where
Fµν = ∂µAν − ∂νAµ + i[Aµ, Aν] , DµXi
= ∂µXi
+ i[Aµ, Xi
] , (131)
and where the constants Cab
i and Ciab are related to the Clifford Dirac matrices
for SO(6)R ∼ SU(4)R which can shown when we consider N = 4 SYM
in four dimension as the dimensional reduction on T6
of D = 10 SYM. By
construction this Lagrangian is invariant under N = 4 Poincare supersym-
metry.
It is instructive to determine the equations of motion that follows from
the Lagrangian density given above. For simplicity we restrict to the case of
the bosonic variables only. Varying with respect to Xi
we obtain
2∂µ(Dµ
Xi
) + 2i(AµDµ
Xi
− iDµ
Xi
Aµ) +
+2g2
(Xj[Xi
, Xj
] − [Xi
, Xj
]Xj) =
= 2DµDµ
Xi
+ 2g2
[Xj, [Xi
, Xj
]] = 0
(132)
31
and varying the action with respect to Aµ we obtain
2
g2
[∂νFνµ
+ iAµFνµ
− iFνµ
Aν] + 2i[Xi, Dµ
Xi
] =
=
2
g2
DνFνµ
+ 2i[Xi, Dµ
Xi
] = 0 .
(133)
Note that Aµ, Xi
belong to the adjoint representation of U(N) so that
A†
= A , (Xi
)†
= Xi
. (134)
It is also useful to expand Aµ, Xi
with the help of the generators of the Lie
group Tα, T†
α = Tα so that
Aµ = Aα
µTα , Xi
= Xiα
Tα . (135)
so that
[Xi
, Xj
] = Xiα
Xjβ
[Tα, Tβ] = Xiα
Xβ
f γ
αβTγ ,
DµXi
= (∂µXiα
+ iAγ
µXiδ
f α
γδ )Tα ,
(136)
where f γ
αβ are structure constants of U(n) algebra defined by the relation
[Tα, Tβ] = f γ
αβTγ , (f γ
αβ)∗
= −f γ
αβ . (137)
Then the equations of motion have the form
Dµ(∂µ
Xiα
+ iAµγ
Xiδ
fα
γδ)Tα + [Xj, (Xiγ
Xjδ
f ω
γδ Tω)] =
= ∂µ(∂µ
Xiα
+ iAµγ
Xiδ
fα
γδ)Tα + iAσ
µ(∂µ
Xω
+ iAµγ
Xi
δf ω
γδ )f α
σωTα + Xβ
j Xiγ
Xjδ
f ω
γδ f α
βωTα = 0
(138)
which has to be valid for all Tα and hence we obtain
∂µ(∂µ
Xiα
+ iAµγ
Xiδ
fα
γδ) + iAσ
µ(∂µ
Xω
+ iAµγ
Xi
δf ω
γδ )f α
σω + Xβ
j Xiγ
Xjδ
f ω
γδ f α
βω = 0
(139)
32
In the same way we obtain the equation of motion for Aα
µ
∂ν(∂ν
Aµα
− ∂µ
Aνα
+ iAνγ
Aµδ
f α
γδ )Tα +
+[Xi, (∂µ
Xiβ
+ iAµγ
Xiδ
f β
γδ )Tβ] =
= ∂ν(∂ν
Aµα
− ∂µ
Aνα
+ iAνγ
Aµδ
f α
γδ )Tα + iXσ
i (∂µ
Xiβ
+ iAµγ
Xiδ
f β
γδ )f α
σβ Tα = 0
(140)
and hence we again find
∂ν(∂ν
Aµα
−∂µ
Aνα
+iAνγ
Aµδ
f α
γδ )+iXσ
i (∂µ
Xiβ
+iAµγ
Xiδ
f β
γδ )f α
σβ = 0 . (141)
Note that by definition we have
Aµ
= ηµν
Aν , Φi
= δij
Φj , ∂µ
= ηµν
∂ν , (142)
where ηµν
= diag(−1, 1, 1, 1) , δij
= diag(1, 1, 1).
Given Lagrangian is also scale invariant. Explicitly, we assign the standard
mass dimensions to the fields and couplings
[Aµ] = [Xi
] = 1 , [λa] =
3
2
, [g] = 0 . (143)
Clearly we see that the Lagrangian density has the scaling dimension [L] = 4
which, together with the scaling dimension [d4
x] = −4 implies that the
action is invariant under scaling transformation. As we know this scaling symmetry
combines with Poincare invariance into larger conformal symmetry
SO(2, 4) ∼ SU(2, 2). This symmetry combines with N = 4 Poincare supersymmetry
into superconformal symmetry given by the supergroup SU(2, 2|4).
It is very remarkable that this theory, upon perturbative quantization,
does not exhibit ultraviolet divergences in the correlation functions of its
canonical fields. Nonperturbatively there could be instanton corretions but
they give finite contributions too and hence it is believed that the theory is
UV finite. As a result the renormalization group β−function of the theory
vanishes identically which implies that the theory is exactly scale invariant
at the quantum level and the superconformal group SU(2, 2|4) is symmetry
that is preserved in quantum theory too.
As we know there are various phases of given theory whose nature is
determined by the potential of the theory. The potential energy term in the
action of N = 4 SYM has the form
g2
i,j
Tr[Xi
, Xj
]2
(144)
33
whose equations of motion have the form
[Xi
, [Xi
, Xj
]] = 0 (145)
that has the solution
[Xi
, Xj
] = 0 , i, j = 1, . . . , 6 . (146)
1
Now we show that there are two solutions of the equations (146)
• The superconformal phase: This phase is characterized by the conditi-
ons
Xi
= 0 , i = 1, . . . , 6 . (147)
We see that the gauge algebra G is unbroken and the superconformal
symmetry SU(2, 2|4) is also unbroken. The physical states and operators
are gauge invariant (say differently, G−singlets) and transform
under unitarity representations of SU(2, 2|4).
• The spontaneously broken or Coulomb phase that is characterized by
the condition
Xi
̸= 0for at least onei (148)
The nature of the resulting theory depends on the degree of residual
symmetry. Generally we have G → U(1)r
where r is the rank of the
group G and the low energy behavior is that of r copies of N = 4 U(1)
theory. Superconformal symmetry is spontaneously broken too due to
the non-zero vacuum expectation value of ⟨Xi
⟩ which also set the scale
of the theory.
3.0.2 Superconformall symmetry
The global continous symmetry group of N = 4 SYM is given supergroup
SU(2, 2|4) with following ingredients 2
:
1
We should also consider the solution that obeys [Xi
, Xj
] = IN×N k , k = constant.
However this is not valid for any finite Xi
since taking the trace of this equation we find
that the left side vanishes identically while the right side implies that k = 0.
2
We shall explain the meaning of the world supergroup. As the name suggests the
supergroup is generalization of ordinary group in the sense that it possesses the same group
structure. The Lie algebra of supergroups is superalgebra that is Z2−graded algebra. The
element of supergroup has the form
A B
C D
, where A, B are Grassmann even matrices,
while B and C are Grassmann odd elements.
34
• Conformal Symmetry forming the group SO(2, 4) ∼ SU(2, 2) that is
generated by translations Pµ
, Loretnz transformations Lµν dilatations
D and special conformal transformations Kµ
,
• R− symmetry that forms the group SO(6)R ∼ SU(4)R that is generated
by TA
, A = 1, . . . , 15
• Poincare supersymmetries generated by the supercharges Qa
α and their
complex conjugates ¯Q˙αa , a = 1, . . . , 4. These charges generate the N =
4 Poincare supersymmetry.
• Conformal symmetries that are generated by the supercharges Sαa and
their complex conjugates ¯Sa
˙α. The presence of these supercharges follows
from the fact that Poincare supersymmetries and the special conformal
transformations Kµ do not commute. Since both are symmetries their
commutator has to be also symmetry that we denote as S generators.
The two bosonic subalgebras SO(2, 4) and SU(4)R commute. The supercharges
Qa
α and Sa
˙α transform under 4 of SU(4)R while ¯Q˙αa and Sαa transform
under 4∗
. Then we can fit these generators in the the supermatrix that represents
the corresponding superalgebra
Pµ Kµ Lµν D Qa
α
¯Sa
˙α
¯Q˙αa Sαa TA (149)
To proceed further note that we have following natural grading of the algebra
that is given by the scaling dimension of the generators
[D] = [Lµν] = [TA
] = 0 , [Pµ
] = 1 , [Kµ] = −1 ,
[Q] = 1/2 , [S] = −1/2 .
(150)
It is also important to stress that the superconformal algebras exist for d ≤ 6
only. Schematically the comutation relations of the superconformal algebra
include in addition to the previous relations, the relations
[D, Q] = −
i
2
Q , [D, S] =
i
2
S , [K, Q] ≃ S , [P, S] ≃ Q ,
{Q, Q} ≃ P, {S, S} ≃ K , {Q, S} ≃ M + D + R .
(151)
35
The exact form of the commutation relations depend on the dimensions of
the space-time where given conformal field theory lives as follows from the
fact that the spinorial representations of the conformal group are different
for different space-time dimensions.
3.1 Superconformal Multiplets of Local Operators
We are interested in the construction of all local, gauge invariant operators
which are polynomial in the canonical fields. As we know that canonical fields
are Xi
, λa and Aµ that have classical dimensions 1, 3/2 and 1, respectively.
It is clear that the gauge invariant operators can be constructed from the
covariant objects Xi
, λa, F±
µν and the covariant derivative Dµ with following
canonical dimensions
[Xi
] = [Dµ] = 1 , [F±
µν] = 2 , [λa] =
3
2
, (152)
where F±
µν are (anti) self-dual gauge field strength. We see that the classical
dimension of the composite operators is positive and that the number of
operators whose dimension is less than a given number is finite. The only
operator with dimension 0 is the unit operator.
Very important object in the theory are superconformal primary operators.
Since the conformal supercharges S have dimension −1/2 we see that
successive application of S to any operator of definite dimension leads to
the operator of dimension 0. On the other hand we have to end there since
otherwise we would generate operators with negative dimension which is impossible
in a unitary representation. Then we define a superconformal primary
operator O as a operator that is non-vanishing operator that obeys
[S, O]± = 0 (153)
In other words we say that superconformal primary operator is the lowest
dimension operator in given superconformal mutiplet. We should stress that
there is an important difference from a conformal primary operator which
is the operator that is annihilated by special conformal generators Kµ
. In
fact, every superconformal primary operator is conformal primary operator
as follows from the fact that
[Kµ
, O] ∼ [{S, S} , O] = 0 (154)
36
since [S, O] = 0. Finally an operator O is called a superconformal descant
operator of an operator O′
, when it is of the form
O = [Q, O′
] , (155)
for some well-defined local polynomial gauge invariant operator O′
. Clearly
if O is a descendant of O′
then these two operators belong to the same
superconformal multiplet. As follows from the commutation relation above
we have
[D, O] = [D, [Q, O′
]] = [Q, [D, O′
]] + [[D, Q] , O′
] =
= △O′ [Q, O′
] +
1
2
[Q, O′
] =
= (△O′ +
1
2
) [Q, O′
] = (△O′ +
1
2
)O ,
(156)
using the fact that
[D, O′
] = △O′ O , [D, Q] =
1
2
Q . (157)
and we see that the dimension of the operator O is △O = △O′ + 1
2
. Further
we see that the operator O cannot be a conformal primary operator because
there is in the same multiplet at least one operator O′
with dimension lower
than O. In other words in given irreducible superconformal multiplet there
is a single superconformal primary operator which is the operator of lowest
dimension and all others are superconformal descendants of this primary.
Let us explicitly construct superconformal primary operator in N = 4
SYM. First of all we know that superconformal primary operator cannot be
given as Q−commutator of another operator. In fact, let us presume that
we have an operator O = [Q, X] for some operator X and calculate the
commutator of this operator with S
[S, [Q, X]] = [Q, [X, S]] + [X, [S, Q]] = · · · + [X, M + D + R] (158)
where the last expression is non-zero since X has definite dimension and
R−charge. It turns out that we have to know how Q transforms the canonical
fields. Schematically we have
{Q, λ} = F+
+ [X, X] , [Q, X] = λ,
Q, ¯λ = DX , [Q, F] = Dλ
(159)
37
From these relations we see that local polynomial operator that contains
any of the elements on the right side of the above structure relations cannot
be primary since then it can be expressed as the commutator with Q with
another operator. In other words chiral primary operators cannot involve
neither the gauginos λ nor the gauge field strengths F±
. Since it has to
be function of the scalars X only it cannot involve neither derivatives nor
commutators of X. As a result we find that the superconformal primary
operators are gauge invariant scalars that involve X only in a symmetrized
way. The symplest operators are the single trace operators of the form
str(Xi1
Xi2
. . . Xin
) , (160)
where ij, j = 1, . . . , n are SO(6)R fundamental representation indices and
where str means the symmetrized trace over the gauge algebra so that the
resulting operator is totally symmetric in the SO(6)R indices ij. Generally
given operator transforms under a reducible representation of SO(6) which, in
our case, is the symmetrized product of fundamentals. Irreducible operators
may be obtained by isolating the traces over SO(6)R indices. Since TrXi
= 0
in SU(N) the simplest operators are
i
TrXi
Xi....Konishi multiplet
Tr(Xi
Xj
+ Xj
Xi
) ∼ supergravity multiplet
(161)
These operators have own image in the AdS space as we will see in following
sections.
Of course, it is also possible to construct more complicated multiple trace
operators that are obtained as products of single trace operators.
3.1.1 N = 4 Chiral or BPS Multiplets of Operators
It turns out that there are special representations that are very important
in the discussion of AdS/CFT correspondence. These representations correspond
the the situation when at least one supercharge Q commutes with the
primary operator. Such representations are usually referred as chiral multiplets
or BPS mutiplets. These representations are shortened and hence
their dimension cannot be renormalized or cannot receive quantum corrections.
It turns out that these half-BPS operators play an important role in
38
the AdS/CFT correpondence. The simplest series is given by single-trance
operators of the form
Ok(x) ≡
1
nk
str X{i1
(x) . . . Xik}
(x) , (162)
where {i1, . . . , ik} means SO(6)R traceless part of the tensor and nk means an
overall normalization of the operator that is fixed by normalizing its 2−point
function.
3.2 Anti-de Sitter Space
3.2.1 Geomery of Anti-de Sitter Space
In order to understand the AdS/CFT correspondence we have to know some
basic geometric facts about Anti-de Sitter space time. Very important is the
relation between conformal compactifications of AdS and of flat space. In
the case of the Euclidean signature metric we can compactify the flat space
Rn
to the n−sphere Sn
by adding a point at infinity so that the conformal
field theory is naturally defined on Sn
. On the other hand (n + 1) dimensional
hyperbolic space which is the Euclidean version of AdS space can be
conformally mapped into the (n + 1) dimensional disk Dn+1. Therefore the
boundary of compactified hyperbolic space is the compactified Euclid space.
It turns out that a similar correspondence holds in case of the Lorentizan
signature metric as well.
Let us say few words about Conformal Structure of Flat Space. Let
us start with the simplest case R1,1
which is two-dimensional Minkowski
space with the metric
ds2
= −dt2
+ dx2
, (−∞ < t, x < +∞) (163)
Let us introduce two coordinates
u± = t ± x , x =
u+ − u−
2
, t =
u+ + u−
2
(164)
so that the line element has the form
ds2
= −du+du− . (165)
Finally we introduce two coordinates τ, θ
u± = tan
τ ± θ
2
(166)
39
so that
du+ =
1
2 cos2 τ+θ
2
(dτ + dθ) , du− =
1
2 cos2 τ−θ
2
(dτ − dθ) (167)
so that the line element has the form
ds2
=
1
4 cos2 τ+θ
2
cos2 τ−θ
2
(−dτ2
+ dθ2
) , (168)
Now we see that the flat Minkowski space is conformally mapped into the
interior of the compact region
|
τ ± θ
2
| < π/2 . (169)
This space has four corners. The corners at τ, θ = (0, ±π) corresponds o the
points at spatial infinity x = ±∞ in the original coordinates. Since light
ray trajectories are invariant under a conformal rescaling of the metric this
mapping gives the causal structure of R1,1
. Note also that the new coordinates
(τ, θ) are well defined at the asymptotic regions of the flat space. In
other words the conformal compactification is useful for a rigorous definition
of asymptotic flatness of space-time when we say that the space-time is
asymptotically flat if it has the same boundary structure as that of the flat
space after conformal transformation.
