MASARYK UNIVERSITY · FACULTY OF SCIENCE
HABILITATION THESIS
Odd Scalar Curvature in
Batalin-Vilkovisky Geometry
BY
Klaus Bering Larsen
Abstract
After a brief introduction to Batalin-Vilkovisky (BV) formalism, we treat aspects of supermathematics
in algebra and differential geometry, such as, stratification theorems, Frobenius theorem
and Darboux theorem on supermanifolds. We use Weinstein’s splitting principle to prove Darboux
theorem for regular, possible degenerate, even and odd Poisson supermanifolds. Khudaverdian’s
nilpotent ∆E operator (which takes semidensities into semidensities of opposite Grassmann-parity)
is introduced on both (i) an atlas of Darboux coordinates and (ii) in arbitrary coordinates. An odd
scalar function νρ is defined and it is shown that it has a geometric interpretation as an odd scalar
curvature.
Keywords: Supermathematics; Supermanifolds; Odd Poisson Geometry; Darboux Theorem;
Antibracket; Batalin-Vilkovisky Geometry; Curvature;
1
Contents
1 Introduction 3
2 Aspects of Supermathematics 6
2.1 Grassmann-numbers and Supernumbers . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Definite Contour Integral for Supernumbers? . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Supermanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Differential Geometric Supermanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Algebro-Geometric/Sheaf-Theoretic Supermanifolds . . . . . . . . . . . . . . . . . . . 8
3 Differential Geometry for Supermanifolds 10
3.1 Stratification Theorem and Frobenius Theorems . . . . . . . . . . . . . . . . . . . . . 10
3.2 Poisson Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 2-form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4 Pre-Symplectic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.5 Symplectic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.6 Stratification Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.7 Darboux Theorem via Weinstein Splitting Method . . . . . . . . . . . . . . . . . . . . 16
3.8 Regular Poisson Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.9 Poisson Structure with Compatible 2-form . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Batalin-Vilkovisky Geometry 20
4.1 Scalars, Densities and Semidensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 Is there a Canonical Density on an Antisymplectic Manifold? . . . . . . . . . . . . . . 20
4.3 The odd Laplacians ∆ρ and ∆F on Scalars . . . . . . . . . . . . . . . . . . . . . . . . 20
4.4 Khudaverdian’s ∆E Operator on Semidensities . . . . . . . . . . . . . . . . . . . . . . 22
4.5 The Odd Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.6 The ∆E Operator in General Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.7 The Nilpotent ∆ Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Odd Scalar Curvature 30
5.1 Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6 Discussions 33
7 Conclusions 33
References 34
2
1 Introduction
Batalin-Vilkovisky (BV) formalism [1, 2, 3] was originally proposed as a recipe to provide a BechiRouet-Stora-Tyutin
(BRST) formulation [4, 5] of an arbitrary local Lagrangian gauge field theory
[6].
In physics, a field ϕa(x) is a map ϕ from an n-dimensional spacetime/world-volume with local coordinates
xµ to a target space/field configuration space. The index a labels the coordinates of the fields
in the target space.
BV formalism often uses deWitt’s condensed notation ϕi ≡ ϕa(x), where all discrete and continuous
indices are collected together into a single index i = (a, x).
In physics the field ϕi therefore contains infinite degrees of freedom. We may therefore formally view
ϕi as coordinates on an infinite-dimensional manifold.
A more rigorous field-theoretic approach (which we shall not pursue here) properly tracks the local
spacetime structure of the field theory jet bundle construction, see e.g., Refs. [7, 8]. The fact that
renormalization naturally fits into the BV formalism contributes to its versatility and state-of-the-art
in gauge field theory quantization, see e.g., Refs. [9, 10, 11].
A gauge theory is a theory with a groupoid of gauge transformations that typically leave the action
invariant up to boundary terms. Gauge symmetry signifies a redundant formulation, where field
configurations on the same gauge orbit are physically equivalent. The redundant formulation can
typically not be removed without rendering the theory non-local or destroy manifest Lorentz symmetry.
A BRST transformation is a nilpotent, Grassmann-odd transformation that encodes the gauge-symmetry.
The corresponding BRST-cohomology is needed in order to consistently define the spectrum of
physical states of the theory. (By the way, BRST symmetry should not be confused with Poincare
supersymmetry, which may or may not be present.)
Besides the original fields ϕi, the BV recipe introduces new fields, such as, e.g., Faddeev-Popov ghosts
ca, and Lagrange multipliers. For each field φα = {ϕi, ca, . . .} of Grassmann-parity εα, there is introduced
an antifield φ∗
α of opposite Grassmann-parity εα +1. Supernumbers and Grassmann-variables
will be discussed in section 2.
The original BV formulation [1, 2, 3] introduced two new interesting mathematical structures.
1. The ∆ operator
∆ =
(−1)εα
σ
→
δℓ
δφα
→
δℓ
δφ∗
α
σ , (1.1)
which is nilpotent and Grassmann-odd
∆2
= 0 , ε(∆) = 1 . (1.2)
2. The antibracket
(φα
, φ∗
β) = δα
β . (1.3)
Here σ = σ(φ) is a density in field configuration space. It is needed in order to ensure that the
∆-operator (1.1) takes scalars in scalars. The antibracket is an antisymplectic structure on the
Grassmann-parity-inverted cotangent bundle.
3
In the BV recipe the quantum action
W = S +
∞
n=1
n
Mn (1.4)
satisfies the quantum master equation (QME)
∆e
i
W
= 0 . (1.5)
To be concrete, the partition function/functional integral/Feynman path integral of the theory is
formally given as
Zψ = σ[dφ] e
i
W
φ∗= δψ
δφ
, (1.6)
where ψ is a gauge-fixing Fermion. It is one of the powerful virtues of the BV formalism, that one
may formally demonstrate that Zψ does not depend on ψ.
The BV formalism leads to two new mathematical disciplines:
• (i) BV algebra, and its generalization, such as, BV homotopy algebra [12, 13, 14, 15, 16, 17, 18,
19, 20, 22, 23], which we shall not discuss further here, and
• (ii) BV geometry [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44,
45, 46, 47, 48, 49], which is the main topic of this thesis, cf. section 4.
At this point, it is natural to generalize the Grassmann-parity-inverted cotangent bundle with Darboux
coordinates {φα; φ∗
β} and density σ2 into a general anti-Poisson supermanifold (M; E) with local
coordinates zA and equipped with a density ρ = ρ(z).
Supermanifolds will be discussed in subsection 2.3. We shall also for simplicity from now on assume
that the manifolds are finite-dimensional, despite that the original motivation in field theory deals
with infinite-dimensional configuration spaces.
Anti-Poisson geometry shares many properties with Grassmann-even Poisson geometry. E.g. they
both sport versions of the Jacobi identity and Darboux’ theorem, cf. section 3.
Somewhat surprisingly, anti-Poisson geometry also has parallels to (pseudo)Riemannian geometry [46]
(paper V). The analogue of the Laplace-Beltrami operator in (pseudo)Riemannian geometry is an odd
Laplacian [29]
∆ρ :=
(−1)εA
2ρ
→
∂ℓ
AρEAB
→
∂ℓ
B ,
→
∂ℓ
A ≡
→
∂ℓ
∂zA
. (1.7)
It is Grassmann-odd but not necessarily nilpotent.
One of our main contributions to the topic was to realize [42, 43] (paper I & II) that for regular BV
geometries (M, E, ρ) there exists a canonical odd scalar νρ (which depends only on the geometric data
E and ρ) such that the ∆ operator
∆ = ∆ρ + νρ (1.8)
is nilpotent,
∆2
= 0 , (1.9)
cf. section 4.7. The nilpotency is important in order to define a chain complex.
4
Furthermore, one of our main conclusions is that the odd scalar
νρ = −
R
8
(1.10)
has a geometric interpretation as odd scalar curvature for an arbitrary connection ∇ that is
1. anti-Poisson,
2. torsionfree, and
3. ρ-compatible,
cf. section 5.
This is a quite remarkable result because there are infinitely many connections that satisfies these 3
conditions.
The QME (1.5) is modified with a νρ-term at the two-loop order O( 2):
1
2
(W, W) = i ∆ρW + 2
νρ . (1.11)
Expanding the quantum action (1.4) in powers of Planck’s constant leads to an infinite tower of
master equations
(S, S) = 0 , (1.12)
(M1, S) = i(∆ρS) , (1.13)
(M2, S) = i(∆ρM1) + νρ −
1
2
(M1, M1) , (1.14)
(Mn, S) = i(∆ρMn−1) −
1
2
n−1
r=1
(Mr, Mn−r) , n ≥ 3 . (1.15)
Eq. (1.12) is known as the classical master equation (CME). Eq. (1.13) determines the one-loop contribution
M1. This is where possible quantum anomalies may appear in quantum field theory. Interestingly,
the odd scalar νρ appears in the eq. (1.14) for the two-loop contribution M2, see e.g., Ref. [45].
General remark about notation. We have two types of grading: A Grassmann grading ε and an exterior
form degree p. The sign conventions are such that two exterior forms ξ and η, of Grassmann parity
εξ, εη and exterior form degree pξ, pη, respectively, commute in the following graded sense:
η ∧ ξ = (−1)εξεη+pξpη
ξ ∧ η (1.16)
inside the exterior algebra. We will often not write the exterior wedges “∧” explicitly.
[A, B] and {A, B} denote the graded commutator [A, B] ≡ AB − (−1)εAεB BA and the graded anticommutator
{A, B} ≡ AB + (−1)εAεB BA, respectively.
We often omit the prefix “super” from various words in supermathematics, such as in, e.g., supermanifold,
supercommutator, etc.
5
2 Aspects of Supermathematics
Before we can begin, we need supermathematics [50, 51, 52, 53, 54, 55, 56, 57], i.e., Grassmann-odd
variables.
From the physics side, this is mainly because matter particles/fields in Nature, (such as, e.g., electrons)
are Fermions, which obey Fermi-Dirac statistics and Pauli’s exclusion principle. In particular, their
multi-particle wave-function needs to be antisymmetric under particle exchange. To properly describe
interactions of Fermionic fields at the classical level, we therefore need to use anticommuting variables.
In other words, we need to use supernumbers.
A second optional source is Poincare supersymmetric field theories.
A third typical (this time mathematical) source of Grassmann-odd variables is exponentiations of
determinants and Pfaffians, e.g.,
Pf(A) = dθn
. . . dθ1
exp
1
2
θi
Aijθj
, ε(Aij) = 0, Aij = −Aji . (2.1)
E.g. Faddeev-Popov ghosts are often introduced in this manner: The argument of the exponentials,
such as, e.g., eq. (2.1), can be interpreted as new action terms in the path integral Z.
The topic of supermathermatics contains several non-intuitive surprises, which we will try to expose
in this chapter.
2.1 Grassmann-numbers and Supernumbers
Here is one approach to Grassmann-numbers and supernumbers [53].
If V1 is an infinite-dimensional∗ C-vector space, then the exterior algebra Λ∞ := •
V1 is (a copy of)
the algebra of supernumbers.
Moreover,
• 0
V1
∼= C is called the body,
• >0
V1 is the soul,
• C1|0 ∼=
even
V1 is the Grassmann-even/Bosonic part, and
• C0|1 ∼=
odd
V1 is the Grassmann-odd/Fermionic part.
Let εz ≡ ε(z) ∈ {0, 1} mod 2 denote the Grassmann-parity of a supernumber z (with definite Grassmann
grading). And let
m(z) ≡ zB (2.2)
denote the body of a supernumber z.
∗
Here we are caught between a rock and a hard place: If V1 is finite-dimensional, there appear unwanted truncation
phenomena. If V1 is infinite-dimensional, delicate problems of analysis arise. (For definiteness, assume that the body
algebra 0
V1
∼= C is equipped with the standard norm, while infinite sums in the soul vector space >0
V1 are formal.)
6
Complex conjugation z → ¯z is an anti-involution zw = ¯w¯z on the algebra of supernumbers. We
define real and imaginary part
Rez :=
z + ¯z
2
, Imz :=
z − ¯z
2i
, (2.3)
of a supernumber z in the standard way. Let from now on F denote the field of either the real or
complex numbers, R or C.
Left (right) differentiation satisfies a left (right) graded Leibniz rule. Left and right differentiation of
a function z → f(z) are connected via the rule
(
→
∂ℓ
∂z
f(z)) = (−1)εz(εf +1)
(f(z)
←
∂r
∂z
) . (2.4)
This is to ensure compatibility
(
→
∂ℓ
∂z
z) = 1 = (z
←
∂r
∂z
) . (2.5)
(The outer parentheses in eqs. (2.4) and (2.5) are supposed to notationally indicate that the differential
operators don’t act beyond the parentheses.)
2.2 Definite Contour Integral for Supernumbers?
It is well-known that if a function z → f(z) has an antiderivative z → F(z), then a contour integral
γ
dz f(z) :=
b
a
dt γ′
(t) F′
(γ(t)) =
b
a
dt (F ◦ γ)′
(t) = F(γ(b)) − F(γ(a)) , t ∈ [a, b] ⊂ R ,
(2.6)
can only depend on the endpoints of the curve. All analytic functions f(x) = ∞
n=0 anxn of a
Grassmann-even supernumber x has an antiderivative F(x) = ∞
n=0 anxn+1/(n + 1). However the
Grassmann-odd function f : θ → θ has no antiderivative, and it is easy to see that the contour integral
γdθ θ depends on the contour beyond the endpoints.
Example 2.1
γ(t) = tθ1 + t(1−t)θ2 , t ∈ [0, 1] , γ(t=0) = 0, γ(t=1) = θ1 , (2.7)
γ
dθ θ =
1
0
dt γ′
(t) γ(t) =
1
0
dt (θ1 + (1−2t)θ2)(tθ1 + t(1−t)θ2) =
1
3
θ1θ2 , (2.8)
which does not just depend on the endpoints.
For this and similar reasons, we give up to try to assign a value to a Grassmann-odd number, whatever
that is supposed to mean. In terms of topology, DeWitt [53] assigned to Grassmann-odd directions
the coarsest topology, i.e., the trivial topology.
The paradigm is that the specific choice of the vector space V1 is meant to be forgotten/is not important.
The soul part is not quantifiable/does not have a value. An element ξ ∈ V1 should be thought
of as an indeterminate.
7
Normally the word ’indeterminate’ is just another word for a ’variable’, but here we use the word
in a stronger sense closer to the etymology of the word: Unlike a variable, which can be determined/evaluated/given
a value later-on by replacing the variable with a number, this is not possible
for the indeterminate ξ ∈ V1.
Put succinctly: There is no physical quantity or physical measurement device in a physical system
that measures a soul-valued output.
But that begs the question: How does the soul then affect a physical system at all? The answer is:
Via Berezin integration:
F0|1
dθ f(θ) :=
→
∂ℓ
∂θ
f(θ) , ε(θ) = 1 ,
R1|0
dx f(x) :=
R
dxB f(xB) ε(x) = 0 . (2.9)
In other words, all souls are ultimately integrated out.
This leads to another question: How do we then understand ’external Grassmann-odd constants’ in a
physical or mathematical model? Answer: The model should be understood as a subsector of a bigger
structure with only intrinsic souls.
2.3 Supermanifolds
There are roughly speaking two equivalent† concepts of supermanifolds M of finite dimensions (n|m),
at least within the applications to physics. Here we will briefly review their main features, cf. the next
two subsections 2.4 and 2.5. Let Fn|m := (F1|0)×m × (F0|1)×n.
2.4 Differential Geometric Supermanifolds
On one hand, there is a differential geometric definition, pioneered by deWitt [53], which considers
atlases of local Fn|m coordinate neighborhoods, which treats Grassmann-even and Grassmann-odd
variables on the same footing.
A map between supermanifolds becomes within local coordinates a map
f : V ⊆ Fn|m
→ V ′
⊆ Fn′|m′
(2.10)
of the form
z′A
= function of the z-coordinates. (2.11)
2.5 Algebro-Geometric/Sheaf-Theoretic Supermanifolds
On the other hand, there is a sheaf-theoretic definition (MB, R) of a ringed space over an underlying
ordinary n-dimensional manifold MB called the body (hence the subscript B), pioneered by
Kostant and Leites [50, 51, 52, 55]. If U ⊆ Rn is an open coordinate neighborhood for the body
MB, then R(U) = RB(U)[θ1, . . . , θm] is the polynomial ring, where θ1, . . . , θm are anti-commuting
†
Note however that the second approach has a vast generalization to category theory and Grothendieck’s schemes.
8
indeterminates. Furthermore, RB(U) is the ring of, e.g., smooth functions C∞(U), real analytic functions
Ran(U), or holomorphic functions H(U). (In this work, we shall assume the category of real
analytic functions, to keep proofs as simple as possible.) This construction treats Grassmann-even
and Grassmann-odd variables very differently.
A morphism between supermanifolds consists within local coordinate neighborhoods an underlying
body map f : U ⊆ Fn → U′ ⊆ Fn′
together with a sheaf map, i.e., an algebra homomorphism
f∗ : R(U′) → R(U), uniquely specified by it action on a local basis
f∗
(xi′
B) = Grassmann-even function of (xB, θ),
f∗
(θa′
) = Grassmann-odd function of (xB, θ). (2.12)
(The sheaf map should be compatible with the underlying body map.)
Given a fixed supermanifold M we can consider the so-called functor of points
Hom(·, M) : SManop
∋ S → Hom(S, M)
S-point of M
∈ Set . (2.13)
In particular, the indeterminates θ1, . . . , θm are viewed as morphisms from S. (Incidentally, that
is almost the perfect metaphor for passing the buck to S if asked What is θi?) Morphisms M →
N between supermanifolds M and N are in bijective correspondence with natural transformations
Hom(·, M) → Hom(·, N), cf. Yoneda’s Lemma.
9
3 Differential Geometry for Supermanifolds
In this section, we would like to mention the generalization to supermanifolds of a couple of wellknown
local theorems from the differential geometry of ordinary manifolds. We need in particular a
superversion of the Darboux theorem to define Khudaverdian’s ∆E operator in the next section 4. In
turn, the Darboux theorem relies on the Frobenius theorem.
Another purpose of this section is to provide proofs that would be accessible for a physics student.
Students of BV formalism will have noticed that Bosonic and Fermionic variables surprisingly often can
be treated in a collective and uniform manner in calculations, merely by keeping track of Grassmannparity
sign factors. This is one of key paradigms that underlies most of the presentation in this
thesis.
One difference between Bosonic and Fermionic directions is that Bosonic directions may feature nontrivial
topology, while Fermionic directions cannot, as we saw in the last section 2. Since the results
will be local, this difference is irrelevant, and we can conveniently to a large extent treat Bosonic and
Fermionic variables on equal footing.
The biggest difference between Bosonic and Fermionic operators is that a Bosonic operator trivially
(super)commutes with itself, while this is not necessarily so for a Fermionic operator.
In the following we need that the rank of a (possible rectangular) super matrix is the dimension (n|m)
of its image. In particular, n + m is the rank of the body of the super matrix.
3.1 Stratification Theorem and Frobenius Theorems
Theorem 3.1 (Stratification theorem for vector field) Given an (n|m)-dimensional supermanifold
M. Let N := n+m. Given a self-(super)commuting vector field X of definite parity with non-zero
rank in a point p ∈ M. Then there exists a local coordinate system (z1, z2, . . . , zN ) such that X =
→
∂ℓ
∂z1 .
Induction Proof: Given a local coordinate system (z1, z2, . . . , zN ). After a possible translation, we
may assume that the point p is at the origin. There exists a coordinate (which we relabel as) z1 such
that m(Xp[z1]) = 0. By a rigid linear transformation, we may assume that Xp[z1] = 1.
Induction assumption: The components XA of the vector field X = XA
→
∂ℓ
A are of the form
XA
= δA
1 + XA
[k] , XA
[k] = O((z)k
) , (3.1)
for some integer k ∈ N.
In the odd case ε(z1) = 1, nilpotency additionally yields
0 = X[XA
] = (
→
∂ℓ
1XA
[k]) + O((z)2k−1
) . (3.2)
Choose
fB
[k+1](z1
, z2
, . . . , zN
) :=



−
z1
0 d˜z1 XB
[k](˜z1, z2, . . . , zN ) if ε(z1) = 0 even ,
−z1XB
[k](z1, z2, . . . , zN ) if ε(z1) = 1 odd .
(3.3)
10
Then
fB
[k+1] = O((z)k+1
) , (
→
∂ℓ
1fB
[k+1]) = −XB
[k] + O((z)2k
) . (3.4)
We now change coordinates
z′B
= zB
+ fB
[k+1] . (3.5)
Then the new components satisfies an even higher induction assumption (3.1):
X′B
= X[z′B
] = (δA
1 + XA
[k])
→
∂ℓ
A(zB
+ fB
[k+1]) = δB
1 + XB
[k] + (
→
∂ℓ
1fB
[k+1]) + O((z)2k
) = δB
1 + O((z)2k
) .
(3.6)
Theorem 3.2 (Abelian Frobenius theorem) If X(1), . . . , X(r) are strongly commuting vector fields,
[X(a), X(b)] = 0 , a, b ∈ {1, . . . , r} , (3.7)
pointwise linearly independent, of definite Grassmann parity, then there exists a local coordinate neighborhood
(z1, z2, . . . , zN ) such that
X(a) =
→
∂ℓ
a , a ∈ {1, . . . , r} . (3.8)
Splitting proof: From the stratification theorem 3.1 we may assume that X(1) =
→
∂ℓ
1. Then the
components XA
(a) cannot depend on the coordinate z1. Define vector fields
Y(a>1) := X(a) − X(a)[z1
]X(1) = X(a) − X1
(a)
→
∂ℓ
(1) ∈ span(
→
∂ℓ
A>1). (3.9)
Straightforward calculations then shows that
[X(1), Y(a=1)] = 0 , (3.10)
[Y(a=1), Y(b=1)] = −X(a)[X1
(b)]
→
∂ℓ
(1) − (−1)εaεb (a ↔ b) = −[X(a), X(b)][z1
] = 0 . (3.11)
So Y(2), . . . , Y(r) is the same type of problem with rank r−1 and dimension N −1 one less each.
Theorem 3.3 (Non-Abelian Frobenius theorem) Given vector fields X(1), . . . , X(r), pointwise
linearly independent, of definite Grassmann parity, such that
∃f(ab)
(c)
: [X(a), X(b)] =
r
c=1
f(ab)
(c)
X(c) , a, b ∈ {1, . . . , r} . (3.12)
Then it is locally integrable, i.e., locally there exists a coordinate system (z1, z2, . . . , zN ) and a matrix
function
S(a)
(b)
, a, b ∈ {1, . . . , r} , (3.13)
such that
X′
(a) =
r
b=1
S(a)
(b)
→
∂ℓ
∂za
. (3.14)
11
Proof: Note that the non-Abelian involution property (3.12) is preserved by taking linear combination
of vector fields
X′
(a) =
r
b=1
S(a)
(b)
X(b) , a ∈ {1, . . . , r} . (3.15)
After such multiplication (3.15) of the vector fields with an invertible matrix S(a)
(b), we may assume
that they are of the form
X′
(a) =
→
∂ℓ
a +
A>r
X′A
(a)
→
∂ℓ
A , a ∈ {1, . . . , r} . (3.16)
Straightforward calculations then shows that
r
c=1
f′
(ab)
(c)
X′
(c) = [X′
(a), X′
(b)] ∈ span(
→
∂ℓ
A>r) ⇒ f′
(ab)
(c)
= 0 . (3.17)
Now use the Abelian Frobenius theorem 3.2.
3.2 Poisson Structure
We next consider a possibly degenerate Poisson manifold (M, E), with Poisson bracket
(f, g) = (f
←
∂r
∂zA
)EAB
(
→
∂ℓ
∂zB
g) . (3.18)
Here the bi-vector
EAB
= (zA
, zB
) (3.19)
is graded skewsymmetric
EAB
= −(−1)(εA+εE)(εB+εE)
EBA
, (3.20)
and has Grassmann parity
ε(EAB
) = εA+εB +εE . (3.21)
It satisfies the Jacobi identity
0 =
f,g,h cycl.
(−1)(εf +εE)(εh+εE)
((f, g), h) . (3.22)
Here εE is the (internal) Grassmann-parity of the Poisson bracket. The case εE = 1 is often called an
antibracket, anti-Poisson bracket, odd Poisson bracket, or a Gerstenhaber bracket.
In general, a Poisson manifold can have singular points where the rank of EAB jumps.
3.3 2-form
Consider a 2-form
E =
1
2
dzA
EAB ∧ dzB
= −
1
2
(−1)εA(1−εE)
EAB dzB
∧ dzA
(3.23)
12
of (internal) Grassmann-parity εE. By definition the tensor EAB has Grassmann parity
ε(EAB) = εA+εB +εE , (3.24)
and is graded skewsymmetric,
EAB = −(−1)εAεB+(1−εE)(εA+εB)
EBA . (3.25)
(Here the exterior form-degree and Grassmann-parity are considered to be non-correlated, independent
gradings. Hopefully, it will not cause confusion that we use the same letter E for both a Poisson bivector
EAB with upper indices and a 2-form EAB with lower indices.)
3.4 Pre-Symplectic Structure
Definition 3.4 A pre-symplectic 2-form is a (not necessarily non-degenerate) closed 2-form.
The closedness relation
dE = 0 (3.26)
reads in components
0 =
A,B,C cycl.
(−1)(εA+1−εE)εC (
→
∂ℓ
AEBC) . (3.27)
3.5 Symplectic Structure
A non-degenerate Poisson manifold (M, E) is called a symplectic manifold. By the non-degeneracy
assumption, there exists an inverse tensor EAB such that
EAB
EBC = δA
C = ECBEBA
. (3.28)
In other words, EAB is a two-form (3.23). The Jacobi identity (3.22) is equivalent to the closedness
relations (3.26) and (3.27).
3.6 Stratification Lemmas
Consider a possibly degenerate Poisson manifold (M, E) with a point p ∈ M. Let there be given a
local coordinate system (z1, z2, . . . , zN ) of definite Grassmann degree in a neighborhood of p ∈ M.
Assumption 3.5 (Seed 1) There exists one local coordinate (which we w.l.o.g. can rename z1), such
that the body
m((z1
, z1
)p)) = 0 (3.29)
does not vanish.
Assumption 3.6 (Seed 2) There exist two different local coordinates (which we can w.l.o.g. can
rename z1 and z2), such that the body
m((z1
, z2
)p) = 0 (3.30)
does not vanish.
13
Remark 3.7 Seed 1 is only possible for an even Poisson bracket with a Grassmann-odd z1; For an
odd Poisson bracket m((z1, z1)p) = 0 automatically.
Remark 3.8 Seed 2 with an even Poisson bracket with two Grassmann-odd variables z1 and z2 can
be traded for a seed 1 by possibly taking a constant linear combination z′1 = az1 + bz2, so that
m((z′1, z′1)p) = 0.
Remark 3.9 For the remaining cases of seed 2 we may assume (possibly after relabeling z1 ↔ z2)
that z1 is a Boson.
Lemma 3.10 (Stratification lemma for seed 1) Given seed 1 then there locally exists a Grassmann-odd
variable Q such that
(Q, Q) = ±1 (3.31)
in a local neighborhood, and such that
m


→
∂ℓ
∂z1
Q
p

 = 0 , (3.32)
i.e., (Q, z2, . . . , zN ) is a local coordinate system.
Lemma 3.11 (Stratification lemma for seed 2) Given seed 2 (excluding the case in remark 3.8,
and possibly after relabeling z1 ↔ z2), then z1 is a Boson and there locally exists a Bosonic variable
S such that
(S, S) = 0 (3.33)
in a local neighborhood, and such that
m


→
∂ℓ
∂z1
S
p

 = 0 , m((S, z2
)p) = 0 , (3.34)
i.e., (S, z2, . . . , zN ) is a local coordinate system.
Induction proof for stratification lemma for seed 1: By a rigid affine transformation v1 :=
az1 + b, we may assume that (v1, v1)p = ±1. By shifting the other coordinates
vA
= zA
− cA
v1
, A ∈ {2, . . . , N} , (3.35)
by an appropriate constant multipla of v1, we may assume that
(vA
, v1
)p = 0, A ∈ {2, . . . , N} . (3.36)
After a possible translation, we may assume that the point p is at the origin.
Induction assumption: There is a Grassmann-odd variable
Q = v1
+ Q[2] , Q[2] = O((v)2
) , (3.37)
such that
(Q, Q) = ±1 + B[k] , B[k] = O((v)k
) , (3.38)
for some integer k ∈ N.
14
We now change the variable
Q′
= Q + f[k+1] , f[k+1] := ∓
1
2
v1
B[k] = O((v)k+1
) . (3.39)
Then
(Q′
, ·) = (v1
+ Q[2] + f[k+1], vA
)
→
∂ℓ
∂vA
= ±
→
∂ℓ
∂v1
+ O((v)0
) , (3.40)
and
(Q′
, Q′
) = (Q, Q) + 2(v1
+ Q[2], vA
)(
→
∂ℓ
∂vA
f[k+1]) + (f[k+1], f[k+1])
= ±1 + B[k] ± 2(
→
∂ℓ
∂v1
f[k+1]) + O((v)k+1
) (3.41)
(3.39)+(3.43)
= ±1 + O((v)k+1
) . (3.42)
The last equality completes the induction step. It follows with the help of the Jacobi identity
0
(3.22)
= (Q′
, (Q′
, Q′
))
(3.40)+(3.41)
= ±(
→
∂ℓ
∂v1
B[k]) + O((v)k
) . (3.43)
Induction proof for stratification lemma for seed 2: There are two remaining possibilities,
cf. eq. (3.30). In the first case, the Poisson bracket, z1 and z2 are all Grassmann-even. Then the lemma
is trivial. Assume from now on the second case: The Poisson bracket and z2 are both Grassmann-odd.
By shifting the coordinates
vA
= zA
− cA
z2
, A ∈ {1, . . . , N}\{2} , (3.44)
by an appropriate constant multipla of z2, we may assume that
(vA
, v1
)p = 0, A ∈ {1, . . . , N}\{2} . (3.45)
(Note that eq. (3.45) with index A = 1 remains linear in cA=1 because we excluded the case of remark
3.8.) By a rigid affine transformation v2 := az2 + b, we may assume that (v1, v2)p = 1. After possible
translation, we may assume that the point p is at the origin.
Induction assumption: There is a variable
S = v1
+ S[2] , S[2] = O((v)2
) , (3.46)
such that
(S, S) = O((v)k
) , (3.47)
for some integer k ∈ N.
We now change the variable
S′
= S + f[k+1] , f[k+1] := −
1
2
v2
(S, S) = O((v)k+1
) . (3.48)
15
Then
(S′
, ·) = (v1
+ S[2] + f[k+1], vA
)
→
∂ℓ
∂vA
=
→
∂ℓ
∂v2
+ O((v)0
) , (3.49)
and
(S′
, S′
) = (S, S) + 2(v1
+ S[2], vA
)(
→
∂ℓ
∂vA
f[k+1]) + (f[k+1], f[k+1])
= (S, S) + 2(
→
∂ℓ
∂v2
f[k+1]) + O((v)k+1
) (3.50)
(3.48)+(3.52)
= O((v)k+1
) . (3.51)
The last equality completes the induction step. It follows with the help of the Jacobi identity
0
(3.22)
= (S′
, (S′
, S′
))
(3.49)+(3.50)
= (
→
∂ℓ
∂v2
(S, S)) + O((v)k
) . (3.52)
3.7 Darboux Theorem via Weinstein Splitting Method
In this subsection we prove a superversion of Weinstein’s splitting proof [58, 59] for Poisson manifolds,
namely theorem 3.12. We believe this is a novel result. Most super Darboux theorems in the literature
only discuss the non-generate symplectic case [39, 60]. In particular, we allow that the rank of the
Poisson structure can jump, i.e., the Poisson structure need not be regular.
Theorem 3.12 (Super Darboux theorem) Given a (n|m)-dimensional supermanifold (M, E) with
a Poisson structure E of (internal) Grassmann-parity εE. Let N := n + m. If the rank of the Poisson
tensor Ep at a point p ∈ M is r, then there exists a local coordinate system (z1, z2, . . . , zN ) around the
point p, such that the first r coordinates (z1, z2, . . . , zr) are Darboux coordinates, i.e., the sub-matrix
(EAB)1≤A,B≤r is a constant invertible matrix.
Remark 3.13 To give a full proof of Darboux theorem 3.12, it is enough to consider seed 1 and 2.
Splitting proof of Darboux theorem 3.12 with seed 1: There exists a Grassmann-odd coordinate
v1 such that
(v1
, v1
) = ±1 , (3.53)
and such that
m


→
∂ℓ
∂z1
v1
p

 = 0 , (3.54)
and so (v1, z2, . . . , zN ) is a coordinate system, cf. the stratification lemma 3.10.
Next X := (v1, ·) is a self-(super)commuting left Hamiltonian vector field. It has non-zero rank in the
point p because of eq. (3.29). Then there exists a new coordinate system (w1, w2, . . . , wN ), such that
→
∂ℓ
∂w1
= X = (v1
, ·) (3.55)
16
because of the stratification theorem 3.1. Then
(v1
, wA≥2
) = 0 . (3.56)
Then
(
→
∂ℓ
∂w1
v1
) = X[v1
] = (v1
, v1
) = ±1 . (3.57)
So (v1, w2, . . . , wN ) is also a coordinate system. From the Jacobi identity
0
(3.22)+(3.56)
= (v1
, (wA
, wB
)) = (v1
, v1
)
=±1
→
∂ℓ
∂v1
(wA
, wB
) +
C≥2
(v1
, wC
)
=0
→
∂ℓ
∂wC
(wA
, wB
)
(3.53)+(3.56)
= ±
→
∂ℓ
∂v1
(wA
, wB
) , A, B ∈ {2, . . . , N} , (3.58)
we see that (wA, wB), A, B ∈ {2, . . . , N}, cannot depend on v1. In conclusion, the coordinate v1 is a
decoupled Darboux coordinate. So we are left with the same problem in one less dimension.
Splitting proof of Darboux theorem 3.12 with seed 2 (excluding the case in remark
3.8): According to the stratification lemma 3.11, there exists a new coordinate system (u1, u2, . . . , uN ),
where u1 is a Boson, such that
(u1
, u1
) = 0 , m (u1
, u2
)p = 0 . (3.59)
The left Hamiltonian vector field X := (u1, ·) is self-(super)commuting because of the stratification
lemma 3.11. It has non-zero rank in the point p because of eq. (3.30). Then there exists a new
coordinate system (v1, v2, . . . , vN ), such that
→
∂ℓ
∂v2
= X = (u1
, ·) (3.60)
because of the stratification theorem 3.1. Since ε(v2) = εE, then
(v2
, v2
) = 0 (3.61)
follows from symmetry (3.20). Then
(u1
, v2
) = X[v2
] =
→
∂ℓ
∂v2
v2
= 1 . (3.62)
There should exists a coordinate vA, A ∈ {1, . . . , N}, such that
m