R1,p
, p ≥ 2
Let us consider higher dimensional Minkowski space-time
ds2
= −dt2
+ dr2
+ r2
dΩ2
p−1 , (170)
where dΩp−1 is the line element on the unit sphere Sp−1
. Introduce variables
u± defined as
u± = t ± r , du± = dt ± dr (171)
we find that the metric element has the form
ds2
= −du+du− +
1
4
(u+ − u−)2
dΩ2
p−1 . (172)
As the next step we introduce variables ˜u± defined as
u± = tan ˜u± , du± =
1
cos2 ˜u±
du± (173)
40
so that
ds2
=
1
cos2 ˜u+ cos2 ˜u−
−d˜u+d˜u− +
1
4
sin2
(˜u+ − ˜u−)dΩ2
p−1 (174)
Finally we introduce variables ˜u± = 1
2
(τ ±θ) so that we obtain the final form
of the metric
ds2
=
1
4 cos2 ˜u+ cos2 ˜u−
−dτ2
+ dθ2
+ sin2
θdΩ2
p−1 . (175)
Note that we have
τ = tan−1
(t + r) + tan−1
(t − r) , θ = tan−1
(t + r) − tan−1
(t − r) . (176)
Now the points (t, r) = (∞, 0), (−∞, 0), (0, ∞) are mapped to the points
(τ, θ) = (π, 0), (−π, 0), (0, π) so that the the original half-plane t ∈ (−∞, ∞), r ≤
0 is mapped to the triangular region in τ, θ plane defined these three points.
However the conformally scaled metric
ds′2
= −dτ2
+ dθ2
sin2
θdΩ2
p−1 (177)
can be analytically continued outside of the triangle and the maximally extended
space with
0 ≤ θ ≤ π, −∞ < τ < +∞ (178)
has the geometry R × Sp
which is Einstein static universe, where the points
θ = 0, π correspond to the north and south poles of Sp
.
Since
∂
∂τ
=
∂
∂u+
∂u+
∂τ
+
∂
∂u−
∂u−
∂τ
=
=
∂
∂u+
1
2 cos2 ˜u+
+
∂
∂u−
1
2 cos2 ˜u−
=
=
1
2
(1 + u2
+)
∂
∂u+
+
1
2
(1 + u2
−)
∂
∂u−
.
(179)
using the fact that cos2
˜u+ = 1
1+u2
+
. Now note that in the original variables
41
we have
∂
∂u+
+
∂
∂u−
=
∂
∂t
,
u2
+
∂
∂u+
+ u2
−
∂
∂u−
=
= (t2
+ r2
)(
∂
∂u+
+
∂
∂u−
+ 2tr(
∂
∂u+
−
∂
∂u−
) =
= (t2
+ r2
)
∂
∂t
+ 2tr
∂
∂r
=
= (−t2
+ r2
)
∂
∂t
+ 2t(r
∂
∂r
+ t
∂
∂t
) .
(180)
Note that we have
Pµ = −i∂µ , Kµ = −i(2xµxν
∂ν − (xρ
xρ)∂µ) (181)
as the generators of translations and special conformal transformations on
R1,p
. If we further identify −i ∂
∂τ
= H, where H is the Hamiltonian corresponding
to the evolution in τ−coordinate so that we have relation
H =
1
2
(P0 + K0) = J0,p+2 . (182)
In other words the generator H = J0,p+2 corresponds to the SO(2) part of
the maximally compact subgroup SO(2) × SO(p + 1) of SO(2, p + 1). Thus
the subgroup SO(2) × SO(p + 1) of the conformal group SO(2, p + 1) can be
identified with the isometry of the Einstein static universe R × Sp
.
Anti-de Sitter Space The (p + 2)-dimensional anti-de Sitter space
AdSp+2 can be represented by the hyperboloid
X2
0 + X2
p+2 −
p+1
i=1
X2
i = R2
(183)
in the flat (p + 3)−dimensional space with the metric
ds2
= −dX2
0 − dX2
p+2 +
p+1
i=1
dX2
i . (184)
42
By construction we see that this space has an isometry SO(2, p + 1) and it
is homogeneous and isotropic. Equation (183) can be solved by setting
X0 = R cosh ρ cos τ , Xp+2 = R cosh ρ sin τ ,
Xi = R sinh ρΩi , (i = 1, . . . , p + 1;
i
Ω2
i = 1) .
(185)
It is easy to see that this ansatz solves (183
X2
0 + X2
p+2 −
p+1
i=1
X2
i =
= R2
cosh2
ρ cos2
τ + R2
cosh2
ρ sin2
τ −
p+1
i=1
R2
sinh2
ρΩ2
i =
= R2
cosh2
ρ − R2
sinh2
ρ = R2
(186)
Then with the help of these variables we find that the line element has the
form
ds2
= −dX2
0 − dX2
p+2 +
p+1
i=1
dX2
i =
= R2
(− cosh2
ρdτ2
+ dρ2
+ sinh2
ρdΩ2
) .
(187)
If we take ρ ≥ 0 and 0 ≤ τ ≤ 2π we find that the solution (185) covers the
whole hyperbolic space. Then (τ, ρ, Ωi) are callet the global coordinates of
AdS. Since for ρ near 0 we have
cosh2
ρ ∼ 1 , sinh2
ρ ∼ ρ2
forρ ≪ 1 (188)
we find that the metric has the form in this region
ds2
= R2
(−dτ2
+ dρ2
+ ρ2
dΩ2
) (189)
we see that hyperboloid has the topology S1
× Rp+1
, where S1
is parameterized
by τ so that it is closed time-like curve, while Rp+1
is parameterized
43
by ρ, Ωi. In order to obtain causal space-time we simply unwrap the circle
S1
so that we take −∞ < τ < ∞ with no identification. In following when
we consider AdSp+2 space we will implicitly presume its universal covering
space.
The isometry group SO(2, p + 1) of AdSp+2 has the maximal compact
subgroup SO(2) × SO(p + 1) where it is clear that SO(2) represents the
constant translations in the τ−direction and SO(p + 1) gives rotation of Sp
.
For some purposes it is useful to introduce the coordinate θ that is related
to ρ by tan θ = sinh ρ , 0 ≤ θ ≤ π/2 so that dρ2
= dθ2
cos2 θ
so that the line
element has the form
ds2
=
R2
cos2 θ
(−dτ2
+ dθ2
+ sin2
θdΩ2
) . (190)
The causal structure of the space-time does not change by a conformal rescaling
on the metric. In other words multiplying the metric by R−2
cos2
θ it
becomes
ds′2
= −dτ2
+ dθ2
+ sin2
θdΩ2
. (191)
This is the metric of the Einstein static universe that appeared with the
dimension lower by one in the conformal compactification of R1,p
. However
it is important to stress that now the coordinate takes value in 0 ≤ θ < π/2
rather than 0 ≤ θ < π that appears in the compactification of R1,p
. In other
words AdSp+2 can be conformally mapped into one half of the Einstein static
universe. In general, if a space-time can be conformally compactified into a
region which has the same boundary structure as one half of the Einstein
static universe the space-time is called asymptotically AdS.
The boundary of AdS is extended in time-like direction τ so that we need
to specify boundary condition on the R × Sp
at θ = π/2 which defines the
boundary of space-time given as points at ρ → ∞. It is really important
that the boundary of AdSp+2 or the boundary of the conformally compactified
AdSp+2 is the same as the conformal compactification of the (p + 1)−
dimensional Minkowski space-time.
In addition to the global parametrization given above there is also another
44
set of coordinates (u, t,⃗x)(0 < u,⃗x ∈ Rp
) that are defined by
X0 =
1
2u
(1 + u2
R2
(1 + ⃗x2
− t2
)) , Xp+2 = Rut ,
Xi
= Ruxi
, (i = 1, . . . , p) ,
Xp+1
=
1
2u
(1 − u2
R2
(1 − ⃗x2
+ t2
)) .
(192)
so that
X2
0 + X2
p+2 −
p+1
i=1
X2
i =
=
1
4u2
(1 + 2u2
R2
(1 + ⃗x2
− t2
) + u4
R4
(1 + ⃗x2
− t2
)2
) +
+R2
u2
t2
− R2
u2
⃗x2
−
1
4u2
(1 − 2u2
R2
(1 − ⃗x2
+ t2
) + u4
R4
(1 − ⃗x2
+ t2
)2
) = R2
(193)
Then the metric has the form
ds2
= R2 du2
u2
+ u2
(−dt2
+ d⃗x2
) . (194)
These coordinates cover one half of the hyperboloid. These coordinates are
called the Poincare coordinates. In this form of the metric the subgroup
ISO(1, p) and SO(1, 1) of the SO(2, p + 1) isometry are manifest where
ISO(1, p) is the Poincare transformation on t(t,⃗x) and SO(1, 1) is
(t,⃗x, u) → (ct, c⃗x, c−1
u) , c > 0 (195)
In the AdS/CFT correspondence this is identified with the dilatation D in
the conformal symmetry group of R1,p
.
Consider again the metric (187) and (194). We clearly see that the norm
of the vector ∂τ is always non-zero in (187) and hence it is natural to call τ
as the global time coordinate of AdS. On the other hand the time-like vector
∂t in (194) becomes null at u = 0. This point is call as Killing horizon.
Euclidean Rotation As it is clear from the metric (187) the metric is
static with respect to the time coordinate τ which is also global coordinate.
Then it is possible to perform Wick rotation in the quantum field theory on
45
AdSp+2.Explicitly, we perform Wick rotation eiτH
→ e−τEH
, which means
that τE = −iτ. Inserting this form to the relation between original variables
X and τ we find
X0 = R cosh ρ cosh τE , Xp+2 = iR cosh ρ sinh τE ,
(196)
Let us define coordinate XE as
XE = −iXp+2 = R cosh ρ sinh τE . (197)
Then the original metric has the form
ds2
= −dX2
0 + dX2
E +
p+1
i=1
dX2
i (198)
and the equation for the hyperboloid has the form
X2
0 − X2
E −
p+1
i=1
X2
i = 0 . (199)
Note also that the same space is obtained by rotating the time coordinate t of
the Poincare coordiantes as tE = −it even though the Poincare coordinates
cover only a part of the entire AdS.
In the coordinates (ρ, τE, ⃗Ωp) and (u, tE,⃗x) the Euclidean metric is expressed
as
ds2
E = R2
(cosh2
ρdτ2
E + dρ2
+ sinh2
ρdΩ2
p) =
= R2
(
du2
u2
+ u2
(dt2
E + d⃗x2
)) .
(200)
It is also very convenient to use another form of the metric when we set
u = 1/y , du = − 1
y2 dy so that we obtain the metric in the form
ds2
= R2
dy2
+ dx2
1 + . . . dx2
p+1
y2
(201)
This Euclidean form of the metric is very useful for analysis of the correlation
functions in field theory. As we know from the standart textbook of quantum
46
field theory the correlation functions < ϕ1 . . . ϕn > defined on the Euclidean
space are related by the Wick rotation to the T-ordered correlation functions
< 0|T(ϕ1 . . . ϕn|0 > in the Minkowski space. The same is true in the anti-de
Sitter space on condition when the theory has a positive definite Hamiltonian
with respect to the global time coordinate τ.
4 The Correspondence
4.1 The Maldacena limit
The space-time metric of N coincident D3-branes has following form
ds2
= 1 +
L4
y4
−1/2
ηµνdxµ
dxν
+ 1 +
L4
y4
1/2
(dy2
+ y2
dΩ2
5) , (202)
where the radius L of the D3-brane is given by
L4
= 4πgsN(α′
)2
. (203)
No we would like to analyze the geometry in more details. For y ≫ L we
have that the metric has the form of the flat space-time R10
. When y < L the
geometry is referred as the throat and naively we should say that it is singular
for y ≪ L. However in this regime it is natural to perform redefinition
u ≡
L2
y
, (204)
where the limit y ≪ L corresponds to the large u ≫ L limit. Then we have
du = −L2
y2 du and the metric has the form
ds2
=
1
1 + u4
L4
ηµνdxµ
dxν
+ 1 +
u4
L4
L4
(
du2
u4
+
1
u2
dΩ2
5) (u ≫ L)
−−−−−→
= L2 1
u2
ηµνdxµ
dxν
+
du2
u2
+ dΩ2
5
(205)
that corresponds to the product geometry. Once component is the five-sphere
S5
with the metric L2
dΩ2
5. The remaining component is the hyperbolic space
47
AdS5 with constant negative curvature metric
ds2
AdS5
=
L2
u2
(du2
+ ηµνdxµ
dxν
) . (206)
In other words, the geometry close the brane (y ∼ 0, u ∼ ∞) is regular
na highly symmetrical and corresponds to the space AdS5 × S5
where both
components have identical radius L.
The Maldacena limit corresponds to keeping fixed gs and N as well as all
physical length scales while letting α′
→ 0. It is very nice that this limit of
string theory exists and is very interesting. In this limit only AdS5 ×S5
region
of the D3-brane geometry survives and contributes to tht string dynamics
of physical processes while the dynamics in the asymptotically flat region
decouples from the theory.
To see this more explicitly consider the Maldacena limit directly on the
string theory non-linear sigma model in the D3-brane background. We restrict
ourselves to the metric part only so that we ignore the contributions from the
tensor field F+
5 . We denote D = 10 coordinates by xM
, M = 0, 1, . . . , 9 and
the metric by GMN (x). The first 4 coordinates coincide with xµ
of the Poincare
invariant D3-brane world-volume while the coordinates on the 5−sphere
are xM
for M = 5, . . . , 9 and x4
= u. The full D3-brane metric has the form
ds2
= 1 +
L4
y4
−1/2
ηµνdxµ
dxν
+ 1 +
L4
y4
1/2
(dy2
+ y2
dΩ2
5) =
= L2
1 +
L4
u4
1/2
du2
u2
+ dΩ2
5 + 1 +
L4
u4
−1/2
1
u2
ηµνdxµ
dxν
≡
≡ L2 ¯GMN (x, L)dxM
dxN
(207)
Let us consider non-linear sigma model
S =
1
4πα′
Σ
d2
σ
√
−γγαβ
GMN (x)∂αxM
∂βxN
, (208)
where γαβ is two dimensional world-sheet metric with inverse γαβ
and where
∂α ≡ ∂
∂σα where σα
, α = 0, 1 are world-sheet coordinates. Finally xM
(σ)
describe an embedding of the string-world sheet into the target space-time.
Now for the metric given in (207) we obtain
S =
L2
4πα′
Σ
d2
σ
√
−γγαβ ¯GMN (x, L)∂αxM
∂βxN
. (209)
48
We see that the overall coupling for the sigma model dynamics is given by
L2
4πα′
=
λ
4π
, λ = gsN . (210)
Keeping N and gs fixed but leting α′
→ 0 implie L → 0 as follows from
previous expression. We also see that under this limit the sigma model action
has a smooth limit given by
SG =
λ
4π Σ
d2
σ
√
−γγαβ ¯GMN (x, 0)∂αxM
∂βxN
. (211)
where the metric ¯GMN (x, 0) is the metric on AdS5 × S5
¯GMN dxM
dxN
=
1
u2
ηµνdxµ
dxν
+
du2
u2
+ dΩ2
5 , (212)
where the metric has unit radius. We also see that the coupling 1/
√
λ replaced
α′
. Now we proceed to the explicit statement of AdS/CFT correspondence.
This conjecture states the equivalence between the following theories
• Type IIB superstring theory on AdS5×S5
where both AdS5 and S5
have
the same radius L, where the 5−form F+
5 has integer flux N = S5 F+
5 ,
where the string coupling is gs
• N = 4 SYM in 4−dimensions with the gauge group SU(N) and YangMills
coupling gY M in its superconformal phase
with the following identifications between the parameters of both theories
gs = g2
Y M , L4
= 4πgsN(α′
)2
. (213)
We should carefully explain what the equivalence or duality means. Briefly,
equivalence includes a precise map between the states and fields on the superstring
side and the local gauge invariant operators on N = 4 SYM side
as well as a correspondence between correlators in both theories.
We should also ask the question about validity of various approximations.
The loop diagrams in field theory can be trust when
g2
Y M N = gsN ∼
L4
(α′)2
≪ 1 . (214)
49
On the other hand the classical gravity description can be used when the
radius of curvature L of AdS and S5
becomes large compared to the string
length,
L4
(α′)2
∼ gsN ∼ g2
Y M N ≫ 1 . (215)
We see that the gravity regime and the perturbative field theory regime are
incompatible which is nice since we can avoid the clear contradiction since
these theories are very different. For that reason we call this correspondence
as duality in the sense that these two theories are conjectured that these
theories are the same but when one theory is weakly coupled while the second
one is strongly coupled. However this fact implies that it is really difficult to
prove this duality since all known calculations tools are defined in the weakly
perturbative regime. On the other hand it makes this duality very useful
since when we presume it validity we can calculate expressions, that cannot
be reached by the perturbative calculus in SYM theory, by calculations in
the weakly coupled dual theory.
The formulation of the conjecture given above is known as the strong
form since it is presumed that it holds for all values of N and of gs = g2
Y M .
On the other hand string theory quantization on a general curved manifold
(including AdS5 × S5
) seems to be very difficult and for that reason it is not
completely clear what the strong equivalence means. On the other hand we
will see that AdS5 × S5
background has very special property which leading
to the integrability of the theory on given background. Before we proceed
to this fundamental property of the theory which was discovered four years
after original formulation of this conjecture we will follow original historical
development and discuss limits in which the Maldacena conjecture becomes
more tractable but still remains non-trivial.
4.1.1 The ’t Hooft Limit
The ’t Hooft limit consists in keeping the ’t Hooft coupling λ = g2
Y M N = gsN
fixed and letting N → ∞. As we know this limit is well defined in YangMills
theory and corresponds to a topological expansion of the field theory’s
Feynman diagrams. In case of AdS space we can proceed as follows. The
string coupling can be expressed from the ’t Hooft coupling as gs = λ/N.
Since λ is being kept fixed we see that ’t Hooft limit corresponds to weak
coupling string perturbatiion theory.
50
This form of the conjecture is weaker than the original version but still
provides powerful correspondence between classical string theory and the
large N limit of gauge theories.
4.1.2 The Large λ limit
When we discussed ’t Hooft limit we kept λ = gsN fixed while N → ∞.
Since N → ∞ we find that the only parameter that is left is λ. Quantum
field perturbation theory corresponds to λ ≪ 1. On the other side of the
correspondence it is natural to take λ ≫ 1 instead. Let us now discuss in
more details the meaning of the expanssion around λ large. Let us perform
an expansion in powers of α′
the effective action so that we have
L = a1α′
R + a2(α′
)2
R2
+ a3(α′
)3
R3
+ . . . . (216)
We are interested at distance scales that are typical of the throat whose scale
is set by the AdS radius L. In other words the scale of the Riemann tensor
is set by
R ∼
1
L2
=
(gsN)−1/2
α′
=
λ−1/2
α′
(217)
and we see that the expansion of the effective action in powers of α′
effectively
corresponds in the expansion in powers of λ−1/2
L = a1λ−1/2
+ a2λ−1
+ a3(α′
)3
λ−3/2
+ . . . (218)
and we see that the expansion in the small α′
is effective replaced expansion
in large λ where the parameter of expansion is λ−1/2
≪ 1.
N = 4 conformal SYM Full Quantum Type IIB string
all N, gY M ⇔ theory on AdS5 × S5
•gs = g2
Y M •L4
= 4πggNα′2
• ’t Hooft limit of N = 4 SYM • Classical Type IIB string theory
λ = g2
Y M N fixed, N → ∞ ⇔ on AdS5 × S5
•1/N expansion •gs string loop expansion
• Large λ limit of N = 4 SYM • Classical Type IIB supergravity
(for N → ∞ ) ⇔ on AdS5 × S5
•λ−1/2
expansion •α′
expansion
51
4.2 Relation between Global Symmetries
It is clear that in order to be duality between boundary theory and the bulk
theory to be valid the global unbroken symmetries of the two theories to
be identical. The continuous global symmetry of N = 4 SYM in its conformal
phase (which is the phase where the vacuum expectation value of scalar
fields is zero) is superconformal group SU(2, 2|4) with the maximal bosonic
subgroup SU(2, 2) × SU(4)R ∼ SO(2, 4) × SO(6)R, where SO(2, 4) is conformal
group in 4−dimensions while SU(4)R is automorphism group of the
N = 4 Poincare supersymmetry algebra. We clearly see that the bosonic
group corresponds to the isometry group of the AdS5 × S5
background. As
we argued before this group can be completed to the full supergroup in case
of SYM theory. In case of the AdS side we find that 16 of 32 Poincare supersymmetries
are preserved by the array of N parallel D3-branes. Further when
we take the near horizon limit of this geometry we find that this AdS limit is
supplemented by another 16 conformal supersymmetries that are broken in
the full D3-brane geometry. Thus the global symmetry SU(2, 2|4) matches
on both sides of the AdS/CFT correspondence.
N = 4 SYM theory also obeys Montonen-Olive or S-duality symmetry
that is realized on the complex constant τ by M¨obius transformations in
SL(2, Z). On the other hand we known that this symmetry is a global discrete
symmetry of Type IIB string theory. This symmetry is unbroken by D3-brane
solution as well that it maps non-trivially the dilaton and axion values that
are constant for given type of solution. On the other hand S-duality is a useful
symmetry in case of the strongest for of the AdS/CFT correspondence only
since when we consider the ’t Hooft limit N → ∞ while keeping λ2
= g2
Y M N
fixed we find that S-duality does not have the right meaning. Explicitly, under
S-duality we have gY M → 1
gY M
and hence λ → N2
λ
which means that in the
limit N → ∞ lambda is no longer fixed and goes to infinity as well.
4.3 Mapping Type IIB Fields and CFT Operators
Since we showed that the global symmetry groups on both sides of the duality
coincide we now have to show that the representations of the group SU(2, 2|4)
also coincide on both sides. We already made remarks considering operators
in SYM theory. We also shown that the significant role is played by the single
color trace operators since all higher trace operators can be construct from
these operators with the help of operator product expansion. Then we should
52
expect that single trace operators correspond to single particle states on the
AdS side. Multiple trace operators should be interpreted as bound states of
these one particle states. Multiple trace BPS operators have the property
that their dimension on the AdS side is simply the sum of the dimensions of
the BPS components.
In order to identify the contents of irreducible representations of SU(2, 2|4)
on the AdS side, we describe all Type IIB massless supergravity and massive
string degrees of freedom by fields ϕ living on AdS5 × S5
. We introduce
coordinates zµ
, µ = 0, 1, . . . , 4 for AdS5 and yu
, u = 1, . . . , 5 and decompose
the metric as
ds2
= gAdS
µν dzµ
dzν
+ gS
uvdyu
dyv
. (219)
Then it is convenient to decompose the field ϕ(z, y) in series on S5
ϕ(z, y) =
k=0
ϕk(z)Yk(y) , (220)
where Yk means the function that belongs to the basis of spherical harmonics
on S5
. For scalars for example, Yk are labelled by the rank k of the totally
symmetric traceless representations of SO(6).
4.3.1 Spherical Harmonics
Here we review few basic facts considering spherical harmonics. The set of
scalar functions on SD
form a vector space that is an infinite dimensional
reducible representation of SO(D + 1). Scalar spherical harmonics form a
complete basis of this space.
It is convenient to regard a function on SD
as a restriction of functions
on the RD+1
in which SD
is embedded. An arbitrary C∞
function on RD+1
may be expanded in polynomials in the Cartesian coordinates xi
, so that it
is sufficient to consider separately functions on RD+1
that are homogeneous
in xi
of degree k. Clearly when we restrict these functions on SD
then these
functions are not all independent. Consider for example r2
xi1
. . . xik where
r2
= D+1
i=1 x2
i . This is a function of degree k + 2 but when we restrict to the
sphere we find that it is identical to xi1
. . . xik (due to the fact that r2
= 1
on sphere) which is a function of degree k. Further, if we wish to restrict
ourselves to functions linearly independent of those of lower degree, we must
consider only functions
Ci1...ik
xi1
. . . xik
(221)
53
where the coeficients Ci1...ik
obey the condition Ci1...ik
δimin
= 0 for any 1 ≤
m, n ≤ k. It is also clear we could demand that Ci1...ik
is symmetric in i1, . . . ik.
In other worlds each independent component of a totally symmetric traceless
tensor of rank k defines spherical harmonic by
Y I
= CI
i1...ik
xi1
. . . xik
. (222)
Now we would like to find the eigenvalues of (∇)2
SD on the sphere. Note that
harmonics as polynomials in RD+1
obey the equation
(∇2
)D+1fk =
∂
∂xj
∂
∂xj
Ci1...ik
xi1
. . . xik
=
= k
∂
∂xj
Ci1...ik
δi1
j xi2
. . . xik
=
= k(k − 1) Ci1i2i3...ik
Ci1...ik
δi1
j δi2j
xi3
. . . xik
= k(k − 1) Ci1...ik
δi1i2
xi3
. . . xik
= 0
(223)
where we firstly used the symmetry of Ci1...ik
and then the property that it
is a traceless tensor. Further note that we can write
(∇2
)D+1 =
1
rD
∂r[rD
∂r] +
1
r2
(∇2
)SD . (224)
Note also that when we express all variables using spherical variables we find
that fk has to depend on r through the overall prefactor rk
. Then we find
(∇2
)D+1fk = 0 =
1
rD
∂r[rD
∂rfk] +
1
r2
(∇2
)SD fk =
=
k
rD
∂r[rD−1
fk] +
1
r2
(∇2
)SD fk =
k(D + k − 1)
rD
rD−2
fk +
1
r2
(∇2
)SD fk = 0 →
(∇2
)SD fk = −k(D + k − 1)fk
(225)
The previous discussion was performed in case of the scalar spherical harmionics.
This discussion can be generalized to the case of the vector and tensor
spherical harmonics.
54
Returning to our analysis we find that the field in the AdS receives contribution
from the momentum along S5
since we can certainly decompose
10-dimensional Laplacian as
1
√
−gMN
∂M [gMN
√
−gMN ∂N ] =
=
1
−gAdS
µν
∂µ[ −gAdS
µν gµν
AdS∂ν] +
1
gS
uv
∂u[ gS
uvguv
∂v] .
(226)
In the same way as in the case of Kaluza-Klein compactification we find that
the field compactified on S5
receives contribution to the mass where the mass
term is given by the expression
m2
= −
1
gS
uv
∂u[ gS
uvguv
∂v] . (227)
For the eigenvalues of the Laplacion on S5
for various spins we find following
relations between mass and scaling dimension
scalars m2
= k(k + 4) ,
spin1/2, 3/2 |m| = 2 + k ,
p − form m2
= (4 + k − p)(k + p) ,
spin 2 m2
= (4 + k)k .
(228)
The question is how these modes are related to the operators on the dual
side.
4.4 The Field↔ Operator Correspondence
As we said above conformal field theory does not have asymptotic states or an
S-matrix so that the natural objects that we have to consider are operators.
For example, in N = 4 SYM we have marginal operator that changes the
values of the coupling constant. As we know the field theory coupling constant
is related to the coupling constant in string theory which, as we know, is
related to the expectation value of the dilaton, where the expectation value
of the dilaton is set by the boundary condition for the dilaton at infinity.
55
In other words changing the gauge theory coupling constant corresponds to
the changing the boundary value of the dilaton. To be more explicitly, let us
denote by O the corresponding operator. Then we consider that we add the
additional term
d4
xϕ0(x)O(x) (229)
to the Lagrangian. Let us presume that such a term was not present in the
original Lagrangian. In opposite situation we could consider ϕ0(x) to be equal
to some coefficient of O(x) in the Lagrangian. Then it is natural to assume
that this will change the boundary condition of the dilaton at the boundary
of AdS, where in the Poincare coordinate system it is ϕ(x, z)|z=0 = ϕ0(x).
To be more explicit, according to the famous paper by E. Witten [8], it is
natural to propose
e d4xϕ0(x)O(x)
CFT
= Zstring[ϕ(x, z)|z=0 = ϕ0(x)] , (230)
where the left hand side is the generating function of correlation functions
in the field theory, ϕ0 is an arbitrary function , where we calculate the correlation
functions of ′ by taking the functional derivatives with respect to ϕ0
and then setting ϕ0 = 0. The right side is the full partition function of string
theory with the boundary condition that the field ϕ has the value ϕ0 on the
boundary AdS. It is also clear that ϕ0 is a function of the four variables that
parameterize the boundary of AdS5.
The expression (230) is general for any field ϕ. In other words every field
propagating on AdS5 is in a one to one correspondence with an operator
in the field theory. Clearly there should be a relation between the mass of
the field ϕ and the scaling dimension of the operator in the conformal field
theory. To see this more explicitly let us consider Euclidean AdS5 or H =
{(z0,⃗z), z0 > 0,⃗z ∈ R4
} with Poincare metric
ds2
=
R2
z2
(dz2
0 + d⃗z2
) . (231)
with boundary ∂H = R4
. It is convenient to represent this space graphically
as a disc whose boundary is a circle. The metric diverges at the boundary
z0 = 0 where the overall scale factor blows up. This scale factor may be
removed by a Weyl rescaling of the metric but such a rescaling is not unique.
A unique well-defined limit to the boundary of AdS5 can only exist if the
boundary theory is scale invariant. For finite values of z0 > 0 the geometry
56
the theory still have 4−dimensional Poincare invariance but it does not have
to be invariant.
We know that superconformal N = 4 Yang-Mills theory is scale invariant
and so that we can say that it can live at the boundary of ∂H. Further,
the dynamical observables of N = 4 SYM are the local invariant polynomial
operators that we introduced in previous sections. These operators naturally
live on the boundary ∂H and are characterized by their dimension, Lorentz
group SO(1, 3) and SU(4)R quantum numbers.
Following discussion given above we decompose all 10−dimensional fields
on Kaluza-Klein towers on S5
so that effective all fields ϕ△(z) propagate
on AdS5. We label these fields by their dimension △ where other quantum
numbers are implicit. We also presume that these fields are asymptotically
free where of course there is a possibility that they interact in the bulk of
AdS. The free field than satisfy the equations
(
1
√
gAdS
∂µ[
√
gAdSgµν
AdS∂νϕ0
△] − m2
△ϕ0
△ = 0 (232)
Let us now concentrate on the z−dependence so that the equation above
takes the form
1
√
gAdS
d
dz
√
gAdSgzz
dϕ0
△
dz
− m2
ϕ0
△ =
(233)
where we also presume an asymptotic behavior ϕ△
0 ∼ zΓ
for Γ = const. Then
the equation above takes the form
1
R2
Γ(Γ − 4)zΓ
− m2
ϕ0
△ = (Γ(Γ − 4) − m2
R2
)ϕ0
△ = 0 (234)
so that we have two roots of the equation above
Γ+ = 2 + 2
√
4 + m2R2 , Γ− = 2 − 2
√
4 + m2R2 . (235)
We see that these two solutions have different behavior when we approach
z0 → 0
ϕ+
0 ∼ z
Γ+
0 = z2+
√
4+m2R2
0 ....normalizable ,
ϕ−
0 ∼ z
Γ−
0 = z2−
√
4+m2R2
0 ...non − normalizable .
(236)
57
There is a relation between the mass of the field ϕ and the scaling dimension
of the operator in the conformal field theory. Let us be more general and
consider the AdSd+1 space where the metric has the form
ds2
=
R2
z2
(dz2
+ (dx1
)2
+ . . . (dxd
)2
) (237)
In this space the equation of motion takes the form
1
√
gAdS
d
dz
√
gAdSgzz dϕ
dz
− m2
ϕ =
=
zd+1
Rd+1
d
dz
[
Rd+1
zd+1
z2
R2
dϕ
dz
] − m2
ϕ = 0
(238)
that for asymptotic ϕ ∼ zΓ
implies
Γ(Γ − d) − m2
R2
= 0 ⇒ Γ± =
d
2
±
d2
4
+ m2R2 .
(239)
We see that the solution with Γ− is blowing up for z → 0 while field with behaves
as zΓ
+ vanishes for z → 0. In other words the second mode corresponds
to the normalizable solution while the first one appears as the source in the
CFT action. In other words we find that the field that should be source in
CFT action has to have following asymptotic behavior
ϕ(⃗x, t) = ϵΓ−
ϕ0(⃗x) = ϵd−△
ϕ0(⃗x) , (240)
and eventually we take the limit ϵ → 0. Note that we wrote Γ− as
Γ− =
d
2
−
d2
4
+ R2m2 = d − (
d
2
+
d2
4
+ R2m2) ≡ d − △ . (241)
We presume that ϕ is dimensionless so that ϕ0 has dimension [length]−Γ−
that implies, through the left side of the equation (230) that the dimension
of the operator O is
−d + [O] + d − △ = 0 ⇒ [O] = △ . (242)
It turns out that in the case of the interacting fields the solutions will
have the same asymptotic behaviors as in the free case. It was argued that
58
the normalizable modes determine the vacuum expectation values of the operators
of associated dimensions and quantum numbers. On the other hand
the non-normalizable solutions do not correspond to the bulk excitations due
to the fact that they are not square normalizable. In fact, they represent the
coupling of external sources to the superagravity or string theory.
5 Explicit Witten’s presentation
When we calculate the correlation functions it is conventient to use the Euclidian
signature of the metric which can be described at several different ways.