→
∂ℓ
∂vA
u1
p

 = 0 . (3.63)
It cannot be v2 because
→
∂ℓ
∂v2
u1
= X[u1
] = (u1
, u1
) = 0 . (3.64)
17
We can w.l.o.g. rename vA as v1. So (u1, v2, . . . , vN ) is a coordinate system.
Consider next the left Hamiltonian vector field Y := (v2, ·). It super-commutes with Y and X because
of eqs. (3.61) and (3.62). So by the Abelian Frobenius theorem 3.2 there exist coordinates (w1, . . . , wN ),
such that
→
∂ℓ
∂w2
= X := (u1
, ·) ,
→
∂ℓ
∂w1
= Y := (v2
, ·) . (3.65)
Then
(u1
, wA≥3
) = 0 , (v2
, wA≥3
) = 0 . (3.66)
One may show that for fixed (w3, . . . , wN ), the map (w1, w2) → (u1, v2) is locally bijective (because
the Jacobian matrix is bijective). Hence (u1, v2, w3, . . . , wN ) is a coordinate system.
From the Jacobi identity
0
(3.22)+(3.66)
= (u1
, (wA
, wB
))
= (u1
, u1
)
=0
→
∂ℓ
∂u1
(wA
, wB
) + (u1
, v2
)
=1
→
∂ℓ
∂v2
(wA
, wB
) +
C≥3
(u1
, wC
)
=0
→
∂ℓ
∂wC
(wA
, wB
)
(3.62)+(3.66)
=
→
∂ℓ
∂v2
(wA
, wB
), A, B ∈ {3, . . . , N} , (3.67)
0
(3.22)+(3.66)
= (v2
, (wA
, wB
))
= (v2
, u1
)
=±1
→
∂ℓ
∂u1
(wA
, wB
) + (v2
, v2
)
→
∂ℓ
∂v2
(wA
, wB
)
=0
+
C≥3
(v2
, wC
)
=0
→
∂ℓ
∂wC
(wA
, wB
)
(3.62)+(3.66)+(3.67)
= ±
→
∂ℓ
∂u1
(wA
, wB
), A, B ∈ {3, . . . , N} , (3.68)
we see that (wA, wB), A, B ∈ {3, . . . , N}, cannot depend on u1 and v2. In conclusion, the coordinates
u1 and v2 are decoupled Darboux coordinates. So we are left with the same problem in two less
dimensions.
3.8 Regular Poisson Structure
Definition 3.14 A Poisson manifold (M, E) is called regular if the Poisson structure E has constant
rank, i.e., the rank r does not jump.
Theorem 3.15 (Darboux theorem for regular Poisson manifolds) A regular N-dimensional Poisson
manifold (M, E) has an atlas of Darboux coordinates
{z1
, . . . , zN
} = {x1
, . . . , xr
, c1
, . . . , cN−r
} (3.69)
such that the sub-block
(xA
, xB
)1≤A,B≤r (3.70)
is constant and invertible, while cA are Casimir coordinates
(cA
, ·) = 0 , A ∈ {1, . . . , N − r} . (3.71)
18
Remark 3.16 One may show that under a coordinate transformation zA → z′B between two Darboux
charts,
→
∂ℓ
∂xA
c′B
= 0 . (3.72)
This shows that the new Casimir coordinates c′B are a function of only the old Casimir coordinates
c′A, i.e., we have an r-dimensional foliation with r-dimensional symplectic leaves. In other words, the
Hamiltonian vector fields form an r-dimensional integrable distribution, cf. the Frobenius theorem 3.3.
3.9 Poisson Structure with Compatible 2-form
Definition 3.17 A globally defined 2-form EAB of a Poisson manifold (M, E) with same internal
Grassmann parity εE is called compatible if
EAB
EBCECD
= EAD
, (3.73)
EABEBC
ECD = EAD . (3.74)
The existence of a compatible 2-form is a relatively mild requirement. However, there could be global
obstructions. The existence of a compatible 2-form is always automatically satisfied for a Dirac bracket
on symplectic manifolds with globally defined second-class constraints [29, 34, 41, 43] (paper II). Note
that the 2-form EAB is neither unique nor necessarily closed.
Obviously, an antisymplectic structure EAB has always a unique compatible 2-form, namely its inverse
structure EAB.
One can define a (1, 1) tensor field as
PA
C ≡ EAB
EBC , (3.75)
or equivalently,
PA
C
≡ EABEBC
= (−1)εA(εC +1)
PC
A . (3.76)
It then follows from either of the compatibility relations (3.73) and (3.74) that PA
B is an idempotent
PA
BPB
C = PA
C . (3.77)
19
4 Batalin-Vilkovisky Geometry
Definition 4.1 A BV manifold (M, E, ρ) is an anti-Poisson manifold (M, E) with odd internal
Grassmann parity εE = 1 equipped with a density ρ.
4.1 Scalars, Densities and Semidensities
Recall that a scalar function f =f(z), a density ρ=ρ(z) and a semidensity σ =σ(z) are by definition
quantities that transform as
f −→ f′
= f , ρ −→ ρ′
=
ρ
J
, σ −→ σ′
=
σ
√
J
, (4.1)
respectively, under general coordinate transformations zA → z′A, where J ≡ sdet∂z′A
∂zB denotes the
Jacobian.
We shall ignore the global issues of orientability of M and the choice of square root for semidensities.
In principle the above f, ρ and σ could either be Bosons or Fermions, however normally we shall
require the densities ρ to be invertible, and therefore Bosons.
The fundamental object in BV geometry is the nilpotent ∆ operator, but first we should introduce
the odd Laplacian ∆F .
4.2 Is there a Canonical Density on an Antisymplectic Manifold?
Recall that for an even symplectic manifold with an even symplectic two-form E = 1
2dzAEABdzB,
there exists a canonical density given by the Pfaffian ρ = Pf(EAB), i.e., there is a natural notion of
volume on an even symplectic manifold. A related fact is the Liouville Theorem for even symplectic
manifolds, which states that Hamiltonian vector fields are divergenceless.
On the other hand, the situation is completely different for an odd symplectic manifold endowed with
an odd antisymplectic two-form E = 1
2 dzAEABdzB. It turns out that there is no canonical choice of
density ρ in this case, as, for instance, the above Pfaffian. This is tied to the fact that there is no
meaningful notion of a superdeterminant/Berezinian for a matrix that is intrinsically Grassmann-odd.
However, the upset runs deeper. In fact, a density ρ can never be a function of the antisymplectic
matrix EAB. Phrased differently, a density ρ always carries information that cannot be deduced from
the antisymplectic structure E alone [35].
Example 4.2 Consider R1|1 endowed with the antisymplectic 2-form E = dx ∧ dθ and ρ = 1. The
antisymplectic structure is invariant under an anti-canonical transformation
x′
=
1
2
x, θ′
= 2θ , (4.2)
but the density ρ′ = 4 in the new coordinates is now 4 times bigger!
4.3 The odd Laplacians ∆ρ and ∆F on Scalars
Given a choice of density ρ, one may define the odd Laplacian [29]
∆ρ :=
(−1)εA
2ρ
→
∂ℓ
AρEAB
→
∂ℓ
B ,
→
∂ℓ
A ≡
→
∂ℓ
∂zA
, (4.3)
20
that takes scalars to scalars of opposite Grassmann parity.
A natural generalization of the odd Laplacian (4.3) is [44] (paper III)
∆F ≡
(−1)εA
2
(
→
∂ℓ
A+FA)EAB
→
∂ℓ
B . (4.4)
where FA = FA(z) is a line bundle connection with Grassmann parity ε(FA) = εA. A line bundle
connection FA transforms under general coordinate transformations zA → z′B as
FA = (
→
∂ℓ
Az′B
)F′
B + (
→
∂ℓ
A ln J) , J ≡ sdet
∂z′B
∂zA
. (4.5)
This transformation property (4.5) guarantees that the expression (4.4) remains invariant under general
coordinate transformations.
The curvature tensor for the line bundle connection FA is
RAB ≡ [
→
∂ℓ
A+FA,
→
∂ℓ
B +FB] = (
→
∂ℓ
AFB) − (−1)εAεB (A ↔ B) . (4.6)
The line bundle is called flat if the curvature vanishes
RAB = 0 . (4.7)
The flatness condition (4.7) is an integrability conditions for the local existence of ρ. The odd Laplacian
∆F reduces to ∆ρ for a flat line bundle of the form
FA = (
→
∂ℓ
A ln ρ) . (4.8)
Recall that the super-commutator of an n-order differential operator and an m-order differential
operator is at most an (n+m−1)-order differential operator. Because the odd Laplacian ∆F is a
second-order differential operator, it follows that the square operator
∆2
F =
1
2
[∆F , ∆F ] (4.9)
(which happens to be half the supercommutator) is at most a third-order operator. The vanishing of
the third-order terms is equivalent to the Jacobi identity (3.22), so ∆2
F is actually at most a secondorder
operator. It can also not have any zero-order terms, since its expression (4.4) has a derivative
to the right.
The vanishing of the second-order terms in ∆2
F means that that the curvature tensor (4.6) (projected
into the range of the anti-Poisson structure) vanish [44] (paper III)
RAD
≡ EAB
RBCECD
(−1)εC = 0 . (4.10)
There exists another descriptive characterization: The second-order terms of ∆2
F vanish if and only if
there is a Leibniz rule for the interplay of the so-called “one-bracket” ∆F and the “two-bracket” (·, ·)
∆F (f, g) = (∆F f, g) − (−1)εf (f, ∆F g) . (4.11)
See Refs. [16, 20] for more details.
For the above reasons, we will from now on make the following equivalent assumption 4.3.
21
Assumption 4.3 (Modular vector field) The square (4.9) of the odd Laplacian ∆F is a linear
derivation, i.e., a first-order differential operator, or equivalently, a vector field [37]
∆2
F (fg) = ∆2
F (f) g + f ∆2
F (g) . (4.12)
This assumption 4.3 is automatically satisfied for a flat line bundle.
Conventionally, one imposes additionally that the ∆F operator should be nilpotent ∆2
F = 0. However,
it is one of the main points of this thesis that this is not necessary, cf. eq. (4.58) below.
The odd Laplacian (4.4) has a geometric interpretation as a divergence of a Hamiltonian vector field
[25, 28]
∆F Ψ = −
1
2
divF (XΨ) , ε(Ψ) = 1 . (4.13)
Here XΨ := (Ψ, ·) denotes a Hamiltonian vector field with a Grassmann-odd Hamiltonian Ψ, and the
divergence divF X of a vector field X, with respect to the density ρ, is
divF X := (−1)εA ((
→
∂ℓ
A + FA)XA
) , ε(X) = 0 . (4.14)
The fact that the odd Laplacian (4.13) is non-zero, shows that antisymplectic manifolds do not have
an analogue of the Liouville Theorem, cf. subsection 4.2.
4.4 Khudaverdian’s ∆E Operator on Semidensities
Khudaverdian [39] only considered the antisymplectic case, but we show here that his construction
actually works for any regular anti-Poisson manifold.
Khudaverdian showed that one may define a Grassmann-odd, nilpotent, second-order operator ∆E
without a choice of density ρ. This ∆E operator does not take scalars to scalars like the odd Laplacian
(4.3), but instead takes semidensities to semidensities of opposite Grassmann parity. Equivalently, the
∆E operator transforms as
∆E −→ ∆′
E =
1
√
J
∆E
√
J (4.15)
under general coordinate transformations zA → z′B, cf. eq. (4.1). Khudaverdian’s construction relies
first of all on an atlas of Darboux charts, which is granted by the Darboux Theorem 3.15, and secondly,
on a Lemma by Batalin and Vilkovisky about the possible form of the Jacobians for anticanonical
transformations, also known as antisymplectomorphisms.
Lemma 4.4 (The Batalin-Vilkovisky Lemma) [3, 6, 37, 38, 39, 41]. Consider a finite anticanonical
transformation between initial Darboux coordinates zA
(i) and final Darboux coordinates zA
(f).
Then the Jacobian J ≡ sdet(∂zA
(f)/∂zB
(i)) satisfies
∆
(i)
1
√
J = 0 . (4.16)
Here ∆
(i)
1 refers to the odd Laplacian (4.3) with ρ=1 in the initial Darboux coordinates zA
(i).
22
A simple proof of the Batalin-Vilkovisky Lemma for finite anticanonical transformations can be found
in Ref. [41].
Definition 4.5 (The ∆E operator in Darboux coordinates) Given Darboux coordinates zA, the
∆E operator is defined on a semidensity σ as [36, 37, 38, 39, 41]
(∆Eσ) := (∆1σ) , (4.17)
where ∆1 is the ∆ρ operator (4.3) with ρ=1.
Remark 4.6 It is important in eq. (4.17) that the formula for the ∆1 operator (4.3) and the semidensity
σ both refer to the same Darboux coordinates zA. The parentheses in eq. (4.17) indicate that
the equation should be understood as an equality among semidensities (in the sense of zeroth-order
differential operators) rather than an identity among differential operators.
Theorem 4.7 The ∆E operator (4.17) does not depend on choice of Darboux coordinates zA, and it
takes semidensities to semidensities. Concretely, this means in terms of formulas, that the right–hand
side of the definition (4.17) transforms as a semidensity
(∆
(f)
1 σ(f)) =
1
√
J
(∆
(i)
1 σ(i)) (4.18)
under an anticanonical transformation between any two Darboux coordinates zA
(i) and zA
(f).
Finite transformation proof of theorem 4.7: One uses the Batalin-Vilkovisky Lemma to argue
that the definition (4.17) does not depend on the choices of Darboux coordinates zA. One calculates:
√
J(∆
(f)
1 σ(f)) =
√
J(∆
(i)
J σ(f)) =
√
J(∆
(i)
J
σ(i)
√
J
) = (∆
(i)
1 σ(i)) −
1
√
J
(∆
(i)
1
√
J)σ(i) = (∆
(i)
1 σ(i)) .
(4.19)
The third equality is a non-trivial property of the odd Laplacian (4.3). The Batalin-Vilkovisky Lemma
is used in the fourth equality.
Infinitesimal transformation proof of theorem 4.7: Strictly speaking, it is enough to consider
infinitesimal anticanonical transformations to justify the definition (4.17). The proof of the
infinitesimal version of the Batalin-Vilkovisky Lemma goes like this: An infinitesimal anticanonical
coordinate transformation δzA = XA is generated by a Grassmann-even vector field X = XA
→
∂ℓ
A that
preserves the antibracket
X[(f, g)] = (X[f], g) + (f, X[g]) , (4.20)
which implies that
EAC
(
→
∂ℓ
CXB
) ≡ (zA
, XB
) = −(−1)(εA+1)(εB+1)
(A ↔ B) . (4.21)
So the Jacobian becomes
ln J ≈ (−1)ǫA
(
→
∂ℓ
AXA
) = div1(X) , (4.22)
and hence
∆1
√
J ≈
1
2
∆1div1(X) =
(−1)εA+εB
4
(
→
∂ℓ
AEAC
→
∂ℓ
C
→
∂ℓ
BXB
)
(4.21)
= 0 . (4.23)
Here we used repeatedly that EAB is constant in Darboux coordinates. The “≈” sign is used to
indicate that equality only holds at the infinitesimal level. (Here we are guilty of mixing active and
passive pictures; the active vector field is properly speaking minus X.)
23
Theorem 4.8 (Nilpotency) Given an atlas of Darboux coordinates, the ∆E operator (4.17) is nilpo-
tent,
∆2
E =
1
2
[∆E, ∆E] = 0 , (4.24)
i.e., it squares to zero, or equivalently, it super-commutes with itself.
Proof of theorem 4.8: The ∆E operator super-commutes with itself, because the zA-derivatives
have no zA’s to act on in Darboux coordinates.
4.5 The Odd Scalar
The plan is now to define the ∆E operator in arbitrary coordinates, but first we need to define an odd
scalar.
Theorem 4.9 (Odd Scalar) [43] (paper II). Given a (not necessarily flat) line bundle connection
FA on an anti-Poisson manifold (M, E) with a compatible 2-form, then the following Grassmann-odd
quantity is a scalar:
νF := ν
(0)
F +
ν(1)
8
−
ν(2)
8
−
ν(3)
24
+
ν(4)
24
+
ν(5)
12
, (4.25)
where
ν
(0)
F :=
(−1)εA
4
(
→
∂ℓ
A +
FA
2
)(EAB
FB) , (4.26)
ν(1)
:= (−1)εA (
→
∂ℓ
B
→
∂ℓ
AEAB
) , (4.27)
ν(2)
:= (−1)εAεC (
→
∂ℓ
DEAB
)EBC(
→
∂ℓ
AECD
) , (4.28)
ν(3)
:= (−1)εB (
→
∂ℓ
AEBC)ECD
(
→
∂ℓ
DEBA
) , (4.29)
ν(4)
:= (−1)εB (
→
∂ℓ
AEBC)ECD
(
→
∂ℓ
DEBF
)PF
A
, (4.30)
ν(5)
:= (−1)εAεC (
→
∂ℓ
DEAB
)EBC(
→
∂ℓ
AECF
)PF
D
= (−1)(εA+1)εB EAD
(
→
∂ℓ
DEBC
)(
→
∂ℓ
CEAF )PF
B . (4.31)
Remark 4.10 Despite this definition (4.25) seems to depend on the choice of compatible 2-form, we
shall later see that it has a geometric meaning as odd scalar curvature.
Sketched proof of theorem 4.9: Under an arbitrary infinitesimal coordinate transformation
δzA = XA, one calculates
δν
(0)
F = −
1
2
∆1div1X , (4.32)
δν(1)
= 4∆1div1X + (−1)εA (
→
∂ℓ
CEAB
)(
→
∂ℓ
B
→
∂ℓ
AXC
) , (4.33)
δν(2)
= (−1)εA (
→
∂ℓ
DEAB
) 2PB
C
(
→
∂ℓ
C
→
∂ℓ
AXD
) + (
→
∂ℓ
B
→
∂ℓ
AXC
)PC
D
, (4.34)
δν(3)
= (−1)εB (
→
∂ℓ
AEBC)ECD
(
→
∂ℓ
DXB
←
∂r
F )EFA
− (−1)(εA+1)(εB+1)
(
→
∂ℓ
DXA
←
∂r
F )EFB
24
−
3
2
(−1)εA PC
D
(
→
∂ℓ
DEAB
)(
→
∂ℓ
B
→
∂ℓ
AXC
) , (4.35)
δν(4)
= −2(−1)εB (
→
∂ℓ
A
→
∂ℓ
BXC
)PC
D
(
→
∂ℓ
DEBF
)PF
A
+(−1)εB (
→
∂ℓ
AEBC)ECD
(
→
∂ℓ
DXB
←
∂r
F )EFA
+(−1)(εB+1)εF PF
A
(
→
∂ℓ
AEBC)ECD
(
→
∂ℓ
DXF
←
∂r
G)EGB
+
1
2
(−1)εA PC
D
(
→
∂ℓ
DEAB
)(
→
∂ℓ
B
→
∂ℓ
AXC
) , (4.36)
δν(5)
= −(−1)εA(εB+1)
(
→
∂ℓ
AEBC)ECD
(
→
∂ℓ
DXA
←
∂r
F )EFB
+2(−1)εB (
→
∂ℓ
A
→
∂ℓ
BXC
)PC
D
(
→
∂ℓ
DEBF
)PF
A
. (4.37)
A proof of eqs. (4.32) and (4.33) can be found in Ref. [42], and eqs. (4.34)–(4.37) are proven in
Ref. [43]. One may verify that while the six constituents ν
(0)
F , ν(1), ν(2), ν(3), ν(4) and ν(5) separately
have non-trivial transformation properties, the linear combination νF in eq. (4.25) is indeed a scalar.
Lemma 4.11 On an antisymplectic manifold (M, E), we have
ν(5)
= ν(2)
= −ν(3)
= −ν(4)
. (4.38)
Proof of lemma 4.11: Straightforward calculation.
Because of lemma 4.11, the theorem 4.9 simplifies in the antisymplectic case.
Theorem 4.12 (Odd Scalar) [42] (paper I). Given a (not necessarily flat) line bundle connection
FA on an antisymplectic manifold (M, E), then the following Grassmann-odd quantity is a scalar:
νF := ν
(0)
F +
ν(1)
8
−
ν(2)
24
, (4.39)
where
ν
(0)
F :=
(−1)εA
4
(
→
∂ℓ
A +
FA
2
)(EAB
FB) , (4.40)
ν(1)
:= (−1)εA (
→
∂ℓ
B
→
∂ℓ
AEAB
) , (4.41)
ν(2)
:= −(−1)εB (zC
, (zB
, zA
))(
→
∂ℓ
AEBC) = (−1)εAεC (
→
∂ℓ
AECD
)(
→
∂ℓ
DEAB
)EBC . (4.42)
Sketched proof of theorem 4.12: One should check that νF is a scalar under general infinitesimal
coordinate transformations. Under an arbitrary infinitesimal coordinate transformation δzA = XA,
one calculates [42] (paper I)
δν
(0)
F = −
1
2
∆1div1X , (4.43)
δν(1)
= 4∆1div1X + (−1)ǫA
(
→
∂ℓ
CEAB
)(
→
∂ℓ
B
→
∂ℓ
AXC
) , (4.44)
δν(2)
= 3(−1)ǫA
(
→
∂ℓ
CEAB
)(
→
∂ℓ
B
→
∂ℓ
AXC
) . (4.45)
25
One easily sees that while the three constituents ν
(0)
F , ν(1) and ν(2) separately have non-trivial transformation
properties, the linear combination νF in eq. (4.25) is indeed a scalar.
Corollary 4.13 (Odd Scalar) [42] (paper I). Given a density ρ on an anti-Poisson manifold (M, E)
with a compatible 2-form, then the following Grassmann-odd quantity is a scalar:
νρ := ν(0)
ρ +
ν(1)
8
−
ν(2)
8
−
ν(3)
24
+
ν(4)
24
+
ν(5)
12
, (4.46)
where
ν(0)
ρ :=
1
√
ρ
(∆1
√
ρ) . (4.47)
Corollary 4.14 (Odd Scalar) [42] (paper I). Given a density ρ on an antisymplectic manifold
(M, E), then the following Grassmann-odd quantity is a scalar:
νρ := ν(0)
ρ +
ν(1)
8
−
ν(2)
24
. (4.48)
4.6 The ∆E Operator in General Coordinates
We now give a definition of the ∆E operator whose definition does not rely on Darboux coordinates.
(However, we should emphasize that we have only been able to prove nilpotency by assuming that the
Poisson manifold is regular.)
Definition 4.15 (The ∆E operator in arbitrary coordinates) [43] (paper II). Given an antiPoisson
manifold (M, E) with a compatible 2-form, then the ∆E operator is defined on a semidensity
σ in an arbitrary coordinate system as
(∆Eσ) := (∆1σ) +
ν(1)
8
−
ν(2)
8
−
ν(3)
24
+
ν(4)
24
+
ν(5)
12
σ , (4.49)
where ∆1 is the ∆ρ operator (4.3) with ρ=1.
Recalling lemma 4.11, we have the following simplification in the antisymplectic case.
Definition 4.16 (The ∆E operator in arbitrary coordinates) [42] (paper I). Given an antisymplectic
manifold (M, E), then the ∆E operator is defined on a semidensity σ in an arbitrary coordinate
system as
(∆Eσ) := (∆1σ) +
ν(1)
8
−
ν(2)
24
σ . (4.50)
Remark 4.17 Notice that in Darboux coordinates, where EAB is constant, i.e., independent of the
coordinates zA, then ν(1), ν(2), ν(3), ν(4) and ν(5) vanish. Hence the definitions 4.15 and 4.16 of the
∆E operator agree with Khudaverdian’s definition 4.5.
26
Theorem 4.18 The ∆E operator (4.49) and (4.50) does not depend on choice of coordinates zA, and
it takes semidensities to semidensities, i.e., the right–hand side of eqs. (4.49) and (4.50) behaves as a
semidensity under general coordinate transformations.
Proof of theorem 4.18: Here we will only explicitly consider the case where the semidensity σ is
invertible to simplify the presentation. (The non-invertible case is fundamentally no different.) In the
invertible case, we customarily write the semidensity σ =
√
ρ as a square root of a density ρ. If we
divide the definition (4.49) and (4.50) with the square root of a density ρ, we obtain the odd scalar
1
√
ρ
(∆E
√
ρ) = νρ , (4.51)
which is independent of coordinate system, cf. corollaries 4.13 and 4.14. This proves the theorem.
Corollary 4.19 The odd scalar is connected to the ∆E operator and the density ρ via
νρ =
1
√
ρ
(∆E
√
ρ) . (4.52)
Now what about nilpotency of the definitions 4.15 and 4.16? Nilpotency is clearly a local statement.
In Darboux coordinates, the nilpotency is obvious. But what about in general coordinates?
More specifically, given an arbitrary coordinate system, could it be that an ingenious repeated use of
the Jacobi identity (3.22) and properties (3.73) and (3.74) of the compatible 2-form, could yield a proof
of the nilpotency of the ∆E operator without resorting to Darboux coordinates? In the antisymplectic
case, this was successfully done in our paper Ref. [44] (paper III). However, a preliminary investigation
strongly suggests that this approach does not generalize to the degenerated case. In the end, we have
not been able to generalize the nilpotency theorem 4.8 to situations where Darboux coordinates do
not exist.
In quantum field theory applications in physics the manifolds typically are infinite dimensional, and
going to Darboux coordinates typically violates locality. Therefore it is of interests of considering
general coordinates, even if Darboux coordinates are formally available.
4.7 The Nilpotent ∆ Operator
Definition 4.20 Given a nilpotent ∆E operator and a density ρ, then a nilpotent ∆ operator, that
takes scalars in scalars, can be defined as
∆ :=
1
√
ρ
∆E
√
ρ = ∆ρ + νρ (4.53)
via conjugation of Khudaverdian’s ∆E operator with the square root of the density.
The second equality in (4.53) is a non-trivial property of the odd Laplacian (4.3).
Nilpotent operators play a prominent role in (co)homology theory. For this reason, it is of interest to
more generally add an odd vector field V = V A
→
∂ℓ
A and an odd scalar function ν to the odd Laplacian
∆F ,
∆ = ∆F + V + ν (4.54)
27
such that the total ∆ operator is nilpotent
∆2
=
1
2
[∆, ∆] = 0 , (4.55)
i.e., supercommutes with itself. In physics, the nilpotency (4.55) encodes Becchi-Rouet-Stora-Tyutin
(BRST) symmetry.
The antibracket (f, g) of two functions f = f(z) and g = g(z) can be defined as a double commutator
[16] with the ∆-operator, acting on the constant unit function 1,
(f, g) = (f
←
∂r
A)EAB
(
→
∂ℓ
Bg) = (−1)εf [[
→
∆, f], g]1
= (−1)εf ∆(fg) − (−1)εf (∆f)g − f(∆g) + (−1)εg fg(∆1) . (4.56)
Let us assume from now on that the line bundle connection F is flat. Recall that then the square ∆2
F
of the odd Laplacian is a first-order operator, cf. assumption 4.3. It follows that the square ∆2 is at
most a second-order operator.
The vanishing of the second-order terms in ∆2 is equivalent to that the odd vector field V preserves
the antibracket
V (f, g) = (V [f], g) − (−1)εf (f, V [g]) , (4.57)
i.e., it belongs to the first Poisson cohomology group. The vector field V has seen applications in
physics in the context of Sp(2)-symmetric BRST/anti-BRST quantization [61]. In the non-degenerate
case, it is locally a Hamiltonian vector field v = (H, ·), and it can be viewed as part of the line bundle
connection FA + V BEBA. For this reason, we shall put V = 0 in what follows.
With V = 0 in eq. (4.54), the nilpotency condition (4.55) becomes equivalent to two conditions (4.58)
and (4.59) as follows.
• At first order,
∆2
F = (ν, ·) , (4.58)
i.e., the modular vector field ∆2
F is a Hamiltonian vector field with the ν as odd Hamiltonian.
• At zeroth order
(∆F ν) = 0 . (4.59)
Equation (4.59) is not an independent condition but it follows instead automatically from the previous
requirements. Proof:
−(∆F ν) =
(−1)εA
2
(
→
∂ℓ
A+FA)(ν, zA
) =
(−1)εA
2
(
→
∂ℓ
A+FA)∆2
F zA
=
(−1)εA+εB
4
(
→
∂ℓ
A+FA)(
→
∂ℓ
B +FB)(zB
, ∆F zA
)
= −
(−1)εA
8
(
→
∂ℓ
A+FA)(
→
∂ℓ
B +FB)∆F (zB
, zA
)
=
(−1)εAεC
16
(
→
∂ℓ
A+FA)(
→
∂ℓ
B +FB)(
→
∂ℓ
C +FC)(zC
, (zB
, zA
))(−1)(εA+1)(εC +1)
= 0 .(4.60)
Here, the ν eq. (4.58) is used in the second equality, the Leibniz rule (4.11) in the fourth equality,
the Jacobi identity (3.22) in the sixth (=last) equality, and the zero curvature condition (4.7) in the
second, fourth and sixth equality.
28
Proposition 4.21 [44] (paper III). For an antisymplectic manifold with a flat line bundle connection
F, the odd scalar νF from eq. (4.39) is a solution to ν to the differential eq. (4.58).
The proof is given in Ref. [44] (paper III). It is equivalent to the proof for the ∆E operator in arbitrary
coordinates.
From proposition 4.21, it follows that the difference ν −νF , must satisfy (ν −νF , ·) = 0, i.e., the
difference ν−νF is a Grassmann-odd constant.
Altogether, it follows for a non-degenerate BV geometry (M, E, ρ), that the ∆ operator (4.54) must
be equal to ∆ρ+νρ up to an odd constant. (The undetermined odd constant comes from the fact that
the square ∆2 = 1
2[∆, ∆] does not change if ∆ is shifted by an odd constant.)
29
5 Odd Scalar Curvature
5.1 Connection
We now introduce a connection ∇ : TM × TM → TM on the tangent bundle. See Ref. [35, 62] for
related discussions. The left covariant derivative (∇AX)B of a left vector field XA is defined as [35]
(∇AX)B
≡ (
→
∂ℓ
AXB
) + (−1)εX (εB+εC )
ΓA
B
C XC
, ε(XA
) = εX + εA , (5.1)
The word “left” implies that XA and (∇AX)B transform with left derivatives
X′B
= XA
(
→
∂ℓ
∂zA
z′B
) , (
→
∂ℓ
∂zA
z′B
)(∇′BX)′C
= (∇AX)B
(
→
∂ℓ
∂zB
z′C
) , (5.2)
under general coordinate transformations zA → z′B. It is convenient to introduce a reordered Christoffel
symbol
ΓA
BC ≡ (−1)εAεB ΓB
A
C (5.3)
to minimize the appearances of sign factors.
Definition 5.1 A connection ΓA
B
C is called anti-Poisson if it preserves the anti-Poisson structure
EAB, i.e., definition [35]
0 = (∇AE)BC
≡ (
→
∂ℓ
AEBC
) + ΓA
B
D EDC
− (−1)(εB+1)(εC +1)
(B ↔ C) . (5.4)
The torsion tensor T : TM × TM → TM is defined as
T(X, Y ) = ∇XY − (−1)εX εY ∇Y X = −(−1)εX εY T(Y, X) . (5.5)
A torsion-free connection with T = 0 has the following symmetry in the lower indices:
ΓA
BC = −(−1)(εB+1)(εC +1)
ΓA
CB . (5.6)
On one hand, a connection ∇ can be used to define a divergence of a Bosonic vector field XA as
str(∇X) ≡ (−1)εA (∇AX)A
= ((−1)εA
→
∂ℓ
A + ΓB
BA)XA
, εX = 0 . (5.7)
On the other hand, the divergence is defined in terms of the line bundle connection F as
divF X ≡ (−1)εA (
→
∂ℓ
A+FA)XA
. (5.8)
See Ref. [63] for a mathematical exposition of divergence operators on supermanifolds.
Definition 5.2 The tangent bundle connection ∇ is called compatible with the line bundle connection
F if their divergences (5.7) and (5.8) are the same, i.e., if
ΓB
BA = (−1)εA FA . (5.9)
30
In the affirmative case, we have a unique divergence operator (and hence a unique notion of volume).
We shall only consider anti-Poisson, torsion-free, and F-compatible connections ∇, i.e., connections
that satisfy the three conditions (5.4), (5.6) and (5.9).
For connections satisfying the three conditions, the odd Laplacian ∆F operator can be written on a
manifestly covariant form
∆F =
(−1)εA
2
∇AEAB
∇B =
(−1)εB
2
EBA
∇A∇B . (5.10)
5.2 Curvature
The Riemann curvature tensor RAB
C
D is defined as the commutator of the ∇ connection
([∇A, ∇B]X)C
= RAB
C
DXD
(−1)εX (εC +εD)
, (5.11)
so that
RAB
C
D = (
→
∂ℓ
AΓB
C
D) + (−1)εBεC ΓA
C
EΓE
BD − (−1)εAεB (A ↔ B) . (5.12)
It is useful to define a reordered Riemann curvature tensor RA
BCD as
RA
BCD ≡ (−1)εA(εB+εC )
RBC
A
D = (−1)εAεB (
→
∂ℓ
BΓA
CD)+ΓA
BEΓE
CD −(−1)εBεC (B ↔ C) . (5.13)
It is interesting to consider the various contractions of the Riemann curvature tensor. There are two
possibilities. Firstly, there is the Ricci two-form
RAB ≡ RAB
C
C(−1)εC = (
→
∂ℓ
AFB) − (−1)εAεB (A ↔ B) . (5.14)
However, the Ricci two-form RAB typically vanishes, cf. eq. (4.7), and even if it does not vanish, its
antisymmetry means that RAB cannot successfully be contracted with the anti-Poisson tensor EAB
to yield a non-zero scalar curvature, cf. eq. (3.20). Secondly, there is the Ricci tensor
RAB ≡ RC
CAB = (−1)εC (
→
∂ℓ
C + FC)ΓC
AB − (
→
∂ℓ
AFB)(−1)εB − ΓA
C
DΓD
CB . (5.15)
Note that when the torsion tensor and Ricci two-form vanish, the Ricci tensor RAB possesses exactly
the same A↔B symmetry (3.20) as the anti-Poisson tensor EAB
RAB = −(−1)(εA+1)(εB+1)
RBA . (5.16)
The odd scalar curvature R is therefore defined in anti-Poisson geometry as the contraction of the
Ricci tensor RAB and the antisymplectic metric EBA,
R ≡ RABEBA
= EAB
RBA . (5.17)
Theorem 5.3 [45] (paper IV). Given an anti-Poisson manifold (M, E) with a compatible 2-form,
and a line bundle connection F, then for an arbitrary, anti-Poisson, torsion-free, and F-compatible
connection ∇, the scalar curvature R does only depend on E and F through the odd scalar νF ,
R = −8νF , (5.18)
even if the line bundle connection F is not flat.
31
Theorem 5.3 is proven in Ref. [45] (paper IV). In particular, one concludes that the scalar curvature
R does not depend on the connection ΓA
BC used.
One can perform various consistency checks on the formalism. Here, let us just mention one. For an
antisymplectic connection ∇, one has
0 = [∇A, ∇B]ECD
= RAB
C
F EFD
− (−1)(εC +1)(εD+1)
(C ↔ D) , (5.19)
or, equivalently,
RC
ABF EFD
= −(−1)εAεB+(εC +1)(εD+1)+(εA+εB)(εC +εD)
RD
BAF EFC
. (5.20)
Contracting the A ↔ C and B ↔ D indices in eq. (5.20) indeed produces the identity R = R. Had
the signs turn out differently, the odd scalar curvature (5.17) would have been stillborn, i.e., always
zero.
32
6 Discussions
One important check of the formulas for the odd scalar in the degenerate and non-degenerate case
comes from conversion [64, 65, 66, 67, 68] of antisymplectic second-class constraints with a Dirac
antibracket [34] into first-class constraints into an extended antisymplectic phase space. Moreover, it
is interesting to check how the construction reacts to reparametrization of the second class constraints.
These investigations were successfully undertaken in our paper [43] (paper II).
7 Conclusions
In this thesis, we have briefly reviewed the Batalin-Vilkovisky (BV) formalism, and treated aspects
of supermathematics in algebra and differential geometry, such as, integration theory, stratification
theorems, Frobenius theorem and Darboux theorem on supermanifolds.
We used Weinstein’s splitting principle to prove Darboux theorem 3.15 for regular, possible degenerate,
even and odd Poisson manifolds.
Khudaverdian’s nilpotent ∆E operator was introduced on both
• (i) an atlas of Darboux coordinates, cf. definition 4.5; and
• (ii) in arbitrary coordinates, cf. definition 4.49.
To express ∆E in arbitrary coordinates (ii) in the degenerate case, we relied on the existence of a
non-unique choice of compatible 2-form EAB. This comes back to haunt us, since we are unable to
prove nilpotency of ∆E without appealing to Darboux coordinates (i). Hence the case (ii) is de facto
not more general than the case (i).
Nevertheless, even in the second scenario (ii) with a compatible 2-form EAB, we were able to define
an odd scalar function νF , cf. theorem 4.9; and show that it has a geometric interpretation as an odd
scalar curvature, cf. theorem 5.3.
Acknowledgement: K.B. would like to thank Igor Batalin, Poul Henrik Damgaard and Hovhannes
Khudaverdian for fruitful discussions. The work of K.B. is supported by the Grant Agency of the
Czech Republic (GACR) under the grant P201/12/G028. K.B. also received financial support from
the faculty of science at Masaryk University under the grant 1102/033.
33
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36
Paper I
A Note on Semidensities
in Antisymplectic Geometry
BY
K. Bering
J. Math. Phys. 47 (2006) 123513
arXiv:hep-th/0604117
arXiv:hep-th/0604117v630Dec2006
A Note on Semidensities in Antisymplectic Geometry
K. Bering1
Institute for Theoretical Physics & Astrophysics
Masaryk University
Kotl´aˇrsk´a 2
CZ-611 37 Brno
Czech Republic
July 24, 2013
Abstract
We revisit Khudaverdian’s geometric construction of an odd nilpotent operator ∆E that sends
semidensities to semidensities on an antisymplectic manifold. We find a local formula for the ∆E
operator in arbitrary coordinates and we discuss its connection to Batalin-Vilkovisky quantization.
MCS number(s): 53A55; 58A50; 58C50; 81T70.
Keywords: Batalin-Vilkovisky Field-Antifield Formalism; Odd Laplacian; Antisymplectic
Geometry; Semidensity; Halfdensity.
1E-mail: bering@physics.muni.cz
1
1 Introduction
Recall that for a symplectic manifold with an even symplectic two-form ω = 1
2dzAωABdzB, there exists
a canonical measure density given by the Pfaffian ρ=Pf(ωAB), i.e. there is a natural notion of volume
in a symplectic manifold. A related fact is the Liouville Theorem, which states that Hamiltonian vector
fields are divergenceless. On the other hand, the situation is completely different for an odd symplectic
manifold, also known as an antisymplectic manifold and endowed with an odd antisymplectic two-form
E = 1
2dΓAEABdΓB. These geometries for instance show up in the Lagrangian quantization method of
Batalin and Vilkovisky [1]. It turns out that there is no canonical choice of measure density ρ in this
case, as, for instance, the above Pfaffian. This is tied to the fact that there is no meaningful notion of
a superdeterminant/Berezinian for a matrix that is intrinsically Grassmann-odd. However, the upset
runs deeper. In fact, a density ρ can never be a function of the antisymplectic matrix EAB. Phrased
differently, a density ρ always carries information that cannot be deduced from the antisymplectic
structure E alone [2]. Within the standard Batalin-Vilkovisky framework, the possible choices of a
density ρ is only partially determined by a requirement of gauge symmetry.
Around 1992 Batalin-Vilkovisky quantization took a more geometric form, in particular with the work
of Schwarz [3]. The concensus was that the geometric setting requires two independent structures: an
odd symplectic, non-degenerate two-form E and a measure density ρ. From these two structures, one
may build a Grassmann-odd, second-order operator ∆ρ, known as the odd Laplacian. Alternatively,
one can view the odd Laplacian ∆ρ itself as the fundamental structure of Batalin-Vilkovisky geometry
[4, 5], which is conventionally required to be nilpotent.
Khudaverdian has constructed [6, 7, 8, 9] a Grassmann-odd, nilpotent, second-order operator ∆E that
does not rely on a choice of density ρ. The caveat is that the ∆E operator is defined on semidensities
rather than on scalars. (The notion of a semidensity is explained in eq. (3.1) below.) In retrospect,
many pieces of Khudaverdian’s construction were known to physicists, see for instance Ref. [10],
p.440. In this short note we reconsider Khudaverdian’s construction and find a local formula for the
∆E operator that applies to arbitrary coordinate systems. The ability to work in any coordinates, not
just Darboux coordinates, is important, since if one first has to search for a set of Darboux coordinates
to the system that one is studying, symmetries (such as, e.g. , Lorentz covariance) or locality that one
would like to preserve during the quantization process, are often lost.
The paper is organized as follows: We consider the antisymplectic structure in Section 2; the odd
Laplacian ∆ρ in Section 3; and in Sections 4 and 5, the ∆E operator using Darboux coordinates
and general coordinates, respectively. Finally, in Section 6 we analyze a modified Batalin-Vilkovisky
scheme based on the ∆E operator.
General remark about notation. We have two types of grading: A Grassmann grading ǫ and an exterior
form degree p. The sign conventions are such that two exterior forms ξ and η, of Grassmann parity
ǫξ, ǫη and exterior form degree pξ, pη, respectively, commute in the following graded sense
η ∧ ξ = (−1)ǫξǫη+pξpη
ξ ∧ η (1.1)
inside the exterior algebra. We will often not write the exterior wedges “∧” explicitly.
2 Antisymplectic Geometry
Consider an antisymplectic manifold (M, E) and let ΓA denote local coordinates of Grassmann parity
ǫA ≡ ǫ(ΓA) (and exterior form degree p(ΓA) = 0). The antisymplectic two-form can locally be written
2
as
E =
1
2
dΓA
EAB dΓB
= −
1
2
EAB dΓB
dΓA
, (2.1)
where EAB =EAB(Γ) is the corresponding matrix representation. Besides carrying gradings ǫ(E) = 1
and p(E) = 2, the antisymplectic two-form E has two defining properties. First, E is closed,
dE = 0 , (2.2)
where the grading conventions for the exterior derivative
d = dΓA
→
∂l
∂ΓA
(2.3)
are ǫ(d) = 0 and p(d) = 1. Secondly, E is non-degenerate, i.e. the antisymplectic matrix EAB has an
inverse matrix EAB,
EAB
EBC = δA
C = ECBEBA
. (2.4)
Instead of the compact exterior form notation E, one may equivalently formulate the above conditions
with all the indices written out explicitly in terms of the matrices EAB or EAB. In detail, the gradings
are
ǫ(EAB) = ǫA + ǫB + 1 = ǫ(EAB) ,
p(EAB) = 0 = p(EAB) ,
(2.5)
the skew-symmetries are
EBA = −(−1)ǫAǫB
EAB ,
EBA
= −(−1)(ǫA+1)(ǫB +1)
EAB
, (2.6)
while the closeness condition and the equivalent Jacobi identity read
cycl. A,B,C
(−1)ǫAǫC
(
→
∂l
∂ΓA
EBC) = 0 , (2.7)
cycl. A,B,C
(−1)(ǫA+1)(ǫC +1)
EAD
(
→
∂l
∂ΓD
EBC
) = 0 , (2.8)
respectively. The inverse matrix EAB with upper indices gives rise to the antibracket [1]
(F, G) = (F
←
∂r
∂ΓA
)EAB
(
→
∂l
∂ΓB
G) , (2.9)
which satisfies a graded skew-symmetry and a graded Jacobi identity as a consequence of eqs. (2.6)
and (2.8). There is an antisymplectic analogue of Darboux’s Theorem that states that locally there
exist Darboux coordinates ΓA ={φα; φ∗
α}, such that the only non-vanishing antibrackets between the
coordinates are (φα, φ∗
β) = δα
β = −(φ∗
β, φα). In Darboux coordinates the antisymplectic two-form is
simply E = dφ∗
α ∧ dφα.
3 Odd Laplacian ∆ρ on Scalars
A scalar function F =F(Γ), a density ρ=ρ(Γ) and a semidensity σ=σ(Γ) are by definition quantities
that transform as
F −→ F′
= F , ρ −→ ρ′
=
ρ
J
, σ −→ σ′
=
σ
√
J
, (3.1)
3
respectively, under general coordinate transformations ΓA → Γ′A, where J ≡ sdet∂Γ′A
∂ΓB denotes the
Jacobian. We shall ignore the global issues of orientation and choice of square root. In principle the
above F, ρ and σ could either be bosons or fermions, however normally we shall require the densities
ρ to be invertible, and therefore bosons.
Given a choice of density ρ one may define the odd Laplacian [4]
∆ρ :=
(−1)ǫA
2ρ
→
∂l
∂ΓA
ρEAB
→
∂l
∂ΓB
, (3.2)
that takes scalars to scalars of opposite Grassmann parity. The odd Laplacian (3.2) has a geometric
interpretation as a divergence of a Hamiltonian vector field [3, 11]
∆ρΨ = −
1
2
divρ(XΨ) , ǫ(Ψ) = 1 . (3.3)
Here XΨ := (Ψ, ·) denotes a Hamiltonian vector field with a Grassmann-odd Hamiltonian Ψ, and the
divergence divρX of a vector field X, with respect to the measure density ρ, is
divρX :=
(−1)ǫA
ρ
→
∂l
∂ΓA
(ρXA
) , ǫ(X) = 0 . (3.4)
The fact that the odd Laplacian (3.3) is non-zero, shows that antisymplectic manifolds do not have
an analogue of the Liouville Theorem mentioned in the Introduction. As a consequence of the Jacobi
identity eq. (2.8), the square operator ∆2
ρ = 1
2[∆ρ, ∆ρ] becomes a linear derivation, i.e. a first-order
differential operator,
∆2
ρ(FG) = ∆2
ρ(F) G + F ∆2
ρ(G) . (3.5)
Conventionally, one imposes additionally that the ∆ρ operator is nilpotent ∆2
ρ = 0, but this is not
necessary for our purposes.
4 Khudaverdian’s ∆E Operator on Semidensities
Khudaverdian showed that one may define a Grassmann-odd, nilpotent, second-order operator ∆E
without a choice of density ρ. This ∆E operator does not take scalars to scalars like the odd Laplacian
(3.2), but instead takes semidensities to semidensities of opposite Grassmann parity. Equivalently, the
∆E operator transforms as
∆E −→ ∆′
E =
1
√
J
∆E
√
J (4.1)
under general coordinate transformations ΓA → Γ′A, cf. eq. (3.1). Khudaverdian’s construction relies
first of all on an atlas of Darboux charts, which is granted by an antisymplectic analogue of Darboux’s
Theorem, and secondly, on a Lemma by Batalin and Vilkovisky about the possible form of the
Jacobians for anticanonical transformations, also known as antisymplectomorphisms.
Lemma 4.1 “The Batalin-Vilkovisky Lemma” [12, 10, 7, 8, 9, 13]. Consider a finite anticanonical
transformation between initial Darboux coordinates ΓA
(i) and final Darboux coordinates ΓA
(f). Then the
Jacobian J ≡ sdet(∂ΓA
(f)/∂ΓB
(i)) satisfies
∆
(i)
1
√
J = 0 . (4.2)
Here ∆
(i)
1 refers to the odd Laplacian (3.2) with ρ=1 in the initial Darboux coordinates ΓA
(i).
4
Given Darboux coordinates ΓA the ∆E operator is defined on a semidensity σ as [6, 7, 8, 9, 13]
(∆Eσ) := (∆1σ) , (4.3)
where ∆1 is the ∆ρ operator (3.2) with ρ=1. It is important in eq. (4.3) that the formula for the ∆1
operator (3.2) and the semidensity σ both refer to the same Darboux coordinates ΓA. The parentheses
in eq. (4.3) indicate that the equation should be understood as an equality among semidensities (in
the sense of zeroth-order differential operators) rather than an identity among differential operators.
One next uses the Batalin-Vilkovisky Lemma to argue that the definition (4.3) does not depend on
the choices of Darboux coordinates ΓA. What this means is, that the right-hand side of the definition
(4.3) transforms as a semidensity
(∆
(f)
1 σ(f)) =
1
√
J
(∆
(i)
1 σ(i)) (4.4)
under an anticanonical transformation between any two Darboux coordinates ΓA
(i) and ΓA
(f). Proof:
√
J(∆
(f)
1 σ(f)) =
√
J(∆
(i)
J σ(f)) =
√
J(∆
(i)
J
σ(i)
√
J
) = (∆
(i)
1 σ(i))−
1
√
J
(∆
(i)
1
√
J)σ(i) = (∆
(i)
1 σ(i)) . (4.5)
The third equality is a non-trivial property of the odd Laplacian (3.2). The Batalin-Vilkovisky Lemma
is used in the fourth equality. Strictly speaking, it is enough to consider infinitesimal anticanonical
transformations to justify the definition (4.3). The proof of the infinitesimal version of the BatalinVilkovisky
Lemma goes like this: An infinitesimal anticanonical coordinate transformation δΓA = XA
is necessarily a Hamiltonian vector field XA = (Ψ, ΓA) ≡ XA
Ψ with an infinitesimal, Grassmann-odd
Hamiltonian Ψ, where ǫ(Ψ) = 1. So
ln J ≈ (−1)ǫA
(
→
∂l
∂ΓA
XA
) = div1(XΨ) = − 2∆1Ψ , (4.6)
and hence
∆1
√
J ≈ − ∆2
1Ψ = 0 , (4.7)
due to the nilpotency of the ∆1 operator in Darboux coordinates. The “≈” sign is used to indicate that
equality only holds at the infinitesimal level. (Here we are guilty of mixing active and passive pictures;
the active vector field is properly speaking minus X.) A simple proof of the Batalin-Vilkovisky Lemma
for finite anticanonical transformations can be found in Ref. [13].
On the other hand, once the definition (4.3) is justified, it is obvious that the ∆E operator supercommutes
with itself, because the ΓA-derivatives have no ΓA’s to act on in Darboux coordinates.
Therefore ∆E is nilpotent,
∆2
E =
1
2
[∆E, ∆E] = 0 . (4.8)
Same sort of reasoning shows that ∆E = ∆T
E is symmetric.
5 The ∆E Operator in General Coordinates
We now give a definition of the ∆E operator that does not rely on Darboux coordinates. We claim
that in arbitrary coordinates the ∆E operator is given as
(∆Eσ) := (∆1σ) +
ν(1)
8
−
ν(2)
24
σ , (5.1)
5
where
ν(1)
:= (−1)ǫA
(
→
∂l
∂ΓB
→
∂l
∂ΓA
EAB
) , (5.2)
ν(2)
:= −(−1)ǫB
(ΓC
, (ΓB
, ΓA
))(
→
∂l
∂ΓA
EBC) = (−1)ǫAǫC
(
→
∂l
∂ΓA
ECD
)(
→
∂l
∂ΓD
EAB
)EBC . (5.3)
Eq. (5.1) is the main result of this paper. Notice that in Darboux coordinates, where EAB is constant,
i.e. independent of the coordinates ΓA, the last two terms ν(1) and ν(2) vanish. Hence the definition
(5.1) agrees in this case with Khudaverdian’s ∆E operator (4.3).
It remains to be shown that the right-hand side of eq. (5.1) behaves as a semidensity under general
coordinate transforms. Here we will only explicitly consider the case where σ is invertible to simplify
the presentation. (The non-invertible case is fundamentally no different.) In the invertible case, we
customarily write the semidensity σ =
√
ρ as a square root of a density ρ, and define a Grassmann-odd
quantity
νρ :=
1
√
ρ
(∆E
√
ρ) = ν(0)
ρ +
ν(1)
8
−
ν(2)
24
, (5.4)
by dividing both sides of the definition (5.1) with the semidensity σ. Here we have defined
ν(0)
ρ :=
1
√
ρ
(∆1
√
ρ) . (5.5)
Hence, to justify the definition (5.1), one should check that νρ is a scalar under general infinitesimal
coordinate transformations. Under an arbitrary infinitesimal coordinate transformation δΓA = XA,
one calculates
δν(0)
ρ = −
1
2
∆1div1X , (5.6)
δν(1)
= 4∆1div1X + (−1)ǫA
(
→
∂l
∂ΓC
EAB
)(
→
∂l
∂ΓB
→
∂l
∂ΓA
XC
) , (5.7)
δν(2)
= 3(−1)ǫA
(
→
∂l
∂ΓC
EAB
)(
→
∂l
∂ΓB
→
∂l
∂ΓA
XC
) , (5.8)
cf. Appendices A–C. One easily sees that while the three constituents ν
(0)
ρ , ν(1) and ν(2) separately
have non-trivial transformation properties, the linear combination νρ in eq. (5.4) is indeed a scalar.
The new definition (5.1) is clearly symmetric ∆E = ∆T
E and one may check that the nilpotency (4.8)
of the ∆E operator (5.1) precisely encodes the Jacobi identity (2.8). The odd Laplacian ∆ρ can be
expressed entirely by the ∆E operator and a choice of density ρ,
(∆ρF) = (∆1F) +
1
√
ρ
(
√
ρ, F) =
1
√
ρ
[
→
∆1, F]
√
ρ =
1
√
ρ
[
→
∆E, F]
√
ρ . (5.9)
Since ν(2) depends on the antisymplectic matrix EAB with lower indices, it is not clear how the formula
(5.1) extends to the degenerate anti-Poisson case.
6 Application to Batalin-Vilkovisky Quantization
It is interesting to transcribe the Batalin-Vilkovisky quantization, based on the odd Laplacian ∆ρ,
into a quantization scheme that is based on the ∆E operator, with the added benefit that no choice of
6
measure density ρ is needed. Since the ∆E operator takes semidensities to semidensities, this suggests
that the Boltzmann factor exp[ i
¯h WE] that appears in the Quantum Master Equation
∆E exp
i
¯h
WE = 0 (6.1)
should now be a semidensity, where
WE = S +
∞
n=1
(i¯h)n
Wn (6.2)
denotes the quantum action. In fact, this was a common interpretation (when restricting to Darboux
coordinates) prior to the introduction of a density ρ around 1992, see for instance Ref. [10], p.440-441.
If one only considers ¯h-independent coordinate transformations ΓA → Γ′A for simplicity, this implies
that the one-loop factor e−W1 is a semidensity, while the rest of the quantum action, i.e. the classical
action S and the higher loop corrections Wn, n ≥ 2, are scalars as usual. For instance, the nilpotent
operator F → eW1 ∆E(e−W1 F) takes scalars F to scalars.
At this stage it might be helpful to compare the above ∆E approach to the ∆ρ formalism. To this
end, fix a density ρ. Then one can define a bona fide scalar quantum action Wρ as
Wρ := WE + (i¯h) ln
√
ρ , (6.3)
or equivalently,
e
i
¯h
WE
=
√
ρ e
i
¯h
Wρ
. (6.4)
This scalar action Wρ satisfies the Modified Quantum Master Equation
(∆ρ + νρ) exp
i
¯h
Wρ = 0 , (6.5)
cf. eq. (5.4), (5.9), (6.1) and (6.4). One may obtain the Quantum Master Equation ∆ρ exp[ i
¯h Wρ] = 0
by additionally imposing the covariant condition νρ = 0, or equivalently ∆E
√
ρ = 0. However this step
is not necessary.
Returning now to the pure ∆E approach with no ρ, the finite ∆E-exact transformations of the form
e
i
¯h
W ′
E = e−[
→
∆E,Ψ]
e
i
¯h
WE
, (6.6)
play an important rˆole in taking solutions WE to the Quantum Master Equation (6.1) into new
solutions W′
E. It is implicitly understood that all objects in eq. (6.6) refer to the same (but arbitrary)
coordinate frame. In general, Ψ is a Grasmann-odd operator that takes semidensities to semidensities.
If Ψ is a scalar function (=zeroth-order operator), one derives
W′
E = eXΨ
WE + (i¯h)
eXΨ − 1
XΨ
∆EΨ . (6.7)
The formula (6.7) is similar to the usual formula in the ∆ρ formalism [13]. One may check that eq.
(6.7) is covariant with respect to general coordinate transformations.
The W-X formulation discussed in Ref. [5] and Ref. [13] carries over with only minor modifications,
since the ∆E operator is symmetric ∆T
E = ∆E. In short, the W-X formulation is a very general fieldantifield
formulation, based on two Master actions, WE and XE, each satisfying a Quantum Master
Equation. At the operational level, symmetric means that the ∆E operator, sandwiched between two
7
semidensities under a (path) integral sign, may be moved from one semidensity to the other, using
integration by part. This is completely analogous to the symmetry of the odd Laplacian ∆ρ = ∆T
ρ
itself. The XE quantum action is a gauge-fixing part,
XE = Gαλα
+ (i¯h)HE + O(λ∗
) , (6.8)
which contains the gauge-fixing constraints Gα in involution,
(Gα, Gβ) = GγUγ
αβ . (6.9)
The gauge-fixing functions Gα implement a generalization of the standard Batalin-Vilkovisky gaugefixing
procedure φ∗
α = ∂Ψ/∂φα. In the simplest cases, the gauge-fixing conditions Gα = 0 are enforced
by integration over the Lagrange multipliers λα. See Ref. [13] for further details on the W-X formulation.
The pertinent measure density in the partition function
Z = [dΓ][dλ] e
i
¯h
(WE+XE)
(6.10)
is now located inside the one-loop parts of the WE and the XE actions. For instance, an on-shell
expression for the one-loop factor e−HE is
e−HE
= J sdet(Fα, Gβ) , (6.11)
where J = sdet(∂¯ΓA/∂ΓB) denotes the Jacobian of the transformation ΓA → ¯ΓA and ¯ΓA ≡ {Fα; Gα}.
The formula (6.11) differs from the original square root formula [14, 15, 13] by not depending on a ρ
density, consistent with the fact that e−HE is no longer a scalar but a semidensity. We recall here the
main point that the one-loop factor e−HE is independent of the Fα’s and the partition function Z is
independent of the Gα’s in involution, cf. eq. (6.9).
To summarize, the density ρ can altogether be avoided in the field-antifield formalism, at the cost of
more complicated transformation rules. We stress that the above transcription has no consequences
for the physics involved. For instance, the ambiguity that existed in the density ρ is still present in
the choice of WE and XE.
Acknowledgement: The Author thanks I.A. Batalin and P.H. Damgaard for discussions. The
Author also thanks the organizers of the workshop “Gerbes, Groupoids, and Quantum Field Theory,
May–July 2006” at the Erwin Schr¨odinger Institute for warm hospitality. This work is supported by
the Ministry of Education of the Czech Republic under the project MSM 0021622409.
A Proof of eq. (5.6)
Consider a general (not necessarily infinitesimal) coordinate transformation ΓA → Γ′A between an
“unprimed” and an “primed” coordinate systems ΓA and Γ′A, respectively, cf. eq. (3.1). The primed
ν
(0)
ρ quantity (5.5) can be re-expressed with the help of the unprimed coordinates as
ν
′(0)
ρ′ :=
1
√
ρ′
(∆′
1 ρ′) =
1
√
ρ′
(∆J ρ′) =
1
√
ρ
(∆1
√
ρ) −
1
√
J
(∆1
√
J) = ν(0)
ρ − ν
(0)
J , (A.1)
where it is convenient (and natural) to introduce the quantity
ν
(0)
J :=
1
√
J
(∆1
√
J) (A.2)
8
with respect to the unprimed reference system. The third equality in eq. (A.1) uses a non-trivial
property of the odd Laplacian (3.2). In the infinitesimal case δΓA = XA, the expression for the
Jacobian J reduces to a divergence ln J ≈ div1X, and one calculates
δν(0)
ρ = ν
′(0)
ρ′ − ν(0)
ρ = − ν
(0)
J = − ∆1(ln
√
J) −
1
2
(ln
√
J, ln
√
J) ≈ −
1
2
∆1div1X , (A.3)
which is eq. (5.6).
B Proof of eq. (5.7)
The infinitesimal variation of ν(1) yields 4 contributions to linear order in the variation δΓA = XA,
δν(1)
= − δν
(1)
I − δν
(1)
II + δν
(1)
III + δν
(1)
IV . (B.1)
They are
δν
(1)
I := (−1)ǫA
(
→
∂l
∂ΓB
XC
)(
→
∂l
∂ΓC
→
∂l
∂ΓA
EAB
) , (B.2)
δν
(1)
II := (−1)ǫA
→
∂l
∂ΓB