Consider a Euclidean space Rd+1
with coordinates y0, . . . , yd and let Bd+1 is
the open unit ball given by the condition
d
i=0
y2
i < R2
. Then AdS can be identified with the unit ball with the metric
ds2
= 4R4
d
i=0 dy2
i
(R2 − |y2|)2
. (243)
With this parameterization is easy to compactify given AdSd+1 when get the
closed unit ball ¯Bd+1 defined by the equation
d
i=0
y2
i ≤ R2
. (244)
Its boundary is the sphere Sd
that is defined by the condition
d
i=0
y2
i = R2
. (245)
Sd
is the Euclidean version of the conformal compactification of Minkowski
space. Since Sd
is the boundary of ¯Bd+1 we find that this statement is the
Euclidean version of the statement that Minkowski space is the boundary of
AdSd+1. We also see that the metric (243) that is defined on Bd+1 cannot be
extended on ¯Bd+1 since the metric is singular at d
i=0 y2
i = R2
. In order to
find a metric that extends on ¯Bd+1 we can take a function f on ¯Bd+1 that is
59
positive on Bd+1 and has a first order zero on the boundary, as for example
f = R2
− |y|2
. Then we consider the line element as
d˜s2
= f2
ds2
, (246)
then we can restrict the metric d˜s2
on Sd
. Since there is not some prefered
choice of f we see that this metric is only well defined up to conformal
transformation. For example, if we consider the function f with another one
f → few
(247)
where w is real function on ¯Bd+1 we find that this replacement will induce
the conformal transformation
d˜s2
→ e2w
d˜s2
(248)
in the metric of Sd
.
In order to have contact with previous definitions let us write the metric
ds2
as
ds2
=
1
(1 − y2
R2 )2
(dy2
+ y2
dΩ) , (249)
where dΩ2
is the metric on the unit sphere. Then we introduce variable r as
y = R tanh
ρ
2
, dy =
R
2 cosh2 ρ
2R
dρ (250)
so that the line element has the form
ds2
= 4 cosh4 ρ
2
(R2 1
4 cosh4 ρ
2
dρ2
+ +
sinh2 ρ
2
cosh2 ρ
2
dΩ) =
= R2
(dρ2
+ sinh2
ρdΩ) .
(251)
5.1 Massless Field Equations
As a simple example we consider the massless equations that allow to reach
the most nice result. In this case the equation of motion takes the form
1
√
gAdS
∂µ[
√
gAdSgµν
AdS∂νϕ] = 0 . (252)
60
The important property about AdSd+1 is that if we have function ϕ(Ω) on the
boundary Sd
then there is a unique extension of ϕ to a function on ¯Bd+1 that
has given boundary values and obeys the field equation, where uniqueness
means that there is non non-zero square-integrable solution of the equation
above which, if it existed, could be added to any given solution above and
hence destroys the uniqueness.
Let us now presume that the boundary value ϕ0 is the source for the
operator O and we have to calculate the classical action for ϕ that obeys
the equation of motion with prescripted boundary conditions. To do this
we try to find Green function which we mean a solution K of the Laplace
equation on Bd+1 whose boundary value is a delta function at a point P on
the boundary. It is convenient to use the representation of Bd+1 as the upper
half space with metric
ds2
=
R2
x2
0
d
i=0
(dxi
)2
, (253)
where the boundary is a copy of Rd
at x0
= 0 together with a single point
P at x0
= ∞, where the point x0
= ∞ consists of a single point since the
metric in xi
direction vanishes as x0 → ∞. Note that without the point P
the boundary would consist the space Rd
. On the other hand we know that
the conformal compactification of Rd
is obtained by adding in a point P at
infinity which in the end gives the sphere Sd
. Since we argued before the the
boundary of the Euclidean AdS is sphere it is natural to include the point
P.
Returning to our example we see that the boundary conditions and metric
are invariant under translations of the xi so that K should obey this symmetry
and to be a function of x0 only. Then the equation of motion has the
form
d
dx0
x−d+1
0
d
dx0
K(x0) = 0 (254)
with the solution
K(x0) = cxd
0 . (255)
where c is a constant. This function grows at infinity so that there is some
sort of singularity at the boundary point P. Let us now consider SO(1, d+1)
transformation that maps P to a finite point. Explicitly we consider the
transformation
xi →
xi
x2
0 + d
j=1 x2
j
, i = 0, . . . , d (256)
61
that transforms the point x0 = ∞ to the origin xi = 0, i = 0, . . . , d. Under
these transformations the function K has the form
K(x) = c
xd
0
(x2
0 + d
j=1 x2
j )d
. (257)
It is easy to see that for x0 → 0 K vanishes except at x1 = x2 · · · = xd = 0.
Further, K is positive. In other words for x0 K is a delta function with
support at xi = 0. Using this Green function the solution of the Laplace
equation on the upper half space with the boundary values ϕ0 has the form
ϕ(x0, xi) = c dx′ xd
0
(x2
0 + |x − x′|2)d
ϕ0(x′
) . (258)
where dx′
= dx′
1dx′
2 . . . dx′
d and |x − x′
|2
= d
j=1(xi − x′
x)2
. Then we find
∂ϕ
∂x0
= dc dx′
[
xd−1
0
(x2
0 + |x − x′|2)d
ϕ0(x′
) −
−
xd+1
0
(x0 + |x − x′|2)d+1
=
= dcxd−1
0 dx
|x − x′
|2
(x2
0 + |x − x′|2)d+1
ϕ0(x′
) .
(259)
In the limit x0 → 0 we can neglect x2
0 in denominator and we obtain
lim
x0→0
∂ϕ
∂x0
= dcxd−1
0 dx
1
|x − x′|2d
ϕ0(x′
) . (260)
Let us now determine the on-shell value of the action for the scalar field
S(ϕ) =
1
2
dd+1
x
√
ggµν
∂µϕ∂νϕ =
= dd+1
x∂µ[ϕ
√
ggµν
∂νϕ] − dd+1
xϕ∂µ[
√
gµν
∂νϕ] =
= lim
ϵ→0 Tϵ
dx
√
hϕ(⃗n · ⃗∇ϕ) ,
(261)
62
where in the first step we used integration by parts and in the second step
we used the fact that the second expression on the second line vanishes since
the field ϕ obeys the equation of motion. Finally the surface Tϵ is the surface
x0 = ϵ, h is its induced metric and n is unit normal vector to Tϵ. From
the definition of the metric we have that hij = 1
x2
0
diag(1, . . . , 1) and hence
√
h = x−d
0 so that
√
h|Tϵ = ϵ−d
. Further, since ⃗n has to obey the equation
nµ
nν
gµν = 1 ⇒ n0
g00n0
= (n0
)2 1
x2
0
= 1 (262)
we find that n0
= x0 that on the boundary has the value nTϵ = ϵ. Then we
obtain
nµ
∂µϕ = n0
∂x0 ϕ = x0
∂ϕ
∂x0
. (263)
Using the value of ∂x0 ϕ determined above and the fact that ϕ → ϕ0 for
x0 → 0 by definition we obtain that the on-shell value of the action has the
form
S(ϕ)on shell
= lim
ϵ→0 Tϵ
dx
√
hϕn0 ∂ϕ
∂x0
) =
= lim
ϵ→0 Tϵ
ϵ−d
ϵ ϵd−1
0 dx′ 1
|x − x′|2d
ϕ0(x′
) =
= cd dxdx′ ϕ0(x)ϕ0(x′
)
|x − x′|2d
(264)
so that the two point function of the operator O is a multiple of |x − x′
|−2d
as expected for a field O of a conformal dimension d.
In what follows we will be more precise with the definition of the correspondence.
As we argued above the correspondence has the general form
exp −Γ[¯ϕ△] = exp
∂H
¯ϕ△O△ (265)
where these expressions are underestood to hold order by order in a perturbative
expansion in the number of fields ¯ϕD. On the AdS side we assume
that we have an action S[ϕ△] that summarizes the dynamics of Type IIB
string theory on AdS5 ×S5
. In the supergravity approximation, S[ϕ△] is just
63
Type IIB supergravity action on AdS5 × S5
. Beyond the supergravity approximation,
S[ϕ△] will include α′
corrections due to the massive string effects.
In this case we have
Γ[¯ϕ△] = extrS[ϕ△] , (266)
where the extremum on the right side is taken over all fields ϕ△ that satisfy
the assymptotic behavior for the boundary fields ¯ϕ△. These sources correspond
to the non normalizable modes and correspond to the sources of the
SYM operators O△.
5.2 Quantum Expansion in 1/N−Witten Diagrams
The actions that we study have an overall coupling constant factor. For
example, in case of the part of Type IIB supergravity for the dilaton Φ and
the acion C in the presence of a metric Gµν in the Einstein fame is given by
S =
1
2κ2
5 H
√
G[−RG + Λ +
1
2
∂µΦ∂µ
Φ +
1
2
e2Φ
∂µC∂µ
C] (267)
Now the dimensional analysis gives that
[κ2
5] = M−3
, [Φ] = [C] = M0
. (268)
We show later that κ2
5 is equal to
κ2
5 =
πR3
2N2
(269)
and hence it implies that for large N κ5 is small and then we can perform
small κ5, or semi-classical expansion of the correlators generated by this
action. As a result we obtain set of rules that has similar form as Feynmann
diagrams which are known as Witten diagrams. The Witten diagram is represented
by a disc whose interior corresponds to the interior of AdS while
the boundary circle corresponds to the boundary of AdS. The graphical rules
are as follows:
• Each external source to ¯ϕ△(⃗xI) is located at the boundary circle of the
Witten diagram at a point ⃗xI .
• We include a propagator from the external source at ⃗xI that points
either to another boundary point or to an interior interaction point.
This second propagator is known as boundary to bulk propagator.
64
(a) (b) (c) (d) (e)
AdS
boundary AdS
• The structure of the interior interaction points is determined from the
integration terms in the bulk action in the same way as in case of
Feynman diagrams.
• Two interior interaction points may be connected by bulk-to bulk propagators
in the same way as in case of ordinary Feynman diagrams.
In the diagram below we see 2−, 3− and 4− point function contributions.
5.3 AdS propagators
In the previous section we determined the propagator of the massless scalar
field. In given section we determine more general form of these propagators.
We work in Euclidean AdSd+1 space that we denote by H with the Poincare
metric
ds2
=
R2
z2
0
(dz2
0 + d⃗z2
) . (270)
Due to the fact that the space H has an isometry SO(1, d + 1) the Green
functions depend upon the SO(1, d+1) invariant distance between two points
in H. The geodetic distance is given by
d(z, w) =
z
w
ds ln
1 + 1 − ζ2
ζ
, ζ =
2z0w0
z2
0 + w2
0 + (⃗z − ⃗w)2
. (271)
To see this it is instructive to solve the geodesic equation in this space. We
do it when we study motion of point particle in AdS that is governed by the
action
S = −m dτ −gMN ∂τ XM ∂τ XN (272)
Varying given action we derive the equation of motion in the form
∂τ
gMN ∂τ XN
−gPR∂τ XP ∂τ XR
−
δgPR
δXM
∂τ XP
∂τ XR
2 −gPR∂τ XP ∂τ XR
= 0 (273)
65
Note that the Minkowski form of AdS metric has the form
ds2
=
R2
z2
(−dt2
+ dxi
dxi + dz2
) (274)
so that we presume that there are no motion in the xi directions and the only
time dependent variable is z0 = z0(t). First of all the equation of motion for
X0
= t has the form
∂τ
gtt∂τ X0
−gPR∂τ XP ∂τ XR
= 0 (275)
We take the ansatz t = τ and hence then this equation implies the conserved
quantity
gtt
−gtt − gzz(∂τ Z)2 − gij∂τ Xi∂τ Xj
= C (276)
on the other hand the equation of motion for Xi
has the form
∂τ
gij∂τ Xj
−gPR∂τ XP ∂τ XR
= 0 (277)
that can be solved by the ansatz Xi
= vi
t + Xi
0 due to the equation (275).
Then instead of solving the equation of motion for z directly we use the
equation (275) to find the differential equation for Z
∂tZ =
(1 − v2)Z2 − 1
CZ
⇒
Z2
= (1 − v2
)(Ct + C0)2
+
1
1 − v2
.
(278)
In Euclidean theory we have similar equation
gzz
gzz + gz0z0 (∂zz0)2 + gijvivj
= C ⇒
1
C2Z2
0
= (1 − v2
− (
dZ0
dz
)2
)
(279)
66
that can be easily solved with the result
1
C2(1 + v2)
= (z + K)2
+ (1 + v2
)(Z0)2
. (280)
that is an equation for ellipse. Note that for v = 0 we find the circle with
radius 1
C2 and we see that the geodesic starts at z = − 1
C2 − K, reaches the
turning point at z = −K and then returns back to the boundary that reaches
at z = 1
C2 − K. It is also clear that we should generalize this equation to the
any point on the boundary performing S0(4) rotation. so that we have the
equation
1
C2
= (⃗Z − ⃗K)2
+ (Z0)2
. (281)
We could be also interested in the distance between two points in the bulk
of AdS. So that we should choose the initial condition that for given vector
⃗Zi the point we are at the point Z0
i and for given ⃗Zf we are at the point Z0
f
so that we find two equations
1
C2
= |⃗Zi − ⃗K|2
+ (Z0
i )2
,
1
C2
= |⃗Zf − ⃗K|2
+ (Z0
f )2
,
(282)
that gives
⃗K =
1
2
(⃗Zf − ⃗Zi)(1 +
(Z0
f )2
− (Z0
i )2
|⃗Zf − ⃗Zi|2
) . (283)
And then C can be found from the expression above using ⃗K.
Generally, using this dependence we can find the geodetic distance between
two points (z0,⃗z) and (w0, ⃗w). However it is much more convenient
to proceed following way. We know that Euclidean Anti-De Sitter space is
described by the equation
−X2
0 + X2
d+1 +
d
i=1
X2
i = −R2
(284)
in the space time Rd,1
with the metric
ds2
= −dX2
0 + dX2
d+1 +
d
i=1
dX2
i = 0 , (285)
67
so that the metric is ηMN = (−1, 1, . . . , 1). Let us introduce the action that
determines the evolution of the particle on given space
S = dτ( ˙XM
ηMN
˙XN
+ Γ(XM
ηMN XN
+ R2
)) (286)
where Γ is Lagrange multiplicator that imposes the constraint that the particle
moves on anti-de Sitter space. Then the equation of motion has the
form
− ¨XN
+ ΓXN
= 0 , (287)
while variation with respect to Γ we obtain the constraint XM
ηMN XN
=
−R2
. Lagrange multiplicator Γ we derive from the first equation when we
multiply it ηMN XN
we obtain
− ¨XM
XM + ΓXM
XM = 0 ⇒ ˙XM ˙XM − ΓR2
= 0 ⇒
⇒ Γ =
1
R2
˙XM ˙XM
(288)
where we used the fact that
d
dτ
(−R2
) =
d
dτ
(XM
ηMN XN
) = 0 ⇒ ˙XM
ηMN XN
= 0 ⇒
¨XM
ηMN XN
+ ˙XM ˙XN = 0
(289)
Then the equation of motion for X has the form
¨XM
−
1
R2
( ˙XN ˙XN )XM
= 0 (290)
Further, using the equation of motion for XM
and multiply it with ˙XM we
obtain
− ¨XM ˙XM + ΓXM ˙XM = 0 ⇒
d
dτ
( ˙XM ˙XM ) = 0
that implies
˙XM ˙XM = const ≡ K2
(291)
Then the equation of motion for XM
has simple form
¨XM
−
K2
R2
XM
= 0 (292)
68
Let us presume an ansatz in the form
XM
= mM
exp
K
R
λ + nM
exp −
K
R
λ , (293)
Then
XM
XM = mM
mM exp 2
K
R
λ + nM
nM exp −2
K
R
λ + 2nN
mM = −R2
(294)
from which we derive the conditions on the vectors n and m
nN
nN = mM
mM = 0 , nM
mN = −
R2
2
. (295)
Now it is easy to determine the geodesic distance between two points X(λ1)
and X(λ2) that is defined as
d =
2
1
ds =
2
1
dλ ˙XM ˙XM = K(λ2 − λ1) . (296)
To proceed further we use following expression
XM
(λ2)XM (λ1) = mM
nM [exp
K
R
(λ2 − λ1) + exp −
K
R
(λ2 − λ1) ]
(297)
from which we obtain
e
K
R
(λ2−λ1)
=
1
R2
(−X(λ2)M
XM (λ1) + (XM (λ2)XM (λ1))2 − R4)
(298)
from which we obtain
K(λ2 − λ1) = R ln
1
R2
(−X(λ2)M
XM (λ1) + (XM (λ2)XM (λ1))4 − R4)
(299)
Let us introduce coordinates u, tE, xi
as
X0 =
1
2u
(1 + u2
(R2
+ ⃗x2
+ t2
E)) , Xd+1 = RutE ,
Xi
= Ruxi
, i = 1, . . . , d − 1 ,
Xd+1 = RutE , Xd
=
1
2u
(1 − u2
(R2
− ⃗x2
− t2
E)) ,
(300)
69
that obeys the equation that defines an embedding of AdS
−X2
0 + X2
d+1 +
d−1
i=1
X2
i + X2
d =
(301)
Finally we introduce z0 = 1
u
so that
X0 =
z0
2
(1 +
1
z2
0
(R2
+ ⃗x2
+ t2
E)) , Xd+1 = R
1
z0
tE ,
Xi
=
R
z0
xi
, i = 1, . . . , d − 1 ,
Xd+1 =
R
z0
tE , Xd
=
z0
2
(1 −
1
z2
0
(R2
− ⃗x2
− t2
E)) ,
(302)
So that we finally obtain
XM
(1)XM (2) = −
1
2
R2 w2
0 + z2
0 + (⃗z − ⃗w)2
z0w0
≡ −
R2
ξ
(303)
and hence the geodesic distance takes the form
d = R ln
1 + 1 − ξ2
ξ
.
(304)
Since we see that the geodesic distance depends on ξ it is more natural to
work with ξ than with the geodetic distance. Sometimes is also used the
notion chordal distance that is defined as
u =
1 − ξ
ξ
. (305)
Then we can find an inverted relation
ξ =
2ed/R
1 + e2d/R
=
1
cosh d/R
⇒ u = cosh
d
R
− 1 . (306)
70
Now we are ready to consider bulk to bulk propagator. Consider scalar field
with the mass m2
= △(△ − d). According to the discussion given previously
we know that
d
2
+
d2
4
+ R2m2 = △ ⇒ m2
=
1
R2
△(△ − d) . (307)
Now the action for the massive field has the form
Sϕ△
=
H
dd+1
z
√
g
1
2
gµν
∂µϕ△∂νϕ△ +
1
2
m2
ϕ2
△ − ϕ△J (308)
The equation of motion for ϕ△ has the form
(□g + m2
)ϕ△ = J , (309)
where the scalar Laplacian has the form
□g = −
1
√
g
∂µ[
√
ggµν
∂ν] = −z2
0∂2
0 + (d − 1)z0∂0 − z2
0
d
i=1
∂2
i . (310)
Let us now presume that the field ϕ△ is given in response to the source by
the equation
ϕ△(z) =
H
dd+1
z′√
g(z′
)G△(z, z′
)J(z′
) , (311)
Inserting this equation to the equation of motion above and using the fact
that
J(z) = dd+1
z′
δ(z − z′
)J(z′
) (312)
which says that G(z, z′
) obeys the equation
(□g + m2
)G△(z, z′
) =
1
√
g
δ(z − z′
) . (313)
Now the Green function G△(z, w) has the form
G△(z, w) = G△(ξ) =
2−△
C△
2(△ − d)
ξ△
F
△
2
,
△
2
+
1
2
; △ −
d
2
+ 1; ξ2
, (314)
where F is hypergeometric function and the overall normalization constant
is defined by
C△ =
Γ(△)
πd/2Γ(△ − d
2
)
. (315)
71
Since 0 ≤ ξ ≤ 1 the hypergeometric function is defined by its Teylor series
for all ξ except at the coincident point ξ = 1, ξ = w. The massive scalar
boundary to bulk propagator This propagator corresponds to the limit
when the source is on the boundary of H.
The action to linearized order is given by
Sϕ△
=
H
dd+1
z
√
z
1
2
gµν
∂µϕ△∂νϕ△ +
1
2
m2
ϕ2
△ −
∂H
dd+1 ¯ϕ△(⃗z) ¯J(⃗z) ,
(316)
where the bulk field ϕ△ is related to the boundary field ¯ϕ△ by the limiting
relation
¯ϕ△(⃗z) = lim
z0→∞
z△−d
0 ϕ△(z0,⃗z) . (317)
The corresponding boundary-to-bulk propagator is the Poisson kernel
K△(z,⃗z) = C△
z0
z2
0 + (⃗z − ⃗x)2
△
. (318)
Then the bulk field that is generated by the boundary source ¯J is given by
ϕ△(z) =
∂H
dd
⃗xK△(z,⃗x) ¯J(⃗x) . (319)
It is also possible to extract these results in slightly diffrent way. Let us
consider the scalar field that again obeys the equation of motion (□g−m2
)ϕ =
0 that has the form
z5
0∂z0 [
1
z3
0
∂ϕ] + z2
0∂i[δij
∂jϕ] − m2
ϕ = 0 . (320)
Let us presume an ansatz ϕ = ei⃗p·⃗x
Z(pz0) so that ∂
∂z0
= ∂
∂m
p, where p2
=
⃗p · ⃗p, m = zp. Then the equation above takes the form
m2 d2
Z
d2m
− 3m
dZ
dm
− m2
Z − m2
R2
Z = 0 . (321)
There are two independent solutions of the equation above which are
Z(m) = u2
I△−2(m)′
, Z(m) = m2
K△−2(m) , (322)
where Iν, Kν are Bessel functions and where
△ = 2 +
√
4 + m2R2 . (323)
72
When we demand the regularity at the interior when z → ∞ we have to
select the second solution since I△−2(m) increases exponentially as m → ∞.
Then the field has the form
ϕ(z0,⃗z) = C(pz0)2
K△−2(pz0)ei⃗p·⃗z
, (324)
where the constant C is determined by imposing the boundary conditions
ϕ(⃗z, z0) = ϕ0(⃗z) = ei⃗p·⃗x
= (ϵp)2
K△−2(pϵ)ei⃗p·⃗x
C ⇒
⇒ C =
1
(ϵp)2K△−2(pϵ)
(325)
and hence if we define the bulk to boundary propagator as K⃗p(⃗z, z0) = ϕp(⃗z,z0)
ϕ0(⃗z)
we find the form of the boulk-to boundary propagator in the form
Kp(⃗z, z0) =
(pz)2
K△−2(pz)
(pϵ)2K△−2(pϵ)
. (326)
As we argued above the boundary valules of string fields act as sources for
the gauge-invariant operators in the field theory. Genearlly we denote the
bulk fileds as ϕ(⃗z, z0) with values ϕ0(⃗z) at z0 = ϵ. The true boundary of AdS
is at z = 0 and ϵ ̸= 0 serves as a cutoff that could be eventually removed.
When we restrict to the supergravity approximation we choose ϕ0 arbitrary
as the boundary conditions at z = ϵ , while the bulk fields have to obey the
equations of motion. Then we evaluate the action on this classical solution
with fixed boundary valules. In other words we have
Wgauge[ϕ0] = ln e d4zϕ0(x)O(z)
= extremumISUGRA[ϕ]ϕ|z0=ϵ=ϕ0 . (327)
In other words the generator of connected Green functions in the gauge theory
in the large λ limit is the on-shell supergravity action.
It is important to stress that it is not clear whether the right side of the
correspondence above is independent on ϵ. Instead we perform this calculation
and then we eventually perform a wave-function renormalization on
either O ir ϕ so that the final answer is independent of the cutoff. Now we
explicitly the two point function. Note that the two point function can be
73
calculated from the generator of connected Green functions when write
dd
xϕ0(x)O(x) =
dd
x dd
k dd
lei⃗k·⃗x
ϕkei⃗l·⃗x
O(⃗l) =
= dd
kϕkO(−⃗k) .
(328)
On the other hand on the right side of the correspondence we have
ISUGRA =
1
2 z0=ϵ
dd
z
√
hϕ(⃗z, z0)nµ
∂µϕ(⃗z, z0) =
= ϵ−d 1
2 z0=ϵ
dd
z dd
k dl
ϕkei⃗k·⃗l
∂z0 Kpϕlei⃗l·⃗x
=
=
1
2
dd
k dd
lϕkδ(p + l)∂z0 Kpϕl .
(329)
To proceed further note that for small x the Bessel function has following
asymptotic behavior
Kα ∼
1
Γ(α + 1)
x
2
α
(330)
and hence we can derive corresponding Green functions.
5.4 Wilson Loops
The Wilson loop is defined as
W[C] = Tr[P exp(i
C
A)] (331)
depends on a loop C that is embedded in four dimensional space. This definition
includes the path ordered integral of the gauge connection along the
contour. The trace is taken over some representation of the gauge group but
we consider the case when the gauge field is in the fundamental representation.
From the exceptational value of the Wilson loop operator < W(C) >
74
we can determine the quark-anti-guark potential. For that reason we consider
rectangular loop with sides of length T and L in Euclidean space. If we
interpret T as the time direction it is clear that for large T the expectation
value behaes as e−TE
where E is the lowest possible energy of the quark-anti
quark configuration.
For example, consider U(1) gauge theory with the gauge transformation
given by
ϕ′
(x) = eiα(x)
ϕ(x) ,
A′
µ(x) = Aµ(x) + ∂µα(x) .
(332)
Through these rules it is clear that the non-local operator ϕ(x)ϕ∗
(y) is not
gauge invariant in general and hence cannot be considered as an observable.
On the other hand it can be shown that following quantity is gauge invariant
ϕ(x)ei P dxµAµ
ϕ∗
(y) , (333)
where P is an arbitrary path from point x to y. To see this note that the
object above transforms under gauge transformation as
ϕ(x)ei P dxµAµ
ϕ∗
(y) → ϕ(x)eiα(x)
ei P dxµ(Aµ+∂µα)
e−iα(y)
ϕ∗
(y) =
= ϕ(x)eiα(x)
ei(α(y)−α(x))
ei P dxµAµ
ϕ∗
(y) =
= ϕ(x)ei P dxµAµ
ϕ∗
(y)
(334)
We see that there is a natural object in the theory that is either Wilson line
WP (x, y) = ei P dxµAµ
(335)
or Wilson loop
WP (x, x) = ei dxµAµ
(336)
The Wilson line represents the coupling of the gauge field to the test charge.
Let us consider a charge particle with world-line y(λ). Then the corresponding
current has the form
Jµ
(x) = dλ
dyµ
dλ
δ(x − yµ
(λ)) . (337)
75
For a given closed path, dy
dλ
can be both positive and negative so that we
have both a positive charge and a negative charge when we accept the fact
that the sign of the charge depends on the sign of dy
dλ
, where dy/dλ > 0 for a
positive charge. We can use Jµ
as the coupling of the gauge field to the point
particle as
δS = d4
xAµJµ
= d4
xAµ(x) dλ
dyµ
dλ
δ(xµ
−yµ
(λ)) = dλ
dyµ
dλ
Aµ(y(λ)) .
(338)
In other words the perturbation of the action corresponds to the exponent
of the Wilson loop and hence we can write that the Wilson loop represents
a partition function in the presence of a test charge
< Wp >=
Z[J]
Z[0]
. (339)
In fact, let us write the Euclidean partition function as
Z = f|e−Hτ
|i , (340)
where |i⟩ , |f⟩ are initial and the final states, respectively. Using the complete
set of energy eigenstates
H |n⟩ = En |n⟩ (341)
we can write
Z =
n,m
⟨f|n⟩ ⟨n| e−Hτ
|m⟩ ⟨m|i⟩ =
n,m
⟨f|m⟩ e−Enτ
⟨m|n⟩ ⟨n|i⟩ =
=
m
⟨f|m⟩ e−Emτ
⟨m|i⟩
(342)
that in the limit τ → ∞ gives
Z
τ→∞
−−−→ e−E0τ
. (343)
In other words in the limit τ → ∞ the Euclidean partition function is dominated
by the ground state and gives the ground state energy. In case when we
can neglect the kinetic energy, it gives the quark-antiquark potential energy
⟨Wp⟩ ≃ e−V (R)τ
, (344)
76
where R is separation between quark-antiquark. When the quarks are confined
as in case of QCD then the potential grows with the separation R so
that V ≃ σR, where σ is called the string tension. Then we obtain
⟨Wp⟩ ≃ e−σRτ
= e−σA
, (345)
where A is the area of the Wilson loop A = Rτ when we consider rectangular
Wilson loop, where the spatial dimension is R while the temporal size has
length τ. This behavior is known as area low. In case of unconfined potential
we have different situation, as for example in case of the Coulomb potential
where the potential decays with the separation. Then in case when τ = R ≫ 1
we find
⟨Wp⟩ ≃ e−O(R)
. (346)
In this way the Wilson loop provides a criterion for the confinement. Now
we are ready to proceed to the discussion of the Wilson loop in AdS/CFT
context. The matter fields in the N = 4 SYM are all in the adjoint representation.
On the other hand quarks correspond to the matter fields in the
fundamental representation. We know that the adjoin representation on the
stack of N D3-branes has the origin in the open strings ending on different
D-branes in the stack. In order to describe probe in the fundamental representation
we consider an infinitely long string. In this case the string can end
on N different branes which means that the string transforms in the fundamental
representation of SU(N) gauge theory. Such a long string represents
a quark. Such a string has an extension and tension so the string has a large
mass. In other words long string represents a heavy quark. This is the natural
object for the discussion of the Wilson loop in AdS/CFT. These open
strings couple to the Wilson loop operator. Then, following the case of the
expectation values of the local operators in QFT and the partition functions
for the supergravity modes on AdS5 × S5
we see that it is natural to propose
that the vacuum expectation value of the Wilson loop in the conformal
field theory is given by the partition function of the fundamental string on
AdS5 × S5
with the condition that we have a string world-sheet that ends
on the loop C. In the supergravity regime, when gsN is large, the leading
contribution to the partition function will be given by the area of the string
world-sheet. This area is measured by AdS metric.
However there is a subtlety in the calculation of this are. It turns out that
the area that is calculated is divergent which is a consequence of the fact that
the string is going all the way to the boundary of AdS. Then if we evaluate
77
the area up to some radial distance U = r we find that for large r it diverges
as r|C|, where |C| is the length of the loop in the field theory. On the other
hand the perturbative computation in the field theory shows that < W > is
finite. Then the divergence in the Wilson loop would imply that there is a
mass renormalization of BPS particle which cannot be certainly true. This
apparent conflict between the divergence of the area of the minimum surface
in AdS and the finiteness of the field theory computation can be explained by
the fact that the appropriate action for the string world-sheet is not the area
itself but its Legendre transformation with respect to the radial coordinate
u. The reason for this procedure is the fact that these string coordinates obey
the Neumann boundary conditions rather than the Dirichlet ones that were
appropriate for the original one. The Legendre transformation subtracts the
divergent term r|C| leaving the resulting action finite.
Using the supergravity approximation we can compute the quark-antiquark
potential in the supergravity approximation. We consider a quark at x =
−L/2 and anti-quark at x = L/2. We consider Nambu-Gotto action for fundamental
string
S =
1
2πα′
dτdσ −aαβ , aαβ = GMN ∂αXM
∂βXN
, (347)
where gMN is AdS5 × S5
metric. The equation of motion that follows from
this action has the form
1
2
∂M gKL∂αXK
∂βXL
(a−1
)βα
√
− det a − ∂α[gMN ∂βxN
(a−1
)βα
√
− det a] , = 0
(348)
where aαβ = GMN ∂αXM
∂βXN
and where we used the fact that
√
det a = exp Tr ln a . (349)
Let us now consider the metric in the form
ds2
AdS =
r2
R2
(−dt2
+ dx2
) +
R2
r2
dr2
(350)
Now we impose the static gauge in the form
τ = t , σ = r , x = x(σ) (351)
78
Of course, strictly speaking there is a subtlety with this gauge since the string
has the turning point at r = rmin which means that our gauge is not well
defined. However this is not problem since we can consider only one half of
the string due to the symmetry. Of course, we could also impose the gauge
t = τ , x = σ, r = r(x), but this would lead to slightly more complicated
result and also this gauge fixing cannot be used for the case of the string
stretched from the boundary to the bulk of AdS. With the help of this gauge
we obtain
aττ = −
r2
R2
, aσσ =
R2
r2
+
r2
R2
x′2
, aτσ = 0 , (352)
where x′
≡ ∂σx. Note that with this ansatz the equation of motion for t are
obeyed while the equation of motion for x gives
∂α[gxx∂σX(a−1
)σα
√
− det a] =
∂σ[gxxx′
(a−1
)σσ
√
− det a] = 0 ⇒
r4
R4
x′
1 + r4
R4 x′2
= K
(353)
The question is how to determine the constant K. Recall that the boundary
is at r = ∞ and that the string has the turning point at r = rmin. At this
turning point, by definition
∂σx|σ=rmin
= ∞ (354)
so that we obtain
lim
r→rmin
r4
R4
x′
1 + r4
R4 x′2
=
r2
min
R2
= K (355)
Now we can solve the equation (357) for x and we obtain
x′2
=
R4
r4
1
r4
r4
min
− 1
. (356)
In order to find the string configuration we can solve the equation above so
that we obtain (when we take the boundary condition x = 0, σ = rmin so
79
that the dependence of x on σ = r is given by the integral
x
0
dx =
r
rmin
R
r′
2
dr′
r′
rmin
4
− 1
. (357)
For r → ∞ we demand the quark to be at x = L and hence we obtain
L =
R2
rmin
∞
1
dy
y2 y4 − 1
=
=
R2
r2
min
√
2π3/2
Γ(1
4
)2
.