(
→
∂l
∂ΓA
XC
)(
→
∂l
∂ΓC
EAB
)


 = δν
(1)
I + δν
(1)
V , (B.3)
δν
(1)
III := (−1)ǫA
→
∂l
∂ΓB
→
∂l
∂ΓA

(XA
←
∂r
∂ΓC
)ECB

 = δν
(1)
IV , (B.4)
δν
(1)
IV := (−1)ǫA
→
∂l
∂ΓB
→
∂l
∂ΓA


EAC
(
→
∂l
∂ΓC
XB
)


 = δν
(1)
I + δν
(1)
V + δν
(1)
V I , (B.5)
δν
(1)
V := (−1)ǫA
(
→
∂l
∂ΓC
EAB
)(
→
∂l
∂ΓB
→
∂l
∂ΓA
XC
) , (B.6)
δν
(1)
V I := (−1)ǫA
→
∂l
∂ΓA


EAC
→
∂l
∂ΓC
→
∂l
∂ΓB
XB
(−1)ǫB


 = 2∆1div1X , (B.7)
where we have noted various relations among the contributions. Altogether, the infinitesimal variation
of ν(1) becomes
δν(1)
= δν
(1)
V + 2δν
(1)
V I , (B.8)
which is eq. (5.7).
C Proof of eq. (5.8)
The infinitesimal variation of
ν(2)
:= (−1)ǫAǫC
(
→
∂l
∂ΓD
EAB
)EBC(
→
∂l
∂ΓA
ECD
) (C.1)
yields 8 contributions to linear order in the variation δΓA = XA, which may be organized as 2 × 4
terms
δν(2)
= 2(−δν
(2)
I − δν
(2)
II + δν
(2)
III + δν
(2)
IV ) , (C.2)
9
due to a (A, B) ↔ (D, C) symmetry in eq. (C.1). They are
δν
(2)
I := (−1)ǫAǫC
(
→
∂l
∂ΓD
EAB
)EBF (XF
←
∂r
∂ΓC
)(
→
∂l
∂ΓA
ECD
) , (C.3)
δν
(2)
II := (−1)ǫAǫC
(
→
∂l
∂ΓD
EAB
)EBC(
→
∂l
∂ΓA
XF
)(
→
∂l
∂ΓF
ECD
) , (C.4)
δν
(2)
III := (−1)ǫAǫC
(
→
∂l
∂ΓD
EAB
)EBC
→
∂l
∂ΓA

(XC
←
∂r
∂ΓF
)EF D

 = δν
(2)
I + δν
(2)
V , (C.5)
δν
(2)
IV := (−1)ǫAǫC
(
→
∂l
∂ΓD
EAB
)EBC
→
∂l
∂ΓA


ECF
(
→
∂l
∂ΓF
XD
)


 = δν
(2)
II + δν
(2)
V I , (C.6)
δν
(2)
V := (−1)ǫAǫC
EF D
(
→
∂l
∂ΓD
EAB
)EBC (
→
∂l
∂ΓA
XC
←
∂r
∂ΓF
) = − δν
(2)
V + δν
(2)
V I , (C.7)
δν
(2)
V I := (−1)ǫA
(
→
∂l
∂ΓC
EAB
)(
→
∂l
∂ΓB
→
∂l
∂ΓA
XC
) , (C.8)
where we have noted various relations among the contributions. The Jacobi identity (2.8) for EAB is
used in the second equality of eq. (C.7). Altogether, the infinitesimal variation of ν(2) becomes
δν(2)
= 3δν
(2)
V I , (C.9)
which is eq. (5.8).
References
[1] I.A. Batalin and G.A. Vilkovisky, Phys. Lett. 102B (1981) 27.
[2] K. Bering, physics/9711010.
[3] A. Schwarz, Commun. Math. Phys. 155 (1993) 249.
[4] I.A. Batalin and I.V. Tyutin, Int. J. Mod. Phys. A8 (1993) 2333.
[5] I.A. Batalin, K. Bering and P.H. Damgaard, Phys. Lett. B389 (1996) 673.
[6] O.M. Khudaverdian, math.DG/9909117.
[7] O.M. Khudaverdian and Th. Voronov, Lett. Math. Phys. 62 (2002) 127.
[8] O.M. Khudaverdian, Contemp. Math. 315 (2002) 199.
[9] O.M. Khudaverdian, Commun. Math. Phys. 247 (2004) 353.
[10] M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, Princeton Univ. Press, 1992.
[11] O.M. Khudaverdian and A.P. Nersessian, Mod. Phys. Lett. A8 (1993) 2377.
[12] I.A. Batalin and G.A. Vilkovisky, Nucl. Phys. B234 (1984) 106.
[13] I.A. Batalin, K. Bering and P.H. Damgaard, Nucl. Phys. B739 (2006) 389.
[14] I.A. Batalin and I.V. Tyutin, Mod. Phys. Lett. A9 (1994) 1707.
10
[15] O.M. Khudaverdian, prepared for the International Workshop On Geometry And Integrable
Models 4-8 October 1994, Dubna, Russia. In Geometry and Integrable models, (P.N. Pyatov and
S. Solodukhin, Eds.), World Scientific, 1996.
11
Paper II
Semidensities,
Second-Class Constraints
and Conversion
in Anti-Poisson Geometry
BY
K. Bering
J. Math. Phys. 49 (2008) 043516
arXiv:0705.3440
arXiv:0705.3440v3[hep-th]25Apr2008
Semidensities, Second-Class Constraints
and Conversion in Anti-Poisson Geometry
K. Bering1
Institute for Theoretical Physics & Astrophysics
Masaryk University
Kotl´aˇrsk´a 2
CZ-611 37 Brno
Czech Republic
February 24, 2013
Abstract
We consider Khudaverdian’s geometric version of a Batalin-Vilkovisky (BV) operator ∆E in the
case of a degenerate anti-Poisson manifold. The characteristic feature of such an operator (aside
from being a Grassmann-odd, nilpotent, second-order differential operator) is that it sends semidensities
to semidensities. We find a local formula for the ∆E operator in arbitrary coordinates.
As an important application of this setup, we consider the Dirac antibracket on an antisymplectic
manifold with antisymplectic second-class constraints. We show that the entire Dirac construction,
including the corresponding Dirac BV operator ∆ED , exactly follows from conversion of the
antisymplectic second-class constraints into first-class constraints on an extended manifold.
MCS number(s): 53A55; 58A50; 58C50; 81T70.
Keywords: Batalin-Vilkovisky Field-Antifield Formalism; Odd Laplacian; Anti-Poisson
Geometry; Semidensity; Second-Class Constraints; Conversion.
1E-mail: bering@physics.muni.cz
1
Contents
1 Introduction 2
2 Anti-Poisson Geometry 4
2.1 Antibracket and Compatible Two-Form . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Odd Laplacian ∆ρ on Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 The ∆E Operator on Semidensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 The ∆E Operator in General Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Nilpotency Condition for the odd Laplacian ∆ρ . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Alternative Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Second-Class Constraints 11
3.1 Review of Dirac Antibracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 The Dirac Operators ∆ED
and ∆ρ,ED
. . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Annihilation Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 Reparametrization of Second-Class Constraints . . . . . . . . . . . . . . . . . . . . . . 14
3.5 Nilpotency Condition for the odd Dirac Laplacian ∆ρ,ED
. . . . . . . . . . . . . . . . . 16
3.6 Dirac Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Conversion of Second-Class into First-Class 16
4.1 Extended Manifold Mext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 First-Class Constraints Ta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.4 The j-Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.5 Discussion of Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.6 The Conversion Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.7 Extended Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 Conclusions 21
A Proof of bi-Darboux Theorem 2.1 22
B Details from the Proof of Lemma 2.9 24
B.1 Proof of eq. (2.38) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
B.2 Proof of eq. (2.39) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
B.3 Proof of eq. (2.40) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
B.4 Proof of eq. (2.41) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
C Proof of Conversion Theorem 4.2 27
C.1 The ¯ Superdeterminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
C.2 Gauge Invariant Function ¯F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
C.3 Gauge Invariant Density ¯ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
C.4 Assembling the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
C.5 Lemma C.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1 Introduction
Consider an antisymplectic manifold (M; E) with coordinates ΓA. Such structure was first used by
Batalin and Vilkovisky to quantize Lagrangian gauge theories [1, 2, 3]. In general, antisymplectic
2
geometry has many of the characteristic features of ordinary symplectic geometry, e.g. the Jacobi
identity and the Darboux Theorem, but there are also important differences: There are no canonical
volume form and no Liouville Theorem in antisymplectic geometry [4]. In the covariant BatalinVilkovisky
(BV) formalism [5, 6] from around 1992 one is (among other things) instructed to make
separate choices of a measure density ρ=ρ(Γ) and a quantum action Wρ =Wρ(Γ). However, the
division into measure and action part is to a large extent an arbitrary division, i.e. it is always possible
to shift parts of the measure ρ into the action Wρ and vice versa. It is only a particular combination
of these two quantities, namely the Boltzmann semidensity
exp[
i
¯h
WE] ≡
√
ρ exp[
i
¯h
Wρ] (1.1)
that enters the physical partition function Z. For instance, if there exist global Darboux coordinates
ΓA ={φα; φ∗
α}, the partition function reads
Z = [dφ] exp[
i
¯h
WE]
φ∗= ∂ψ
∂φ
, (1.2)
where ψ=ψ(φ) is the gauge fermion. (More generally, the partition function Z is described by the
so-called W-X formalism [7, 8].) The field-antifield formalism was reformulated in Ref. [9] entirely in
the minimal language of semidensities, which skips ρ altogether. According to this minimal approach,
the Boltzmann semidensity exp[ i
¯h WE] should satisfy the Quantum Master Equation
∆E exp[
i
¯h
WE] = 0 (1.3)
to ensure independence of gauge-fixing. Here ∆E is Khudaverdian’s BV operator, which takes semidensities
to semidensities, cf. Ref. [8, 10, 11, 12, 13] and Definition 2.3 below. Of course, the density
ρ may always be re-introduced to compare with the 1992 formulation. In doing so, for an arbitrary
choice of ρ,
1. the Boltzmann semidensity exp[ i
¯h WE] descend to a Boltzmann scalar exp[ i
¯h Wρ] = exp[ i
¯h WE]/
√
ρ,
2. the ∆E operator descend to a (not necessarily nilpotent) odd Laplacian ∆ρ, which takes scalars
to scalars, cf. Definition 2.2 below; and
3. the Quantum Master Eq. (1.3) descend to the Modified Quantum Master Equation
(∆ρ + νρ) exp[
i
¯h
Wρ] = 0 , (1.4)
where νρ is an odd scalar, cf. Definition 2.8 below.
We emphasize that this construction works for any ρ. However, to arrive at the 1992 formulation
[5, 6], which has νρ =0 and a nilpotent odd Laplacian ∆2
ρ =0, one should impose conditions on ρ.
The paper is organized as follows. Anti-Poisson geometry is reviewed in Section 2. The notions of
compatible two-form fields and bi-Darboux coordinates are introduced in Subsection 2.1. A new Theorem
2.1 provides necessary and sufficient conditions for the existence of bi-Darboux coordinates. The
definition of the ∆E operator for a degenerate anti-Poisson structure E is given using both Darboux
and general coordinates in Subsection 2.3 and 2.4, respectively. The ∆E formula in general coordinates
does require the existence of a compatible two-form fields, however, it does not matter which
compatible two-form field that is used (in case there is more than one choice), cf. Lemma 2.7. All
information about how the ∆E operator acts on semidensities can be packed into a Grassmann-odd
3
scalar quantity νρ, which already appeared in eq. (1.4) above. The odd scalar νρ is important, because
in practice it is easier to handle a scalar object rather than the full second-order differential operator
∆E, and hence many of the ensuring arguments is performed using νρ. The Dirac antibracket is an
important application of the geometric setup from Section 2, since it always admits a compatible twoform
field. Antisymplectic second-class constraints and the Dirac antibracket [6, 8, 14] are reviewed
in Subsection 3.1. A Proposition 3.1 in Subsection 3.2 provides a useful formula for the corresponding
Dirac odd scalar νρ,ED
. Subsection 3.4 discusses the stability of the Dirac construction under
reparameterizations of the second-class constraints. In Section 4 the Dirac construction is derived via
conversion [15, 16, 17, 18, 19] of the antisymplectic second-class constraints into first-class constraints
on an extended manifold. As an application of the construction to Batalin-Vilkovisky quantization,
the corresponding Dirac and extended partition functions are provided in Subsections 3.6 and 4.7,
respectively. Finally, Section 5 contains our conclusions.
General remark about notation. We have two types of grading: A Grassmann grading ε and an exterior
form degree p. The sign conventions are such that two exterior forms ξ and η, of Grassmann parity
εξ, εη and exterior form degree pξ, pη, respectively, commute in the following graded sense
η ∧ ξ = (−1)εξ
εη+pξpη
ξ ∧ η (1.5)
inside the exterior algebra. We will often not write the exterior wedges “∧” explicitly.
2 Anti-Poisson Geometry
2.1 Antibracket and Compatible Two-Form
We consider an anti-Poisson manifold (M; EAB) with a (possibly degenerate) antibracket
(F, G) = (F
←
∂r
A)EAB
(
→
∂l
BG) = − (−1)(εF +1)(εG+1)
(G, F) ,
→
∂l
A ≡
→
∂l
∂ΓA
, (2.1)
Here the ΓA’s denote local coordinates of Grassmann parity εA ≡ ε(ΓA), and EAB =EAB(Γ) is the
local matrix representation of the anti-Poisson structure E. The Jacobi identity
cycl. F,G,H
(−1)(εF +1)(εH +1)
(F, (G, H)) = 0 (2.2)
reads in local coordinates
cycl. A,B,C
(−1)(εA+1)(εC +1)
EAD
(
→
∂l
DEBC
) = 0 . (2.3)
The main new feature (as compared to Ref. [9]) is that the anti-Poisson structure EAB could be
degenerate. There is an anti-Poisson analogue of Darboux’s Theorem that states that locally, if the
rank of EAB is constant, there exist Darboux coordinates ΓA ={φα; φ∗
α; Θa}, such that the only nonvanishing
antibrackets between the coordinates are (φα, φ∗
β) = δα
β = −(φ∗
β, φα). In other words, the
Jacobi identity is the integrability condition for the Darboux coordinates. The variables φα, φ∗
α and
Θa are called fields, antifields and Casimirs, respectively.
We shall assume that the anti-Poisson manifold (M; EAB) admits a globally defined odd two-form
field EAB with lower indices that is compatible with the anti-Poisson structure EAB in the sense that
EAB
EBCECD
= EAD
,
4
EABEBC
ECD = EAD . (2.4)
As always, the matrices EAB and EAB are assumed to have the Grassmann gradings
ε(EAB
) = εA + εB + 1 = ε(EAB) , (2.5)
and the skew-symmetries
EBA
= −(−1)(εA+1)(εB +1)
EAB
,
EBA = −(−1)εAεB EAB . (2.6)
The odd two-form field can be written as
E =
1
2
dΓA
EAB dΓB
= −
1
2
EAB dΓB
dΓA
. (2.7)
The two-form field EAB would be closed if
dE = 0 , (2.8)
or equivalently, with all the indices written out, if
cycl. A,B,C
(−1)εAεC (
→
∂l
AEBC) = 0 . (2.9)
A closed degenerate two-form is called a pre-antisymplectic structure. In the non-degenerate case,
the matrix EAB from eq. (2.4) would be a closed antisymplectic two-form field and the inverse of
the anti-Poisson structure EAB. In the degenerate case, there is in general not a unique matrix EAB
fulfilling eqs. (2.4), (2.5) and (2.6), and there is no reason for it to be closed. In Darboux coordinates
ΓA ={φα; φ∗
α; Θa}, there is still a freedom in a compatible two-form
E = dφ∗
α ∧ dφα
+ dΘa
Maα ∧ dφα
+ dφ∗
αNα
a ∧ dΘa
+ dΘa
MaαNα
b ∧ dΘb
(2.10)
given by two arbitrary matrices Maα =Maα(Γ) and Nα
a =Nα
a(Γ). A Darboux coordinate system
ΓA ={φα; φ∗
α; Θa} is called a bi-Darboux coordinate system, if the two-form is just E = dφ∗
α ∧ dφα,
i.e. if both the matrices Maα =0 and Nα
a =0 in eq. (2.10) are equal to zero. In short, the ΓA’s are
bi-Darboux coordinates, if both matrices EAB and EAB with upper and lower indices are on standard
form.
Theorem 2.1 Given an anti-Poisson manifold (M; EAB) with a compatible two-form field EAB. Then
there locally exist bi-Darboux coordinates if and only if the two-form field EAB is closed.
There is a similar Bi-Darboux Theorem for even Poisson structures. A proof of Theorem 2.1 is given
in Appendix A. One can define a projection as
PA
C ≡ EAB
EBC , (2.11)
or equivalently,
PA
C
≡ EABEBC
= (−1)εA(εC +1)
PC
A . (2.12)
It follows from property (2.4) that
PA
BPB
C = PA
C . (2.13)
In the non-degenerate case PA
B = δA
B = PB
A.
5
2.2 Odd Laplacian ∆ρ on Scalars
Recall that a scalar function F =F(Γ), a density ρ=ρ(Γ) and a semidensity σ=σ(Γ) are by definition
quantities that transform as
F −→ F′
= F , ρ −→ ρ′
=
ρ
J
, σ −→ σ′
=
σ
√
J
, (2.14)
respectively, under general coordinate transformations ΓA → Γ′A, where J ≡ sdet∂Γ′A
∂ΓB denotes the
Jacobian. We shall ignore the global issues of orientation and choice of square root. Also we assume
that densities ρ are invertible.
Definition 2.2 Given a choice of a density ρ, the odd Laplacian ∆ρ is defined as [6]
∆ρ ≡
(−1)εA
2ρ
→
∂l
AρEAB
→
∂l
B . (2.15)
This Grassmann-odd, second-order operator takes scalar functions to scalar functions. In situations
with more than one anti-Poisson structure EAB, we shall sometimes use the slightly longer notation
∆ρ ≡ ∆ρ,E to acknowledge that it depends on two inputs: ρ and EAB. The odd Laplacian ∆ρ
“differentiates” the antibracket (·, ·), i.e. the following Leibniz-type rule holds
∆ρ(F, G) = (∆ρF, G) + (−1)(εF +1)
(F, ∆ρG) . (2.16)
For further information on this important operator, see Ref. [8, 9] and Subsection 2.5 below.
2.3 The ∆E Operator on Semidensities
There is an another important Grassmann-odd, nilpotent, second-order operator ∆E that depends only
on the anti-Poisson structure EAB. Contrary to the odd Laplacian ∆ρ ≡ ∆ρ,E of last Subsection 2.2,
the ∆E operator does not rely on a choice of density ρ. The caveat is that while the odd Laplacian ∆ρ
takes scalars to scalars, the ∆E operator takes semidensities to semidensities of opposite Grassmann
parity. Equivalently, the ∆E operator transforms as
∆E −→ ∆′
E =
1
√
J
∆E
√
J (2.17)
under general coordinate transformations ΓA → Γ′A, cf. eq. (2.14). It is defined as follows:
Definition 2.3 Let there be given an anti-Poisson manifold (M; E). In Darboux coordinates ΓA, the
∆E operator is defined on a semidensity σ as [8, 10, 11, 12, 13]
(∆Eσ) ≡ (∆1σ) , (2.18)
where ∆1 denotes the expression (2.15) for the odd Laplacian ∆ρ=1 with ρ replaced by 1.
It is implicitly understood in eq. (2.18) that the formula for the ∆1 operator (2.15) and the semidensity
σ both refer to the same Darboux coordinates ΓA. The parentheses in eq. (2.18) indicate that
the equation should be understood as an equality among semidensities (in the sense of zeroth-order
differential operators) rather than an identity among differential operators. The Definition 2.3 does
not depend on the Darboux coordinate system being used, due to the following Lemma 2.4:
6
Lemma 2.4 When using the Definition 2.3, the (∆Eσ) transforms as a semidensity under (anticanonical)
transformations between sets of Darboux coordinates.
Thus the ∆E operator is a well-defined operator on an open cover of Darboux neighborhoods. Within
this cover, the ∆E is indirectly defined in non-Darboux coordinates by use of the transformation
property (2.17). Lemma 2.4 was first proven in the non-degenerate case in Ref. [13] and in the
degenerate case in Ref. [8]. We shall also give an independent proof in the next Subsection 2.4, cf.
Lemma 2.6 below. In some cases the ∆E operator may be extended to singular points (i.e. points
where the rank of the anti-Poisson tensor EAB jumps) by continuity.
Working in Darboux coordinates, it is obvious that the ∆E operator super-commutes with itself,
because the ΓA-derivatives have no ΓA’s to act on when EAB is on Darboux form. Therefore ∆E is
nilpotent,
∆2
E =
1
2
[∆E, ∆E] = 0 . (2.19)
Same sort of reasoning shows that ∆E = ∆T
E is symmetric.
2.4 The ∆E Operator in General Coordinates
We now give a definition of the ∆E operator that does not refer to Darboux coordinates.
Definition 2.5 Given an anti-Poisson manifold (M; EAB) that admits a compatible two-form field
EAB. In arbitrary coordinates ΓA, the ∆E operator is defined as
(∆Eσ) ≡ (∆1σ) +
ν(1)
8
−
ν(2)
8
−
ν(3)
24
+
ν(4)
24
+
ν(5)
12
σ , (2.20)
where
ν(1)
≡ (−1)εA (
→
∂l
B
→
∂l
AEAB
) , (2.21)
ν(2)
≡ (−1)εAεC (
→
∂l
DEAB
)EBC(
→
∂l
AECD
) , (2.22)
ν(3)
≡ (−1)εB (
→
∂l
AEBC)ECD
(
→
∂l
DEBA
) , (2.23)
ν(4)
≡ (−1)εB (
→
∂l
AEBC)ECD
(
→
∂l
DEBF
)PF
A
, (2.24)
ν(5)
≡ (−1)εAεC (
→
∂l
DEAB
)EBC(
→
∂l
AECF
)PF
D
= (−1)(εA+1)εB EAD
(
→
∂l
DEBC
)(
→
∂l
CEAF )PF
B . (2.25)
Notice that in Darboux coordinates, where EAB is constant, i.e. independent of the coordinates ΓA,
the last five terms ν(1), ν(2), ν(3), ν(4) and ν(5) become zero. Hence the new Definition 2.5 agrees in
Darboux coordinates with the previous Definition 2.3. The benefit of the new Definition 2.5 is that
one now have an explicit formula for ∆E in an arbitrary coordinate system. The full justification of
Definition 2.5 is provided by the following Lemma 2.6 and Lemma 2.7.
Lemma 2.6 When using the new Definition 2.5, the (∆Eσ) transforms as a semidensity under general
coordinate transformations.
7
Lemma 2.7 When using the new Definition 2.5, the (∆Eσ) does not depend on the compatible twoform
field EAB used.
The explicit formula (2.20) and Lemma 2.6 are the main results of Section 2.
Proof of Lemma 2.7: The two-form field EAB enters only the Definition 2.5 via ν(2), ν(3), ν(4) and
ν(5). Assuming the Lemma 2.6, i.e. that the behavior (2.17) under general coordinate transformations
has already been established, one may, in particular, go to Darboux coordinates, where ν(2), ν(3), ν(4)
and ν(5) vanish identically.
To prove Lemma 2.6 we shall first reformulate it as an equivalent Lemma 2.9, cf. below. We shall also
only explicitly consider the case where σ is invertible to simplify the presentation. (The non-invertible
case is fundamentally no different.) In the invertible case, we customarily write the semidensity σ =
√
ρ
as a square root of a density ρ, and define a Grassmann-odd quantity νρ as follows.
Definition 2.8 The odd scalar νρ is defined as
νρ ≡
1
√
ρ
(∆E
√
ρ) = ν(0)
ρ +
ν(1)
8
−
ν(2)
8
−
ν(3)
24
+
ν(4)
24
+
ν(5)
12
, (2.26)
where ν(1), ν(2), ν(3), ν(4), ν(5) are given in eqs. (2.21)–(2.25), and the quantity ν
(0)
ρ is given as
ν(0)
ρ ≡
1
√
ρ
(∆1
√
ρ) . (2.27)
In situations with more than one anti-Poisson structure EAB, we shall sometimes use the slightly
longer notation νρ ≡ νρ,E. By dividing both sides of the definition (2.20) with the semidensity σ, one
may reformulate the content of Lemma 2.6 as:
Lemma 2.9 The Grassmann-odd quantity νρ is a scalar, i.e. it does not depend on the coordinate
system.
We shall give two independent proofs of this important Lemma 2.9; one relying on Darboux Theorem
and the other using infinitesimal coordinate transformations.
Proof of Lemma 2.9 using a Darboux coordinate patch: It is enough to consider how
νρ behaves on coordinate transformations ΓA
0 → ΓA between Darboux coordinates ΓA
0 and general
coordinates ΓA. (An arbitrary coordinate transformation between two general coordinate patches can
always be split into two successive coordinate transformations of the above kind by inserting a third
Darboux coordinate patch in between.) The idea is now to first consider the expression (2.26) for νρ in
the ΓA coordinate system, and decompose it in building blocks that refer to the Darboux coordinates
ΓA
0 , e.g.
EAD
= (ΓA
←
∂r
∂ΓB
0
)EBC
0 (
→
∂l
∂ΓC
0
ΓD
) , EAD = (
→
∂l
∂Γa
ΓB
0 )E 0
BC(ΓC
0
←
∂r
∂ΓD
) , ρ =
ρ0
J
. (2.28)
8
Here J ≡ sdet(∂ΓA/∂ΓB
0 ) denotes the Jacobian of the coordinate transformations ΓA
0 → ΓA. Recall
that the two-form field E 0
BC is not necessarily constant in the Darboux coordinates ΓA
0 , cf eq. (2.10).
By straightforward calculation, one gets
ν(0)
ρ ≡
1
√
ρ
(∆1,E
√
ρ) =
1
√
ρ
(∆J,E0
√
ρ) =
1
√
ρ0
(∆1,E0
√
ρ0) −
1
√
J
(∆1,E0
√
J) , (2.29)
ν(1)
=
8
√
J
(∆1,E0
√
J) − (−1)εB (
→
∂l
∂ΓA
0
ΓB
,
→
∂l
∂ΓB
ΓA
0 ) , (2.30)
ν(2)
= −(−1)εB (
→
∂l
∂ΓA
0
ΓB
,
→
∂l
∂ΓB
ΓA
0 ) − 2(−1)εB (
→
∂l
∂ΓA
0
ΓB
,
→
∂l
∂ΓB
ΓC
0 )P0,A
C , (2.31)
ν(3)
= 3(−1)εB (
→
∂l
∂ΓA
0
ΓB
,
→
∂l
∂ΓB
ΓC
0 )P0,A
C
−(−1)(εA+1)(εC +1)
(ΓA
0
←
∂r
∂ΓB
)(ΓB
←
∂r
∂ΓC
0
, E 0
AD)EDC
0 , (2.32)
ν(4)
= (−1)εB (
→
∂l
∂ΓA
0
ΓB
,
→
∂l
∂ΓB
ΓC
0 )P0,A
C + 2(−1)εAεC P0,B
A (
→
∂l
∂ΓB
0
ΓC
, ΓA
0
←
∂r
∂ΓD
)PD
C
−(−1)(εA+1)(εC +1)
(ΓA
0
←
∂r
∂ΓB
)(ΓB
←
∂r
∂ΓC
0
, E 0
AD)EDC
0 , (2.33)
ν(5)
= −2(−1)εB (
→
∂l
∂ΓA
0
ΓB
,
→
∂l
∂ΓB
ΓC
0 )P0,A
C − (−1)εAεC P0,B
A (
→
∂l
∂ΓB
0
ΓC
, ΓA
0
←
∂r
∂ΓD
)PD
C . (2.34)
The last equality in eq. (2.29) is a non-trivial property of the odd Laplacian. It is now easy to check
that all but one of the above terms on the right-hand sides of eqs. (2.29)–(2.34) cancel in the pertinent
linear combination (2.26), i.e.
νρ = ν(0)
ρ +
ν(1)
8
−
ν(2)
8
−
ν(3)
24
+
ν(4)
24
+
ν(5)
12
=
1
√
ρ0
(∆1,E0
√
ρ0) . (2.35)
The surviving term, on the other hand, is just the definition for νρ in the Darboux coordinates ΓA
0 .
Proof of Lemma 2.9 using infinitesimal coordinate transformations: Under an arbitrary
infinitesimal coordinate transformation δΓA = XA, one calculates
δν(0)
ρ = −
1
2
∆1div1X , (2.36)
δν(1)
= 4∆1div1X + (−1)εA (
→
∂l
CEAB
)(
→
∂l
B
→
∂l
AXC
) , (2.37)
δν(2)
= (−1)εA (
→
∂l
DEAB
) 2PB
C
(
→
∂l
C
→
∂l
AXD
) + (
→
∂l
B
→
∂l
AXC
)PC
D
, (2.38)
δν(3)
= (−1)εB (
→
∂l
AEBC)ECD
(
→
∂l
DXB
←
∂r
F )EF A
− (−1)(εA+1)(εB+1)
(
→
∂l
DXA
←
∂r
F )EF B
−
3
2
(−1)εA PC
D
(
→
∂l
DEAB
)(
→
∂l
B
→
∂l
AXC
) , (2.39)
δν(4)
= −2(−1)εB (
→
∂l
A
→
∂l
BXC
)PC
D
(
→
∂l
DEBF
)PF
A
9
+(−1)εB (
→
∂l
AEBC)ECD
(
→
∂l
DXB
←
∂r
F )EF A
+(−1)(εB +1)εF PF
A
(
→
∂l
AEBC)ECD
(
→
∂l
DXF
←
∂r
G)EGB
+
1
2
(−1)εA PC
D
(
→
∂l
DEAB
)(
→
∂l
B
→
∂l
AXC
) , (2.40)
δν(5)
= −(−1)εA(εB+1)
(
→
∂l
AEBC)ECD
(
→
∂l
DXA
←
∂r
F )EF B
+2(−1)εB (
→
∂l
A
→
∂l
BXC
)PC
D
(
→
∂l
DEBF
)PF
A
. (2.41)
A proof of eqs. (2.36) and (2.37) can be found in Ref. [9], and eqs. (2.38)–(2.41) are proven in Appendix
B. One may verify that while the six constituents ν
(0)
ρ , ν(1), ν(2), ν(3), ν(4) and ν(5) separately
have non-trivial transformation properties, the linear combination νρ in eq. (2.26) is indeed a scalar.
The new Definition 2.5 is clearly symmetric ∆E = ∆T
E. To check explicitly in general coordinates that
∆E is nilpotent is a straightforward (but admittedly tedious) exercise. However, since we have just
proven that ∆E behaves covariantly under general coordinate transformations, our previous proof of
nilpotency from last Subsection 2.3 using Darboux coordinates suffices. To summarize:
Theorem 2.10 The ∆E operator (2.20) is nilpotent (2.19) if and only if the antibracket (2.1) satisfies
the Jacobi identity (2.3).
In the rest of the paper we will always assume that the Jacobi identity (2.3) is satisfied, and hence
that the ∆E operator (2.20) is nilpotent.
2.5 Nilpotency Condition for the odd Laplacian ∆ρ
At this point it is instructive to recall the nilpotency condition for the odd Laplacian ∆ρ, although
we shall not assume that it is satisfied. It follows from the Jacobi identity (2.3) alone, that ∆2
ρ is a
linear derivation, i.e. a first-order differential operator. The interplay between the two second-order
differential operators ∆E and ∆ρ is perhaps best summarized by the following operator identity:
∆ρ + νρ =
1
√
ρ
∆E
√
ρ , (2.42)
cf. eq. (5.9) of Ref. [9]. In words: Apart from the νρ term the odd Laplacian ∆ρ is the ∆E operator
dressed with a
√
ρ factor. From this operator identity (2.42) and the nilpotency (2.19) of the ∆E
operator, one derives the explicit form of the linear derivation:
∆2
ρ = (νρ, · ) . (2.43)
Therefore the nilpotency condition for ∆ρ reads [8, 11]
∆2
ρ = 0 ⇔ νρ is a Casimir. (2.44)
Let us also mention for later that if one acts with the operator identity (2.42) on a scalar function√
F, one gets
νρF = νρ +
1
√
F
(∆ρ
√
F) . (2.45)
10
2.6 Alternative Expressions
It is convenient to introduce
ν(23)
≡ ν(2) + ν(3) = (−1)εB (
→
∂l
APB
C
)(
→
∂l
CEBA
) , (2.46)
ν(35)
≡ ν(3) + ν(5) = (−1)εB (
→
∂l
APB
C
)PC
D
(
→
∂l
DEBA
)
= (−1)εB (
→
∂l
APB
C)ECD
(
→
∂l
DPB
A
) , (2.47)
ν(45)
≡ ν(4) + ν(5) = (−1)εB PA
D
(
→
∂l
DPB
C
)(
→
∂l
CEBA
) , (2.48)
ν
(23)
(45) ≡ ν(23) − ν(45) = (−1)εB(εD+1)
(
→
∂l
APB
C)ECD
(
→
∂l
BPD
A
) . (2.49)
Then the ∆E operator (2.20) may be re-written as
(∆Eσ) = (∆1σ) +
ν(1)
8
−
ν(2)
24
−
ν(23)
12
+
ν(35) + ν(45)
24
σ , (2.50)
= (∆1σ) +


ν(1)
8
−
ν(2) + ν(23) − ν(35) + ν
(23)
(45)
24

 σ . (2.51)
In the closed case (2.8) one may show that
ν(35)
+ ν(45)
= 0 , (2.52)
so that the ∆E operator (2.50) simplifies to
(∆Eσ) = (∆1σ) +
ν(1)
8
−
ν(2)
8
−
ν(3)
12
σ . (2.53)
In the non-degenerate case, which is automatically closed, one also has
ν(23)
= 0 , (2.54)
so that the ∆E operator (2.50) simplifies even further to
(∆Eσ) = (∆1σ) +
ν(1)
8
−
ν(2)
24
σ , (2.55)
in agreement with eq. (5.1) in Ref. [9].
3 Second-Class Constraints
3.1 Review of Dirac Antibracket
One of the most important examples of degenerate anti-Poisson structures is provided by the Dirac
antibracket [6, 8, 14]. Consider a manifold (M; E) with a non-degenerate anti-Poisson structure EAB
(called an antisymplectic phase space), and let a submanifold ˜M ≡ {Γ ∈ M|Θ(Γ) = 0} be the zerolocus
of a set of constraints Θa =Θa(Γ) with Grassmann parity ε(Θa)=εa. (In this Subsection, the
defining set of constraints is kept fixed for simplicity. We will consider reparametrizations of the
11
constraints in Subsection 3.4.) Assume that the Θa constraints are second-class in the antibracket
sense, i.e. the antibracket matrix
Eab
≡ (Θa
, Θb
) (3.1)
of the Θa constraints has by definition an inverse matrix Eab,
EabEbc
= δc
a . (3.2)
The Dirac antibracket is defined completely analogous to the usual Dirac bracket for even Poisson
brackets [6],
(F, G)D ≡ (F, G) − (F, Θa
)Eab(Θb
, G) , (3.3)
or in coordinates,
EAB
(D) ≡ EAB
− (ΓA
, Θa
)Eab(Θb
, ΓB
) . (3.4)
The Dirac antibracket satisfies a strong Jacobi identity
F,G,H cycl.
(−1)(εF +1)(εH +1)
((F, G)D, H)D = 0 . (3.5)
The adjective “strong” stresses the fact that the Jacobi identity holds off-shell with respect to the
second-class constraints Θa, i.e. everywhere in the phase space M. There is a canonical Dirac two-form
given by
ED
≡ E −
1
2
dΘa
Eab ∧ dΘb
, (3.6)
or in local coordinates
E
(D)
AB ≡ EAB − (
→
∂l
AΘa
)Eab(Θb
←
∂r
B) . (3.7)
The two-form field E
(D)
AB is compatible with the Dirac bracket, i.e. it satisfies the property (2.4), but
it is not necessarily closed. Local coordinates ΓA = {γA; Θa}, where the second-class constraints Θa
are part of the coordinates, are called unitarizing coordinates. In the physics terminology, the secondclass
constraints Θa represent unphysical degrees of freedom, which can be eliminated from the system,
i.e. put to zero, to reveal a reduced submanifold ˜M, whose coordinates γA constitute the true physical
degrees of freedom. Notation: We use capital roman letters A, B, C, . . . from the beginning of the
alphabet as upper index for both the full and the reduced variables ΓA and γA, respectively. A tilde
“∼” over an object will denote the corresponding reduced object.
Unitarizing coordinates ΓA = {γA; Θa}, where the second-class variables Θa and the physical variables
γA are perpendicular to each other in the antibracket sense
(γA
, Θa
) = 0 , (3.8)
are called transversal coordinates. One may prove that transversal coordinate systems exist locally,
although one might have to reparametrize the Θa constraints in order to get to them, cf. Subsection 3.4
below.
3.2 The Dirac Operators ∆ED
and ∆ρ,ED
The next step is to build Khudaverdian’s BV operator ∆ED
for the degenerate Dirac antibracket
structure (3.3), and, if a density ρ is available, the odd Laplacian ∆ρ,ED
. In other words, one should
substitute E → ED everywhere in the previous Section 2. Some facts about the ∆ED
operator are
12
immediately clear. First of all, it is covariant under general coordinate transformations, cf. Subsection
2.4. Furthermore, it is strongly nilpotent
∆2
ED
= 0 , (3.9)
due to the strong Jacobi identity (3.5) and Theorem 2.10. The following Proposition 3.1 expresses the
Dirac odd scalar νρ,ED
in terms of the non-degenerate antisymplectic structure and the second-class
constraints Θa.
Proposition 3.1 The Dirac odd scalar νρ,ED
is given by
νρ,ED
= νρ −
ν
(6)
ρ,D
2
−
ν
(7)
ρ,D
2
−
ν
(8)
D
8
+
ν
(9)
D
24
, (3.10)
where νρ ≡ νρ,E is the odd scalar for the non-degenerate antisymplectic structure E, and
ν
(6)
ρ,D ≡ (∆ρΘa
)Eab(∆ρΘb
)(−1)εb , (3.11)
ν
(7)
ρ,D ≡ (−1)εa+εb (Θa
, Eab(∆ρΘb
)) = (Θa
, (∆ρΘb
)Eba) , (3.12)
ν
(8)
D ≡ (−1)εb (Θa
, (Θb
, Eba)) , (3.13)
ν
(9)
D ≡ (−1)(εa+1)(εd
+1)
(Θd
, Eab)Ebc
(Ecd, Θa
)
= −(−1)εb (Θa
, Ebc
)Ecd(Θd
, Eba) . (3.14)
Proof of Proposition 3.1: Since both sides of eq. (3.10) are scalars under general coordinate
transformations, it is sufficient to work in Darboux coordinates for the non-degenerate EAB structure.
By straightforward calculation, one gets
ν
(0)
ρ,D = ν(0)
ρ − (∆1Θa
)Eab(Θb
, ln
√
ρ) −
(−1)εa
2
√
ρ
(Θa
, Eab(Θb
,
√
ρ)) , (3.15)
ν
(1)
D = −4(∆1Θa
)Eab(∆1Θb
)(−1)εb − 4(−1)εa+εb (Θa
, Eab(∆1Θb
))
−ν
(8)
D − (−1)εa (
→
∂l
AΘa
, Eab)(Θb
, ΓA
) , (3.16)
ν
(2)
D = (−1)εbεc Eca(Θa
, Θb
←
∂r
B)EBC
(D)(
→
∂l
CΘc
, Θd
)Edb
= −(−1)εa (Θb
, Θa
←
∂r
A)(ΓA
, Eab)D = − (−1)εa (Θb
, Θa
←
∂r
A)(ΓA
, Eab) −
ν
(9)
D
3
, (3.17)
ν
(3)
D = 0 , (3.18)
ν
(4)
D = 0 , (3.19)
ν
(5)
D = 0 . (3.20)
The pertinent linear combination (2.26) of eqs. (3.15)–(3.20) yields the eq. (3.10).
3.3 Annihilation Relations
The fact that the Θa constraints are null-directions for the Dirac construction is reflected slightly
differently in 1) the Dirac antibracket (·, ·)D, 2) the Dirac odd Laplacian ∆ρ,ED
, and 3) the ∆ED
13
operator. Explicitly, for a scalar function F, a density ρ and a semidensity σ, one has
(F, Θa
)D = 0 , (3.21)
(∆ρ,ED
Θa
) =
(−1)εA
2ρ
→
∂l
Aρ(ΓA
, Θa
)D = 0 , (3.22)
[
→
∆ED
, Θa
]σ = [
→
∆1,ED
, Θa
]σ = (∆1,ED
Θa
)σ + (−1)εa (Θa
, σ)D = 0 , (3.23)
respectively. Eqs. (3.21)–(3.23) generalize to
(F, f(Θ))D = 0 , (3.24)
(∆ρ,ED
f(Θ)) = 0 , (3.25)
[
→
∆ED
, f(Θ)]σ = 0 , (3.26)
for an arbitrary function f(Θ) of the constraints Θa. (In other words: f is here assumed not to depend
on the physical variables γA.) Note however, that if Θa is not among the defining set of constraints,
but only a linear combination of those (i.e. the coefficients in the linear combination could involve
the physical variables γA), the last equality in each of the above eqs. (3.21)–(3.26) becomes weak,
i.e. there could be off-shell contributions, cf. next Subsection 3.4 and Ref. [8].
3.4 Reparametrization of Second-Class Constraints
A general and tricky feature of the Dirac construction, is, that it changes if one uses another defining
set of second-class constraints
Θa
−→ Θ′a
= Λa
b(Γ) Θb
. (3.27)
However, the dependence is so soft that physics, which lives on-shell, is not affected [8]. We shall here
clarify in exactly what sense the ∆ED
operator remains invariant on-shell under reparametrization of
the constraints.
To warm up, let us recall that the Dirac antibrackets (F, G)D and (F, G)′
D, defined using the primed
and unprimed constraints Θ′a and Θa, respectively, are the same on-shell
(F, G)′
D ≈ (F, G)D . (3.28)
Here the symbol “≈” is the Dirac weak equivalence symbol, which denotes equivalence modulo terms
of order O(Θ). More generally,
F′
≈ F ∧ G′
≈ G ⇒ (F′
, G′
)′
D ≈ (F, G)D . (3.29)
Hence the reduced bracket
( ˜F, ˜G)∼ ≡ (F, G)D|Θ=0 , (3.30)
is independent of both the choice of constraints Θa and the representatives F = F(Γ), G = G(Γ) on
M. Here ˜F ≡ F|Θ=0 = ˜F(γ) and ˜G ≡ G|Θ=0 = ˜G(γ) are functions on the physical submanifold ˜M.
On the other hand, to have a well-defined notion of reduced densities and semidensities on the physical
submanifold ˜M, it is necessary to let the densities and semidensities transform as
ρ′
≈ ρ Λ , σ′
≈ σ
√
Λ , (3.31)
14
under reparametrization of the defining set of constraints Θa → Θ′a = Λa
b Θb. Here
Λ ≡ sdet(Λa
b) (3.32)
denotes the superdeterminant of the reparametrization matrix Λa
b = Λa
b(Γ). The reduction
˜ρ ≡ ρ|Θ=0 , ˜σ ≡ σ|Θ=0 , (3.33)
is then by definition performed in a unitarizing coordinate system ΓA = {γA; Θa}, where it is implicitly
understood that the Θa coordinates coincide with the defining set of constraints. Similar to the Dirac
antibracket (·, ·)D, we imagine that the densities and semidensities refer to an internal defining set of
Θa constraints. If one chooses another defining set of constraints Θ′a, and an accompanying unitarizing
coordinate system Γ′A = {γ′A; Θ′a}, the superdeterminant factor Λ in the reparametrization rule (3.31)
is designed to cancel the Jacobian factor J from the coordinate transformation (2.14) on-shell, so that
the reduced definition (3.33) stays the same.
Similarly, it is necessary that the ∆ED
operator, which takes semidensities to semidensities, transforms
as
∆′
ED
≈
√
Λ ∆ED
1
√
Λ
(3.34)
as an operator identity. Stated more precisely, the odd scalar νρ,ED
from Definition 2.8 should be
invariant on-shell
νρ′,E ′
D
≈ νρ,ED
(3.35)
under reparametrization of the constraints. This is the core issue at stake. To prove that it indeed
holds, first note that it is enough to check the claim (3.35) if the set of unprimed constraints Θa happens
to belong to a set of transversal coordinates ΓA = {γA; Θa}. (If this is not the case, one can always
locally find a transversal coordinate system, and split the above reparamerization into two successive
reparamerizations that both involve the transversal coordinates.) Transversal coordinates will simplify
considerably the ensuing calculations. In general, the on-shell change of νρ,ED
depends on how the
Dirac antibracket (·, ·)D changes up to the second order in Θa, cf. eq. (4.39) in Ref. [8]. Explicitly,
one may show that the quantities ν
(0)
ρ,D, ν
(1)
D , ν
(2)
D , ν
(3)
D , ν
(4)
D and ν
(5)
D , defined in eqs. (2.21)–(2.25) and
(2.27), transform as
ν
′(0)
ρ,D ≡
1
√
ρ′
(∆1,E ′
D
ρ′) ≈
1
√
Λρ
(∆ 1
Λ
,ED
Λρ) = ν
(0)
ρ,D −
√
Λ(∆1,ED
1
√
Λ
) , (3.36)
ν
′(1)
D ≈ ν
(1)
D + 8
√
Λ(∆1,ED
1
√
Λ
) − (−1)εb (
→
∂l
∂Θ′a
Θb
,
→
∂l
∂Θb
Θ′a
)D , (3.37)
ν
′(2)
D ≈ ν
(2)
D − (−1)εb (
→
∂l
∂Θ′a
Θb
,
→
∂l
∂Θb
Θ′a
)D , (3.38)
ν
′(3)
D ≈ ν
(3)
D , (3.39)
ν
′(4)
D ≈ ν
(4)
D , (3.40)
ν
′(5)
D ≈ ν
(5)
D . (3.41)
The last equality in eq. (3.36) is a non-trivial property of the odd Laplacian. It is now easy to see
that the relevant linear combination νρ,ED
of ν
(0)
ρ,D, ν
(1)
D , ν
(2)
D , ν
(3)
D , ν
(4)
D and ν
(5)
D is invariant on-shell.
15
3.5 Nilpotency Condition for the odd Dirac Laplacian ∆ρ,ED
One of the surprising conclusions of Ref. [8] was that one cannot maintain a strong nilpotency of the
Dirac odd Laplacian ∆ρ,ED
under reparametrization of the second-class constraints. This is consistent
with our new results. Using the terminology of last Subsection 3.4, one would say that the effect
is caused by the off-shell variations of the odd scalar νρ,ED
and the Dirac antibracket (·, ·)D, cf. the
following calculation:
∆2
ρ′,E ′
D
= (νρ′,E ′
D
, · )′
D ≈ (νρ,ED
, · )D = ∆2
ρ,ED
. (3.42)
Here use is made of eqs. (2.43), (3.29) and (3.35). This should be compared to the situation with
the ∆ED
operator where the strong nilpotency (3.9) is manifest from the onset, regardless of which
defining set of Θa constraints is used.
3.6 Dirac Partition Function
As an application of the ∆ED
operator, it is interesting to consider the first-level Dirac partition
function in the λ∗
α =0 gauge. A review of the first-level formalism can be found in Ref. [8]. The
partition function reads
ZD = [dΓ][dλ] exp[
i
¯h
(WED
+ XED
)]
λ∗=0 a
δ(Θa
) , (3.43)
where WED
=WED
(Γ) and XED
=XED
(Γ; λ, λ∗) satisfy the Quantum Master Equations
∆ED
exp[
i
¯h
WED
] = 0 , (3.44)
((−1)εα
→
∂l
∂λα
→
∂l
∂λ∗
α
+ ∆ED
) exp[
i
¯h
XED
] = 0 . (3.45)
The formula (3.43) for the Dirac partition function ZD differs from the original formula [8, 14] by
not depending on a ρ. Instead, the partition function ZD is invariant under general coordinate transformations
and under reparametrization of the Θa constraints because the Boltzmann semidensities
exp[ i
¯h WED
] and exp[ i
¯h XED
] transform according to (2.14) and (3.31). Given an arbitrary density ρ,
it is possible to introduce Boltzmann scalars
exp[
i
¯h
Wρ] ≡ exp[
i
¯h
WED
]/
√
ρ , (3.46)
exp[
i
¯h
Xρ] ≡ exp[
i
¯h
XED
]/
√
ρ , (3.47)
which satisfy corresponding Modified Quantum Master Equations similar to eq. (1.4).
4 Conversion of Second-Class into First-Class
Originally, the conversion of second-class constraints into first-class constraints was developed for even
Poisson geometry [15, 16, 17, 18]. Later it was adapted to anti-Poisson geometry in Ref. [19], more
precisely to the Dirac antibracket (·, ·)D and odd Laplacian ∆ρ,ED
. In this Section 4 we develop the
anti-Poisson conversion method further and show that the Dirac ∆ED
operator from last Section 3
can also be derived via conversion.
16
4.1 Extended Manifold Mext
As in Section 3 the starting point is a general non-degenerate antisymplectic manifold (M; E) with a
set of globally defined second-class constraints Θa = Θa(Γ), which have Grassmann parity ε(Θa)=εa.
We now consider a cartesian product Mext ≡ M × V , where (V ; ω) is a vector space with a constant
and non-degenerate antisymplectic metric, and such that the dimension of V is equal to the number
of Θa constraints. We will often identify M with M × {0} ⊆ Mext. The extended manifold Mext has
antisymplectic structure Eext ≡ E ⊕ ω.
Assume that points (i.e. vectors) in the vector space V are described by a set of coordinates Φa with
Grassmann parity ε(Φa)=εa+1. For each set of local coordinates ΓA for the manifold M, the extended
manifold Mext will have local coordinates ΓA
ext ≡{ΓA; Φa}. Notation: We use capital roman letters A,
B, C, . . . from the beginning of the alphabet as upper index for both the original and the extended
variables ΓA and ΓA
ext, respectively. In detail, the extended antibracket (·, ·)ext on Mext reads
(ΓA
, ΓB
)ext ≡ (ΓA
, ΓB
) = EAB
, (4.1)
(ΓA
, Φa)ext ≡ 0 , (4.2)
(Φa, Φb)ext ≡ ωab , ε(ωab) = εa+εb+1 , (4.3)
where, in particular, the antisymplectic matrix ωab = −(−1)εaεb ωba does not depend on ΓA nor on Φa.
In other words, up to a constant matrix, the Φa coordinates are global Darboux coordinates for the
vector space V .
4.2 First-Class Constraints Ta
One next seeks Abelian first-class constraints Ta =Ta(Γ; Φ) such that
(Ta
, Tb
)ext = 0 , Ta
|Φ=0 = Θa
. (4.4)
Eq. (4.4) is the defining relation for the conversion of second-class constraints Θa into first-class
constraint Ta. The first-class constraints Ta are treated as power series expansions in the Φa variables
Ta
= Θa
+
Xab
L Φb
ΦbXba
R
+
1
2