(358)
Note that this equation gives
rmin ≃
R2
L
. (359)
Further, for r ≫ rmin we find from (357)
x =
R2
rmin
r/rmin
1
dy
y2 y4 − 1
=
=
R2
rmin
∞
1
dy
y2 y4 − 1
−
R2
rmin
∞
r/rmin
dy
y2 y4 − 1
=
= L −
R2
rmin
∞
r/rmin
dy
y4
⇒
x − L ≃
R2
rmin
y−3
|∞
r/rmin
≃ −r−3
(360)
using
∞
1
=
r/rmin
1
+
∞
r/rmin
(361)
and we used the fact that in the second integral r/rmin ≫ 1 and hence y ≫ 1
in the whole integration interval. We see that for large r the string quickly
approaches the point x = L.
80
Now we evaluate the action for this string configuration in order to evaluate
the quark potential. We obtain
S = −
1
2πα′
dτ
∞
rmin
dr
r2
1 +
R4
r4
x′2 =
= −
1
2πα′
dτ
∞
rmin
r
rmin
2
r
rmin
4
− 1
.
(362)
Since
S = − dτV (363)
and since we have two half of the string we find
V =
2
2πα′
rmin
∞
1
dyy2
y4 − 1
(364)
This potential diverges which has natural physical interpretation since in this
case the quark is a probe with infinite heavy mass. On the other hand we
would like to separate the finite contribution by substracting this large mass
contribution. This string configuration is described by the ansatz x′
= 0 so
that the corresponding string action has the form
Sstr = −
1
2πα′
dτ dσ −gtt(grr + gxxx′2) =
= −
1
2πα′
dτ
∞
0
dr = −
1
2πα′
rm dτ[
1
0
dy +
∞
1
dy] =
= −
1
2πα′
rm −
1
2πα′
∞
1
dy ≡ − dτV0
(365)
and hence the finite potential has the form
Vfin = V − V0 =
2
2πα′
rmin[
∞
1
dy[
y2
y4 − 1
− 1] − 1] . (366)
81
We see that this potential is proportional to rmin but rmin but we know that
rmin ≃ R2
L
so that we know that E ≃ 1
L
that has the form of the Coulomb
potential. In fact, explicit calculation gives
Vfin = −
4π2
Γ(1
4
)4
λ1/2
L
. (367)
Let us outline given procedure:
• It is very simple to add probe to the original system.
• The procedure how the Wilson loop is calculated using AdS/CFT correspondence
can be generalized to the more general background as for
example asymptotically AdS spacetimes.
• This probe analysis can be extended to the dynamical strings in given
background.
It is also very interesting to see how the confining phase can be described
using the dual geometry. Recall that confining phase is characterized by linear
dependence of the potential. More explicitly, the AdS/CFT correspondence
can be extended to the backgrounds that are dual to the field theories at finite
temperature. At finite temperature the large N and strong coupling limit of
d = 4, N = 4 SYM is dual to the near-horizon geometry of near extremal D3branes
in Type IIB string theory which is given by a Schwarzschild-anti-de
Sitter Type IIB supergravity compactification
ds2
=
U2
R2
(−Hdt2
+ dx2
) + R2 1
H
dU2
U2
+ R2
dΩ2
5) , (368)
where
H = 1 −
U4
0
U4
, U4
0 =
27
π4
3
g4
eff
µ
N2
, R2
= 4πgN , geff = g2
Y M N .
(369)
where the parameter µ is interpreted as the free energy density on the near
extremal D3-brane and hence
µ ≈
4π2
45
N2
T4
. (370)
82
Note that in the field theory limit α′
→ 0 µ remains finite.
At finite critical temperature T = Tc pure SU(N) theory exhibits a deconfinement
phase transition. It turns out that the relevant order parameter
is the Wilson-Polyakov loop
P(x) =
1
N
TrP exp i
1
T
0
A0(x)dt (371)
Below the critical temperature T < Tc we find that < P >= 0 which means
that QCD confines. Above T > Tc we have < P > is non-zero and takes the
value in ZN which is the center of SU(N).
When we study the field theory at finite temperature we transform from
the Minkowski metric to the Euclidean one. First of all let us consider the
analytic continuation to Euclidean signature tE = it where the metric has
the form
ds2
E = f(r)dt2
E +
dr2
f(r)
+ . . . . (372)
Now we would like to analyse the region near the horizon r ≃ r0, where by
definition f(r0) = 0. Introducing coordinate ˜r as r = r0 + ˜r we can write
f(˜r + r0) = f(r0) + f′
(r0)˜r (373)
so that the metric has the form
ds2
E = f′
(r0)˜rdt2
E +
1
f′(r0)˜r
d˜r2
+ . . . (374)
Finally we introduce the coordindate ρ defined as
ρ = 2
˜r
f′
(r0)
, dρ =
1
f′(r0)˜r
(375)
so that the relevant part of the line element has the form
ds2
= dρ2
+ ρ2
d
f′
(r0)
2
tE
2
. (376)
Now we see that the metric has the same form as a plane in polar coordinates
if f′
(r0)tE/2 has the period 2π. Then the period of tE is equal to
f′
(r0)β/2 = 2π ⇒ β =
4π
|f′(r0)|
. (377)
83
On the other hand the compactified Euclidean time direction means that we
have the system at finite temperature which is equal to
T =
1
β
=
f′
(r0)
4π
. (378)
For example, in case of Schwarzschild black hole we have f(r) = 1 − r0
r
so
that f′
= r0
r2 = 1
r0
and hence
T =
1
4πr0
. (379)
In our case the relevant part of the metric is
ds2
=
U2
R2
(Hdt2
ER2 1
H
dU2
U2
= |U =
R2
Y
| =
R2
Y 2
[H(Y )dt2
E +
1
H(Y )
dY 2
] (380)
where the horizon is localized at Y0 = R2
U0
and the function H(Y ) is
H(Y ) = 1 −
U4
0
R8
Y 4
, H′
(Y ) = −4
U4
0
R8
U3
, |H′
(Y0)| = 4
U0
R2
(381)
and hence
T =
H′
(Y0)
4π
=
U0
πR2
. (382)
Let us now consider string as a probe in given background
S = −
1
2πα′
dτ
√
det h , h = gMN ∂αXM
∂βXN
. (383)
We again impose the static gauge
tE = τ , U = σ , x = x(U) (384)
so that the equation of motion for x gives
∂u[
√
gttgxxx′
guu + gxxx′2
] = 0 ⇒
x′2
=
C2
guu
gttguu − C2
=
x′2
=
R4
C2
U4H(U4
R4 H − C2)
(385)
84
As in previous case we determine the integration constant C as the point
where x′
= ∞ that gives
√
gttgxx|(Umin) = C ⇒ C2
=
1
R4
(U4
min − U4
0 ) . (386)
Then we obtain
x′
= R2
1 −
U4
0
U4
min
1
(U4 − U4
0 )( U4
U4
min
− 1)
(387)
and hence we obtain
x =
R2
Umin
√
ϵ
U/Umin
1
dy
(y4 − 1)(y4 − 1 + ϵ)
,
(388)
where
ϵ = 1 −
U4
0
U4
min
. (389)
We can again find relation between Umin and L when as
L =
R2
Umin
√
ϵ
∞
1
dy
(y4 − 1)(y2 − 1 + ϵ)
. (390)
We calculate potential in the same way as in case of zero energy however we
have to take into account that now tE coordinate is compactified with period
of β. Further, we have to substract the contributions of two heavy strings
which however do not end at U = 0 but at the horizon U = U0. In other
words, this contribution is equal to
S =
1
2πα′
dτ
∞
U0
dU (391)
and hence the finite potential is equal to
V =
2
2πα′
∞
Umin
dU
U4 − U4
0
U4 − U4
min
−
2
2πα′
Umin
U0
dU −
2
2πα′
∞
Umin
dU =
=
2
2πα′
Umin
∞
1
y4 − 1 + ϵ
y4 − 1
− 1 − Umin + U0
(392)
85
This is the potential between quark and antiquark at finite temperature that
is function of L and T. However in order to find this we should eliminate
Umin that appears in given expression and that can be determined from the
equation above. This can be done only numerically so that it is instructive
to analyze the behavior if given potential at different limits. First of all the
region ϵ ≈ 1 corresponds to U0 ≪ 1 which means that T ≪ 1 and hence
we are at the low temperature region. At the same way the case ϵ ≈ 0
corresponds to the high temperature region TL ≫ 1. For small temperature
we can determine the dependence of L on rmin as in the free zero temperature
case r2
min ∼ R2
L
. Then we find that the lowest correction to the potential is
proportional to (TL)4
, where we have to take into account that the theory is
in conformal phase so that the temperature has to appear in the combination
TL. In principle we can calculate this energy from the result given above
however it is instructive to proceed in slightly different way. Let us consider
the pure AdS spacetime with the metric
ds2
=
r
R
2
(−dt2
+ dx2
+ . . . ) (393)
where the line element comes with the factor r2
. We also see that we measure
the gauge theory time and distance using t and x which differs from the
proper time and distance of the AdS space-time.
Let us denote the quark-antiquark separation as △x = 2L. The quarkantiquark
pair is represented by a string that connects the pair. As we know
string has tension which implies that the string wants to minimalize its length
so that naively we would say that the state with the lowest energy is the
straight string at r = ∞, where we recall that the boundary is at r = ∞.
However this is not true since we have to take into account the fact that the
metric comes with the factor r2
. It turns out that state with less eneergy
corresponds to the string that goes inside the AdS space-time (r ̸= ∞) since
the line element has the factor r2
so that the string is shorther at the origin
where the line element is △xr2
. In fact, we can roughly divide the string that
connects two quarks into two parts. The first one where the string extends
vertically and the part where the string extend horizontally. In other words
we can approximate the string by rectangular string where only the horizontal
string contributes to the quark potential since this part of the string changes
as we vary the separation of the quarks 2L, while the vertical part of the
string does not change very rapidly and it simply corresponds to the quark
86
mass. As we also showed above the turning point occurs at
rmax ∼
R2
L
. (394)
Now from the line element we find that the length of the horizontal string is
r
R
2L so that the string energy is approximately
E(r) ∼
r
R
L . (395)
It is also important to stress that given energy does not correspond to the
gauge theory energy which follows from the fact that the time-like direction
comes with the prefactor r2
too. In fact, the gauge theory time is the coordinate
time t since the proper time τr defined as −dτ2
= ds2
is related to the
gauge theory time by
τr =
r
R
t (396)
so that we have an inverse relation between the proper energy and the gauge
theory energy as
Et =
r
R
E(r) . (397)
Then the potential in the gauge theory is given as
Et =
rm
R
E(r) =
rm
R
2
L ∼
R2
L
. (398)
We see that in this picture we derived the Coulomb potential E = 1
L
and
not the confining potential E ∼ L. We see that even if the quark-antiquark
are connected by string it still corresponds to the unconfinning potential.
We also see that this potential is proportional to R2
, which, according to
AdS/CFT relations, means that R2
∼ λ1/2
which means that it represents
non-perturbative effect. In fact, this potential can be also calculated nonperturbatively
from the field theory point of view and it behaves as λ1/2
at
strong coupling.
The previous discussion can be generalized to the more general form of
the metric where
ds2
= g00dt2
+ gxxdx2
+ . . . . (399)
Now the proper time is related to the coordinate time through the relation
τ =
√
−g00t and hence the gauge energy is related to the proper energy of
87
the string as
Et =
√
−g00|rmax E(r) =
1
2πα′
√
−g00gxx|rmax L (400)
where the metric is again evaluated at rmin.
Confining Phase
We would like to see whether confining phase can be described using
AdS/CFT correspondence as well. As we know the pure AdS space-time
corresponds to the N = 4 SYM which is not QCD. Further, N = 4 SYM
is scale invariant and hence the confining phase does not exist even at zero
temperature. Then in order to describe theory that has similar properties ad
QCD we have to modify AdS geometry in some way.
In fact, there are number of models that deform the AdS space-time
with corresponding modification of the dual theory. Let us however consider
following model. The AdS spacetime extends from r = ∞ to r = 0 but let us
presume that we cut off the AdS spacetime at r = rc. Let us also presume
that confinement occurs at some low energy phase Λ. As we argued above r−
coordinate has the interpretation as the gauge theory energy scale so that
the confinement means that the AdS space-time is modofied deep insider
AdS. It is also clear that there is no modification of the potential when the
string is far enough from the cutoff r = rc. In this case we find the pure AdS
space-time. On the other hand if the separation of the quarks is large enough
we find new effect. To see this note that in the AdS space-time the turning
point behaves as rmax ∼ 1/L. But in the cutoff AdS space-time the string
reaches r = rc for large enough L. Since this is the end of the space-time
string cannot go further and hence the energy of the horizontal string is
Et =
√
−g00|r=rc L ∼ r2
c L (401)
that is really the energy of the confining potential.
The same arguments can be used in case of non-zero temperature AdS
space-time which is known as the plasma phase. As we argued above the dual
geometry is AdS black hole with the horizon at r = r0. Again, if the string
is far enough from the black hole the geometry is approximately the AdS
space-time with corresponding Coulomb potential. However when the string
reaches the horizon the new effect occurs. As we know the line element is
ds2
= −
r
L
2
(1 −
r0
r
4
)dt2
+ . . . (402)
88
so that g00 = 0 at horizon r = r0. Then we however find that Et = 0
which means that the horizontal string has no contribution to the energy.
In other words there is no force where the quark separation is large enough.
This is known as the Debye screening in AdS/CFT. In other words we have
following picture. When the separation of the string is small we have standard
Coulomb like behavior while for large separations of the quarks we find that
these quarks become free due to the screening by the thermal bath.
Despite of this fact we would like to see how we can find simple model
that can describe the confinement. Let us consider AdS black hole with line
element in the form
ds2
=
r
R
2
(−hdt2
+ dx2
+ dy2
+ dz2
) + R2 dr2
hr2
, (403)
where
h = 1 −
r0
r
4
. (404)
Let us now compactify z− direction so that 0 ≤ z < l so that we have solution
with asymptotic geometry R1,2
× S1
. However there is another solution with
this asymptotic geometry. To see this we perform double Wick rotation
z′
= it , z = it′
(405)
of the black hole and we obtain the metric
ds2
5 =
r
R
2
(−dt′2
+ dx2
+ dy2
+ hdz′2
) + R2 dr2
hr2
(406)
that has the same asymptotic structure R1,2
× S1
. This geometry is known
as AdS soliton. Clearly they are the same as Euclidean geometries but they
have different Lorentzian interpretation. The AdS solition is not a black hole
since due to the factor h in front of dz′2
we should say that the space-time
ends at r = r0. Then from the analysis performed above we obtain the energy
of the quark-antiquark potential in the form
Et =
√
−gt′t′ gxxL =
r0
R
2
L (407)
that is confining potential.
89
6 Thermal N = 4 SYM and AdS/CFT Corre-
spondence
In this section we discuss aspects of the AdS/CFT correspondence at finite
temperature. As we show in previous lecture the thermal field theory could
be dual to the background with thermal properties. In this lecture we will be
more precise with this statement.
We start with the Schwarzschild-AdS black hole (SAdS). It is possible to
consider either Schwarzschild black hole in AdS with spherical horizon or we
can consider AdS black hole with planar horizon known also as a AdS black
branes.
The SAdS5 black hole is a solution of the Einstein equation with a negative
cosmological spatime with the metric
ds2
5 = −
r
L
2
h(r)dt2
+
dr2
r
L
2
h(r)
+
r
L
2
(dx2
+ dy2
+ dz2
) ,
h(r) = 1 −
r
r0
4
.
(408)
The horizon is located at r = r0. The coordinates (x, y, z) represent R3
coordinates and we see that the horizon extends in (x, y, z)−directions. Note
that this is different from the case of Schwarzschild black hole where this line
element has the form r2
dΩ2
. Note that the presence of the black hole in the
AdS space-time means that now SAdS5 is not the space-time with constant
curvature as opposite to the case of AdS5 with curvature singularity at r = 0.
As we know AdS space-time is invariant under scaling
xµ
→ axµ
, r →
r
a
. (409)
Now under this scaling the relevant part of the metric element of SAdS scale
as
ds2
5 → −
r
L
2
1 −
ar0
r
4
dt2
(410)
which means that we derive the same SAdS when we scale the horizon as
r0 → 1
a
r0. In other words black holes with different horizon radii are all
equivalent. Further, we show previously that the temperature of the black
90
hole is T ∝ r0so that we see that we can change the temperature by scaling.
In other words all temperatures are equivalent and the physics is the same,
of course with exception of zero temperature. This also means that there is
no characteristic temperature which could correspond to a phase translation.