Y abc
L ΦcΦb
ΦbY bac
M Φc
ΦbΦcY cba
R



+
1
6
Zabcd
L ΦdΦcΦb + O(Φ4
) . (4.5)
The expressions Xab
L Φb ≡ ΦbXba
R and Y abc
L ΦcΦb ≡ ΦbY bac
M Φc ≡ ΦbΦcY cba
R inside the curly brackets
“{ }” of eq. (4.5) reflect various (equivalent) ways of ordering the Φa variables. The rules for shifting
between the ordering prescriptions are
Xab
L = (−1)(εa+1)(εb+1)
Xba
R , (4.6)
(−1)(εa+1)(εb+1)
Y bac
L = Y abc
M = (−1)(εb+1)(εc+1)
Y acb
R . (4.7)
One may show that a solution Ta to the system (4.4) exists, but that it is not unique. For instance,
the condition on the Xab =Xab(Γ) structure functions reads
Ead
≡ (Θa
, Θd
) = − Xab
L ωbcXcd
R . (4.8)
The matrices Xab
L and Xab
R are necessarily invertible with inverse matrices XL
ab =(−1)εaεb XR
ab, since
both Eab ≡(Θa, Θb) and ωab ≡(Φa, Φb)ext in eq. (4.8) are invertible. One may view Xab as a Grassmannodd
vielbein between the curved second-class matrix Eab and the flat metric ωab. At the next order
in Φa, the condition on the Y abc =Y abc(Γ) structure functions reads
(Θa
, Xcb
R ) + Xad
L ωdeY ecb
R + (Xac
L , Θb
) + Y acd
L ωdeXeb
R = 0 , (4.9)
and so forth.
17
4.3 Gauge Invariance
The idea is now to view the first-class constraints Ta as generators of gauge symmetry and Φa =0 as
a particular gauge. We start by defining gauge-invariant observables on the extended manifold Mext.
Definition 4.1 A scalar function ¯F = ¯F(Γ; Φ), a density ¯ρ= ¯ρ(Γ; Φ) or a semidensity ¯σ= ¯σ(Γ; Φ) on
the extended manifold Mext is called a gauge-invariant extension of a scalar function F =F(Γ), a
density ρ=ρ(Γ) or a semidensity σ=σ(Γ) on the original manifold M, if the following conditions are
satisfied
( ¯F, Ta
)ext = 0 , ¯F Φ=0 = F , (4.10)
(∆¯ρTa
) = 0 , ¯ρ|Φ=0 = ρj , (4.11)
[
→
∆Eext
, Ta
]¯σ = 0 , ¯σ|Φ=0 = σ j , (4.12)
respectively, where the j-factor is defined in eq. (4.13) below.
4.4 The j-Factor
The factor
j ≡ ¯|Φ=0 = sdet(ωacXcb
R ) (4.13)
is defined as the Φ=0 restriction of the superdeterminant
¯ ≡ sdet(Φa, Tb
)ext = [d ¯C][dC] exp
i
¯h
¯Ca
(Φa, Tb
)extCb , ε( ¯Ca
) = εa+1 = ε(Ca) . (4.14)
The j-factor (4.13) is independent of the choice of Xab structure functions because of eq. (4.8). It is a
density for the vector space V such that the corresponding volume form j[dΦ] on V is independent of
the choice of coordinates Φa. In this way the multiplication with j in eq. (4.11) transforms a density
ρ on the manifold M into a density ρj for the extended manifold Mext ≡M × V . The j-factor is
unique up to an overall constant and can be physically explained as a Faddeev-Popov determinant,
see Subsection 4.7.
Below we shall overwhelmingly justify the j-factor in Definition 4.1, in particular, through the Conversion
Theorem 4.2, but let us start by briefly mentioning a curious implication. Consider what
happens to the set of vielbein solutions Xab
L to eq. (4.8) under reparametrizations of the defining set
of second-class constraints Θa → Θ′a = Λa
b Θb. It is natural to expect that there exists a bijective
map Xab
L → X′ab
L between the solutions such that
X′ac
L ≈ Λa
bXbc
L , (4.15)
where “≈” denotes weak equivalence, cf. Subsection 3.4. According to such map, the j-factor would
transform as
j′
≈ Λ j . (4.16)
Recalling the transformation rule (3.31) for ρ, this implies that the density ¯ρ|Φ=0 =ρj on Mext changes
with the square of Λ,
¯ρ′
Φ=0 ≈ Λ2
¯ρ|Φ=0 . (4.17)
So while the j-factor does indeed cancel the effect of changing the Φa coordinates, it doubles the effect
of changing the second-class constraints Θa! Nevertheless, this doubling phenomenon fits nicely with
the rest of the conversion construction, cf. Subsection 4.7 below.
18
4.5 Discussion of Gauge Invariance
Let us now justify the conditions (4.10)–(4.12). The first condition (4.10) is simply the antisymplectic
definition of gauge invariance. As an example of condition (4.10), note that a first-class constraint
Ta = ¯Θa is a gauge-invariant extension of the corresponding second-class constraint Θa. The other
two conditions (4.11) and (4.12) are a priori less obvious, but there are many reasons to impose them:
1. The three conditions (4.10)–(4.12) are covariant with respect to coordinate changes.
2. The conditions (4.10)–(4.12) are consistent with each others, say, if one considers a density
ρ′ = ρF, or a semidensity σ =
√
ρ.
3. The conditions (4.10)–(4.12) are natural counterparts of the annihilation properties (3.21)–(3.23).
4. One may show that there exist unique gauge-invariant extensions ¯F, ¯ρ and ¯σ satisfying the
condition (4.10), (4.11) and (4.12), respectively.
5. The extended antibracket (·, ·)ext, the extended odd Laplacian ∆¯ρ ≡ ∆¯ρ,Eext
, the extended ∆Eext
operator and the extended odd scalar ν¯ρ ≡ ν¯ρ,Eext
are compatibly with the gauge-invariance
conditions (4.10)–(4.12), i.e.
( ¯F ¯G, Ta
)ext = ¯F( ¯G, Ta
)ext + (−1)εF εG ¯G( ¯F, Ta
)ext = 0 , (4.18)
(( ¯F, ¯G)ext, Ta
)ext = ( ¯F, ( ¯G, Ta
)ext)ext + (−1)(εa+1)(εG+1)
(( ¯F, Ta
)ext, ¯G)ext = 0 ,(4.19)
(∆¯ρ
¯F, Ta
)ext = ∆¯ρ( ¯F, Ta
)ext + (−1)εF ( ¯F, ∆¯ρTa
)ext = 0 , (4.20)
[
→
∆Eext
, Ta
](∆Eext
¯σ) = (∆Eext
Ta
∆Eext
¯σ) = (∆Eext
[Ta
,
→
∆Eext
]¯σ) = 0 , (4.21)
(ν¯ρ, Ta
)ext = (∆2
¯ρTa
) = 0 . (4.22)
Here use is made of the ordinary Leibniz rule, the Jacobi identity (2.2), the BV Leibniz rule
(2.16), the eq. (2.19) and the eq. (2.43), respectively.
6. The conditions (4.10)–(4.12) imply the Conversion Theorem 4.2 below.
4.6 The Conversion Map
The gauge-invariant extension map
F(M) ∋ F
∼=
−→ ¯F ∈ F(Mext)inv (4.23)
(which is also known as the conversion map) is an isomorphism of functions on M to gauge-invariant
function on Mext, cf. point 4 of the last Subsection 4.5. The inverse conversion map is simply the
restriction to M,
F(Mext)inv ∋ ¯F
∼=
−→ ¯F Φ=0 ∈ F(M) . (4.24)
The following Theorem 4.2 is the heart of the conversion method. It shows that the inverse conversion
map transforms the extended model into the Dirac construction.
Theorem 4.2 The restrictions to M of the extended antibracket (·, ·)ext, the extended odd Laplacian
∆¯ρ ≡ ∆¯ρ,Eext
, the extended ∆Eext
operator and the extended odd scalar ν¯ρ ≡ ν¯ρ,Eext
reproduce the
19
corresponding Dirac constructions:
( ¯F, ¯G)ext Φ=0 = (F, G)D , (4.25)
(∆¯ρ,Eext
¯F)
Φ=0
= (∆ρ,ED
F) , (4.26)
(∆Eext
¯σ)
Φ=0
= j(∆ED
σ) , (4.27)
ν¯ρ,Eext Φ=0
= νρ,ED
. (4.28)
In principle, it is enough to prove eq. (4.28), since eq. (4.28) ⇔ eq. (4.27) ⇒ eq. (4.26) ⇒ eq. (4.25).
Nevertheless, we shall give independent proofs of eqs. (4.25), (4.26) and (4.28) in Appendix C. The
following Corollary 4.3 restates the conclusions of Conversion Theorem 4.2 using the forward conversion
map.
Corollary 4.3
(FG)−
= ¯F ¯G , (4.29)
((F, G)D)−
= ( ¯F, ¯G)ext , (4.30)
(∆ρ,ED
F)−
= (∆¯ρ,Eext
¯F) , (4.31)
( j∆ED
σ)−
= (∆Eext
¯σ) , (4.32)
(νρ,ED
)−
= ν¯ρ,Eext
. (4.33)
In particular, eqs. (4.25) and (4.30) show that the conversion map is an isomorphism in the sense
of anti-Poisson algebras between the Dirac anti-Poisson algebra (F(M); (·, ·)D) and the anti-Poisson
algebra (F(Mext)inv; (·, ·)ext) of gauge-invariant functions on Mext.
4.7 Extended Partition Function
The first-level partition function in the λ∗
α = 0 gauge reads
Zext = [dΓext][dλ] exp[
i
¯h
(WEext
+ XEext
)]
λ∗=0
1
sdet(χa, Tb)ext c
δ(Tc
)
d
δ(χd) , (4.34)
where WEext
=WEext
(Γext) and XEext
=XEext
(Γext; λ, λ∗) satisfy the Quantum Master Equations
∆Eext
exp[
i
¯h
WEext
] = 0 , (4.35)
((−1)εα
→
∂l
∂λα
→
∂l
∂λ∗
α
+ ∆Eext
) exp[
i
¯h
XEext
] = 0 , (4.36)
and they are gauge invariant in the sense of condition (4.12):
[
→
∆Eext
, Ta
] exp[
i
¯h
WEext
] = 0 , exp[ i
¯h WEext
]
Φ=0
= j exp[
i
¯h
WED
] , (4.37)
[
→
∆Eext
, Ta
] exp[
i
¯h
XEext
] = 0 , exp[ i
¯hXEext
]
Φ=0
= j exp[
i
¯h
XED
] . (4.38)
20
Here the Boltzmann semidensities exp[ i
¯hWED
] and exp[ i
¯h XED
] satisfy the Quantum Master Equations
(3.44) and (3.45), respectively. It is an important fact that in the gauge χa =Φa, the expression (4.34)
for the extended partition function reduces to the Dirac partition function (3.43), i.e.
Zext = ZD . (4.39)
Given a density ρ=ρ(Γ) on M, and a density ¯ρ= ¯ρ(Γext) on Mext that satisfies eq. (4.11), it is possible
to introduce Boltzmann scalars
exp[
i
¯h
W¯ρ] ≡ exp[
i
¯h
WEext
]/
√
¯ρ , (4.40)
exp[
i
¯h
X¯ρ] ≡ exp[
i
¯h
XEext
]/
√
¯ρ , (4.41)
which satisfy corresponding Modified Quantum Master Equations similar to eq. (1.4). The Quantum
Actions W¯ρ and X¯ρ defined this way are automatically gauge invariant
(W¯ρ, Ta
)ext = 0 , W¯ρ
Φ=0
= Wρ , (4.42)
(X¯ρ, Ta
)ext = 0 , X¯ρ
Φ=0
= Xρ . (4.43)
Here Wρ and Xρ are defined in eqs. (3.46) and (3.47), respectively.
5 Conclusions
We have shown for a general degenerate anti-Poisson manifold (under the relatively mild assumption
of a compatible two-form field) how to define in arbitrary coordinates the ∆E operator, which takes
semidensities to semidensities, cf. Lemma 2.6. A large class of such degenerate antibrackets are
provided by the Dirac antibracket construction. We have given a formula for the Dirac ∆ED
operator,
cf. Proposition 3.1, and shown in Subsection 3.4 that it is on-shell invariant under reparametrizations
of the second-class constraints. Finally, we showed that the Dirac ∆ED
operator also follows from the
antisymplectic conversion scheme, cf. Conversion Theorem 4.2.
Let us conclude with the following remark. It is often pointed out that the antibracket (·, ·) is a
descendant of the odd Laplacian ∆ρ. It measures the failure of the odd Laplacian ∆ρ to act as a linear
derivation, i.e. to satisfy the Leibniz rule. It can be written as a double-commutator [7, 20, 21]
(F, G) = (−1)εF [[
→
∆ρ, F], G]1 . (5.1)
In turn, the odd Laplacian ∆ρ is a descendant of the ∆E operator [8, 9]
(∆ρF) =
1
√
ρ
[
→
∆E, F]
√
ρ . (5.2)
That is, one has schematically the following hierarchy:
∆E operator
⇓
Odd Laplacian ∆ρ ⇐ Density ρ
⇓
Antibracket (·, ·)
(5.3)
21
Whereas the ∆E operator is manifestly nilpotent, cf. Theorem 2.10, there is no fundamental reason
to require the odd Laplacian ∆ρ to be nilpotent. (Of course, if ∆ρ is not nilpotent, the Boltzmann
scalar exp[ i
¯h Wρ] would in general have to satisfy a Modified Quantum Master Equation with a nontrivial
νρ term, cf. eq. (1.4). See also the recent preprint [22].) The Dirac odd Laplacian ∆ρ,ED
offers
more evidence that nilpotency of the odd Laplacian is not fundamental, at least not in its strong
formulation, since in this case the nilpotency can only be maintained weakly under reparametrizations
of the second-class constraints Θa, cf. Ref. [8] and Subsection 3.5.
Acknowledgement: The Author thanks I.A. Batalin and P.H. Damgaard for discussions. The
Author also thanks the Erwin Schr¨odinger Institute for warm hospitality. This work is supported by
the Ministry of Education of the Czech Republic under the project MSM 0021622409.
A Proof of bi-Darboux Theorem 2.1
If there exists an atlas of bi-Darboux coordinates, the two-form E = dφ∗
α ∧ dφα is obviously closed.
Now consider the other direction. Assume that the two-form E is closed. Then there locally exists a
pre-antisymplectic one-form potential ϑ such that
dϑ = E . (A.1)
Independently one knows that locally there exist Darboux coordinates ΓA ={φα; φ∗
α; Θa}. Since the
two-form E is assumed to be compatible with the anti-Poisson structure, it must be of the form (2.10).
It is always possible to organize the pre-antisymplectic one-form potential as
ϑ ∼ φ∗
αdφα
+ ϑAdγA
+ ϑ′
adΘa
, (A.2)
where γA ={φα; φ∗
α} collectively denotes the fields and the antifields without the Casimirs. The symbol
“∼” denotes equality modulo exact terms, whose precise expressions are irrelevant, since we are
ultimately only interested in the two-form E. It follows from eqs. (2.10), (A.1) and (A.2) that
(
→
∂l
∂γA
ϑB) = (−1)εAεB (A ↔ B) , (A.3)
and hence there locally exists a fermionic function Ψ′ such that
ϑA = (
→
∂l
∂γA
Ψ′
) . (A.4)
Defining
ϑa ≡ ϑ′
a − (
→
∂l
∂Θa
Ψ′
) , (A.5)
the pre-antisymplectic one-form potential (A.2) reduces to
ϑ ∼ φ∗
αdφα
+ ϑadΘa
. (A.6)
We would like to show that the second term ϑadΘa in eq. (A.6) vanishes under a suitable anticanonical
transformation. Eqs. (A.1) and (A.6) imply that the matrices Maα and Nα
a in eq. (2.10) are
−Maα = (ϑa
←
∂r
∂φα
) = (ϑa, φ∗
α) , (A.7)
Nα
a = (
→
∂l
∂φ∗
α
ϑa) = (φα
, ϑa) , (A.8)
22
and that the pre-antisymplectic potential components ϑa =ϑa(Γ) satisfy a flatness condition:
Fab ≡ (
→
∂l
∂Θa
ϑb) − (−1)εaεb (
→
∂l
∂Θb
ϑa) + (ϑa, ϑb) = 0 . (A.9)
Put more illuminating, the condition (A.9) implies that the vector fields
Da ≡
→
∂l
∂Θa
+ adϑa (A.10)
commute
[Da, Db] = adFab = 0 . (A.11)
Here the adjoint action “ad” refers to the antibracket (adF)G ≡ (F, G), where F and G are functions.
In other words, adF denotes the Hamiltonian vector field with Hamiltonian F. The vector fields Da
are not Hamiltonian, although they do preserve the antibracket
Da(F, G) = (Da[F], G) + (−1)εa(εF +1)
(F, Da[G]) , (A.12)
i.e. they are generators of anticanonical transformations that do not leave the Casimirs invariant.
It is an important fact that the Da are covariant derivatives in the Casimir directions with a Lie
algebra valued gauge potential adϑa. Here the Lie algebra is (a subalgebra of) the space Γ(TM) of
vector fields, equipped with the commutator [·, ·], i.e. the Lie bracket of vector fields. An infinitesimal
variation δϑa of the pre-antisymplectic potential components ϑa must satisfy
Da[δϑb] = (−1)εaεb (a ↔ b) (A.13)
in order to respect the flatness condition (A.9). The last eq. (A.13) implies in turn, that the only
allowed infinitesimal variations δϑa are infinitesimal gauge transformations
δϑa = Da[δΨ] , (A.14)
where δΨ is an infinitesimal fermionic gauge generator. The infinitesimal gauge transformation of the
gauge potential adϑa is
ad(δϑa) = [Da, ad(δΨ)] , (A.15)
where use is made of eq. (A.12). Despite the appearance, the eq. (A.15) is exactly the standard formula
δAµ = Dµε for infinitesimal non-Abelian gauge transformations. Any discrepancy is merely in notation,
not in content. So one can take advantage of well-known facts about non-Abelian gauge theory
and e.g. Wilson-lines. In particular, the infinitesimal transformations (A.14) and (A.15) generalize to
finite gauge transformations. The field strength (or curvature) is zero, cf. eq. (A.11), so the gauge
potential adϑa is pure gauge. This means that there locally exists a gauge where the gauge potential
vanishes identically,
adϑa = 0 . (A.16)
An infinitesimal gauge transformation (A.14) may be implemented with the help of a Hamiltonian
vector field ad(δΨ) with infinitesimal Hamiltonian δΨ. Using the active picture, the Lie derivative of
the pre-antisymplectic one-form potential with respect to the Hamiltonian vector field ad(δΨ) is
Lad(δΨ)ϑ = [iad(δΨ), d]ϑ ∼ iad(δΨ)E = (δΨ
←
∂r
∂γA
)dγa
+ (δΨ, ϑa)dΘa
∼ − Da[δΨ]dΘa
. (A.17)
i.e. by flowing along the Hamiltonian vector field ad(δΨ), one may mimic (minus) the infinitesimal
gauge transformation (A.14). More generally, finite gauge transformations of ϑa are in one-to-one
23
correspondence with anticanonical transformations that leave the Casimirs invariant. In particular,
one may go to the trivial gauge (A.16) where the ϑa themselves are Casimirs. The flatness condition
(A.9) then reduces to
(
→
∂l
∂Θa
ϑb) = (−1)εaεb (a ↔ b) , (A.18)
so there exists a fermionic Casimir function Ψ=Ψ(Θ) such that
ϑa = (
→
∂l
∂Θa
Ψ) , (A.19)
and hence the second term in eq. (A.6) is just an exact term,
ϑadΘa
= dΨ ∼ 0 . (A.20)
This shows that there locally exists an anticanonical transformation that leaves the Casimirs invariant,
such that the two-form reduces to E = dφ∗
α ∧ dφα.
B Details from the Proof of Lemma 2.9
B.1 Proof of eq. (2.38)
The infinitesimal variation of ν(2) in eq. (2.22) yields 8 contributions to linear order in the variation
δΓA = XA, which may be organized as 2 × 4 terms
δν(2)
= 2(−δν
(2)
I − δν
(2)
II + δν
(2)
III + δν
(2)
IV ) , (B.1)
due to a (A, B) ↔ (D, C) symmetry in eq. (2.22). They are
δν
(2)
I ≡ (−1)εAεC (
→
∂l
DEAB
)EBF (XF
←
∂r
C)(
→
∂l
AECD
) , (B.2)
δν
(2)
II ≡ (−1)εAεC (
→
∂l
DEAB
)EBC(
→
∂l
AXF
)(
→
∂l
F ECD
) , (B.3)
δν
(2)
III ≡ (−1)εAεC (
→
∂l
DEAB
)EBC
→
∂l
A (XC
←
∂r
F )EF D
= δν
(2)
I + δν
(2)
V , (B.4)
δν
(2)
IV ≡ (−1)εAεC (
→
∂l
DEAB
)EBC
→
∂l
A ECF
(
→
∂l
F XD
) = δν
(2)
II + δν
(2)
V I , (B.5)
δν
(2)
V ≡ (−1)εAεC EF D
(
→
∂l
DEAB
)EBC(
→
∂l
AXC
←
∂r
F ) = − δν
(2)
V + δν
(2)
V II , (B.6)
δν
(2)
V I ≡ (−1)εA (
→
∂l
DEAB
)PB
C
(
→
∂l
C
→
∂l
AXD
) , (B.7)
δν
(2)
V II ≡ (−1)εA PC
D
(
→
∂l
DEAB
)(
→
∂l
B
→
∂l
AXC
) , (B.8)
where we have noted various relations among the contributions. The Jacobi identity (2.3) for EAB is
used in the second equality of eq. (B.6). Altogether, the infinitesimal variation of ν(2) becomes
δν(2)
= 2δν
(2)
V I + δν
(2)
V II , (B.9)
which is eq. (2.38).
24
B.2 Proof of eq. (2.39)
The infinitesimal variation of ν(3) in eq. (2.23) yields 6 contributions to linear order in the variation
δΓA = XA,
δν(3)
= δν
(3)
I + δν
(3)
II + δν
(3)
III − δν
(3)
IV − δν
(3)
V − δν
(3)
V I . (B.10)
They are
δν
(3)
I ≡ (−1)εB (
→
∂l
AEBC)(XC
←
∂r
F )EF D
(
→
∂l
DEBA
) , (B.11)
δν
(3)
II ≡ (−1)εB (
→
∂l
AEBC)ECD
→
∂l
D (XB
←
∂r
F )EF A
= δν
(3)
V II + δν
(3)
V III , (B.12)
δν
(3)
III ≡ (−1)εB (
→
∂l
AEBC)ECD
→
∂l
D EBF
(
→
∂l
F XA
) = δν
(3)
IV + δν
(3)
IX , (B.13)
δν
(3)
IV ≡ (−1)εB ECD
(
→
∂l
DEBA
)(
→
∂l
AXF
)(
→
∂l
F EBC) , (B.14)
δν
(3)
V ≡ (−1)εB ECD
(
→
∂l
DEBA
)
→
∂l
A (
→
∂l
BXF
)EF C = δν
(3)
V II + δν
(3)
X , (B.15)
δν
(3)
V I ≡ (−1)εB ECD
(
→
∂l
DEBA
)
→
∂l
A EBF (XF
←
∂r
C) = δν
(3)
I − δν
(3)
XI , (B.16)
δν
(3)
V II ≡ (−1)εA(εB+1)
(
→
∂l
AEBC)ECD
(
→
∂l
DEAF
)(
→
∂l
F XB
)
= −(−1)εB(εC +1)
ECD
(
→
∂l
DEBA
)(
→
∂l
AECF )(XF
←
∂r
B) , (B.17)
δν
(3)
V III ≡ (−1)εB (
→
∂l
AEBC)ECD
(
→
∂l
DXB
←
∂r
F )EF A
, (B.18)
δν
(3)
IX ≡ (−1)εA(εB+1)
(
→
∂l
AEBC)ECD
(
→
∂l
DXA
←
∂r
F )EF B
, (B.19)
δν
(3)
X ≡ (−1)εA PC
D
(
→
∂l
DEAB
)(
→
∂l
B
→
∂l
AXC
) , (B.20)
δν
(3)
XI ≡ (−1)εB (εC+1)
ECD
(
→
∂l
DEBA
)(
→
∂l
A
→
∂l
CXF
)EF B = − δν
(3)
X − δν
(3)
XI , (B.21)
where we have noted various relations among the contributions. The Jacobi identity (2.3) for EAB is
used in the second equality of eq. (B.21). Altogether, the infinitesimal variation of ν(3) becomes
δν(3)
= δν
(3)
V III + δν
(3)
IX −
3
2
δν
(3)
X , (B.22)
which is eq. (2.39).
B.3 Proof of eq. (2.40)
The infinitesimal variation of ν(4) in eq. (2.24) yields 6 contributions to linear order in the variation
δΓA = XA,
δν(4)
= − δν
(4)
I − δν
(4)
II + δν
(4)
III + δν
(4)
IV + δν
(4)
V − δν
(4)
V I . (B.23)
They are
δν
(4)
I ≡ (−1)εB ECD
(
→
∂l
DEBF
)PF
A
→
∂l
A (
→
∂l
BXG
)EGC = − δν
(4)
V II + δν
(4)
V III , (B.24)
δν
(4)
II ≡ (−1)εB ECD
(
→
∂l
DEBF
)PF
A
→
∂l
A EBG(XG
←
∂r
C) = δν
(4)
III − δν
(4)
IX , (B.25)
25
δν
(4)
III ≡ (−1)εB PF
A
(
→
∂l
AEBC)(XC
←
∂r
G)EGD
(
→
∂l
DEBF
) , (B.26)
δν
(4)
IV ≡ (−1)εB PF
A
(
→
∂l
AEBC)ECD
→
∂l
D (XB
←
∂r
G)EGF
= δν
(4)
X + δν
(4)
XI , (B.27)
δν
(4)
V ≡ (−1)εB PF
A
(
→
∂l
AEBC)ECD
→
∂l
D EBG
(
→
∂l
GXF
) = δν
(4)
V I + δν
(4)
XII , (B.28)
δν
(4)
V I ≡ (−1)εB PG
A
(
→
∂l
AEBC)ECD
(
→
∂l
DEBF
)(
→
∂l
F XG
) , (B.29)
δν
(4)
V II ≡ (−1)εB(εC +1)
ECD
(
→
∂l
DEBF
)PF
A
(
→
∂l
AECG)(XG
←
∂r
B) , (B.30)
δν
(4)
V III ≡ (−1)εB (
→
∂l
A
→
∂l
BXC
)PC
D
(
→
∂l
DEBF
)PF
A
, (B.31)
δν
(4)
IX ≡ (−1)εB(εC +1)
ECD
(
→
∂l
DEBF
)PF
A
(
→
∂l
A
→
∂l
CXG
)EGB = − δν
(4)
V III − δν
(4)
XIII , (B.32)
δν
(4)
X ≡ (−1)(εB+1)εF PF
A
(
→
∂l
AEBC)ECD
(
→
∂l
DEF G
)(
→
∂l
GXB
) = − δν
(4)
V II , (B.33)
δν
(4)
XI ≡ (−1)εB (
→
∂l
AEBC)ECD
(
→
∂l
DXB
←
∂r
F )EF A
, (B.34)
δν
(4)
XII ≡ (−1)(εB+1)εF PF
A
(
→
∂l
AEBC)ECD
(
→
∂l
DXF
←
∂r
G)EGB
, (B.35)
δν
(4)
XIII ≡ (−1)(εA+1)εB EAD
(
→
∂l
DEBC
)(
→
∂l
C
→
∂l
AXG
)EGB = − δν
(4)
XIII − δν
(4)
XIV , (B.36)
δν
(4)
XIV ≡ (−1)εA PC
D
(
→
∂l
DEAB
)(
→
∂l
B
→
∂l
AXC
) , (B.37)
where we have noted various relations among the contributions. The Jacobi identity (2.3) for EAB is
used in the second equality of eqs. (B.32) and (B.36). Altogether, the infinitesimal variation of ν(4)
becomes
δν(4)
= − δν
(4)
V III + δν
(4)
IX + δν
(4)
XI + δν
(4)
XII = − 2δν
(4)
V III + δν
(4)
XI + δν
(4)
XII +
1
2
δν
(4)
XIV , (B.38)
which is eq. (2.39).
B.4 Proof of eq. (2.41)
The infinitesimal variation of ν(5) in eq. (2.25) yields 8 contributions to linear order in the variation
δΓA = XA,
δν(5)
= δν
(5)
I + δν
(5)
II + δν
(5)
III − δν
(5)
IV − δν
(5)
V − δν
(5)
V I + δν
(5)
V II − δν
(5)
V III . (B.39)
They are
δν
(5)
I ≡ (−1)(εA+1)εB (XA
←
∂r
G)EGD
(
→
∂l
DEBC
)(
→
∂l
CEAF )PF
B , (B.40)
δν
(5)
II ≡ (−1)(εA+1)εB (
→
∂l
CEAF )PF
BEAD
→
∂l
D (XB
←
∂r
G)EGC
= δν
(5)
V III + δν
(5)
IX , (B.41)
δν
(5)
III ≡ (−1)(εA+1)εB (
→
∂l
CEAF )PF
BEAD
→
∂l
D EBG
(
→
∂l
GXC
) = δν
(5)
IV − δν
(5)
X , (B.42)
δν
(5)
IV ≡ (−1)(εA+1)εB EAD
(
→
∂l
DEBC
)(
→
∂l
CXG
)(
→
∂l
GEAF )PF
B , (B.43)
δν
(5)
V ≡ (−1)(εA+1)εB PF
BEAD
(
→
∂l
DEBC
)
→
∂l
C (
→
∂l
AXG
)EGF = δν
(5)
I + δν
(5)
XI , (B.44)
26
δν
(5)
V I ≡ (−1)(εA+1)εB PF
BEAD
(
→
∂l
DEBC
)
→
∂l
C EAG(XG
←
∂r
F ) = δν
(5)
V II − δν
(5)
XII , (B.45)
δν
(5)
V II ≡ (−1)(εA+1)εB EAD
(
→
∂l
DEBC
)(
→
∂l
CEAF )(XF
←
∂r
G)PG
B , (B.46)
δν
(5)
V III ≡ (−1)(εA+1)εB EAD
(
→
∂l
DEBC
)(
→
∂l
CEAF )PF
G(XG
←
∂r
B) , (B.47)
δν
(5)
IX ≡ (−1)(εA+1)εB EAD
(
→
∂l
DXB
←
∂r
G)EGC
(
→
∂l
CEAF )PF
B
= (−1)εBεF PF
D
(
→
∂l
DXB
←
∂r
G)EGC
(
→
∂l
CEF A
)EAB = δν
(5)
XI + δν
(5)
XII , (B.48)
δν
(5)
X ≡ (−1)(εA+1)εC (
→
∂l
CEAB)EBF
(
→
∂l
F XC
←
∂r
G)EGA
, (B.49)
δν
(5)
XI ≡ (−1)(εA+1)εB EAD
(
→
∂l
DEBC
)(
→
∂l
C
→
∂l
AXG
)EGB , (B.50)
δν
(5)
XII ≡ (−1)εB (
→
∂l
A
→
∂l
BXC
)PC
D
(
→
∂l
DEBF
)PF
A
, (B.51)
where we have noted various relations among the contributions. The Jacobi identity (2.3) for EAB is
used in the third equality of eq. (B.48). Altogether, the infinitesimal variation of ν(5) becomes
δν(5)
= δν
(5)
IX − δν
(5)
X − δν
(5)
XI + δν
(5)
XII = − δν
(5)
X + 2δν
(5)
XII , (B.52)
which is eq. (2.41).
C Proof of Conversion Theorem 4.2
C.1 The ¯ Superdeterminant
Even-though it is only the j-factor (4.13) and not the whole ¯ superdeterminant (4.14) that enters the
conversion map, it is nevertheless convenient to organize the discussion in terms of coefficient functions
for (the logarithm of) the ¯ superdeterminant
ln ¯ ≡ ¯n = n +
na
LΦa
Φana
R
+
1
2
nab
L ΦbΦa + O(Φ3
) , n ≡ ln j . (C.1)
By combining eqs. (4.5), (4.14) and (C.1), one finds the first-order coefficient functions na to be
na
L = (−1)εb XR
bcY cba
M = (−1)εb
+1
XL
bcY cba
L
na
R = Y abc
M XL
cb(−1)εb = Y abc
R XR
cb(−1)εb
+1
.
(C.2)
The second-order coefficient functions read
ncd
L = (−1)εb+1
XL
baZabcd
L + (−1)(εa+1)εc XR
abY bce
R XR
ef Y fad
M . (C.3)
In particular, the contracted second-order coefficient function is
(−1)εc+1
ncd
L ωdc = z(1)
− y(2)
, (C.4)
where we have introduced the following short-hand notation
y(2)
≡ (−1)εaεf XR
abY bfc
M ωcdY dae
M XL
ef , (C.5)
z(1)
≡ (−1)εb+εcXL
baZabcd
L ωdc . (C.6)
Since there is not a unique choice of the structure functions Xab, Y abc, Zabcd, etc, one must apply the
Ta involution relation (4.4) to eliminate their appearances. We have to wait until Subsection C.5 to
27
completely eliminate all Y abc appearances, but we can do a first step in this direction. The quadratic
Y abc dependence inside the odd y(2) variable (C.5) can be related to a linear Y abc dependence inside
a new y(1) variable as follows
0
(4.4)
=
1
2
(−1)εa+1
Y bac
R XR
cd
→
∂l
∂Φd
XL
ae(Te
, Tf
)extXR
fb
Φ=0
=
1
2
(−1)εcεe XR
cd
→
∂l
∂Φd
(Te
, Tf
)ext
Φ=0
XR
fbY bca
M XL
ae
= (−1)εb Y cba
R (XR
ab, Θd
)XR
dc + (−1)εaεf XR
abY bfc
M ωcdY dae
M XL
ef = y(1)
+ y(2)
, (C.7)
where
y(1)
≡ (−1)εb Y cba
R (XR
ab, Θd
)XR
dc . (C.8)
The only way the Zabcd
L structure functions enters the discussion is through the odd z(1) variable (C.6).
It can be eliminated using the following equation
0
(4.4)
= XR
dc
→
∂l
∂Φc
(−1)εa XR
ab
→
∂l
∂Φb
(Ta
, Td
)ext Φ=0
= (−1)εa XR
ab
→
∂l
∂Φb
(Ta
, Td
)ext
←
∂r
∂Φc
XL
cd(−1)εd
Φ=0
= (n, n) + na
Lωabnb
R + 2(−1)εb+εc XL
ba(Y abc
L , Θd
)XR
dc + (−1)εaεd XR
ab(Xbd
R , Xac
L )XL
cd
+2(−1)εb+εc XL
baZabcd
L ωdc + (−1)εaεf XR
abY bfc
M ωcdY dae
M XL
ef
= (n, n) + na
Lωabnb
R + 2(na
R, Θb
)XR
ba + (−1)εb (Xab
L , XL
ba) + 2z(1)
+ y(1)
. (C.9)
C.2 Gauge Invariant Function ¯F
The gauge-invariant extension ¯F is a power series expansion in the Φa variables, e.g. ,
¯F = F +
ΦaFa
R
Fa
LΦa
+
1
2
ΦaΦbFba
R + O(Φ3
) . (C.10)
The coefficient functions for ¯F are uniquely determined by gauge invariance condition (4.10). The
first-order coefficient functions read
Fa
R = −ωabXL
bc(Θc, F) = Xab
R Ebc(Θc, F) ,
Fa
L = −(F, Θc)XR
cbωba = (F, Θc)EcbXba
L ,
(C.11)
The contracted second-order coefficient function (−1)εa+1ωabFba
R is determined by the following cal-
culation
0
(4.10)
= (−1)εa XR
ab
→
∂l
∂Φb
(Ta
, ¯F)ext Φ=0
= XL
ba(Θa
, Fb
R)(−1)εb
+1
+ (n, F) + na
LωabFb
R + (−1)εa+1
ωabFba
R . (C.12)
C.3 Gauge Invariant Density ¯ρ
The (logarithm of the) gauge-invariant density ¯ρ is a power series expansion in the Φa variables, e.g. ,
ln
√
¯ρ ≡ ¯ℓ = ℓ +
ℓa
LΦa
Φaℓa
R
+
1
2
ΦaΦbℓba
R + O(Φ3
) , ℓ ≡ ln ρj . (C.13)
28
The coefficient functions for ¯ρ are uniquely determined by the gauge invariance condition (4.11). The
first-order coefficient functions ℓa can be found from the following Lemma C.1.
Lemma C.1
1
2
nc
L − ℓc
L = (∆ρΘa
)XR
abωbc
+
1
2
(−1)εa (Θa
, XR
ab)ωbc
=
(−1)εA
2ρ
→
∂l
Aρ(ΓA
, Θa
)XR
abωbc
, (C.14)
1
2
nc
R − ℓc
R = ωcb
XL
ba(∆ρΘa
)(−1)εa +
1
2
ωcb
(XL
ba, Θa
)(−1)εa
= ωcb
XL
ba(Θa
, ΓA
)ρ
←
∂r
A
(−1)εA
2ρ
. (C.15)
Proof of Lemma C.1: Combine
0
(4.11)
= (∆¯ρTa
)
Φ=0
= (∆ρjΘa
) +
1
2
(−1)εb+1
ωbcY cba
R + ℓc
LωcbXba
R (C.16)
and
0
(4.4)
= (−1)εb XR
bc
→
∂l
∂Φc
(Tb
, Ta
)ext Φ=0
= (−1)εc+1
XL
cb(Θb
, Xca
R ) + (n, Θa
) + nc
LωcbXba
R + (−1)εb+1
ωbcY cba
R . (C.17)
The contracted second-order coefficient function (−1)εa+1ωabℓba
R is determined by the following calcu-
lation
0
(4.11)
= XR
ab
→
∂l
∂Φb
(∆¯ρTa
)
Φ=0
= (∆ρjXab
L )XL
ba(−1)εa +
1
2
(−1)εb+εcXL
baZabcd
L ωdc
+XL
ba(Θa
, ℓb
L) + na
Lωabℓb
R + (−1)εa+1
ωabℓba
R . (C.18)
C.4 Assembling the Proof
Proof of eq. (4.25):
( ¯F, ¯G)ext Φ=0 = (F, G) + Fa
LωabGb
R
(C.11)
= (F, G)D . (C.19)
Proof of eq. (4.26):
(∆¯ρ,Eext
¯F)
Φ=0
= (∆ρjF) +
1
2
(−1)εa+1
ωabFba
R + ℓa
LωabFb
R
(C.12)
= (∆ρjF) −
1
2
(n, F) + (ℓa
L −
1
2
na
L)ωabFb
R −
1
2
XL
ba(Θa
, Fb
R)(−1)εb
+1
(C.14)
= (∆ρF) − (∆ρΘa
)XR
abFb
R −
1
2
(−1)εa (Θa
, XR
abFb
R)
= (∆ρF) − (∆ρΘa
)Eab(Θb
, F) −
1
2
(−1)εa (Θa
, Eab(Θb
, F))
= (∆ρ,ED
F) . (C.20)
29
Proof of eq. (4.28): Using eq. (2.45) it follows that
νρj − νρ
(2.45)
=
1
√
j
(∆ρ j) =
1
2
(∆ρn) +
1
8
(n, n) =
1
2
(∆ρjn) −
1
8
(n, n)
=
1
2
(∆ρjXab
L )XL
ba(−1)εa −
1
4
(−1)εb (Xab
L , XL
ba) −
1
8
(n, n) , (C.21)
so that
ν¯ρ,Eext Φ=0
= νρj +
1
2
(−1)εa+1
ωabℓba
R +
1
2
ℓa
Lωabℓb
R
(C.21)
= νρ −
1
8
(n, n) +
1
2
(∆ρjXab
L )XL
ba(−1)εa −
1
4
(−1)εb (Xab
L , XL
ba)
+
1
2
(−1)εa+1
ωabℓba
R +
1
2
ℓa
Lωabℓb
R
(C.18)
= νρ −
1
8
(n, n) −
1
2
na
Lωabℓb
R +
1
2
ℓa
Lωabℓb
R
−
1
4
(−1)εb (Xab
L , XL
ba) −
z(1)
4
−
1
2
XL
ba(Θa
, ℓb
L)
(C.9)
= νρ +
1
2
(
1
2
na
L − ℓa
L)ωab(
1
2
nb
R − ℓb
R) +
1
2
(Θa
,
1
2
nb
L − ℓb
L)XL
ba
−
1
8
(−1)εb (Xab
L , XL
ba) +
y(1)
8
(C.14)
= νρ −
ν
(6)
ρ,D
2
+
1
2
(Θa
, XR
ab)ωbc
(∆ρΘc
)(−1)εc +
1
8
(−1)εa+εd(Θa
, XR
ab)ωbc
(XL
cd, Θd
)
+
1
2
(Θa
,
1
2
nb
L − ℓb
L)XL
ba −
1
8
(−1)εb (Xab
L , XL
ba) +
y(1)
8
= νρ −
ν
(6)
ρ,D
2
−
ν
(7)
ρ,D
2
+
1
8
(−1)εa+εd(Θa
, XR
ab)ωbc
(XL
cd, Θd
)
+
1
4
(−1)εb (Θa
, (Θb
, XR
bc))ωcd
XL
da −
1
8
(−1)εb (Xab
L , XL
ba) +
y(1)
8
= νρ,ED
−
ν
(9)
D
24
−
x(1)
8
+
y(1)
8
(C.23)
= νρ,ED
, (C.22)
where the last equality in eq. (C.22) follows from Lemma C.2 below, and the odd quantity x(1) is
defined in eq. (C.25).
C.5 Lemma C.2
It turns out that the most difficult part in the proof of eq. (4.28) is to eliminate the Y abc dependence
from the odd y(1) quantity (C.8). Lemma C.2 gives a formula for y(1) that are manifestly independent
of Y abc.
Lemma C.2
y(1)
=
ν
(9)
D
3
+ x(1)
. (C.23)
30
Proof of Lemma C.2: We first decompose the odd ν
(9)
D quantity (3.14) as
ν
(9)
D ≡ (−1)(εa+1)(εd
+1)
(Θd
, Eab)Ebc
(Ecd, Θa
) = − x(1)
− 2x(2)
− x(3)
, (C.24)
where
x(1)
≡ (−1)(εa+1)(εd
+1)
(Θd
, XR
ab)ωbc
(XL
cd, Θa
) , (C.25)
x(2)
≡ (−1)εb(εd+1)
(Θd
, Xbc
L )(XL
cd, Θa
)Eab = (−1)εb(εd+1)
Eba(Θa
, XR
dc)(Xcb
R , Θd
) , (C.26)
x(3)
≡ (−1)εbεe Eef (Θf
, Xbc
L )ωcd(Xde
R , Θa
)Eab . (C.27)
Secondly, we define
x(4)
≡ (−1)εc Eab(Θb
, Xcd
L )(XL
dc, Θa
)
= −(−1)εc Eab(Θb
, Xcd
R )(XR
dc, Θa
) = x(3)
− x(1)
. (C.28)
The third (=last) equality in eq. (C.28) is a non-trivial assertion. To prove it, we define the following
quantities:
x(5)
≡ (−1)εd (Θf
, XR
dc)Xcb
R Eba(Θa
, Ede
)Eef
= −(−1)εb
εe Eef (Θf
, Xbc
L )XL
cd(Ede
, Θa
)Eab = x(2)
+ x(3)
, (C.29)
x(6)
≡ (−1)εb
εe Eef (Θf
, Xbc
L )XL
cd(Θd
, Eea
)Eab = − x(1)
− x(2)
, (C.30)
x(7)
≡ (−1)εd (Θf
, XR
dc)Xcb
R Eba(Ead
, Θe
)Eef = − x(4)
+ x(8)
, (C.31)
x(8)
≡ (−1)(εa+1)(εd+1)
Eab
(XR
bd, Θe
)Eef (Θf
, XR
ac)ωcd
= 0 , (C.32)
where eq. (4.8) is used in the second equality of eqs. (C.29), (C.30) and (C.31). Remarkably the
quantity x(8) vanishes due to an antisymmetry under the index permutation ace ↔ bdf. One may now
check that the Jacobi identity
cycl. a,b,c
(−1)(εa+1)(εc+1)
(Eab
, Θc
) = 0 (C.33)
yields eq. (C.28):
0 = x(5)
+ x(6)
+ x(7)
= − x(1)
+ x(3)
− x(4)
. (C.34)
Thirdly, we define
y(3)
≡ (−1)εc EabY bcd
L ωdeXef
R (XR
fc, Θa
) = y(1)
+ x(4)
+ x(2)
, (C.35)
y(4)
≡ (−1)(εa+1)εd ωab
XL
bcY cde
L ωef Xfg
R (XR
ga, Θh
)XR
hd = y(1)
− x(1)
+ x(2)
, (C.36)
where eq. (4.9) is used in the second equality of eqs. (C.35) and (C.36). Note that x(1) to x(8) are
manifestly independent of the Y abc structure functions. We shall soon see that this is also the case for
the variables y(1) to y(4). It turns out to be possible to rewrite y(3) as
y(3)
=
1
2
(−1)(εa+1)(εc+εf
+1)+εc
EabY bcd
L XL
de(Eef
, Θa
)XR
fc
= (−1)εaεc EabY bcd
L XL
de(Eea
, Θf
)XR
fc = − y(1)
− y(4)
. (C.37)
Here the Jacobi identity (C.33) is used in the second equality of eq. (C.37). Altogether, eqs. (C.35),
(C.36) and (C.37) yields
3y(1)
= x(1)
− 2x(2)
− x(4)
. (C.38)
Now Lemma C.2 follows by combining eqs. (C.24), (C.28) and (C.38).
31
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[21] K. Bering, Commun. Math. Phys. 274 (2007) 297; arXiv:hep-th/0603116.
[22] I.A. Batalin and K. Bering, J. Math. Phys. 49 (2008) 033515, arXiv:0708.0400.
32
Paper III
Odd Scalar Curvature
in Field-Antifield Formalism
BY
I.A. Batalin and K. Bering
J. Math. Phys. 49 (2008) 033515
arXiv:0708.0400
arXiv:0708.0400v6[hep-th]24Jan2008
Odd Scalar Curvature in Field-Antifield Formalism
Igor A. Batalinab
and Klaus Beringac
a
The Niels Bohr Institute
Blegdamsvej 17
DK-2100 Copenhagen
Denmark
b
I.E. Tamm Theory Division
P.N. Lebedev Physics Institute
Russian Academy of Sciences
53 Leninisky Prospect
Moscow 119991
Russia
c
Institute for Theoretical Physics & Astrophysics
Masaryk University
Kotl´aˇrsk´a 2
CZ-611 37 Brno
Czech Republic
January 5, 2014
Abstract
We consider the possibility of adding a Grassmann-odd function ν to the odd Laplacian. Requiring
the total ∆ operator to be nilpotent leads to a differential condition for ν, which is integrable. It
turns out that the odd function ν is not an independent geometric object, but is instead completely
specified by the antisymplectic structure E and the density ρ. The main impact of introducing the
ν term is that it makes compatibility relations between E and ρ obsolete. We give a geometric
interpretation of ν as (minus 1/8 times) the odd scalar curvature of an arbitrary antisymplectic,