This is a consequence of the fact that N = 4 SYM is scale invariant and there
is no dimensionful quantity except the temperature. Of course, this is valid
for the SYM in flat space-time but the situation could be different when we
consider theory on S3
.
In other words we see that the black hole metric in AdS is invariant under
the scaling
xµ
→ axµ
, , r →
1
a
r , r0 →
1
a
r0 . (411)
6.1 Thermodynamic quantities of AdS black hole
Here we determine thermodynamic quantities of the SAdS5 black hole that,
in the AdS/CFT correspondence they are interpreted as thermodynamic
quantities of dual N = 4 SYM at strong coupling. Recall that we have
the relation between SYM and the bulk theory:
N2
=
π
2
R3
G5
, λ =
R
ls
4
. (412)
As we know the temperature of the black hole is given by the relation
T =
f′
(r0)
4π
=
=
2r
R2
1
4π
(1 − (
r0
r
)4
+
1
4πR2
(
r
R
)2 r4
0
r5
=
=
r0
4πR2
.
(413)
Now since the black hole is infinite extended in all three spatial dimensions
its entropy is divergent and hence it is more appropriate to use the entropy
density s. Let us introduce infrared cutoff so that we impose upper limit on
spatial coordinates 0 ≤ x, y, z ≤ Lx, Ly, Lz, so that the gauge theory volume
is V3 = LxLyLz. Now it is known from the physics of the black hole that
the entropy of the black hole is proportional to the area of the horizon. Note
91
that the spatial section of the black hole space-time in AdS is equal to (at
constant r)
dσ2
=
r
R
2
(dx2
+ dy2
+ dz2
) (414)
so that area of the horizon is (evaluated at r = r0)
A =
hor
d3
σ =
r0
R
2
LxLyLz (415)
when we introduced an infrared cutoff. Then the entropy of the black hole is
equal to
S =
A
4G5
=
1
4G5
r0
R
3
V3 (416)
and hence the entropy density is equal to
s =
S
V3
=
1
4G5
r0
R
3
(417)
If we now express r0 using T and further using R3
= 2G5
π
N2
we obtain
s =
π2
2
N2
T3
. (418)
We will discuss the physical interpretation of this result in the dual theory
below. Finally the rest of the thermodynamical quantities can be determined
using thermodynamic relations. From the first law
dϵ = Tds (419)
we obtain
ds =
3
2
π2
N2
T2
dT , dϵ =
3
2
π2
N2
T3
dT ⇒ ϵ =
3
8
π2
N2
T4
. (420)
Finally from the Euler relation
ϵ = Ts − P (421)
we obtain the pressure P as
P =
1
8
π2
N2
T4
(422)
92
that implies an important relation between the pressure and the energy
density
P =
1
3
ϵ (423)
It is very interesting to discuss the dependence of the energy on temperature.
The dependence ϵ ∝ T4
is manifestation of the Stefan-Boltzman law which
follows from the fact that N = 4 SYM is scale invariant and there is no
dimensionfull quantity except of temperature. Then since the energy density
has a quantity of the mass dimension [ϵ] = M−4
it is clear that it has to be
proportional to T4
. However clearly this dimensional analysis cannot determine
the numerical factor so that using this black hole calculation we derive
non-trivial result that should be compared with the calculation performed in
case of the free gas.
We should also stress that the fact that the black hole obeys the StefanBoltzmann
law is rather non-trivial. For example, in case of the five-dimensional
Schwarzschild black hole we find M ∝ 1
T2 and hence the Stefan-Boltzmann
law is not reproduced.
Further, the Schwarzschild black hole has a negative heat capacity
C =
dM
dT
∝ −
1
T3
(424)
and hence there is no stable equilibrium. This is well known fact that Schwarzschild
black hole in the flat assymptotic spacetime emits the Hawking
radiation which means that the black hole loses its mass due to the radiation.
This also leads to the growing of the temperature of the black hole and the
black hole does not reach an equilibrium.
However in case of SAdS the situation is very different. Let us be more
explicit and write
(5 − dimensional) Schwarzschild : M ≃
1
G5T2
,
(5 − dimensional) Schwarzschild − AdS : ϵ =
(πR)3
4G5
T4
.
(425)
We see that in case of Schwarzschild black hole we have two dimensionful
quantities G5 and T. On the other hand in case of AdS black hole there
is another dimensionful quantity which is the radius of AdS R that we can
93
combine with G5 to give a dimensionless quantity N. Clearly this cannot be
done in case of Schwarzschild black hole and hence Stefan-Boltzmann law
cannot appear in this case.
N2
dependence The entropy density is proportional to N2
. The fact
that the entropy scales with N2
implies that the theory is in unconfinement
phase. More precisely, the entropy counts the number of degrees of freedom.
SU(N) theory is gauge theory with N × N hermitean matrices so that we
have N2
degrees of freedom approximately. In unconfined phase these degrees
of freedom contribute to the entropy. In the confined phase only SU(N)
gauge singlets contribute to the entropy and consequently the entropy is not
proportional to N2
.
Traceless stress energy tensor As we saw the energy density and the
pressure satisfies the relation
ϵ = 3P . (426)
that implies that the energy momentum tensor is traceless. In fact, this is
also consequence of the scale invariance of N = 4 SYM.
Free gas result It is instructive to compare these results with the free gas
calculation. Let us consider the partition function of free particles. Denote
the energy level as ωi and ni as the number of particles that occupy the
energy level ωi. The total energy of the microstate is determined by numbers
n1, n2, . . . and is given by
E(n1, n2, . . . ) =
i
niωi . (427)
The partition function is given by
Z = e−βE(n1,n2,... )
=
n1 n2
. . . e−β(n1ω1+n2ω2+... )
. (428)
where β = 1/T. For bosons ni takes the value from 0 to ∞ so that
ZB =
∞
n1=0
e−βn1ω1
× · · · =
∞
i=1
1
1 − e−βωi
(429)
using
∞
n=1
e−βnω
= 1 + e−βω
+ (e−βω
)2
+ · · · =
1
1 − e−βω
. (430)
94
Then the thermodynamic quantities are evaluated as
ln ZB = −
∞
i=1
ln(1 − e−βωi
) →
→ −V
d3
q
(2π)3
ln(1 − e−βω
) =
−
V
(2π)3
4π
∞
0
dqq2
ln(1 − e−βq
) =
−
V
2π2
1
β3
∞
0
dxx2
ln(1 − e−x
) =
= V
π2
90β3
.
(431)
In previous calculation we replaced the sum by integral and used the dispersion
relation ω = |q| that is valid for massless particles. We also used the
relation ∞
0
dxx2
ln(1 − e−x
) = −
π2
45
(432)
Now, with the help of the partition function we determine corresponding
thermodynamic quantities
ϵB = −
1
V
∂ ln ZB
∂β
==
π2
30β4
,
sB =
1
V
ln ZB + βϵB =
2π2
45β3
.
(433)
95
In case of fermions we hav that ni takes the value either 0 or 1 so that
ZF = (1 + e−βω1
)× =
∞
i=1
(1 + e−βωi
) ,
ln ZF =
∞
i=1
ln(1 + e−βω
) →
→ V
d3
q
(2π)3
ln(1 + e−βω
) =
=
V
(2π)3
4π
∞
0
dqq2
ln(1 + e−βq
) =
= V
7π2
720β3
,
(434)
where we used an integral
∞
0
dxx2
ln(1 + e−x
) =
7π4
360
. (435)
Using this result we find
ln ZF =
7
8
ln ZB (436)
Using these results we can determine the entropy of the SYM at the free gas
approximation. Firstly, using the form of the entropy for the bosonic degree
of freedom we find the photon contribution to the entropy density
sphoton =
2π2
45
T3
× 2 , (437)
where the last factor 2 comes from the polarization of photons. However this
is true for the photon gas only but in case of N = 4 SYM we have more
degress of freedom. In fact, the number of the bosonic degrees of freedom is
equal to
Nbos = (2 + 6) × (N2
− 1) , (438)
where the factor 2 corresponds two states of the gauge field and the factor 6
corresponds to the number of the scalar fields. Finally the factor (N2
− 1) is
the number of degrees of freedom for the field in the adjoint representation
96
of SU(N). In case of fermions we find that their number is the same as the
number of bosons due to the supersymmetry:
NF = NB . (439)
On the other hand we know that fermions contribute 7/8 of the bosons
entropy so that
Ndof = Nbos +
7
8
Nfer = (2 + 6) ×
15
8
(N2
− 1) = 15(N2
− 1) . (440)
Consequently the entropy density in the free gas is equal to
sfree =
2π2
45
Ndof T3
=
2π2
3
(N2
− 1)T3
≃
2π2
3
N2
T3
. (441)
We see that this expression has the same functional form as in the case of
the entropy calculated using the black hole but the coeficients are different.
In fact, they are related by the formula
sBH =
3
4
sfree . (442)
Now we see that there is famous 3/4 discrepancy that has following origin.
The black hole computation corresponds to the strong coupling result that is
different from the free gas result. In other words AdS/CFT predicts that the
entropy of N = 4 SYM at strong coupling becomes 3/4 of the free gas result.
It is very difficult to check this prediction but it was shown that similar
behavior can be obtained in lattice simulations.
6.2 The AdS black hole with spherical horizon
The AdS black hole with spherical horizon is given by
ds2
5 = −
r2
R2
+ 1 −
r4
0
R2r2
dt2
+
dr2
r2
R2 + 1 −
r4
0
R2r2
+ r2
dΩ2
3 , (443)
and we see that for r0 = 0 the metric reduces to the metric in the global
coordinates. The horizon is located at r = r+ where
r2
+
R2
+ 1 −
r4
0
R2r2
+
= 0 ⇒
r4
+ + R2
r2
+ − r4
0 = 0 ⇒
r2
+ =
1
2
( R4 + 4r4
0 − R2
) .
(444)
97
It turns out that spherical black holes with different horizon radii are not
equivalent. This has very important consequence for the phase structure of
the dual SYM theory where, however we have to presume that it is now
defined on the compact space. Such a theory has very rich physics as phase
transition.
7 Phase transition in AdS/CFT correspon-
dence
It is well known that when we change parameters of thermodynamic system,
as for example temperature, this system could perform a transition to a
macroscopically different state that is more stable. This phenomena is known
as phase transition. It turns out that in a phase transition a thermodynamic
potential such as free energy become non-analytic. More precisely, the nth
order phase transition is the transition when the analyticity is broken in the
n-th derivative of the thermodynamic potential. Explicitly
• First order phase transitino when F is continuous and F′
is disconinuous.
Note that F′
means the derivative with resepct to any independent
variable of thermodynamic potential.
• Second-order phase transition F and F′
are continuous but F′′
is dis-
continuous.
Very nice example of such a system is a ferromagnet that has sponaneous
magnetization M below te transition temperatur Tc and the magnetization
vanishes at T = Tc. Macroscopic variable such as M that characterizes two
phases is called the order parameter.
We introduce the free energy F = F(T, M) for such a system. However it
turns out that it is useful to use magnetic field as the control parameter and
use the another thermodynamic potential which is Gibbs energy defined as
Legendre transformation of the free energy
G(T, H) = F − MH. (445)
so that
M = −
∂G
∂H
. (446)
98
It turns out that in case of ferromagnet M is coninuous at the transition point
that is defined as the point H = 0. This fact implies that given transition is
the second-order phase transition.
After this brief introduction to the process of the phase transition we will
study phase transitions in AdS/CFT case. We know that in N = 4 there
is no characteristic temperature and hence there is no phase transition in
given theory. However it is interesting to consider theories and black holes
that have properties of the phase transitions from following reasons:
• There are gauge theories with rich phase structure, as for example
QCD. These phenomena can be nicely described using AdS/CFT corre-
spondence.
• This is even more useful in case of the strongly coupled systems where
the dual gravity description is very useful tool.
It is clear that it is very useful to study the phase space phenomena in
the AdS space and try to find their interpretations in the dual theory. We
start with the calculation of the black hole partition function that, in the
saddle-point approximation is given as the evaluation of the action on the
classical solution. Then we can interpret the exponential of given function as
the partition function of the dual gauge theory
Zgauge ≃ e−SE
onshell , (447)
where SE
onshell is on-shell action that we derive when we insert the solution
of the equation of motion into it. However it is important to stress that it is
possible to have two solutions corresponding to given boundary conditions.
Then we have to take these solutions so that
Zgauge ≃ e−SE
onshell1 + e−SE
onshell2 (448)
where Sonshell1,2 are actions evaluated on the solutions 1, 2 respectively. Clearly
dominant contributions give the solution with lower free energy.
Let us consider compactified N = 4 SYM compactified on S1
with periodicity
l. The N = 4 SYM on R4
is scale invariant and there is no dimensionful
parameter except of temperature. Further, as we know, we can always change
the temperature by scaling and all temperatures are equivalent. However
introducing the scale l has crucial consequence for given theory. It is not
possible to change the temperature by a scaling and it is paremeterized by
99
a dimensionless parameter Tl. It turns out that given theory possesses the
first order transition at Tl = 1.
As we showed previously there are two possible dual geometries that
approach R1,2
× S1
asymptotically. The first one is the SAdS5 black hole
ds2
5 =
r
R
2
(−hdt2
+ dx2
+ dy2
+ dz2
) + R2 dr2
hr2
,
h = 1 −
r0
r
4
, 0 ≤ z < l .
(449)
The second solution is AdS soliton
ds2
5 =
r
R
2
(−dt′2
+ dx2
+ dy2
+ hdz′2
) + R2 dr2
hr2
,
l =
πR2
r0
.
(450)
that, as we know, is derived from the black hole by the double Wick rotation
z′
= it , z = it′
.
The SAdS black hole describes the plasma phase wheras AdS soliton
describes the confining phase which we also saw when we analyzed Wilson
lines in these backgrounds. Explicitly, we found that Wilson loop describes
linear potential in the background of AdS soliton. Further, AdS soliton does
not have entropy due to the fact that it is not black hole.
At high temperature the AdS soliton undergoes a first-order phase transition
to the SAdS black hole. This phase transition then describes a confinement/deconfinement
transition in the dual gauge theory, where however
we should stress that this transition does not correspond to the usual confinement
in QCD from following reason. We still work with the background
that is dual N = 4 SYM which however is defined on compact space.
To see how the phase transition comes from let us evaluate the free energy
of these space-times. We start with the partition function of AdS black hole.
The gravitational action has following form
SE = Sbulk + SGH + SCT , (451)
where Sbulk, SGH and SCT are bulk action, the Gibbons-Hawking action and
the counterterm action respectively. The bulk action is the five dimensional
100
action that has the form
S = −
1
16πG5
d5
x
√
−g(R − 2Λ) ,
2Λ = −
12
L2
.
(452)
The Gibbons-Hawking action is added to the EH action in order to have a
well-defined variational problem
S = −
2
16πG5
d4
x
√
γK , (453)
which is a surface term evaluated at u = 0. Further,γ is 4−dimensional metric
at the surface surface and K is the trace of the extrinsic curvature.
The surface term does not affect the equation of motion but its presence
determines the value of the on-shell action. In fact, in some cases there is no
bulk contribution to the cases so that the on-shell action is entirely determined
by Gibbons-Hawking action.
Now we discuss these two actions evaluated on the classical solution corresponding
o the SAdS. Then we will discuss the meaning of the counterterm
action. Let us consider following background
ds2
5 =
r
L
2
(hdt2
E + dx2
+ dy2
+ dz2
) + L2 dr2
hr2
, h = 1 −
r0
r
4
,
= (454)
Let us introduce coordinate
u =
r0
r
, h = 1 − u4
, dr
= −
r0
u2
du (455)
so that the line element takes the form
ds2
5 =
r0
L
2 1
u2
(hdt2
E + dx2
+ dy2
+ dz2
) + R2 du2
hu2
, h = 1 − u4
. (456)
In the coordinate r the black hole horizon is located at r = r0 and AdS
boundary is located at r = ∞. In the coordinate u the horizon is located at
u = 1 and the AdS boundary is located at u = 0.
101
Let us now return to the bulk action. The variation of the action with
respect to gMN implies following equation of motion
RMN −
1
2
g)MN R + ΛgMN = 0 . (457)
Taking the trace of given equation gives
R −
5
2
R + 5Λ = 0 ⇒ R =
10
3
Λ (458)
that for given Λ gives
R = −
20
L2
. (459)
Then on-shell action is
Sbulk = −
1
16πG5
d5
x
√
g
4
3
Λ =
1
2πG5L2
d5
x
√
g . (460)
For the black hole spacetime we have
g =
r6
L6
=
r8
0
L6u8
(461)
and hence we find
Sbulk =
1
2πG5L2
β
0
dt d3
x
1
0
du
r4
0
L3u4
=
= lim
u→0
βV3r4
0
8πGL5
1
u4
− 1 |u=0 ≡
βV3
16πG5
r4
0
L4
ˆSbulk ,
(462)
where V3 is the volume of the spatial section. The bulk action diverges as
u → 0 since it is proportional to the space-time volume.