torsion-free and ρ-compatible connection. We show that the total ∆ operator is a ρ-dressed version
of Khudaverdian’s ∆E operator, which takes semidensities to semidensities. We also show that
the construction generalizes to the situation where ρ is replaced by a non-flat line bundle connection
F. This generalization is implemented by breaking the nilpotency of ∆ with an arbitrary
Grassmann-even second-order operator source.
PACS number(s): 02.40.-k; 03.65.Ca; 04.60.Gw; 11.10.-z; 11.10.Ef; 11.15.Bt.
Keywords: BV Field-Antifield Formalism; Odd Laplacian; Antisymplectic Geometry; Semidensity;
Antisymplectic Connection; Odd Scalar Curvature.
bE-mail: batalin@lpi.ru cE-mail: bering@physics.muni.cz
1
1 Introduction
Conventionally [1, 2, 3, 4] the geometric arena for quantization of Lagrangian theories in the fieldantifield
formalism [5, 6, 7] is taken to be an antisymplectic manifold (M; E) with a measure density ρ.
Each point in the manifold M with local coordinates ΓA and Grassmann parity εA ≡ ε(ΓA) represents
a field-antifield configuration ΓA ={φα; φ∗
α}, the antisymplectic structure E provides the antibracket
(·, ·), and the density ρ yields the path integral measure. However, up until recently, it has been
necessary to impose a compatibility condition [2, 8] between the two geometric structures E and ρ to
ensure nilpotency of the odd Laplacian
∆ρ ≡
(−1)εA
2ρ
→
∂l
AρEAB
→
∂l
B ,
→
∂l
A ≡
→
∂l
∂ΓA
. (1.1)
In this paper, we show that the compatibility condition between E and ρ can be omitted if one adds
an odd scalar function ν to the odd Laplacian ∆ρ,
∆ = ∆ρ + ν (1.2)
such that the total ∆ operator is nilpotent
∆2
= 0 . (1.3)
Nilpotency is important for the field-antifield formalism in many ways, for instance in securing that
the physical partition function Z is independent of gauge-choice, see Appendix A. (More precisely,
what is really vital is the nilpotency of the underlying ∆E operator, cf. Sections 8-9.) In physics terms,
the addition of the ν function to the odd Laplacian ∆ρ implies that the quantum master equation
∆e
i
¯h
W
= 0 (1.4)
is modified with a ν term at the two-loop order O(¯h2
):
1
2
(W, W) = i¯h∆ρW + ¯h2
ν , (1.5)
and ∆ρ is in general no longer a nilpotent operator. It turns out that the zeroth-order ν term is uniquely
determined from the nilpotency requirement (1.3) apart from an odd constant. One particular solution
to the zeroth-order term, which we call νρ, takes a special form [9]
νρ ≡ ν(0)
ρ +
ν(1)
8
−
ν(2)
24
, (1.6)
where ν
(0)
ρ , ν(1) and ν(2) are defined as
ν(0)
ρ ≡
1
√
ρ
(∆1
√
ρ) , (1.7)
ν(1)
≡ (−1)εA (
→
∂l
B
→
∂l
AEAB
) , (1.8)
ν(2)
≡ (−1)εAεC (
→
∂l
DEAB
)EBC(
→
∂l
AECD
) (1.9)
= −(−1)εB (
→
∂l
AEBC)ECD
(
→
∂l
DEBA
) . (1.10)
Here, ∆1 in eq. (1.7) denotes the expression (1.1) for the odd Laplacian ∆ρ=1 with ρ replaced by 1. In
particular, the odd scalar νρ is a function of E and ρ, so there is no call for new independent geometric
2
structures on the manifold M. In Sections 2–6 we show that ∆ρ+ν is the only possible ∆ operator
within the set of all second-order differential operators. The now obsolete compatibility condition
[2, 8] between E and ρ can be recast as νρ = odd constant, thereby making contact to the previous
approach [2], which uses the odd Laplacian ∆ρ only. The explicit formula (1.6) for νρ is proven in
Section 7 and Appendix B. The formula (1.6) first appeared in Ref. [9]. That paper was devoted to
Khudaverdian’s ∆E operator [10, 11, 12, 13], which takes semidensities to semidensities. This is no
coincidence: At the bare level of mathematical formulas the construction is intimately related to the
∆E operator, as shown in Sections 8-9. However the starting point is different. On one hand, Ref. [9]
studied the ∆E operator in its minimal and purest setting, which is a manifold with an antisymplectic
structure E but without a density ρ. On the other hand, the starting point of the current paper is a
∆ operator that takes scalar functions to scalar functions, and this implies that a choice of ρ (or F,
cf. below) should be made. Later in Sections 10 and 11 we interpret the odd νρ function as (minus
1/8 times) the odd scalar curvature R of an arbitrary antisymplectic, torsion-free and ρ-compatible
connection,
νρ = −
R
8
. (1.11)
One of the main priorities for the current article is to ensure that all arguments are handled in completely
general coordinates without resorting to Darboux coordinates at any stage. This is important
to give a physical theory a natural, coordinate-independent, geometric status in the antisymplectic
phase space. We shall also throughout the paper often address the question of generalizing the density
ρ to a non-flat line bundle connection F. It is well-known [2] that a density ρ gives rise to a flat line
bundle connection
FA = (
→
∂l
A ln ρ) . (1.12)
In fact, several mathematical objects, for instance the odd Laplacian ∆ρ and the odd scalar νρ,
can be formulated entirely using F instead of ρ. Surprisingly, many of these objects continue to be
well-defined for non-flat F’s as well, where the nilpotency (and the ordinary physical description) is
broken down. In Section 5 we shall therefore temporarily digress to contemplate a modification of the
nilpotency condition that addresses these mathematical observations. Finally, Section 12 contains our
conclusions.
General remark about notation. We have two types of grading: A Grassmann grading ε and an exterior
form degree p. The sign conventions are such that two exterior forms ξ and η, of Grassmann parity
εξ, εη and exterior form degree pξ, pη, respectively, commute in the following graded sense:
η ∧ ξ = (−1)εξεη+pξpη
ξ ∧ η (1.13)
inside the exterior algebra. We will often not write the exterior wedges “∧” explicitly.
2 General Second-Order ∆ operator
We here introduce the setting and notation more carefully, and argue that the ∆ operator must be
equal to ∆ρ+νρ up to an odd constant. (The undetermined odd constant comes from the fact that the
square ∆2 = 1
2[∆, ∆] does not change if ∆ is shifted by an odd constant.) Consider now an arbitrary
Grassmann-odd, second-order, differential operator ∆ that takes scalar functions to scalar functions.
In this paper, we shall only discuss the non-degenerate case, where the second-order term in ∆ is
of maximal rank, and hence provides for a non-degenerated antibracket (·, ·), cf. the Definition (2.6)
below. (The non-degeneracy assumption is motivated by the fact that it is satisfied for currently known
applications. The degenerate case may be dealt with via for instance the antisymplectic conversion
3
mechanism [14, 15].) Due to the non-degeneracy assumption, it is always possible to organize ∆ as
∆ = ∆F + ν , (2.1)
where ν is a zeroth-order term and ∆F is an operator with terms of second and first order [2]
∆F ≡
(−1)εA
2
(
→
∂l
A+FA)EAB
→
∂l
B . (2.2)
Here, EAB =EAB(Γ), FA =FA(Γ) and ν =ν(Γ) is a (2, 0)-tensor, a line bundle connection, and a scalar,
respectively. We shall sometimes use the slightly longer notation ∆F ≡ ∆F,E to acknowledge that it
depends on two inputs: F and E. The line bundle connection FA transforms under general coordinate
transformations ΓA → Γ′B as
FA = (
→
∂l
∂ΓA
Γ′B
)F′
B + (
→
∂l
∂ΓA
ln J) , J ≡ sdet
∂Γ′B
∂ΓA
. (2.3)
These transformation properties guarantee that the expressions (2.1) and (2.2) remain invariant under
general coordinate transformations. The Grassmann-parities are
ε(EAB
) = εA +εB +1 , ε(FA) = εA , ε(ν) = 1 . (2.4)
One may, without loss of generality, assume that the (2, 0)-tensor EAB has a Grassmann-graded
skewsymmetry
EAB
= − (−1)(εA+1)(εB+1)
EBA
. (2.5)
The antibracket (f, g) of two functions f = f(Γ) and g = g(Γ) is defined via a double commutator∗
[16] with the ∆-operator, acting on the constant unit function 1,
(f, g) ≡ (−1)εf [[
→
∆, f], g]1 ≡ (−1)εf ∆(fg) − (−1)εf (∆f)g − f(∆g) + (−1)εg fg(∆1)
= (f
←
∂r
A)EAB
(
→
∂l
Bg) = − (−1)(εf +1)(εg+1)
(g, f) , (2.6)
where use is made of the skewsymmetry (2.5) in the third equality. By the non-degeneracy assumption,
there exists an inverse matrix EAB such that
EAB
EBC = δA
C = ECBEBA
. (2.7)
Since the tensor EAB possesses a graded A↔B skewsymmetry (2.5), the inverse tensor EAB must be
skewsymmetric,
EAB = −(−1)εAεB EBA . (2.8)
In other words, EAB is a two-form
E =
1
2
dΓA
EAB ∧ dΓB
. (2.9)
The Grassmann parity is
ε(EAB) = εA +εB +1 . (2.10)
∗
Here, and throughout the paper, [A, B] and {A, B} denote the graded commutator [A, B] ≡ AB − (−1)εA
εB BA and
the graded anticommutator {A, B} ≡ AB + (−1)εAεB BA, respectively.
4
3 Nilpotency Conditions: Part I
The square ∆2 = 1
2[∆, ∆] of an odd second-order operator (2.1) is generally a third-order differential
operator, which we, for simplicity, imagine has been normal ordered, i.e. with all derivatives standing
to the right. Nilpotency (1.3) of the ∆ operator leads to conditions on EAB, FA and ν. Let us
therefore systematically, over the next four Sections 3–6, discuss order by order the consequences of
the nilpotency condition ∆2 = 0, starting with the highest (third) order terms, and going down until
we reach the zeroth order.
The third-order terms of ∆2 vanish if and only if the Jacobi identity
cycl. f,g,h
(−1)(εf
+1)(εh
+1)
(f, (g, h)) = 0 (3.1)
for the antibracket (·, ·) holds. We shall always assume this from now on. Equivalently, the two-form
EAB is closed,
dE = 0 . (3.2)
In terms of the matrices EAB and EAB, the Jacobi identity (3.1) and the closeness condition (3.2)
read
cycl. A,B,C
(−1)(εA+1)(εC +1)
EAD
(
→
∂l
DEBC
) = 0 , (3.3)
cycl. A,B,C
(−1)εAεC (
→
∂l
AEBC) = 0 , (3.4)
respectively. By definition, a non-degenerate tensor EAB with Grassmann-parity (2.10), skewsymmetry
(2.8), and closeness relation (3.4) is called an antisymplectic structure.
Granted the Jacobi identity (3.1), the second-order terms of ∆2 can be written on the form
1
4
RAB
→
∂l
B
→
∂l
A , (3.5)
where RAB with upper indices is a shorthand for
RAD
≡ EAB
RBCECD
(−1)εC , (3.6)
and RAB with lower indices is the curvature tensor for the line bundle connection FA:
RAB ≡ [
→
∂l
A +FA,
→
∂l
B +FB] = (
→
∂l
AFB) − (−1)εAεB (A ↔ B) . (3.7)
Remarkably, the two tensors RAB and RAB carry opposite symmetry:
RAB = −(−1)εAεB RBA , (3.8)
RAB
= (−1)εAεB RBA
. (3.9)
It follows that in the non-degenerate case, the second-order terms of ∆2 vanish if and only if the line
bundle connection FA has vanishing curvature
RAB = 0 . (3.10)
5
The zero curvature condition (3.10) is an integrability condition for the local existence of a density ρ,
FA = (
→
∂l
A ln ρ) . (3.11)
Under the F ↔ ρ identification (3.11) the ∆F operator (2.2) just becomes the ordinary odd Laplacian
∆ρ from eq. (1.1),
∆F = ∆ρ . (3.12)
Conventionally the field-antifield formalism requires the F ↔ ρ identification (3.11) to hold globally.
Nevertheless, we shall present many of the constructions below using F rather than ρ, to be as general
as possible.
There exists a descriptive characterization: Granted the Jacobi identity (3.1), the second-order terms
of ∆2 vanish if and only if there is a Leibniz rule for the interplay of the so-called “one-bracket”
Φ1
∆ ≡∆−(∆1)=∆F and the “two-bracket” (·, ·)
∆F (f, g) = (∆F f, g) − (−1)εf (f, ∆F g) . (3.13)
See Ref. [16, 17] for more details.
4 A Non-Zero F-Curvature?
In eq. (3.10) of the previous Section 3 we learned that the nilpotency condition (1.3) completely kills
the line bundle curvature R. Nevertheless, several constructions continue to be well-defined for nonzero
R. For instance, both the important scalars νF and R fall into this category, cf. eqs. (7.1) and
(11.7) below. Another example, which turns out to be related to our discussion, is the Grassmannodd
2-cocycle of Khudaverdian and Voronov [8, 11, 18]. It is defined using two (possibly non-flat) line
bundle connections F(1) and F(2) as follows:
ν(F(1)
; F(2)
, E) ≡
1
4
divF (12) X(12) ≡
(−1)εA
4
(
→
∂l
A +
F(1) + F(2)
2
)(EAB
(F
(1)
B −F
(2)
B )) , (4.1)
where the divergence “div” is defined in eq. (10.13),
F(12)
≡
F(1) + F(2)
2
, (4.2)
and
XA
(12) ≡ EAB
(F
(1)
B −F
(2)
B ) . (4.3)
It is clear from Definition (4.1) that ν(F(1); F(2), E) behaves as a scalar under general coordinate transformations.
This is because the average F(12) is again a line bundle connection, and X(12) is a vector
field since the difference F
(1)
B −F
(2)
B is a co-vector (=one-form), cf. eq. (2.3). That ν(F(1); F(2), E) is
a 2-cocycle
ν(F(1)
; F(2)
, E) + ν(F(2)
; F(3)
, E) + ν(F(3)
; F(1)
, E) = 0 (4.4)
follows easily by rewriting Definition (4.1) as
ν(F(1)
; F(2)
, E) = ν
(0)
F (1) − ν
(0)
F (2) , (4.5)
where ν
(0)
F generalizes eq. (1.7):
ν
(0)
F ≡
(−1)εA
4
(
→
∂l
A +
FA
2
)(EAB
FB) . (4.6)
6
Note that Definitions (4.1) and (4.6) continue to make sense for non-flat F’s. We should stress that
ν
(0)
F itself is not a scalar, but we shall soon see that it can be replaced in eq. (4.5) by a scalar νF , cf.
eq. (7.1) below. In other words, ν(F(1); F(2), E) is a 2-coboundary.
The F-curvature RAB is also an interesting geometric object in its own right. It can be identified
with a Ricci two-form of a tangent bundle connection ∇, cf. eq. (11.4) in Section 11 below. The Ricci
two-form
R =
1
2
dΓA
RAB ∧ dΓB
(−1)εB (4.7)
is closed
dR = 0 , (4.8)
due to the Bianchi identity
cycl. A,B,C
(−1)εAεC (
→
∂l
ARBC) = 0 , (4.9)
so the two-form (4.7) defines a cohomology class.
5 Breaking the Nilpotency
Due to the above mathematical reasons we shall digress in this Section 5 to contemplate how a nonzero
F-curvature could arise in antisymplectic geometry, although we should stress that it remains
unclear if it is useful in physics. Nevertheless, the strategy that we shall adapt here is to append a
general Grassmann-even (possibly degenerate) second-order operator source 1
2 ∆R to the right-hand
side of the nilpotency condition (1.3):
∆2
=
1
2
∆R . (5.1)
A covariant and general way of realizing the second-order ∆R operator is to write
∆R ≡ ∆F,R + VR + nR , (5.2)
where
∆F,R ≡
(−1)εA
2
(
→
∂l
A +FA)RAB
→
∂l
B (5.3)
is an Grassmann-even Laplacian based on FA and RAB. We have included a Grassmann-even vector
field
VR ≡ V A
R
→
∂l
A (5.4)
and a scalar function nR to give a systematic treatment. Note that the vector field VR is the difference
of the subleading connection terms inside ∆R and ∆F,R. We shall show below that the nR term is
completely determined by consistency, while VR in principle can be any locally Hamiltonian vector
field subjected to the following restriction: Both V A
R and nR should be proportional to the R-source
(or its derivatives) in order to restore nilpotency (1.3) in the limit R → 0.
The new condition (5.1) still imposes the Jacobi identity (3.1) for the antibracket (·, ·) at the third
order, since the modification is just of second order. (We mention, for later, that the Jacobi identity
alone guarantees the existence of a nilpotent ∆E operator and its quantization scheme, cf. Sections 8-9,
regardless of how the nilpotency (5.1) of ∆ is broken at lower orders.) The second-order terms in eq.
(5.1) implies that the F-curvature RAB defined in eq. (3.7) should be identified with the principal
7
symbol RAB appearing inside the ∆F,R operator (5.3), thereby justifying the notation. Note that the
Leibniz rule (3.13) is no longer valid. To see this, it is useful to define an even R-bracket [19]
(f, g)R ≡ [[
→
∆R, f], g]1 ≡ ∆R(fg) − (∆Rf)g − f(∆Rg) + fg(∆R1)
= (f
←
∂r
A)RAB
(
→
∂l
Bg) = (−1)εf εg
(g, f)R . (5.5)
It turns out that the R-bracket (·, ·)R measures the failure of the Leibniz rule:
1
2
(f, g)R = (−1)εf ∆F (f, g) − (−1)εf (∆F f, g) + (f, ∆F g) . (5.6)
Note that this R-bracket (·, ·)R does not satisfy a Jacobi identity. (In fact, we shall see that the
closeness relation (4.8) for RAB will instead lead to a compatibility relation (5.8) below.) Since
∆2
F − 1
2∆F,R is a first-order operator, cf. eqs. (2.1) and (5.1), the commutator
1
2
[∆F,R, ∆F ] = [∆F , ∆2
F −
1
2
∆F,R] (5.7)
becomes a second-order operator at most. (We shall improve this estimate in Lemma 5.1 below.) This
fact already implies that the two brackets (·, ·) and (·, ·)R are compatible in the sense that
cycl. f,g,h
(−1)εf (εh+1)
((f, g), h)R =
cycl. f,g,h
(−1)εf (εh+1)+εg
((f, g)R, h) . (5.8)
Phrased differently, one may define a one-parameter family of antisymplectic two-forms
E(θ) ≡ E + θR ≡ E + Rθ =
1
2
dΓA
EAB(θ) ∧ dΓB
, dE(θ) = 0 , (5.9)
which depends on a Grassmann-odd parameter θ. In components it reads
EAB(θ) = EAB + RABθ , (5.10)
EAB
(θ) = EAB
+ (−1)εA θRAB
= EAB
+ RAB
θ(−1)εB . (5.11)
There exists locally an antisymplectic one-form potential
U(θ) ≡ UA(θ)dΓA , UA(θ) ≡ UA + FAθ ,
dU(θ) = E(θ) ,
→
∂l
AUB(θ) − (−1)εAεB (A ↔ B) = EAB(θ) .
(5.12)
We will now improve the estimate from eq. (5.7):
Lemma 5.1 The commutator [∆F , ∆F,R] is always a first-order operator at most.
Proof of Lemma 5.1: Note that the commutator [∆F , ∆F,R] appears inside the square
(∆F (θ))2
= ∆2
F + θ[∆F,R, ∆F ] = ∆2
F + [∆F , ∆F,R]θ (5.13)
of the Grassmann-odd second-order operator
∆F (θ) ≡ ∆F + θ∆F,R ≡ ∆F + ∆F,Rθ =
(−1)εA
2
(
→
∂l
A +FA)EAB
(θ)
→
∂l
B . (5.14)
8
One knows from the general discussion in the previous Section 3 that the third-order terms in the
square (5.13) vanish because EAB(θ) satisfies the Jacobi identity (3.3). Moreover, the second-order
terms in the square (5.13) are of the form
(−1)εC
4
EAB
(θ) RBC ECD
(θ)
→
∂l
D
→
∂l
A =
1
4
RAB
→
∂l
B
→
∂l
A , (5.15)
cf. eqs. (3.5) and (3.6). It is easy to see that the two θ-dependent terms inside the left-hand side of eq.
(5.15) cancel against each other. In fact, each of the two terms vanish separately due to skewsymmetry:
(−1)εC +εF EAB
RBCECD
RDF EF G
= RAC
ECDRDG
= (−1)(εA+1)(εG+1)
(A ↔ G) . (5.16)
Therefore, the θ-dependent part of the square (5.13) must be of first order at most.
(One may also give a proof of Lemma 5.1 based on Lemma B.1 in Appendix B.) Lemma 5.1 implies
(for instance via the technology of Ref. [16]) that
∆F,R(f, g) − (∆F,Rf, g) − (f, ∆F,Rg) = (−1)εf ∆F (f, g)R − (−1)εf (∆F f, g)R
−(f, ∆F g)R , (5.17)
(∆2
F −
1
2
∆F,R)(f, g) = ((∆2
F −
1
2
∆F,R)f, g) + (f, (∆2
F −
1
2
∆F,R)g) . (5.18)
More generally, there exists a superformulation
∆(θ) ≡ ∆ + θ∆R ≡ ∆ + ∆Rθ =
(−1)εA
2
(
→
∂l
A +FA(θ))EAB
(θ)
→
∂l
B + ν(θ) , (5.19)
where
ν(θ) ≡ ν + θnR ≡ ν + nRθ , (5.20)
and
FA(θ) ≡ FA + 2EABV B
R θ ≡ FA − 2V B
R EBAθ . (5.21)
The nilpotency condition
∆(θ) −
1
2
∂
∂θ
2
= 0 (5.22)
precisely encodes the deformed condition (5.1) and its consistency relation
0 = [∆, [∆, ∆]] = [∆, ∆R] = [∆F +ν, ∆F,R+VR+nR]
= [∆F , ∆F,R] + [∆F , VR] + [∆F , nR] − [∆F,R+VR, ν] . (5.23)
Note in the last line of eq. (5.23) that the first term [∆F , ∆F,R] and the two last terms [∆F , nR] and
[∆F,R+VR, ν] are all of first order. Hence, the second term [∆F , VR] must be of first order as well.
This in turn implies that VR should be a generating vector field for an anticanonical transformation:
VR(f, g) = (VR(f), g) + (f, VR(g)) . (5.24)
Since the antibracket is non-degenerated, it follows that VR must be a locally Hamiltonian vector field,
which we, for simplicity, will assume is a globally Hamiltonian vector field
VR = − 2(νR, ·) , (5.25)
with some Fermionic globally defined Hamiltonian νR. The factor “−2” in eq. (5.25) is chosen for
later convenience. The Hamiltonian νR in eq. (5.25) should be considered as an additional geometric
9
input, which labels the different ways (5.1) of breaking the nilpotency of ∆. It is a priori only defined
in eq. (5.25) up to an odd constant. We fix this constant by requiring that
νR → 0 for R → 0 . (5.26)
Altogether, the Hamiltonian νR does not contribute to the curvature
→
∂l
AFB(θ) − (−1)εAεB (A ↔ B) = RAB (5.27)
of the line bundle connection
FA(θ) = FA + 4(
→
∂l
AνR)θ . (5.28)
Now let us continue the investigation of the deformed condition (5.1). The first-order terms of eq.
(5.1) cancel if and only if
∆2
F −
1
2
∆F,R = (ν−νR, ·) . (5.29)
This is a differential equation for the function ν =ν(Γ), or, equivalently, for the difference ν−νR. It
now becomes clear that the νR function provides an auxiliary curvature background for the ν function.
Since we assume that νR is given, we will now focus on the difference ν−νR rather than on ν itself.
The Frobenius integrability condition for eq. (5.29) comes from the fact that the operator ∆2
F − 1
2∆F,R
differentiates the antibracket, cf. eq. (5.18). This implies that the difference ν−νR can be written as
a contour integral
(ν−νR)(Γ) = (ν−νR)(Γ0) +
Γ
Γ0
((∆2
F −
1
2
∆F,R)ΓA
)EAB
Γ→Γ′
dΓ′B
(5.30)
that is independent of the curve (aside from the two endpoints). It only depends on E, F, and an
odd integration constant (ν−νR)(Γ0). In particular, we conclude that the difference ν−νR does not
introduce any new geometric structures. The first-order commutator from Lemma 5.1 can now be
expressed in terms of the difference ν−νR as follows:
1
2
[∆F,R, ∆F ] = [∆F , ∆2
F −
1
2
∆F,R] = ∆F (ν−νR, ·) − (ν−νR, ∆F (·))
= (∆F (ν−νR), ·) −
1
2
(ν−νR, ·)R . (5.31)
Here, eq. (5.29) is used in the second equality and the deformed Leibniz rule (5.6) is used in the third
(=last) equality.
Finally, the zeroth-order terms of eq. (5.1) cancel if and only if
nR = 2(∆F ν) , (5.32)
so this fixes completely the Grassmann-even function nR. One can show that if the Hamiltonian vector
field V A
R vanishes in the flat limit R → 0, then the nR function, defined via eq. (5.32), automatically
does the same, cf. eq. (6.2) below. The nilpotency-breaking operator ∆R will therefore vanish for
R → 0, as it should.
6 Nilpotency Conditions: Part II
After this digression into non-zero R curvature, let us now return to the nilpotent (and ordinary
physical) situation ∆2 = 0, where R, V A
R and nR are all zero. Not much changes for the condition
10
(5.29) for the first-order terms other that one should remove the νR function and the ∆F,R operator
from the Frobenius integrability condition (5.18), the differential eq. (5.29), and the contour integral
(5.30). (Of course, now the Frobenius integrability condition is just an easy consequence of the Leibniz
rule (3.13) applied twice.) The condition (5.32) for the zeroth-order terms becomes
(∆F ν) = 0 . (6.1)
Equation (6.1) is not an independent condition but it follows instead automatically from the previous
requirements. Proof:
−(∆F ν) =
(−1)εA
2
(
→
∂l
A +FA)(ν, ΓA
) =
(−1)εA
2
(
→
∂l
A +FA)∆2
F ΓA
=
(−1)εA+εB
4
(
→
∂l
A +FA)(
→
∂l
B +FB)(ΓB
, ∆F ΓA
)
= −
(−1)εA
8
(
→
∂l
A+FA)(
→
∂l
B +FB)∆F (ΓB
, ΓA
)
=
(−1)εAεC
16
(
→
∂l
A +FA)(
→
∂l
B +FB)(
→
∂l
C +FC)(ΓC
, (ΓB
, ΓA
))(−1)(εA+1)(εC +1)
= 0 . (6.2)
Here, the ν eq. (5.29) is used in the second equality, the Leibniz rule (3.13) in the fourth equality,
the Jacobi identity (3.1) in the sixth (=last) equality, and the zero curvature condition (3.10) in the
second, fourth and sixth equality.
7 An Explicit Solution νF
Remarkably, the integral (5.30) can be performed.
Proposition 7.1 The odd quantity
νF ≡ ν
(0)
F +
ν(1)
8
−
ν(2)
24
(7.1)
is a solution to the differential eq. (5.29) for the difference ν−νR, even if the line bundle connection
F is not flat.
Here, ν
(0)
F , ν(1) and ν(2) are given by eqs. (4.6), (1.8) and (1.9), respectively. Proposition 7.1 is proven
in Appendix B by repeated use of the Jacobi identity (3.3) and the closeness relation (3.4). Notice
that under the F ↔ ρ identification (3.11), the F-dependent Definitions (4.6) and (7.1) reduce to their
ρ counterparts (1.6) and (1.7),
νF = νρ , ν
(0)
F = ν(0)
ρ . (7.2)
Notation: νF or νρ with subscript “F” or “ρ” denotes one particular solution (7.1) or (1.6) to the
difference ν−νR in eq. (5.29), respectively.
Proposition 7.2 The νF quantity (7.1) is invariant under general coordinate transformations, i.e. it
is a scalar, even if the line bundle connection F is not flat.
11
Proof of Proposition 7.2: Under an arbitrary infinitesimal coordinate transformation δΓA = XA,
one calculates [9]
δν
(0)
F = −
1
2
∆1div1X , (7.3)
δν(1)
= 4∆1div1X + (−1)εA (
→
∂l
CEAB
)(
→
∂l
B
→
∂l
AXC
) , (7.4)
δν(2)
= 3(−1)εA (
→
∂l
CEAB
)(
→
∂l
B
→
∂l
AXC
) , (7.5)
where ∆1 and div1 denote the expressions (1.1) and (10.14) for the odd Laplacian ∆ρ=1 and the
divergence divρ=1 with ρ replaced by 1. One easily sees that while the three constituents ν
(0)
F , ν(1)
and ν(2) separately have non-trivial transformation properties, the linear combination νF in eq. (7.1)
is indeed a scalar. Proposition 7.2 also follows from the identification of νF as an odd scalar curvature,
cf. eq. (11.8) below.
The difference ν−νR is only determined up to an odd integration constant because the defining relation
(5.29) is a differential relation. The explicit solution νF in (7.1) provides us with an opportunity to
fix this odd integration constant once and for all. Out of all the solutions to the difference ν−νR, we
choose the νF solution (7.1), i.e. we identify from now on
ν ≡ νF + νR . (7.6)
We do this for two reasons. Firstly, any odd constants inside the νF expression (7.1) can only arise
implicitly through E and F, which means that if E and F do not carry any odd constants, then
the νF solution (7.1) will be free of odd constants as well. Similarly, the νR part does not contain
odd constants because of the boundary condition (5.26). Secondly, the expression νF is the only
solution that has an interpretation as an odd scalar curvature, cf. eq. (11.8) below. This completes
the reduction of a general second-order ∆ operator to
∆ = ∆F + ν = ∆F + νF + νR −→ ∆ρ + νρ for R → 0 . (7.7)
8 The ∆E operator
Let us briefly outline the connection to Khudaverdian’s ∆E operator [10, 11, 12, 13], which takes
semidensities to semidensities. The ∆E operator was defined in Ref. [9] as
∆E ≡ ∆1 +
ν(1)
8
−
ν(2)
24
, (8.1)
where ∆1 denotes the expression (1.1) for the odd Laplacian ∆ρ=1 with ρ replaced by 1. Some of
the strengths of Definition (8.1) are that it works in any coordinate system and that it is manifestly
independent of ρ or F. However, it is a rather lengthy calculation to demonstrate in a ρ-less
or F-less environment that ∆E has the pertinent transformation property under general coordinate
transformations, and that it is nilpotent
∆2
E = 0 , (8.2)
cf. Ref. [9]. Once we are given a density ρ, the situation simplifies considerably. Then, the ∆E operator
becomes just the operator ∆ ≡ ∆ρ+νρ conjugated with the square root of ρ:
∆E =
√
ρ∆
1
√
ρ
. (8.3)
12
Proof of eq. (8.3): Let σ denote an arbitrary semidensity. Then, it follows from the explicit νρ
formula (1.6) that
(∆Eσ) = (∆1σ) + (
ν(1)
8
−
ν(2)
24
)σ = (∆1σ) − (∆1
√
ρ)
σ
√
ρ
+ νρσ
=
√
ρ(∆1
σ
√
ρ
) + (
√
ρ,
σ
√
ρ
) + νρσ =
√
ρ(∆ρ
σ
√
ρ
) + νρσ =
√
ρ(∆
σ
√
ρ
) . (8.4)
It is remarkable that the
√
ρ-conjugated ∆ operator
√
ρ∆ 1√
ρ does not depend on ρ at all! On the other
hand, it is obvious that the operator
√
ρ∆ 1√
ρ is nilpotent and that it satisfies the required transformation
law under general coordinate transformations, i.e. that it takes semidensities to semidensities.
This is because the ∆ operator itself is a nilpotent operator and ∆ takes scalar functions to scalar
functions. Let us also mention that
νρ = (∆1) =
1
√
ρ
(∆E
√
ρ) . (8.5)
The right-hand side of eq. (8.5) served as a definition of the odd scalar νρ in Ref. [9].
More generally, the operators ∆E and ∆ ≡ ∆F +νF +νR are linked via
∆E = ∆ −
(−1)εA
2
FA(ΓA
, ·) − ν
(0)
F − νR . (8.6)
Equation (8.6) may be viewed as a generalization of eq. (8.3) to non-flat F’s, or, equivalently, to
non-nilpotent ∆’s, cf. eq. (3.10). It might be worth emphasizing that ∆E is nilpotent even in this
situation, since ∆E only depends on E.
9 F-Independent Formalism
There exists [9, 15] a manifestly F-independent quantization scheme based on the ∆E operator. Since
we will demand that the quantization is covariant with respect to the antisymplectic phase space,
it will be necessary to use first-level formalism or one of its higher-level generalizations [2, 20]. See
Ref. [8] for a review of the multi-level formalism. It turns out to be most efficient to use the second-level
formalism in order not to deal directly with weak quantum master equations [21]. Let ΓA denote all the
zeroth- and first-level fields and antifields, and let λα denote the second-level Lagrange multipliers for
the first-level gauge-fixing constraints. Assume also that there is no dependence on the corresponding
second-level antifields λ∗
α. The second-level partition function
Z = [dΓ][dλ] e
i
¯h
(WE+XE)
(9.1)
contains two Boltzmann semidensities: a gauge-generating semidensity e
i
¯h
WE and a gauge-fixing semidensity
e
i
¯h
XE , where WE and XE denote the corresponding quantum actions. The two Boltzmann
semidensities are both required to satisfy strong quantum master equations
∆Ee
i
¯h
WE = 0 , ∆Ee
i
¯h
XE = 0 , (9.2)
or equivalently,
1
2
(WE, WE) = i¯h∆1WE + ¯h2
∆E1 ,
1
2
(XE, XE) = i¯h∆1XE + ¯h2
∆E1 , (9.3)
13
where
∆E1 =
ν(1)
8
−
ν(2)
24
. (9.4)
The caveat is that the quantum actions WE and XE are not scalars. They obey non-trivial transformation
laws under general coordinate transformations, since they are logarithms of semidensities. It
is shown in Appendix A that the partition function (9.1) is independent of the gauge choice XE.
If we are given a density ρ, we may introduce a nilpotent ∆ operator (8.3) and Boltzmann scalars
e
i
¯h
W
and e
i
¯h
X
by dressing appropriately with square roots of ρ:
√
ρ∆ = ∆E
√
ρ , e
i
¯h
WE =
√
ρe
i
¯h
W
, e
i
¯h
XE =
√
ρe
i
¯h
X
. (9.5)
Then ∆ = ∆ρ+νρ and the two scalar actions W and X will satisfy the strong quantum master eq.
(1.4) from the Introduction, which in non-exponential form reads
1
2
(W, W) = i¯h∆ρW + ¯h2
νρ ,
1
2
(X, X) = i¯h∆ρX + ¯h2
νρ . (9.6)
The partition function (9.1) then reduces to the familiar W-X form:
Z = ρ[dΓ][dλ] e
i
¯h
(W +X)
. (9.7)
Conversely, since the partition function (9.7) via the above identifications (9.5) can be written in the
manifestly ρ-independent form (9.1), one may state that in this sense the partition function (9.7) does
not depend on ρ. The point is that the well-known ambiguity in the choice of measure that exists in the
field-antifield formalism has been fully transcribed into an ambiguity in the choice of the Boltzmann
semidensity e
i
¯h
WE . Put differently, if one splits the Boltzmann semidensity e
i
¯h
WE into a Boltzmann
scalar e
i
¯h
W
and a density ρ as done in eq. (9.5), the measure ambiguity sits inside the scalar e
i
¯h
W
, not
in ρ, as ρ actually drops out of Z.
More generally, imagine that we are given a non-nilpotent operator ∆ ≡ ∆F +νF +νR with a non-flat
line bundle connection F that satisfies the deformed nilpotency condition (5.1). We can still define
the partition function in this situation via the above quantization scheme (9.1) based on the nilpotent
∆E operator. Such an approach will of course be manifestly F-independent by construction.
10 Connection
We now introduce a connection ∇ : TM × TM → TM. See Ref. [19, 22] for related discussions. The
left covariant derivative (∇AX)B of a left vector field XA is defined as [19]
(∇AX)B
≡ (
→
∂l
AXB
) + (−1)εX (εB+εC)
ΓA
B
CXC
, ε(XA
) = εX + εA , (10.1)
The word “left” implies that XA and (∇AX)B transform with left derivatives
X′B
= XA
(
→
∂l
∂ΓA
Γ′B
) , (
→
∂l
∂ΓA
Γ′B
)(∇′BX)′C
= (∇AX)B
(
→
∂l
∂ΓB
Γ′C
) , (10.2)
under general coordinate transformations ΓA → Γ′B. It is convenient to introduce a reordered Christoffel
symbol
ΓA
BC ≡ (−1)εAεB ΓB
A
C (10.3)
14