Let us now perform the calculation in case of Gibbons-Hawking action.
Let us write the line element for the SAdS black hole in the form
ds2
5 = guudu2
+ γµνdxµ
dxν
. (463)
We see that the 4−dimensional metric defined on the surface u = const is
given by γµν. Clearly the normal vector to this surface has the only one
non-zero component nu
. If we require that its norm is one we find
nu
nu
guu = 1 ⇒ nµ
= −
1
√
guu
, (464)
102
where we chosen the sign − in front of the square root since the normal
nµ
points in the direction od decreasing u. For diagonal metric the extrinsic
curvature is given by
Kµν =
1
2
nu
∂uγµν . (465)
Its trace is defined as
K = γµν
Kµν =
1
2
nu
∂uγµνγµν
. (466)
Since det γ = exp Tr ln γ we obtain
∂u
√
g =
1
2
∂uγµνγνµ√
g (467)
Then comparing there two contributions we obtain
K = nµ 1
√
γ
∂u
√
γ . (468)
For AdS black hole we have
guu =
L
√
hu
, γ =
r0
L
4 1
u4
√
h (469)
and hence we obtain
lim
u→0
√
γK =
8
u4
− 4 (470)
and hence we find that SGH is equal to
SGH =
1
16πG
βr4
0V3
L4
[−
8
u4
+ 4]|u=0 . (471)
We see that the bulk action together with GH action diverges in the limit
u → 0. For that reason we add another surface term to the action known as
counterterm action.
In order to explain this notation note that in AdS/CFT correspondence
this divergence is interpreted as the ultraviolet divergence of the dual field
theory. The standard procedure how to deal with divergences in QFT is to
add a finite number of local counterterms to we add these terms to the gravitational
action in order to have finite partition function. This procedure is
known as the holographic renormalization. Explicitly, we choose the counterterm
action in the following way
103
• It is written in terms of boundary metric γµν and the quantities calculated
from it, as for example Ricci scalar R.
• It consists only a finite number of terms.
• The coeficients of these terms are chosen n order to cancel divergences.
In case of 5− dimensional space-time the counterterm action is given b
SCT =
1
16πG5
d4
x
√
γ
6
L
+
L
2
R
(472)
so that for the black hole solution we find
SCT =
1
16πG5
βV3
L4
6
√
h
u4
|u=0 →
1
16πG5
βV3
L4
6
u4
− 3 . (473)
In summary, we have following result
Sbulk =
βV3
16πG5
r4
0
L4
2
u4
− 2 |u=0 ,
SGH =
βV3
16πG5
r4
0
L4
−
8
u4
+ 3 |u=0 ,
SCF =
βV3
16πG5
r4
0
L4
6
u4
− 3 |u=0 .
(474)
In summary we obtain
SE(on − shell) = Sbulk + SGH + SCT = −
βV3
16πG5
r4
0
L5
. (475)
The on-shell partition function is related to the partition function Z and the
free energy F as
Z = e−SE(on−shell)
, F = −
1
β
ln Z ==
1
β
SE(on − shell) . (476)
so that the free energy of the black hole has the form
FBH = −
V3
16πG5
r4
0
L5
= −
(πLT)4
16πG5L
V3 = −
1
8
π2
N2
c T4
V3 , (477)
104
using T = r0
πL2 . In case of the AdS soliton we can proceed in the same way. In
fact, the AdS soliton has the same Euclidean geometry so the free energy for
the AdS soliton takes the same form with the important difference that now
r0 is not related to the temperature T but it is related to the S1
periodicity
l
Fsoliton = −
V3
16πG5
r4
0
L5
= −
V3L3
16πG5
π4
l4
. (478)
using the relation between periodicity l and r0
l =
πL2
r0
. (479)
Then the free energy difference is equal to
△F = FBH − Fsoliton = −
V3L3
16πG5
π4
(T4
−
1
l4
)
(480)
and hence we have
• At low temperature T < 1/l where Fsoliton < FBH the stable solution
is AdS soliton that describes confining phase.
• At high temperature T > 1/l we have FBH < Fsoliton and the stable
solution is the black hole that describes unconfining phase.
We also see one important point. Black hole has entropy while the AdS
soliton doesn’t we see that entropy is discontinuous at the point Tl = 1. Since
S = −∂T F is discontinous we find that this is first-order phaseIntransition.
The phase transition that was described above is generally called the
Hawking-Page transition. In fact, original Hawking-Page transition uses SAdS
with spherical horizon where the line element has the form
ds2
5 = −
r2
L2
+ 1 −
r4
0
L2r2
dt2
+
dr2
r2
L2 + 1 −
r4
0
L2r2
+ r2
dΩ2
3 . (481)
In the limit r → ∞ we find that the line element has the form
ds2
=
r2
L2
(−dt2
+ L2
dΩ2
3) (482)
105
so that the dual gauge theory is N = 4 SYM on S3
with the radius L.
Since there is a characteristic scale L we see that the theory is parameterized
by a dimensionless parameter TL and the theory possesses the first
order transition. The horizon is located at r = r+ that is solution of the
equation
r2
+
L2
+ 1 −
r4
0
L2r2
+
= 0 → r4
0 = r4
+ + L2
r2
+ . (483)
and the temperature is given by
T =
2r2
+ + L2
2πr+L2
. (484)
For given value of T we have two values of r+.
8 Advanced Topics: Gravity And Entaglement
Let us outline basic idea of AdS/CFT correspondence. The basic idea is that
certain non-gravitational system on fixed space-time is equivalent to quantum
gravitational theories whose states correspond to different spacetimes with
given asymptotic behavior. In other words, each sat in the non-gravitational
theory corresponds to the state in the dual gravitational theory and each
observable in non-gravitational theory corresponds to some observable in
the gravitatonal theory. Of course, the interpretation of these states and
observables can be completely different on both sides of the correspondence.
However there is one quantity that has the same interpretation wich is the
energy: Energy of a CFT state corresponds to the total energy of the spacetime
which is measured by the ADM mass. As we know from the previous
parts of this lecture such a classical example of the non-gravitational system
is a conformal field theory defined on some fixed space-time B where B is
either Minkowski space Rd−1,1
or a sphere with a time direction Sd−1
× R.
If we claim that CFT is holographic theory then there is a correspondence
between various quantum states of the CFT and states in the dual theory,
where these various states can describe different spacetime geometries. But
what is important is that the assymptotic behavior of these geometries is the
same. For example, for CFT on Minkowski space Rd−1,1
the vacuum state of
the theory corresponds to (d + 1)−dimensional AdS with a Minkowsi space
106
boundary that can be described by the metric
ds2
=
L2
z2
(dz2
+ dxµdxµ
) , (485)
where the boundary is localized at z = 0. Then more general excited states
of the CFT are dual to different geometries that approach this geometry as
z → 0. This requirement implies that these geometries can be described as
ds2
=
L2
z
2
(dx2
+ Γµν(x, z)dxµ
dxν
) , (486)
where for small z we have
Γµν(x, z) = ηµν + O(zd
) , (487)
that implies that limz→0 Γµν(x, z) = ηµν and really the space-time asymptotically
approaches AdS space-time. For CFT defined on a more general spacetime
B the definition is the same with the exception that ηµν in (487) is
replaced with metric on B.
Let us now discuss various states of CFT and their relation to the dual
geometries. For small perturbations to the vacuum state of CFT the corresponding
geometries are represented by small perturbations to AdS and this
is well known from the previous parts of this lecture. On the other hand
high-energy excited states have corresponding dual space-times with different
geometries and even topologies. One such a famous example is the case
of the thermal state of the CFT. As we know for the Minkowski-space CFT
the geometry dual to a thermal state is the planar AdS black hole. The case
when the CFT is defined on the sphere is more interesting and was analyzed
in previous part of the lecture.
One can ask the question whether all CFT are holographic. It turns out
that we do not have set of necessary and sufficient conditions which tell us
whether any particular CFT is holographic. We believe that in order to have
gravity that looks like Einstein gravity coupled to matter requires that we
have a CFT with a large number of degrees of freedom and strong coupling.
8.1 Entropy and Geometry
Now we return to the problem of the black holes in AdS space-time. As we
known AdS/CFT correspondence a Schwarzschild black hole in AdS spacetime
can be identified with a high-energy therma state of the corresponding
107
CFT on a sphere. Such a theory has a discrete spectrum of energy eingenvalues
|Ei⟩ and the thermal state corresponds to the corresponding canonical
ensemble. However we can say more since we know that any CFT state is
dual to asymptotically AdS space-time with or without a black hole. Further,
for any subsystem of the CFT, entropy of the subsystem corresponds to a
area of a particular surface in the corresponding space-time. In order to understand
this relation better it is useful to review description of quantum
subsystems.
8.2 Quantum Subsystems and Entaglement
Let us have a quantum system and imagine that it has its subsystem A. Let
us denote the complement of this subsystem as ¯A. Then we can decompose
the Hilbert space as
H = HA ⊗ H ¯A (488)
In other words the Hilbert space is given as a tensor product of Hilbert spaces
of corresponding subsystems.
Let us now consider a state |Ψ⟩ ∈ H of the full system. Then we can ask
the question, what is the state of the subsystem A? We could presume that
it is possible to find some state ψA
∈ HA that carries all information about
the subsystem. We could demand that for every operator OA that acts on
HA alone
ψA
OA ψA
= ⟨Ψ| OA ⊗ IB |Ψ⟩ , (489)
where IB is identity element acting on the subspace HB. However it turns
out that for general |Ψ⟩ there does not exist such a state ψA
. In other
words, the state of the subsystem is not described by any single state in the
Hilbert space HA. In fact, previous formula is valid when we can express |Ψ⟩
as |Ψ⟩ = ψA
ψB
since then
⟨Ψ| OA ⊗ I |Ψ⟩ = ψB
ψA
(OA ⊗ IB) ψA
ψB
=
= ⟨ψ|A OA |ψ⟩A ⟨ψB| OB |ψ⟩B = braψAOA |ψ⟩A .
(490)
8.2.1 Ensembles of quantum states
The right way how to describe given subsystem is to use an idea of an Ensemble
of states that is known as a Mixed state which is opposite to the pure
108
state which is single Hilbert space vector. Explicitly, we consider a collection
{(ψi, pi)} of orthogonal states and corresponding probabilities. We define expectation
value of an operator O in the ensemble to be the average of the
expectation values for the individual states weighted by the probabilities
⟨O⟩ensemble =
i
pi ⟨ψi| O |ψi⟩ . (491)
For system consisting from more parts with state|Ψ⟩ or more generally an
ensamble of states for the full system we can always find ensamble of states
for subsystem A such that all expectation values of OA are reproduced
⟨Ψ| (OA ⊗ IB) |Ψ⟩ =
i
pi ψA
i OA ψA
i (492)
Now pi represent classical uncertainty about system in the state |ψi⟩. We
prove this prescription below.
Given such an ensemble we can define an operator
ρA ≡
i
pi ψA
i ψA
i (493)
that is known as Density matrix for subsystem. The density matrix is an
Hermitian operator with unit trace and non-negative eigenvalues pi. In fact,
it is easy to see that ψA
i is eigenvector of this operator since
ρA ψA
i =
j
pj ψA
j ψA
j |ψA
i = pi ψA
i (494)
using the fact that ψA
i are orthogonal. In other words density matrix provides
a mathematical description of ensamble of states.
In order to compute exceptation value of operator using density of matrix
wc calculate the trace this operator
⟨OA⟩ =
j
ψA
j OAρA ψA
j =
j
ψA
j OA
i
pk ψA
i ψA
i |ψA
j =
j
ψA
j OA
i
pk ψA
i δi
j =
i
pi ψA
i OA ψA
i
(495)
109
Now we are ready to calculate density matrix for subsystem. Let Ψ is the
state of the full system that, when {|ψn⟩} is basis of subsystem A and {|ψN ⟩}
is basis for the complement of A that we denote as B. Then the state |Ψ⟩
can be written as
|Ψ⟩ =
n,N
cN,n |ψn⟩ ⊗ |ψN ⟩ (496)
Since |Ψ⟩ is normalized ⟨Ψ|Ψ⟩ = 1 we obtain
⟨Ψ|Ψ⟩ =
N,n M,n
c∗
N,ncM,m ⟨ψn| ⊗ ⟨ψN | |ψm⟩ ⊗ |ψM ⟩ =
=
N,n M,n
c∗
N,ncM,mδn
mδM
N =
n,N
|cn,N |2
= 1
(497)
Then the operator
ρ ≡ |Ψ⟩ ⟨Ψ| (498)
represents density matrix for the whole system. Then the expectation value
of operator OA ⊗ IB on subsystem we have
Tr((OA ⊗ IB)ρ) =
k,K
⟨ψk| ⊗ ⟨ψK| ((OA ⊗ IB)ρ) |ψk⟩ ⊗ |ψK⟩ =
=
k,K
⟨ψk| ⊗ ⟨ψK| (OA ⊗ IB)
n,N m,M
c∗
n,N cm,M |ψn⟩ ⊗ |ψN ⟩ ⟨ψm| ⊗ ⟨ψM | |ψk⟩ ⊗ |ψK⟩ =
=
m,M
⟨ψm| ⊗ ⟨ψM | (OA ⊗ IB)
n,N
c∗
n,N cm,M |ψn⟩ ⊗ |ψN ⟩ =
=
n,m N
c∗
n,N cm,N ⟨ψn| OA |ψ⟩m =
k
⟨ψk|
n,m N
OA |ψn⟩ ⟨ψm| |ψk⟩ =
= Tr(OAρA) .
(499)
where the reduced density operator is defined as
ρA =
n,m N
c∗
n,N cm,N |ψm⟩ ⟨ψn| (500)
110
Note that it can be defined as partial trace over the subspace B
TrB(ρ) =
K
⟨ψK|
n,N m,M
c∗
n,N cm,M |ψn⟩ ⊗ |ψN ⟩ ⟨ψm| ⊗ ⟨ψM | |ψK⟩ =
=
n,N m,N
c∗
n,N cm,N |ψn⟩ ⟨ψn|
(501)
The previous result shows that quantum subsystem can be represented as
ensamble. We will say about ensamble below.
8.3 Thermodynamics ensambles and entropy
We firstly define microcanonical ensamble. This ensamble is defined as ensamble
of states with energy in the small region [E, E +dE]. More explicitly, this
ensamble is collection of states
|Ei⟩ , i = 1, . . . , n, pi =
1
n
(502)
This ensamble is used when we calculate expected values of quntities when
only the overall energy for some closed state is known. Then the ignorance
about microstate can be expressed by entropy which is given as
S = ln n , (kB ≡ 1) (503)
where n is number of microstates with the total energy E. We can think
about each state to give individual contribution to entropy
Sstate =
1
n
ln n = −p ln p , p =
1
n
. (504)
If we now presume that the individual contribution of the state is the same
for more general ensambles with different probability pi we can define the
total entropy as extensive quantity given as a sum of particular entropies
S = −
i
pi ln pi = −TrρA ln ρA , (505)
which is valid for any ensamble. We can take this formula as general definition
of entropy for arbitrary ensamble that describes any quantum subsystem.
111
As the second ensamble familiar from standard statistical physics is thermal
or canonical ensamble. This represents a system A which weakly interacts
with the thermal bath which we denote as B. Note that A with B consists
closed system. In terms of the energy eigenstates
|Ei⟩ , HA |E⟩i = Ei |E⟩i (506)
this ensamble is given as collections of states and corresponding probabilities
|Ei⟩ , pi = e−βEi
/Z (507)
where
Z =
i
e−βEi
. (508)
8.4 Entanglement
There are several possibilities how to define entanglement. One such a possibility
is to say that the state of system, which consists from subsystem and
complement cannot be write as product state in the sense that
|Ψ⟩ = |ψA⟩ ⊗ |ψB⟩ (509)
Another possibility is to say that subsystem A is entangled with the rest of
the system if the ensamble that describes this subsystem has probabilities
pi ̸= 1.
It is also important to say that entanglement is not the same for all
systems, in other words, some systems are more entangled than the others.
For example, let us consider two spins A, B where each spin is in the ⟨↑| or
⟨↓|. The Hilbert space of this system is
H = HA ⊗ HB (510)
where HA, HB are Hilbert spaces of particle A and B, respectively. An orthonormal
basis of this systems consists four elements
|↑⟩A ⊗ |↑⟩B , |↑⟩B ⊗ |↓⟩B , |↓⟩A ⊗ |↑⟩B , |↓⟩A ⊗ |↓⟩B . (511)
Then the entangled state of two spins has the form
A |↑⟩ ⊗ (512)
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