to minimize the appearances of sign factors. On an antisymplectic manifold (M; E), it is furthermore
possible to define a Christoffel symbol with three lower indices
ΓABC ≡ EADΓD
BC(−1)εB . (10.4)
Let us also define
γABC ≡ ΓABC −
1
3
(EA{B
←
∂r
C}) ≡ ΓABC −
1
3
(EAB
←
∂r
C + EAC
←
∂r
B(−1)εBεC ) . (10.5)
γABC is not a tensor but it still has some useful properties, see eqs. (10.8) and (10.11) below. One can
think of γABC as parametrizing all the possible connections ∇ on (M; E).
An antisymplectic connection ΓA
B
C satisfies by definition [19]
0 = (∇AE)BC
≡ (
→
∂l
AEBC
) + ΓA
B
DEDC
− (−1)(εB +1)(εC +1)
(B ↔ C) , (10.6)
so that the antisymplectic metric EAB is covariantly preserved. In terms of the two-form EAB, the
antisymplectic condition reads
0 = (∇AE)BC ≡ (
→
∂l
AEBC) − ((−1)εAεB ΓBAC − (−1)εBεC (B ↔ C)) . (10.7)
Written in terms of the γABC symbol, the antisymplectic condition (10.7) becomes a purely algebraic
equation, due to the closeness relation (3.4):
γABC = (−1)εAεB+εBεC +εCεA γCBA . (10.8)
A torsion-free connection has the following symmetry in the lower indices:
ΓA
BC = −(−1)(εB +1)(εC +1)
ΓA
CB , (10.9)
ΓABC = (−1)εBεC ΓACB , (10.10)
γABC = (−1)εBεC γACB . (10.11)
Note that (−1)εAεB γBAC = γABC = (−1)εBεC γACB is totally symmetric for an antisymplectic torsionfree
connection. (Similar results hold for even symplectic structures.)
A connection ∇ can be used to define a divergence of a Bosonic vector field XA as
str(∇X) ≡ (−1)εA (∇AX)A
= ((−1)εA
→
∂l
A + ΓB
BA)XA
, εX = 0 . (10.12)
On the other hand, the divergence is defined in terms of F or ρ as
divF X ≡ (−1)εA (
→
∂l
A +FA)XA
, (10.13)
divρX ≡
(−1)εA
ρ
→
∂l
A(ρXA
) . (10.14)
See Ref. [23] for a mathematical exposition of divergence operators on supermanifolds. Under the
F ↔ ρ identification (3.11), the last two Definitions (10.13) and (10.14) agree:
divF X = divρX . (10.15)
15
In order to have a unique divergence operator (and hence a unique notion of volume), it is necessary
to impose the following compatibility condition between FA and the Christoffel symbols ΓA
BC:
ΓB
BA = (−1)εA FA . (10.16)
We shall only consider antisymplectic, torsion-free, and F-compatible connections ∇, i.e. connections
that satisfy the three conditions (10.6), (10.9) and (10.16). The first and third condition ensure
the compatibility with E and F, respectively. The second (the torsion-free condition) guarantees
compatibility with the closeness relation (3.4). It can be demonstrated that connections satisfying
these three conditions exist locally for N > 1, where 2N denotes the number of antisymplectic variables
ΓA, A = 1, . . . , 2N. (There are counterexamples for N =1 where ∇ need not exist.) For connections
satisfying the three conditions, the ∆F operator can be written on a manifestly covariant form
∆F =
(−1)εA
2
∇AEAB
∇B =
(−1)εB
2
EBA
∇A∇B . (10.17)
11 Curvature
The Riemann curvature tensor RAB
C
D is defined as the commutator of the ∇ connection
([∇A, ∇B]X)C
= RAB
C
DXD
(−1)εX (εC +εD)
, (11.1)
so that
RAB
C
D = (
→
∂l
AΓB
C
D) + (−1)εBεC ΓA
C
EΓE
BD − (−1)εAεB (A ↔ B) . (11.2)
It is useful to define a reordered Riemann curvature tensor RA
BCD as
RA
BCD ≡ (−1)εA(εB+εC)
RBC
A
D = (−1)εAεB (
→
∂l
BΓA
CD)+ΓA
BEΓE
CD −(−1)εBεC (B ↔ C) . (11.3)
It is interesting to consider the various contractions of the Riemann curvature tensor. There are two
possibilities. Firstly, there is the Ricci two-form
RAB ≡ RAB
C
C(−1)εC = (
→
∂l
AFB) − (−1)εAεB (A ↔ B) . (11.4)
However, the Ricci two-form RAB typically vanishes, cf. eq. (3.10), and even if it does not vanish, its
antisymmetry (3.8) means that RAB cannot successfully be contracted with the antisymplectic metric
EAB to yield a non-zero scalar curvature, cf. eq. (2.5). Secondly, there is the Ricci tensor
RAB ≡ RC
CAB = (−1)εC (
→
∂l
C + FC)ΓC
AB − (
→
∂l
AFB)(−1)εB − ΓA
C
DΓD
CB . (11.5)
Note that when the torsion tensor and Ricci two-form vanish, the Ricci tensor RAB possesses exactly
the same A↔B symmetry (2.5) as the antisymplectic metric EAB with upper indices
RAB = − (−1)(εA+1)(εB+1)
RBA . (11.6)
The odd scalar curvature R is therefore defined in antisymplectic geometry as the contraction of the
Ricci tensor RAB and the antisymplectic metric EBA,
R ≡ RABEBA
= EAB
RBA . (11.7)
Proposition 11.1 For an arbitrary, antisymplectic, torsion-free, and F-compatible connections ∇,
the scalar curvature R does only depend on E and F through the odd scalar νF ,
R = − 8νF , (11.8)
even if the line bundle connection F is not flat.
16
Proposition 11.1 is shown in Appendix C. In particular, one concludes that the scalar curvature R
does not depend on the connection ΓA
BC used.
One can perform various consistency checks on the formalism. Here, let us just mention one. For an
antisymplectic connection ∇, one has
0 = [∇A, ∇B]ECD
= RAB
C
F EF D
− (−1)(εC +1)(εD+1)
(C ↔ D) , (11.9)
or, equivalently,
RC
ABF EF D
= − (−1)εAεB+(εC+1)(εD+1)+(εA+εB)(εC +εD)
RD
BAF EF C
. (11.10)
Contracting the A ↔ C and B ↔ D indices in eq. (11.10) indeed produces the identity R = R. Had
the signs turn out differently, the odd scalar curvature (11.7) would have been stillborn, i.e. always
zero.
12 Conclusions
In this paper, we have first of all analyzed a general non-degenerate, second-order ∆ operator, and
found that nilpotency determines the ∆ operator uniquely (after dismissing an odd constant). The
result is that ∆ has to be ∆ρ+νρ, where ∆ρ is the odd Laplacian, and νρ is an odd scalar function
(=zeroth-order operator) that only depends on the density ρ and the antisymplectic structure E.
Secondly, we have shown that several constructions in antisymplectic geometry can be extended to a
non-flat line bundle connection F, which replaces ρ. We did this by breaking the nilpotency ∆2 = 1
2 ∆R
by a general second-order operator ∆R, which acts as a source for the F-curvature R. In this more
general case, the ∆ operator takes the form ∆F +νF +νR, where ∆F and νF are generalizations of
the odd Laplacian ∆ρ and the odd scalar νρ, respectively. The νR term is an auxiliary curvature
background encoded in the ∆R operator. Thirdly, we have identified the νF function with (minus
1/8 times) the odd scalar curvature R of an arbitrary antisymplectic, torsion-free, and F-compatible
connection.
One may summarize by saying that two notions of curvature play an important rˆole in this paper: 1) a
line bundle curvature RAB defined in eq. (3.7) and 2) an odd scalar curvature R defined in eq. (11.7).
The former provides a natural framework for several mathematical constructions, but it remains currently
unclear if it would be useful in physics. On the other hand, the field-antifield formalism naturally
embraces the latter type of curvature both physically and mathematically. Concretely, we saw that
the odd scalar curvature R manifests itself via a zeroth-order term νF in the ∆ operator, which could
potentially be used in a physical application some day. Altogether, the odd scalar curvature R and
νF represent an important milestone in our understanding of the symmetries and the supergeometric
structures behind the powerful field-antifield formalism.
Acknowledgements: We would like to thank P.H. Damgaard for discussions and the Niels Bohr
institute for warm hospitality. The work of I.A.B. is supported by grants RFBR 05-01-00996, RFBR
05-02-17217 and LSS-4401.2006.2. The work of K.B. is supported by the Ministry of Education of the
Czech Republic under the project MSM 0021622409.
17
A Independence of Gauge-Fixing in the F-Independent Formalism
In this Appendix A, we prove in two different ways that the partition function (9.1) is independent of
gauge-fixing. Let us introduce the following shorthand notation
w ≡ e
i
¯h
WE , x ≡ e
i
¯h
XE , (A.1)
for the two Boltzmann semidensities, so that the partition function (9.1) simply becomes
Z = [dΓ][dλ] w x . (A.2)
The Boltzmann semidensities w and x are ∆E-closed because of the two quantum master eqs. (9.2).
Since the ∆E operator is nilpotent, one may argue on general grounds that an arbitrary infinitesimal
variation of x should be ∆E-exact, which may be written as
δx = [
→
∆E, δΨ]x ≡ ∆E(δΨ x) + δΨ(∆Ex) , (A.3)
if one assumes that x is invertible and satisfies the quantum master eq. (9.2). Phrased equivalently,
the variation δXE of the quantum action is BRST-exact,
δXE = (XE, δΨ) +
¯h
i
∆1(δΨ) = σXE
(δΨ) , (A.4)
where σXE
=(XE, ·) + ¯h
i ∆1 is a quantum BRST-operator. One may now proceed in at least two ways.
One axiomatic way [24] uses that the ∆E operator (8.1) is symmetric,
∆T
E = ∆E , (A.5)
i.e. stabile under integration by part. Then, an infinitesimal variation (A.3) of the gauge-fixing Boltzmann
semidensity x changes the partition function as
δZ = [dΓ][dλ] w δx = [dΓ][dλ] w [
→
∆E, δΨ]x
= [dΓ][dλ] [(∆Ew) δΨ x + w δΨ (∆Ex)] = 0 , (A.6)
where the symmetry property (A.5) is used in the third equality and the two quantum master equations
(9.2) in the fourth (= last) equality. Notice how this proof requires very little knowledge of the detailed
form of ∆E. Another proof [2, 5, 21] uses an intrinsic infinitesimal redefinition of the integration
variables,
δΓA
=
i
2¯h
(ΓA
, XE −WE)δΨ +
1
2
(ΓA
, δΨ) =
w
2x
(ΓA
,
x δΨ
w
) , δλα
= 0 , (A.7)
to induce the allowed variation (A.3) of x. Now it is instructive to write the path integral integrand
as a volume form Ω ≡ wx[dΓ][dλ] with measure density wx. The Lie-derivative is
δΩ = (divwxδΓ)Ω . (A.8)
In detail, the field-antifield redefinition (A.7) yields the following logarithmic variation of Ω:
divwxδΓ ≡
(−1)εA
wx
→
∂l
A(wx δΓA
) =
(−1)εA
2wx
→
∂l
Aw2
(ΓA
,
x δΨ
w
) =
w
x
∆w2
x δΨ
w
=
1
x
∆1(x δΨ) − (∆1w)
δΨ
w
=
1
x
∆1(x δΨ) + (
ν(1)
8
−
ν(2)
24
)δΨ =
1
x
∆E(x δΨ)
18
=
1
x
[
→
∆E, δΨ]x = δ ln x . (A.9)
Here, a non-trivial property of the odd Laplacian (1.1) is used in the fourth equality, the two quantum
master equations (9.2) are used in the fifth and seventh equality, and the formula (A.3) for the allowed
variation of x is used in the eighth (=last) equality. If one reads the above eq. (A.9) in the opposite
direction, one sees that all allowed variations (A.3) of the gauge-fixing Boltzmann semidensity x can
be reproduced by an intrinsic field-antifield redefinition (A.7),
δZ = [dΓ][dλ] w δx = Ω δ ln x = Ω divwxδΓ = δΩ = 0 . (A.10)
One concludes that the partition function Z = Ω must be independent of the gauge-fixing x part
since an intrinsic redefinition of dummy integration variables cannot change the value of the path
integral.
B Proof of Proposition 7.1
In this Appendix B, we show that the νF expression (7.1) satisfies the differential eq. (5.29) for the
difference ν−νR. We start by recalling that the ∆F operator (2.2) is
∆F ≡ ∆1 + V , (B.1)
where ∆1 denotes the expression (1.1) for the odd Laplacian ∆ρ=1 with ρ replaced by 1, and where
we, for convenience, have defined
V ≡
(−1)εA
2
FA(ΓA
, ·) . (B.2)
Lemma B.1 The square of the ∆F operator is
∆2
F ≡ ∆2
1 + [∆1, V ] + V 2
= ∆2
1 +
1
2
∆F,R + (ν
(0)
F , ·) . (B.3)
Proof of Lemma B.1: One finds by straightforward calculations that
4V 2
= (−1)εA+εB FA(ΓA
, FB(ΓB
, ·))
= (−1)εA FBFA(ΓA
, (ΓB
, ·)) + (−1)εA+εB FAEAC
(
→
∂l
CFB)(ΓB
, ·)
=
(−1)εA
2
FBFA((ΓA
, ΓB
), ·) + (−1)εA FAEAC
[FC
←
∂r
B + RCB(−1)εB ](ΓB
, ·)
=
(−1)εA
2
(FAEAB
FB, ·) + (−1)εA+εC FAEAB
RBC(ΓC
, ·) , (B.4)
and
2[∆1, V ] = (−1)εA ∆1FA(ΓA
, ·) + (−1)εA FA(ΓA
, ∆1(·))
= (−1)εA (∆1FA)(ΓA
, ·) + (FA, (ΓA
, ·)) + FA∆1(ΓA
, ·) + (−1)εA FA(ΓA
, ∆1(·))
=
(−1)εA+εB
2
(
→
∂l
BEBC
→
∂l
CFA)(ΓA
, ·) + (FA
←
∂r
B)(ΓB
, (ΓA
, ·)) + FA(∆1ΓA
, ·)
=
(−1)εB
2
(
→
∂l
BEBC
[FC
←
∂r
A + RCA(−1)εA ])(ΓA
, ·)
+
1
2
[FA
←
∂r
B + (−1)εB
→
∂l
AFB + (−1)(εA+1)εB RBA](ΓB
, (ΓA
, ·)) + FA(∆1ΓA
, ·)
19
=
(−1)εC
2
ECB
(
→
∂l
BFC, ·) + (∆1ΓC
)(FC , ·) +
(−1)εA+εB
2
(
→
∂l
BEBC
RCA)(ΓA
, ·)
+
(−1)εB
2
(
→
∂l
AFB)((ΓB
, ΓA
), ·) −
(−1)(εA+1)(εC +1)
2
ECB
RBA
→
∂l
C(ΓA
, ·) + FA(∆1ΓA
, ·)
=
(−1)εB
2
(EBA
→
∂l
AFB, ·) + (FA∆1ΓA
, ·) +
(−1)εA+εC
2
→
∂l
AEAB
RBC(ΓC
, ·)
=
(−1)εA
2
(
→
∂l
A(EAB
FB), ·) +
(−1)εA+εC
2
→
∂l
AEAB
RBC(ΓC
, ·) , (B.5)
where the Jacobi identity (3.1) has been applied in the third and fifth equality of eqs. (B.4) and (B.5),
respectively.
(As an aside, we mention that Lemma B.1 can be used to prove Lemma 5.1 in Section 5.) When one
compares Lemma B.1 with the ν differential eq. (5.29), one sees the first clue that the νF expression
(7.1) is a solution. More precisely, Lemma B.1 has extracted the ν
(0)
F part for us. Next task is to
uncover the ν(1) term (1.8).
Lemma B.2
8(∆2
1ΓA
) = (ν(1)
, ΓA
) − (−1)εC (
→
∂l
BECD
)(
→
∂l
D
→
∂l
CEBA
) . (B.6)
Proof of Lemma B.2: Combine
(
→
∂l
B∆1EBA
) − 2(∆2
1ΓA
) = [
→
∂l
B, ∆1]EBA
=
1
2
(−1)εC (
→
∂l
BECD
)(
→
∂l
D
→
∂l
CEBA
) + (
→
∂l
B∆1ΓC
)
→
∂l
CEBA
,
(B.7)
and
(
→
∂l
B∆1EBA
) =
→
∂l
B∆1(ΓB
, ΓA
) =
→
∂l
B(∆1ΓB
, ΓA
) − (−1)εB
→
∂l
B(ΓB
, ∆1ΓA
)
=
1
2
(ν(1)
, ΓA
) + (
→
∂l
C∆1ΓB
)(
→
∂l
BECA
) − 2(∆2
1ΓA
) . (B.8)
So far, we have reproduced the ν
(0)
F and the ν(1) part of the νF solution to the ν differential eq. (5.29).
Finally, we should extract the ν(2) term (1.9). The prefactor 1/24 in the νF formula (7.1) hints that
such a calculation is going be lengthy. Rewrite first Lemma B.2 as
8(∆2
1ΓB
)EBA = (
→
∂l
Aν(1)
) − νI
A , (B.9)
where
νI
A ≡ (−1)εD (
→
∂l
CEDF
)(
→
∂l
F
→
∂l
DECB
)EBA = νII
A + νIII
A , (B.10)
νII
A ≡ (−1)εBεD (
→
∂l
DEBC
)(
→
∂l
CEDF
)(
→
∂l
F EBA) = − νII
A − νIV
A , (B.11)
νIII
A ≡ (−1)εD (
→
∂l
CEDF
)
→
∂l
F ((
→
∂l
DECB
)EBA)
20
= −(−1)(εB+εC)εD (
→
∂l
CEDF
)
→
∂l
F (ECB
→
∂l
DEBA) = νII
A + νV
A , (B.12)
νIV
A ≡ (−1)εC εD (
→
∂l
AEBC)(
→
∂l
DECF
)(
→
∂l
F EDB
) , (B.13)
νV
A ≡ (−1)εC EBF
(
→
∂l
F ECD
)(
→
∂l
D
→
∂l
CEBA) = − 2νV I
A , (B.14)
νV I
A ≡ (−1)εB(εC +1)
ECF
(
→
∂l
F EBD
)(
→
∂l
D
→
∂l
CEBA) = νV
A + νV II
A , (B.15)
νV II
A ≡ (−1)εC (
→
∂l
A
→
∂l
BECD)EDF
(
→
∂l
F ECB
) . (B.16)
Here, the Jacobi identity (3.3) is used in the second equality of eq. (B.14), and the closeness relation
(3.4) is used in the second equalities of eqs. (B.11) and (B.15). Altogether eqs. (B.10)–(B.16) yield
νI
A = νII
A + νIII
A = 2νII
A + νV
A = − νIV
A + νV
A = − νIV
A −
2
3
νV II
A . (B.17)
Ultimately, we would like to show that νI
A is equal to (
→
∂l
Aν(2))/3. The achievement in eq. (B.17) is
more modest: The free “A” index on the νI
A expression has been moved to a derivative
→
∂l
A in νIV
A and
νV II
A . On the other hand, differentiation with respect to ΓA of the two expressions (1.9) and (1.10)
for the ν(2) quantity (1.9) yields two more relations
νIV
A + 2νV III
A = (
→
∂l
Aν(2)
) = νV III
A − νV II
A − νIX
A , (B.18)
where
νV III
A ≡ (−1)εC εF (
→
∂l
A
→
∂l
BECD
)EDF (
→
∂l
CEF B
) , (B.19)
νIX
A ≡ (−1)εC (
→
∂l
AEDF
)(
→
∂l
F ECB
)(
→
∂l
BECD)
= −(−1)εBεG (
→
∂l
AEDF
)(
→
∂l
F EBC
)ECG(
→
∂l
BEGH
)EHD
= (−1)εBεG (
→
∂l
AEHD)EDF
(
→
∂l
F EBC
)ECG(
→
∂l
BEGH
) = νIV
A − νX
A , (B.20)
νX
A ≡ (−1)εBεG+(εB+εC)(εD+1)
(
→
∂l
AEHD)EBF
(
→
∂l
F ECD
)ECG(
→
∂l
BEGH
)
= (−1)(εB +1)εD+εC(εB+εH +1)
(
→
∂l
AEHD)EBF
(
→
∂l
F EDC
)(
→
∂l
BEHG
)EGC = 0 . (B.21)
Here, the Jacobi identity (3.1) is used in the fourth equality of eq. (B.20). Remarkably, the νX
A term
vanishes due to an antisymmetry under the index permutation FDC ↔ BHG. Altogether, νIX
A = νIV
A
and
νI
A = − νIV
A −
2
3
νV II
A = − νIV
A −
2
3
(νV III
A − νIV
A −
→
∂l
Aν(2)
) =
1
3
(
→
∂l
Aν(2)
) . (B.22)
Combining eqs. (B.3), (B.9) and (B.22) shows that the νF expression (7.1) satisfies the ν differential
eq. (5.29).
C Proof of Proposition 11.1
In this Appendix C, we prove that the odd scalar curvature R is minus eight times the odd scalar νF .
The odd scalar curvature
R ≡ RABEBA
= RI + RII − RIII − RIV (C.1)
21
inherits four terms RI, RII, RIII and RIV from the expression (11.5) for the Ricci tensor RAB. They
are defined as
RI ≡ (−1)εA (
→
∂l
AΓA
BC)ECB
= RV − RV I , (C.2)
RII ≡ (−1)εA FAΓA
BCECB
= − (−1)εB FA(
→
∂l
B + FB)EBA
, (C.3)
RIII ≡ (−1)εB EBA
(
→
∂l
AFB) , (C.4)
RIV ≡ ΓA
C
DΓD
CBEBA
= − RIV − RV I , (C.5)
RV ≡ (−1)εA
→
∂l
A(ΓA
BCECB
) = − (−1)εB
→
∂l
A(
→
∂l
B + FB)EBA
= −ν(1)
− (−1)εA
→
∂l
A(EAB
FB) , (C.6)
RV I ≡ ΓA
BC(ECB
←
∂r
A) . (C.7)
Here, the antisymplectic and the torsion-free conditions (10.6) and (10.9) are used in the second
equality of eq. (C.5), and a contracted version of the antisymplectic condition (10.6)
(−1)εB (
→
∂l
B +FB)EBA
+ (−1)εA ΓA
BCECB
= 0 (C.8)
is used in the second equalities of eqs. (C.3) and (C.6). Inserting back in eq. (C.1), one finds that
R = − 8ν
(0)
F − ν(1)
−
1
2
RV I , (C.9)
where ν
(0)
F and ν(1) are given in eqs. (4.6) and (1.8). Now it remains to eliminate RV I from eq. (C.9).
Note that RV I only depends on the torsion-free part of the connection ΓA
BC, so one does in principle
not need the torsion-free condition (10.9) from now on. One calculates that
1
2
RV I = −
1
2
(−1)εA(εD+1)
ΓB
A
CECD
(
→
∂l
AEDF )EF B
= − (−1)εA ΓB
A
CECD
(
→
∂l
DEAF )EF B
= ΓA
BCECD
(
→
∂l
DEBF
)EF A = − ν(2)
− RV I . (C.10)
Here, the closeness relation (3.4) is used in the second equality and the antisymplectic condition (10.6)
in the fourth equality. In other words,
RV I = −
2
3
ν(2)
. (C.11)
Combining eqs. (C.9) and (C.11) yields the main result of Proposition 11.1:
R = − 8νF . (C.12)
References
[1] A. Schwarz, Commun. Math. Phys. 155 (1993) 249.
[2] I.A. Batalin and I.V. Tyutin, Int. J. Mod. Phys. A8 (1993) 2333.
[3] O.M. Khudaverdian and A.P. Nersessian, Mod. Phys. Lett. A8 (1993) 2377.
[4] H. Hata and B. Zwiebach, Ann. Phys. (N.Y.) 229 (1994) 177.
[5] I.A. Batalin and G.A. Vilkovisky, Phys. Lett. 102B (1981) 27.
22
[6] I.A. Batalin and G.A. Vilkovisky, Phys. Rev. D28 (1983) 2567 [E: D30 (1984) 508].
[7] I.A. Batalin and G.A. Vilkovisky, Nucl. Phys. B234 (1984) 106.
[8] I.A. Batalin, K. Bering and P.H. Damgaard, Nucl. Phys. B739 (2006) 389.
[9] K. Bering, J. Math. Phys. 47 (2006) 123513, arXiv:hep-th/0604117.
[10] O.M. Khudaverdian, arXiv:math.DG/9909117.
[11] O.M. Khudaverdian and Th. Voronov, Lett. Math. Phys. 62 (2002) 127.
[12] O.M. Khudaverdian, Contemp. Math. 315 (2002) 199.
[13] O.M. Khudaverdian, Commun. Math. Phys. 247 (2004) 353.
[14] I.A. Batalin and R. Marnelius, Nucl. Phys. B511 (1998) 495.
[15] K. Bering, arXiv:0705.3440.
[16] K. Bering, P.H. Damgaard and J. Alfaro, Nucl. Phys. B478 (1996) 459.
[17] K. Bering, Commun. Math. Phys. 274 (2007) 297.
[18] O.M. Khudaverdian and Th. Voronov, Amer. Math. Soc. Transl. 2.212 (2004) 179.
[19] K. Bering, arXiv:physics/9711010.
[20] I.A. Batalin and I.V. Tyutin, Mod. Phys. Lett. A8 (1993) 3673; ibid. A9 (1994) 1707;
Amer. Math. Soc. Transl. 2.177 (1996) 23.
[21] I.A. Batalin, R. Marnelius and A. Semikhatov, Nucl. Phys. B446 (1995) 249.
[22] B. Geyer and P.M. Lavrov, Int. J. Mod. Phys. A19 (2004) 3195.
[23] Y. Kosmann-Schwarzbach and J. Monterde, Ann. Inst. Fourier (Grenoble) 52 (2002) 419.
[24] I.A. Batalin, K. Bering and P.H. Damgaard, Phys. Lett. B389 (1996) 673.
23
Paper IV
Odd Scalar Curvature
in Anti-Poisson Geometry
BY
I.A. Batalin and K. Bering
Phys. Lett. B663 (2008) 132
arXiv:0712.3699
arXiv:0712.3699v3[hep-th]11Apr2008
Odd Scalar Curvature in Anti-Poisson Geometry
Igor A. Batalin1
I.E. Tamm Theory Division
P.N. Lebedev Physics Institute
Russian Academy of Sciences
53 Leninsky Prospect
Moscow 119991
Russia
and
Klaus Bering2
Institute for Theoretical Physics & Astrophysics
Masaryk University
Kotl´aˇrsk´a 2
CZ-611 37 Brno
Czech Republic
January 31, 2017
Abstract
Recent works have revealed that the recipe for field-antifield quantization of Lagrangian gauge
theories can be considerably relaxed when it comes to choosing a path integral measure ρ if a zeroorder
term νρ is added to the ∆ operator. The effects of this odd scalar term νρ become relevant at
two-loop order. We prove that νρ is essentially the odd scalar curvature of an arbitrary torsion-free
connection that is compatible with both the anti-Poisson structure E and the density ρ. This
extends a previous result for non-degenerate antisymplectic manifolds to degenerate anti-Poisson
manifolds that admit a compatible two-form.
PACS number(s): 02.40.-k; 03.65.Ca; 04.60.Gw; 11.10.-z; 11.10.Ef; 11.15.Bt.
Keywords: BV Field-Antifield Formalism; Odd Laplacian; Anti-Poisson Geometry; Semidensity;
Connection; Odd Scalar Curvature.
1E-mail: batalin@lpi.ru 2E-mail: bering@physics.muni.cz
1
1 Introduction
The main purpose of this Letter is to report on new geometric insights into the field-antifield formalism.
In general, the field-antifield formalism [1, 2, 3] is a recipe for constructing Feynman rules
for Lagrangian field theories with gauge symmetries. The field-antifield formalism is in principle able
to handle the most general gauge algebra, i.e. open gauge algebras of reducible type. The input is
usually a local relativistic field theory, formulated via a classical action principle in a geometric configuration
space. In the field-antifield scheme, the original field variables are extended with various
stages of ghosts, antighosts and Lagrange multipliers — all of which are then further extended with
corresponding antifields; the gauge symmetries are encoded in a nilpotent Fermionic BRST symmetry
[4, 5]; and the original action is deformed into a BRST-invariant master action, whose Hessian has
the maximal allowed rank. The full quantum master action
W = S +
∞
n=1
¯hn
Mn (1.1)
is determined recursively order by order in ¯h from a consistent set of quantum master equations
(S, S) = 0 , (1.2)
(M1, S) = i(∆ρS) , (1.3)
(M2, S) = i(∆ρM1) + νρ −
1
2
(M1, M1) , (1.4)
(Mn, S) = i(∆ρMn−1) −
1
2
n−1
r=1
(Mr, Mn−r) , n ≥ 3 . (1.5)
Here (·, ·) is the antibracket (or anti-Poisson structure), ∆ρ is the odd Laplacian and νρ is an odd
scalar, which become relevant in perturbation theory at loop order 0, 1, and 2, respectively. It has
only recently been realized that the field-antifield formalism can consistently accommodate a non-zero
νρ term, thereby providing a more flexible framework for field-antifield quantization [6, 7, 8].
The classical master equation (1.2) is a generalization of Zinn-Justin’s equation [9], which allows to
set up consistent renormalization (if the field theory is renormalizable). If the theory is not anomalous
at the one-loop level, there will exist a local solution M1 to the next equation (1.3), and so forth.
Although the field-antifield formalism in its basic form is only a formal scheme — i.e. particularly, it
assumes that results from finite dimensional analysis are directly applicable to field theory, which has
infinitely many degrees of freedom — it has nevertheless been successfully applied to a large variety
of physical models. It has mainly been used in a truncated form of the full set of quantum master eqs.
(1.2) – (1.5), where all the following quantities
(S, S), (∆ρS), νρ, M1, M2, M3, . . . , (1.6)
are set identically equal to zero. One can for instance mention the AKSZ paradigm [10, 11] as a broad
example that uses the truncated field-antifield formalism (1.6) to quantize supersymmetric topological
field theories [12, 13, 14, 15]. Currently, very few scientific works describe solutions with non-zero
Mn’s, primarily due to the singular nature of the odd Laplacian ∆ρ in field theory (again because of
the infinitely many degrees of freedom). Nevertheless, it should be fruitful to study generic solutions of
the full quantum master equation. See the original paper [1] for an interesting solution with M1 = 0.
Finally, it has in many cases been explicitly checked that the field-antifield formalism produces the
same result as the Hamiltonian formulation [16, 17, 18]. The formalism has also influenced work
in closed string field theory [19] and several branches of mathematics. The geometry behind the
field-antifield formalism was further clarified in Ref. [20, 21, 22, 23].
2
In this Letter we shall only explicitly consider the case of finitely many variables. Our main result
concerns the odd scalar νρ, which is a certain function of the anti-Poisson structure EAB and the
density ρ, cf. eq. (6.1) below. It turns out that νρ has a geometric interpretation as (minus 1/8 times)
the odd scalar curvature R of any connection ∇ that satisfies three conditions; namely that ∇ is
1) anti-Poisson, 2) torsion-free and 3) ρ-compatible. This is a rather robust conclusion as we shall
prove in this Letter that it even holds for degenerate antibrackets. (Degenerate anti-Poisson structures
appear naturally from for instance the Dirac antibracket construction for antisymplectic second-class
constraints [7, 21, 24, 25].)
2 Anti-Poisson structure EAB
An anti-Poisson structure is by definition a possibly degenerate (2, 0) tensor field EAB with upper
indices that is Grassmann-odd
ε(EAB
) = εA + εB + 1 , (2.1)
that is skewsymmetric
EAB
= − (−1)(εA+1)(εB+1)
EBA
, (2.2)
and that satisfies the Jacobi identity
cycl. A,B,C
(−1)(εA+1)(εC +1)
EAD
(
→
∂ℓ
DEBC
) = 0 . (2.3)
3 Compatible two-form EAB
In general, an anti-Poisson manifold could have singular points where the rank of EAB jumps, and it
is necessary to impose a regularity criterion to proceed. We shall here assume that the anti-Poisson
structure EAB admits a compatible two-form field EAB, i.e. that there exists a two-form field EAB
with lower indices that is Grassmann-odd
ε(EAB) = εA + εB + 1 , (3.1)
that is skewsymmetric
EAB = − (−1)εAεB EBA , (3.2)
and that is compatible with the anti-Poisson structure in the sense that
EAB
EBCECD
= EAD
, (3.3)
EABEBC
ECD = EAD . (3.4)
This is a relatively mild requirement, which is always automatically satisfied for a Dirac antibracket
on antisymplectic manifolds with antisymplectic second-class constraints [7, 21, 24, 25]. Note that the
two-form EAB is neither unique nor necessarily closed. One can define a (1, 1) tensor field as
PA
C ≡ EAB
EBC , (3.5)
or equivalently,
PA
C
≡ EABEBC
= (−1)εA(εC +1)
PC
A . (3.6)
It then follows from either of the compatibility relations (3.3) and (3.4) that PA
B is an idempotent
PA
BPB
C = PA
C . (3.7)
3
4 The ∆E Operator
An anti-Poisson structure with a compatible two-form field EAB gives rise to a Grassmann-odd, secondorder
∆E operator that takes semidensities to semidensities. It is defined in arbitrary coordinates as
[7]
∆E ≡ ∆1 +
ν(1)
8
−
ν(2)
8
−
ν(3)
24
+
ν(4)
24
+
ν(5)
12
, (4.1)
where ∆1 is the odd Laplacian
∆ρ ≡
(−1)εA
2ρ
→
∂ℓ
AρEAB
→
∂ℓ
B , (4.2)
with ρ = 1, and where
ν(1)
≡ (−1)εA (
→
∂ℓ
B
→
∂ℓ
AEAB
) , (4.3)
ν(2)
≡ (−1)εAεC (
→
∂ℓ
DEAB
)EBC(
→
∂ℓ
AECD
) , (4.4)
ν(3)
≡ (−1)εB (
→
∂ℓ
AEBC)ECD
(
→
∂ℓ
DEBA
) , (4.5)
ν(4)
≡ (−1)εB (
→
∂ℓ
AEBC)ECD
(
→
∂ℓ
DEBF
)PF
A
, (4.6)
ν(5)
≡ (−1)εAεC (
→
∂ℓ
DEAB
)EBC(
→
∂ℓ
AECF
)PF
D
= (−1)(εA+1)εB EAD
(
→
∂ℓ
DEBC
)(
→
∂ℓ
CEAF )PF
B . (4.7)
It is shown in Ref. [7] that the ∆E operator defined in eq. (4.1) does not depend on the choice of
local coordinates, it does not depend on the choice of compatible two-form field EAB, and it does
map semidensities into semidensities. Moreover, the Jacobi identity (2.3) precisely ensures that ∆E is
nilpotent
∆2
E =
1
2
[∆E, ∆E] = 0 . (4.8)
Earlier works on the ∆E operator include Ref. [6, 25, 26, 27, 28, 29].
5 The ∆ Operator
Classically, the field-antifield formalism is governed by the anti-Poisson structure EAB, or equivalently,
the antibracket
(f, g) ≡ (f
←
∂r
A)EAB
(
→
∂ℓ
Bg) = − (−1)(εf
+1)(εg+1)
(g, f) . (5.1)
Quantum mechanically, the field-antifield recipe instructs one to choose an arbitrary path integral
measure ρ, and to use it to build a nilpotent, Grassmann-odd, second-order ∆ operator that takes
scalar functions into scalar functions. It is natural to build the ∆ operator by conjugating the ∆E
operator (4.1) with appropriate square roots of the density ρ as follows:
∆ ≡
1
√
ρ
∆E
√
ρ . (5.2)
In this way the ∆ operator trivially inherits the nilpotency property from the ∆E operator,
∆2
=
1
√
ρ
∆2
E
√
ρ = 0 . (5.3)
In physical applications the nilpotency (5.3) of ∆ is important for the underlying BRST symmetry of
the theory.
4
6 The Odd Scalar νρ
The odd scalar function νρ is defined as
νρ ≡ (∆1) =
1
√
ρ
(∆E
√
ρ) = ν(0)
ρ +
ν(1)
8
−
ν(2)
8
−
ν(3)
24
+
ν(4)
24
+
ν(5)
12
, (6.1)
where ν(1), ν(2), ν(3), ν(4), ν(5) are given in eqs. (4.3)–(4.7), and the quantity ν
(0)
ρ is given as
ν(0)
ρ ≡
1
√
ρ
(∆1
√
ρ) . (6.2)
The second-order ∆ operator (5.2) decomposes as
∆ = ∆ρ + νρ , (6.3)
where ∆ρ is the odd Laplacian (4.2). The nilpotency of ∆ implies that
∆2
ρ = (νρ , · ) , (6.4)
(∆ρνρ) = 0 . (6.5)
The possibility of a non-trivial νρ has only recently been observed, cf. Ref. [6, 7, 8]. In the past, the
odd scalar term νρ was not present due to a certain compatibility relation between E and ρ, which
was unnecessarily imposed, and which (using our new terminology) made νρ vanish. In terms of the
quantum master equation
∆e
i
¯h
W
= 0 , (6.6)
the odd scalar νρ enters at the two-loop order O(¯h2
)
1
2
(W, W) = i¯h∆ρW + ¯h2
νρ , (6.7)
which in turn leads to the set of eqs. (1.2) – (1.5).
7 Connection
In the next two Sections 7 and 8 we will briefly state our sign conventions and definitions for the
covariant derivative and the curvature in the presence of Fermionic degrees of freedom. A more
complete treatment can be found in Ref. [8, 30]. Other references include Ref. [31]. Our convention
for the left covariant derivative (∇AX)B of a left vector field XA is [30]
(∇AX)B
≡ (
→
∂ℓ
AXB
) + (−1)εX (εB+εC)
ΓA
B
CXC
, ε(XA
) = εX + εA . (7.1)
A connection ΓA
B
C is called anti-Poisson if it preserves the anti-Poisson structure EAB, i.e.
0 = (∇AE)BC
≡ (
→
∂ℓ
AEBC
) + ΓA
B
DEDC
− (−1)(εB +1)(εC +1)
(B ↔ C) . (7.2)
It is useful to define a reordered Christoffel symbol ΓA
BC as
ΓA
BC ≡ (−1)εAεB ΓB
A
C . (7.3)
5
A torsion-free connection ΓA
BC has the following symmetry in the lower indices:
ΓA
BC = − (−1)(εB +1)(εC +1)
ΓA
CB . (7.4)
A connection ΓA
BC is called ρ-compatible if
ΓB
BA = (ln ρ
←
∂r
A) . (7.5)
There are in principle two definitions for the divergence divX of a Bosonic vector field X with εX =0.
The first divergence definition depends on the density ρ
divρX ≡
(−1)εA
ρ
→
∂ℓ
A(ρXA
) , (7.6)
while the second definition depends on the connection ∇
div∇X ≡ str(∇X) ≡ (−1)εA (∇AX)A
= ((−1)εA
→
∂ℓ
A + ΓB
BA)XA
. (7.7)
The ρ-compatibility condition (7.5) precisely ensures that the two definitions (7.6) and (7.7) coincide,
and hence that there is a unique notion of volume [32]. We shall only consider torsion-free connections
∇ that are anti-Poisson and ρ-compatible, i.e. connections that satisfy the above three conditions
(7.2), (7.4) and (7.5). Then the odd Laplacian ∆ρ can be written on a manifestly covariant form
∆ρ =
(−1)εA
2
∇AEAB
∇B =
(−1)εB
2
EBA
∇A∇B . (7.8)
8 Curvature
The Riemann curvature tensor is
RA
BCD ≡ (−1)εAεB (
→
∂ℓ
BΓA
CD) + ΓA
BEΓE
CD − (−1)εBεC (B ↔ C) . (8.1)
(Note that the ordering of indices on the Riemann curvature tensor is slightly non-standard to minimize
appearances of sign factors.) The Ricci tensor is
RAB ≡ RC
CAB =
(−1)εC
ρ
(
→
∂ℓ
CρΓC
AB)−(
→
∂ℓ
A ln ρ
←
∂r
B)−ΓA
C
DΓD
CB = −(−1)(εA+1)(εB +1)
RBA . (8.2)
9 Odd Scalar Curvature
The odd scalar curvature R is defined as the Ricci tensor RAB contracted with the anti-Poisson tensor
EAB,
R ≡ RABEBA
= EAB
RBA , ε(R) = 1 . (9.1)
We now assert that the odd scalar curvature
R = − 8νρ (9.2)
of an arbitrary connection ∇ that is anti-Poisson, torsion-free and ρ-compatible, is equal to (minus
eight times) the odd scalar νρ. In particular one sees that the odd scalar curvature R carries no
information about the connection ∇ used, and it depends only on E and ρ. Equation (9.2) was proven
for the non-degenerated case in Ref. [8]. The degenerated case is proven in Appendix A.
6
Acknowledgement: We would like to thank P.H. Damgaard for discussions. K.B. thanks the Lebedev
Physics Institute and the Niels Bohr Institute for warm hospitality. The work of I.A.B. is supported
by grants RFBR 05-01-00996, RFBR 05-02-17217 and LSS-4401.2006.2. The work of K.B. is supported
by the Ministry of Education of the Czech Republic under the project MSM 0021622409.
A Proof of the Main Eq. (9.2)
Equation (C.9) in Ref. [8] yields that the odd scalar curvature R can be written as
R = − 8ν(0)
ρ − ν(1)
−
1
2
RI , (A.1)
where ν
(0)
ρ , ν(1) and RI are defined in eqs. (6.2), (4.3) and (A.2), respectively. Since the expression
(A.2) below for RI only depends on the torsion-free part of the connection, one does in principle not
need the torsion-free condition (7.4) from now on. The heart of the proof consists of the following ten
“one-line calculations”:
RI ≡ ΓA
BC(ECB
←
∂r
A) = ΓA
BC((ECD
EDF EF B
)
←
∂r
A) = 2RII + RIII , (A.2)
RII ≡ ΓA
BCPC
D(EDB
←
∂r
A) = − RIV − ν(2)
, (A.3)
RIII ≡ (−1)εA(εC +1)
ΓF
A
BEBC
(
→
∂ℓ
AECD)EDF
= 2RIII + RV , (A.4)
RIV ≡ ΓA
BCECD
(
→
∂ℓ
DEBF
)EF A = RV I − RIV , (A.5)
RV ≡ (−1)εAεC ΓF
A
BPB
C(
→
∂ℓ
AECD
)PD
F
= RV II − ν(5)
, (A.6)
RV I ≡ ΓA
BC(ECB
←
∂r
D)PD
A = 2RV III + RIX , (A.7)
RV II ≡ (−1)(εA+1)(εC +1)
EABΓB
CDEDF
(
→
∂ℓ
F EAG
)PG
C
= RIV − RV III , (A.8)
RV III ≡ ΓA
BCPC
D(EDB
←
∂r
F )PF
A = − RIV − ν(5)
, (A.9)
RIX ≡ (−1)εA(εC +1)
ΓG
A
BEBC
PA
D
(
→
∂ℓ
DECF )EF G
= − RX − ν(4)
, (A.10)
RX ≡ (−1)εA ΓF
A
BEBC
(
→
∂ℓ
CEAD)EDF
= − RIII − ν(3)
. (A.11)
Here we have used the upper compatibility relation (3.3) for the two-form EAB in the second equality
of eqs. (A.2), (A.7), (A.8), (A.9) and (A.10); the lower compatibility relation (3.4) for the two-form
EAB in the second equality of eq. (A.4); the anti-Poisson property (7.2) for the connection ∇ in the
second equality of eqs. (A.3), (A.6), (A.9), (A.10) and (A.11); and the Jacobi identity (2.3) in the
second equality of eqs. (A.5) and (A.8). From these ten relations (A.2)–(A.11), the quantity RIII can
be determined as follows:
−RIII = RV = RV II − ν(5)
= (RIV − RV III) + (RIV + RV III) = 2RIV
= RV I = 2RV III + RIX = − 2(RIV + ν(5)
) + (RIII + ν(3)
− ν(4)
)
= 2RIII + (ν(3)
− ν(4)
− 2ν(5)
) , (A.12)
so that
RIII =
1
3
(−ν(3)
+ ν(4)
+ 2ν(5)
) . (A.13)
Next, RI can be expressed in terms of RIII:
1
2
RI = RII +
1
2
RIII = − (RIV + ν(2)
) +
1
2
RIII = RIII − ν(2)
. (A.14)
7
Inserting eqs. (A.13) and (A.14) into eq. (A.1) yields the main eq. (9.2):
R = − 8ν(0)
ρ − ν(1)
−
1
2
RI = − 8ν(0)
ρ − ν(1)
+ ν(2)
+
1
3
(ν(3)
− ν(4)
− 2ν(5)
) = − 8νρ . (A.15)
References
[1] I.A. Batalin and G.A. Vilkovisky, Phys. Lett. 102B (1981) 27.
[2] I.A. Batalin and G.A. Vilkovisky, Phys. Rev. D28 (1983) 2567 [E: D30 (1984) 508].
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[4] C. Becchi, A. Rouet and R. Stora, Phys. Lett. B54 (1974) 344; Commun. Math. Phys. 42 (1975)
127; Ann. Phys. (NY) 98 (1976) 287.
[5] I.V. Tyutin, Lebedev Institute preprint 39 (1975).
[6] K. Bering, J. Math. Phys. 47 (2006) 123513, arXiv:hep-th/0604117.
[7] K. Bering, arXiv:0705.3440, to appear in J. Math. Phys.
[8] I.A. Batalin and K. Bering, J. Math. Phys. 49 (2008) 033515, arXiv:0708.0400.
[9] J. Zinn-Justin, in Trends in elementary particle theory (eds. H. Rollnik and K. Dietz), Lecture
Notes in Physics 37 (1975).
[10] M. Alexandrov, M. Kontsevich, A. Schwarz and O. Zaboronsky, Int. J. Mod. Phys. A12 (1997)
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[16] G.V. Grigoryan, R.P. Grigoryan and I.V. Tyutin, Sov. J. Nucl. Phys. 53 (1991) 1058.
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8
[26] O.M. Khudaverdian, arXiv:math.DG/9909117.
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9
Paper V
A Comparative Study
of Laplacians and
Schroedinger-LichnerowiczWeitzenboeck
Identities
in Riemannian and
Antisymplectic Geometry
BY
I.A. Batalin and K. Bering
J. Math. Phys. 50 (2009) 073504
arXiv:0809.4269
arXiv:0809.4269v4[hep-th]11Jul2009
A Comparative Study of Laplacians and
Schr¨odinger–Lichnerowicz–Weitzenb¨ock Identities
in Riemannian and Antisymplectic Geometry
Igor A. Batalina,b
and Klaus Beringa,c
a
The Niels Bohr Institute
The Niels Bohr International Academy
Blegdamsvej 17
DK–2100 Copenhagen
Denmark
b
I.E. Tamm Theory Division
P.N. Lebedev Physics Institute
Russian Academy of Sciences
53 Leninsky Prospect
Moscow 119991
Russia
c
Institute for Theoretical Physics & Astrophysics
Masaryk University
Kotl´aˇrsk´a 2
CZ–611 37 Brno
Czech Republic
June 14, 2013
Abstract
We introduce an antisymplectic Dirac operator and antisymplectic gamma matrices. We explore
similarities between, on one hand, the Schr¨odinger–Lichnerowicz formula for spinor bundles in
Riemannian spin geometry, which contains a zeroth–order term proportional to the Levi–Civita
scalar curvature, and, on the other hand, the nilpotent, Grassmann–odd, second–order ∆ operator
in antisymplectic geometry, which in general has a zeroth–order term proportional to the odd
scalar curvature of an arbitrary antisymplectic and torsionfree connection that is compatible with
the measure density. Finally, we discuss the close relationship with the two–loop scalar curvature
term in the quantum Hamiltonian for a particle in a curved Riemannian space.
MSC number(s): 15A66; 53A55; 53B20; 58A50; 58C50.
Keywords: Dirac Operator; Spin Representations; BV Field–Antifield Formalism; Antisymplectic
Geometry; Odd Laplacian.
bE–mail: batalin@lpi.ru cE–mail: bering@physics.muni.cz
1
Contents
1 Introduction 3
1.1 General Remarks About Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 General Theory 7
2.1 Connection ∇(Γ) = d + Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 The Riemann Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 The Ricci Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.6 The Ricci Two–Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.7 Covariant Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.8 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Riemannian Geometry 10
3.1 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Laplacian ∆ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Two–cocycle ν(ρ′; ρ, g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4 Scalar νρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.5 ∆ And ∆g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.6 Levi–Civita Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.7 The Riemann Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.8 Scalar Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.9 The ∆ Operator At ρ = ρg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.10 Particle In Curved Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.11 First–Order SAB Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.12 ΓA Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.13 C Versus Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.14 Hodge ∗ Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.15 Hodge–Dirac Operator D(T) = d + λδ . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.A Appendix: Is There A Second–Order Formalism? . . . . . . . . . . . . . . . . . . . . . 28
4 Antisymplectic Geometry 29
4.1 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Odd Laplacian ∆ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Odd Scalar νρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 ∆ And ∆E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.5 Antisymplectic Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.6 The Riemann Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.7 Odd Scalar Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.8 First–Order SAB Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.9 ΓA Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.10 Dirac Operator D(T) = d + θδ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.A Appendix: Is There A Second–Order Formalism? . . . . . . . . . . . . . . . . . . . . . 36
4.B Appendix: What Is An Antisymplectic Clifford Algebra? . . . . . . . . . . . . . . . . . 37
2
5 General Spin Theory 38
5.1 Spin Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2 Spin Connection ∇(ω) = d + ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Spin Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.4 Covariant Tensors with Flat Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.5 Local Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6 Riemannian Spin Geometry 41
6.1 Spin Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2 Levi–Civita Spin Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.3 First–Order sab Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.4 γa Matrices And Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.5 Dirac Operator D(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.6 Second–Order σab Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.7 Lichnerowicz’ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.8 Clifford Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.9 Intertwining Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.10 Schr¨odinger–Lichnerowicz’ Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7 Antisymplectic Spin Geometry 50
7.1 Spin Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.2 First–Order sab Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.3 γa Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.4 Dirac Operator D(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.A Appendix: Shifted s′ab
+ Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
8 Conclusions 53
1 Introduction
What do Riemannian and antisymplectic geometry have in common? The short answer is that out
of the 2 × 2 = 4 classical classes of even and odd, Riemannian and symplectic geometries, they are
the only two possibilities that possess non–trivial Laplacians, scalar curvatures and Weitzenb¨ock–type
identities, cf. Table 1. Our present investigation is partly spurred by the following remarkable fact.
On one hand, one has the nilpotent, Grassmann–odd ∆ operator, which plays a fundamental rˆole in
antisymplectic geometry, and which helps encode the BRST symmetry in the field–antifield formalism
[1, 2, 3]. It can be written as [4]
2∆ = 2∆ρ −
R
4
(antisymplectic) (1.0.1)
where ∆ρ is the odd Laplacian, and R is the odd scalar curvature of an arbitrary antisymplectic,
torsionfree and ρ–compatible connection ∇(Γ) =d+Γ. On the other hand, on a Riemannian spin
manifold, one has the Schr¨odinger–Lichnerowicz formula [5, 6]
D(σ)
D(σ)
= ∆(σ)
ρg
−
R
4
(Riemannian) (1.0.2)
where D(σ) is the Dirac operator, ∆
(σ)
ρg
is the spinor Laplacian, and R is the scalar Levi–Civita
curvature. The formula (1.0.1) has been multiplied with a factor of 2 to ease comparison with formula
(1.0.2) because of the standard practice to normalize odd Laplacians with an internal factor 1/2. In
3
both formulas (1.0.1) and (1.0.2), the coefficient in front of the zeroth–order scalar curvature term is
exactly the same, namely minus a quarter! Of course, there are crucial differences between eqs. (1.0.1)
and (1.0.2). The second–order operators in eq. (1.0.1) act on scalar functions, while the Dirac operator
D(σ) and the Laplacian ∆
(σ)
ρg
in eq. (1.0.2) act on spinors, as the index “σ” is meant to indicate. (The
subscript ρg ≡
√
g refers to the canonical Riemannian density.)
Our investigation can roughly be divided in three parts. The first part (which is mainly covered in
Subsections 3.1–3.5, 3.9 and 4.1–4.4) is to define a Grassmann–even Riemannian analogue of the odd
∆ operator (1.0.1), that takes scalars in scalars:
∆ρg
−
R
4
(Riemannian) . (1.0.3)
Here ∆ρg
is the Laplace–Beltrami operator and R is the Levi–Civita scalar curvature. The zeroth–
order term −R/4 in the even operator (1.0.3) is special in several ways (as compared to other choices
of the zeroth–order term). For instance, the even operator (1.0.3) with this particular zeroth–order
term −R/4 is closely related to the quantum Hamiltonian ˆH for a particle moving in the Riemannian
manifold [7, 8, 9, 10, 11, 12, 13, 14, 15, 16], cf. Subsection 3.10. Central to our investigation is the
fact that the zeroth–order term −R/4 also possesses a special mathematical property. To see this
property, one notes that it is possible to uniquely identify how all zeroth–order terms depend on
the canonical Riemannian density ρg, due to a classification of scalar invariants, see Proposition 3.2.
Therefore it is possible to consistently replace all the appearances of ρg with an arbitrary density
ρ. One may now show that the ρ–lifted version of the operator (1.0.3) is the unique operator such
that the
√
ρ–conjugated operator is independent of ρ. That’s the special property. This has parallels
to antisymplectic geometry, where the odd ∆ operator (1.0.1) shares a similar characterization. In
antisymplectic geometry, the
√
ρ–conjugated operator
∆E =
√
ρ∆
1
√
ρ
(antisymplectic) (1.0.4)
is precisely Khudaverdian’s ∆E operator [17, 18, 19, 20, 21, 22, 23]. The ∆E operator (1.0.4) is
distinguished by being nilpotent and independent of ρ. In fact, when one tracks the equations in
detail, it is possible to see that the same coefficient −1/4 in front of the odd and even scalar curvature
terms in eqs. (1.0.1) and (1.0.3) is not a coincidence, but indeed follows from the same underlying
principle of ρ–independence. Thus it establishes a bridge between the odd and even operators (1.0.1)
and (1.0.3).
We should also mention that the even operator (1.0.3) is often compared with the conformally covariant
Laplacian
∆ρg
−
(N − 2)R
(N − 1)4
(Riemannian) (1.0.5)
where N =dim(M) is the dimension of the Riemannian manifold M. The zeroth–order term −R/4
corresponds to N =∞.
The second part (which is covered in Subsections 6.4–6.10) is to check within Riemannian geometry, if
there is a bridge between the even operator (1.0.3) that acts on scalar functions, and the square
of the Dirac operator (1.0.2) that acts on the spinor bundle S. There is a well–defined group–
theoretical procedure how to compare scalars and spinors. Firstly, the Dirac operator is extended
to a Dirac operator that acts on the bispinor bundle S ⊗ ST . The Clebsch–Gordan decomposition
S ⊗ ST = 1 ⊕ . . ., in turn, contains a singlet representation, i.e., a scalar invariant, which is denoted as
||s . Thus one just has to project the square of the bispinor Dirac operator to the singlet representation
to obtain an operator that acts on scalars. Somewhat surprisingly, the operator turns out to be just
4
the bare Laplace–Beltrami operator ∆ρg
with no zeroth–order term at all, cf. Theorem 6.6. Roughly
speaking, after the projection to the singlet state ||s , the −R/4 curvature term in the spinor sector S
is canceled by an opposite amount +R/4 in the transposed spinor sector ST . So we have to conclude
for the second part, that the above group–theoretical procedure yields no relation between the even
operator (1.0.3) that acts on scalar functions, and the square of the Dirac operator (1.0.2), despite
the fact that they both contain the same −R/4 term!
The third part develops the antisymplectic side. It is spurred by the following questions.
1. Do there exist antisymplectic Clifford algebras and spinors?
2. Does there exists a natural spinor generalization ∆(σ) of the odd ∆ operator (1.0.1), which takes
antisymplectic spinors to antisymplectic spinors?
3. Can the odd ∆(σ) operator from question 2 be written as a square
∆(σ) ?
= D(σ)
⋆ D(σ)
(antisymplectic) (1.0.6)
of an antisymplectic Dirac operator D(σ) =γA∇
(σ)
A , where “⋆” is a Fermionic multiplication,
ε(⋆) = 1, and γA are antisymplectic γ matrices?
The answers, which will be derived in detail in Sections 4 and 7, are, by most standards, “no” to
question 3, and “yes, there exists a first–order formalism, but there is no second–order formalism”
to question 1 and 2. Here the first– and second–order formalism refer to the realizations of the Lie–
algebras of infinitesimal frame and coordinate changes in terms of first– and second–order differential
operators, respectively. The obstacle in eq. (1.0.6) lies in the definition of the ⋆ multiplication. We
shall, however, introduce a Fermionic nilpotent parameter θ that can be though of as the inverse
⋆−1, but since such θ parameter by definition is not invertible, the ⋆ multiplication itself becomes
meaningless. The trick is therefore, roughly speaking, to multiply both side of eq. (1.0.6) with θ≡⋆−1,
cf. Theorem 4.4 and Theorem 7.1.
At the coarsest level, the main text is organized into 3 × 2 = 6 sections. The three Sections 2–4 are
devoted to general (=not necessarily spin) manifolds, while the next three Sections 5–7 deal exclusively
with spin manifolds. Sections 3 and 6 consider the Riemannian case, and Sections 4 and 7 consider
the antisymplectic case, while Sections 2 and 5 consider the general theory that is common for both
Riemannian and antisymplectic case. The general theory Sections 2 and 5 explain differential geometry,
such as, connections, torsion tensors, vielbeins, flat and curved exterior forms, etc., in the context of
supermanifolds, where sign factors are important. The Riemannian curvature tensor, the Ricci tensor
and the scalar curvature are considered in Subsections 2.4–2.6, 3.7–3.8 and 4.6–4.7. Finally, Section 8
has our conclusions.
1.1 General Remarks About Notation
Adjectives from supermathematics such as “graded”, “super”, etc., are implicitly implied. The sign
conventions are such that two exterior forms ξ and η, of Grassmann–parity εξ, εη and of form–degree
pξ, pη, commute in the following graded sense:
η ∧ ξ = (−1)εξεη+pξpη
ξ ∧ η ≡ (−1)εξ·εη ξ ∧ η (1.1.1)
inside the exterior algebra. The pair (ε, p) acts as a 2–dimensional vector–valued Grassmann–parity
ε :=
ε
p (mod 2)
, (1.1.2)
5
Table 1: The 2 × 2 = 4 classical geometries and their symmetries [18]. Only even Riemannian and
antisymplectic geometries have non–trivial Laplacians, scalar curvatures and Weitzenb¨ock–type iden-
tities.
Even Geometry Odd Geometry
g = Y AgAB ∨ Y B g = Y AgAB ∨ Y B
Riemannian ε(gAB) = εA + εB ε(gAB) = εA + εB + 1
Covariant gBA = −(−1)(εA+1)(εB +1)gAB gBA = (−1)εAεB gAB
Metric Symmetric Symmetric
No Closeness Relation No Closeness Relation
Inverse ε(gAB) = εA + εB ε(gAB) = εA + εB + 1
Riemannian gBA = (−1)εAεB gAB gBA = (−1)(εA+1)(εB+1)
gAB
Contravariant Symmetric Skewsymmetric
Metric Even Laplacian No Laplacian
ω = 1
2CAωAB ∧ CB E = 1
2CAEAB ∧ CB
Symplectic ε(ωAB) = εA + εB ε(EAB) = εA + εB + 1
Covariant ωBA = (−1)(εA+1)(εB+1)
ωAB EBA = −(−1)εAεB EAB
Two–Form Skewsymmetric Skewsymmetric
Closeness Relation Closeness Relation
Inverse ε(ωAB) = εA + εB ε(EAB) = εA + εB + 1
Symplectic ωBA = −(−1)εAεB ωAB EBA = −(−1)(εA+1)(εB+1)
EAB
Contravariant Skewsymmetric Symmetric
Tensor No Laplacian Odd Laplacian
as indicated in the second equality of eq. (1.1.1). The first component carries ordinary Grassmann–
parity ε, while the second component carries form–parity, i.e., form degree modulo two. The exterior
wedge symbol “∧” is often not written explicitly, as it is redundant information that can be deduced
from the Grassmann– and form–parity. The commutator [F, G] and anticommutator {F, G}+ of two
operators F and G are
[F, G] := FG − (−1)εF εG+pF pG GF ≡ FG − (−1)εF ·εG GF , (1.1.3)
{F, G}+ := FG + (−1)εF εG+pF pG GF ≡ FG + (−1)εF ·εG GF . (1.1.4)
The commutator (1.1.3) fulfills the Jacobi identity
cycl. F,G,H
(−1)εF ·εH [F, [G, H]] = 0 . (1.1.5)
The transposed of a product of operators is:
(FG)T
= (−1)εF εG+pF pG GT
FT
≡ (−1)εF ·εG GT
FT
. (1.1.6)
Covariant and exterior derivatives will always be from the left, while partial derivatives can be from
either left or right. We shall sometimes use round parenthesis “()” to indicate how far derivatives act,
see e.g., eqs. (2.3.3), (3.3.2), (3.4.2) and (3.4.3) below.
6
2 General Theory
2.1 Connection ∇(Γ)
= d + Γ
Let there be given a manifold M with local coordinates zA of Grassmann–parity ε(zA) = εA (and
form–degree p(zA) = 0). Assume that M is endowed with a measure density ρ. Let Γ(TM) denote
the set of sections in the tangent bundle TM, i.e., the set of vector fields on M. Let M be endowed
with a tangent bundle connection ∇(Γ) = d + Γ = dzA ⊗ ∇
(Γ)
A : Γ(TM) × Γ(TM) → Γ(TM)
∇
(Γ)
A =
→
∂ℓ
∂zA
+ ∂r
B ΓB
AC
→
dzC
. (2.1.1)
Here ∂r
A ≡(−1)εA ∂ℓ
A are not usual partial derivatives. In particular, they do not act on the Christoffel
symbols ΓB
AC in eq. (2.1.1). Rather they are a dual basis to the one–forms
→
dzA:
→
dzA
(∂r
B) = δA
B , ε(
→
dzA
) = εA = ε(∂r
A) . (2.1.2)
Phrased differently, the ∂r
A are merely bookkeeping devices, that transform as right partial derivatives
under general coordinate transformations. (To be able to distinguish them from true partial derivatives,
the differentiation variable zA on a true partial derivative ∂/∂zA is written explicitly.) For fixed
index “A” in eq. (2.1.1), the Christoffel symbol ΓB
AC is a matrix with respect to index “B” and index
“C”, and ∂r
B ΓB
AC
→
dzC is the corresponding linear operator: TM → TM. (We shall often refer to a
linear operator by its matrix, and vice–versa.)
The form–parities p(
→
dzA)=p(∂r
A) are either all 0 or all 1, depending on applications, whereas a
1–form dzA with no arrow “→” always carries odd form–parity p(dzA)=1 (and Grassmann–parity
ε(dzA)=εA).
2.2 Torsion
The torsion tensor T(Γ) : Γ(TM) × Γ(TM) → Γ(TM) is defined as
T(Γ)
≡
1
2
dzA
∧ ∂r
B T(Γ)B
AC dzC
:= [∇(Γ) ∧, Id]
= [dzA
→
∂ℓ
∂zA
+ dzA
∂r
B ΓB
AD
→
dzD ∧, ∂r
C dzC
] = dzA
∧ ∂r
B ΓB
AC dzC
. (2.2.1)
where it is implicitly understood that there are no contractions with base manifold indices, in this
case index “A” and index “C”. As expected, the torsion tensor is just an antisymmetrization of the
Christoffel symbol ΓB
AC with respect to the lower indices,
T(Γ)A
BC := ΓA
BC + (−1)(εB+1)(εC +1)
(B ↔ C) . (2.2.2)
In particular, the Christoffel symbol
ΓA
BC = −(−1)(εB +1)(εC +1)
(B ↔ C) (2.2.3)
is symmetric with respect to the lower indices when the connection is torsionfree.
7
2.3 Divergence
A connection ∇(Γ) can be used to define a divergence of a Bosonic vector field XA as
str(∇(Γ)
X) ≡ (−1)εA (∇
(Γ)
A X)A
= ((−1)εA
→
∂ℓ
∂zA
+ ΓB
BA)XA
, εX = 0 . (2.3.1)
On the other hand, the divergence is defined in terms of ρ as
divρX :=
(−1)εA
ρ
→
∂ℓ
∂zA
(ρXA
) . (2.3.2)
See Ref. [24] for a mathematical exposition of divergence operators on supermanifolds. The ∇(Γ)
connection is called compatible with the measure density ρ if
ΓB
BA = (−1)εA (
→
∂ℓ
∂zA
ln ρ) . (2.3.3)
In this case, the two definitions (2.3.1) and (2.3.2) of divergence agree, cf. Ref. [4].
2.4 The Riemann Curvature
We discuss in this Subsection 2.4 the Riemann curvature tensor on a supermanifold [25]. See Ref. [12]
and Ref. [26] for related discussions. The Riemann curvature R(Γ) is defined as (half) the commutator
of the ∇(Γ) connection (2.1.1),
R(Γ)
=
1
2
[∇(Γ) ∧, ∇(Γ)
] = −
1
2
dzB
∧ dzA
⊗ [∇
(Γ)
A , ∇
(Γ)
B ]
= −
1
2
dzB
∧ dzA
⊗ ∂r
D RD
ABC
→
dzC
, (2.4.1)
where it is implicitly understood that there are no contractions with base manifold indices, in this
case index “A” and index “B”. (For a torsionfree connection such contractions vanish, and there is
no ambiguity.)
RD
ABC =
→
dzD
[∇
(Γ)
A , ∇
(Γ)
B ]∂r
C
= (−1)εDεA (
→
∂ℓ
∂zA
ΓD
BC) + ΓD
AE ΓE
BC − (−1)εAεB (A ↔ B) , (2.4.2)
Note that the order of indices in the Riemann curvature tensor RD
ABC is non–standard. This is to
minimize appearances of Grassmann sign factors. Alternatively, the Riemann curvature tensor may
be defined as
R(X, Y )Z = [∇
(Γ)
X , ∇
(Γ)
Y ] − ∇
(Γ)
[X,Y ] Z = Y B
XA
RAB
D
CZC
∂ℓ
D , (2.4.3)
where X = XA∂ℓ
A, Y = Y B∂ℓ
B and Z = ZC∂ℓ
C are left vector field of even Grassmann– and form–
parity. The Riemann curvature tensor RAB
D
C reads in local coordinates
RAB
D
C = (−1)εD(εA+εB)
RD
ABC = (
→
∂ℓ
∂zA
ΓB
D
C) + (−1)εBεD ΓA
D
E ΓE
BC − (−1)εAεB (A ↔ B) .
(2.4.4)
8
Here we have introduced a reordered Christoffel symbol
ΓA
B
C := (−1)εAεB ΓB
AC . (2.4.5)
It is sometimes useful to reorder the indices in the Riemann curvature tensors as
RABC
D
= ([∇A, ∇B]∂ℓ
C)D
= (−1)εC (εD+1)
RAB
D
C . (2.4.6)
Note that all expressions (2.4.2), (2.4.4) and (2.4.6) of Riemann curvature tensor are antisymmetric
under an (A ↔ B) exchange of index “A” and “B”. The first Bianchi identity reads (in the torsionfree
case):
0 =
cycl. A,B,C
(−1)εAεC RABC
D
. (2.4.7)
We have exceptionally used the convention p(∂ℓ
A)=0 in eqs. (2.4.3) and (2.4.6).
2.5 The Ricci Tensor
The Ricci tensor is defined as
RAB := RC
CAB . (2.5.1)
The Ricci tensor becomes symmetric
RAB =
(−1)εC
ρ
→
∂ℓ
∂zC
(ρΓC
AB) − (
→
∂ℓ
∂zA
ln ρ
←
∂r
∂zB
) − ΓA
D
C ΓC
DB
= −(−1)(εA+1)(εB+1)
(A ↔ B) , (2.5.2)
when the ∇(Γ) connection is torsionfree T(Γ) =0 and ρ–compatible (2.3.3).
2.6 The Ricci Two–Form
The Ricci two–form is defined as
RAB := RAB
C
C(−1)εC = −(−1)εAεB (A ↔ B) . (2.6.1)
The Ricci two–form vanishes
RAB = 0 , (2.6.2)
when the ∇(Γ) connection is torsionfree T(Γ) =0 and ρ–compatible (2.3.3).
2.7 Covariant Tensors
Let
Ωmn(M) := Γ m
(T∗
M) ⊗ n
(T∗
M) (2.7.1)
be the vector space of (0, m+n)–tensors ηA1···AmB1···Bn
(z) that are antisymmetric with respect to the
first m indices A1 . . . Am, and symmetric with respect to the last n indices B1 . . . Bn. As usual, it is
practical to introduce a coordinate–free notation
η(z; C; Y ) =
1
m!n!
CAm
∧ · · · ∧ CA1
ηA1···AmB1···Bn
(z) ⊗ Y Bn
∨ · · · ∨ Y B1
. (2.7.2)
9
Here the variables Y A are symmetric counterparts to the one–form basis CA ≡ dzA.
CA ∧ CB = −(−1)εAεB CB ∧ CA , ε(CA) = εA , p(CA) = 1 ,
Y A ∨ Y B = (−1)εAεB Y B ∨ Y A , ε(Y A) = εA , p(Y A) = 0 .
(2.7.3)
The covariant derivative can be realized on covariant tensors η ∈ Ωmn(M) by a linear differential
operator
∇
(T)
A =
→
∂ℓ
∂zA
− ΓA
B
C TC
B , (2.7.4)
where
TA
B := CA
→
∂ℓ
∂CB
+ Y A
→
∂ℓ
∂Y B
(2.7.5)
are themselves linear differential operators. They are generators of the general linear (= gl) Lie–
algebra,
[TA
B, TC
D] = δC
B TA
D − (−1)(εA+εB)(εC +εD)
δA
D TC
B . (2.7.6)
It is important for the implementation (2.7.4) to make sense that η carries no explicit indices, i.e., all
indices should be paired as indicated in eq. (2.7.2). The Lie–algebra (2.7.6) reflects infinitesimal
coordinate transformation, i.e., diffeomorphism invariance.
2.8 Coordinate Transformations
Consider for simplicity a one–form η = ηA(z)CA ∈ Ω10(M). The covariant derivative reads
(∇Aη)C = (
→
∂ℓ
∂zA
ηC) − ηB ΓB
AC . (2.8.1)
Under a coordinate transformation zA → z′A one has
ηA = η′
B(z′B
←
∂r
∂zA
) , (2.8.2)
C′A
= (z′A
←
∂r
∂zB
)CB
= CB
(
→
∂ℓ
∂zB
z′A
) , (2.8.3)
(−1)εAεB (z′B
←
∂r
∂zD
)ΓD
AC = (
→
∂ℓ
∂zA
z′B
←
∂r
∂zC
) + (
→
∂ℓ
∂zA
z′D
)Γ′
D
B
E(z′E
←
∂r
∂zC
) , (2.8.4)
so that the covariant derivative transforms covariantly,
(∇Aη)D = (
→
∂ℓ
∂zA
z′B
)(∇′Bη′
)C(z′C
←
∂r
∂zD
) . (2.8.5)
3 Riemannian Geometry
3.1 Metric
Let there be given a (pseudo) Riemannian metric, i.e., a covariant symmetric (0, 2) tensor field
g = Y A
gAB ∨ Y B
∈ Ω02(M) , (3.1.1)
10
of Grassmann–parity
ε(gAB) = εA + εB , ε(g) = 0 , p(gAB) = 0 , (3.1.2)
and of symmetry
gBA = −(−1)(εA+1)(εB+1)
gAB . (3.1.3)
We shall not need nor discuss positivity/reality/Hermiticity–conditions in this paper (except for the
application to a particle in a curved space, cf. Subsection 3.10). The symmetry (3.1.3) becomes more
transparent if one reorders the Riemannian metric as
g = Y B
∨ Y A
˜gAB , (3.1.4)
where
˜gAB := gAB(−1)εB . (3.1.5)
Then the symmetry (3.1.3) simply reads
˜gBA = (−1)εAεB ˜gAB . (3.1.6)
The Riemannian metric gAB is assumed to be non–degenerate, i.e., there exists an inverse contravariant
symmetric (2, 0) tensor field gAB such that
gAB gBC
= δC
A . (3.1.7)
The inverse gAB has Grassmann–parity
ε(gAB
) = εA + εB , (3.1.8)
and symmetry
gBA
= (−1)εAεB gAB
. (3.1.9)
The canonical density on a Riemannian manifold is
ρg :=
√
g := sdet(gAB) . (3.1.10)
This should be compared with the antisymplectic case, where the density ρ is kept arbitrary, since
there is no canonical choice [25]. To ease comparison, we shall temporarily allow for arbitrary densities
ρ in the Riemannian case as well.
3.2 Laplacian ∆ρ
A Laplacian ∆ρ, which takes scalar functions to scalar functions, can be constructed from the inverse
metric gAB and a (not necessarily canonical) density ρ,
∆ρ :=
(−1)εA
ρ
→
∂ℓ
∂zA
ρgAB
→
∂ℓ
∂zB
, ε(∆ρ) = 0 , p(∆ρ) = 0 . (3.2.1)
A metric bracket (f, g) of two functions f =f(z) and g=g(z) can be defined via a double commutator
with the Laplacian, acting on the constant unit function 1,
(f, g) :=
1
2
[[
→
∆ρ, f], g]1 ≡
1
2
∆ρ(fg) −
1
2
(∆ρf)g −
1
2
f(∆ρg) +
1
2
fg(∆ρ1)
= (f
←
∂r
∂zA
)gAB
(
→
∂ℓ
∂zB
g) = (−1)εf
εg
(g, f) . (3.2.2)
There are no closeness relations (resp. Jacobi identities) associated with the Riemannian gAB metric
(3.1.4) (resp. metric (·, ·) bracket (3.2.2)) in contrast to symplectic situations. In fact, even if
such closeness relations and Jacobi identities were to be artificially enforced in one coordinate patch,
they would not transform covariantly under general coordinate transformations zA → z′B. See also
Subsection 3.1 in Ref. [27].
11
3.3 Two–cocycle ν(ρ′
; ρ, g)
It is possible to introduce a Riemannian analogue of the two–cocycle of Khudaverdian and Voronov
[18, 21, 4]. The two–cocycle ν(ρ′; ρ, g) is a function of a measure density ρ′ with respect to a reference
system (ρ, g),
ν(ρ′
; ρ, g) :=
ρ
ρ′
(∆ρ
ρ′
ρ
) = ν
(0)
ρ′ − ν(0)
ρ , (3.3.1)
where
ν(0)
ρ :=
1
√
ρ
(∆1
√
ρ) = −
√
ρ(∆ρ
1
√
ρ
) = (∆1 ln
√
ρ) + (ln
√
ρ, ln
√
ρ) . (3.3.2)
Here ∆1 is the Laplacian (3.2.1) with ρ=1. The expression (3.3.1) acts as a scalar under general
coordinate transformations, and satisfies the following two–cocycle condition:
ν(ρ1; ρ2, g) + ν(ρ2; ρ3, g) + ν(ρ3; ρ1, g) = 0 . (3.3.3)
In fact, it is a two–coboundary, because we shall prove in the next Subsection 3.4, that there exists a
scalar νρ, such that
ν(ρ′
; ρ, g) = νρ′ − νρ . (3.3.4)
3.4 Scalar νρ
A Grassmann–even function νρ can be constructed from the metric g and a (not necessarily canonical)
density ρ as
νρ := ν(0)
ρ +
ν(1)
4
−
ν(2)
8
−
ν(3)
16
, (3.4.1)
where ν
(0)
ρ is given by eq. (3.3.2), and
ν(1)
:= (−1)εA (
→
∂ℓ
∂zA
gAB
←
∂r
∂zB
)(−1)εB , (3.4.2)
ν(2)
:= −(−1)εC (zC
, (zB
, zA
))(
→
∂ℓ
∂zA
gBC)
= −(−1)(εA+1)(εD+1)
(
→
∂ℓ
∂zD
gAB
)gBC (gCD
←
∂r
∂zA
) , (3.4.3)
ν(3)
:= (−1)εA (gAB, gBA
) . (3.4.4)
Here (·, ·) is the metric bracket (3.2.2).
Lemma 3.1 The even quantity νρ is a scalar, i.e., it does not depend on the coordinate system.
Proof of Lemma 3.1: Under an arbitrary infinitesimal coordinate transformation δzA = XA, one
calculates (by using methods similar to the antisymplectic case [22])
δν(0)
ρ = −
1
2
∆1div1X , (3.4.5)
δν(1)
= 2∆1div1X + (−1)ǫC
(
→
∂ℓ
∂zC
gAB
)(
→
∂ℓ
∂zB
→
∂ℓ
∂zA
XC
) , (3.4.6)
12
δν(2)
= 2(−1)ǫC
(
→
∂ℓ
∂zC
gAB
)(
→
∂ℓ
∂zB
→
∂ℓ
∂zA
XC
) + 2(−1)ǫA
gAB(gBC
,
→
∂ℓ
∂zC
XA
) , (3.4.7)
δν(3)
= −4(−1)ǫA
gAB(gBC
,
→
∂ℓ
∂zC
XA
) . (3.4.8)
One easily sees that while the four constituents ν
(0)
ρ , ν(1), ν(2) and ν(3) separately have non–trivial
transformation properties, the linear combination νρ in eq. (3.4.1) is indeed a scalar.
Spurred by what happens in the antisymplectic case [4], we would like to classify which zeroth–order
term ν one could add to the Laplacian (3.2.1). The following Proposition 3.2 is designed to answer
this question.
Proposition 3.2 (Classification of 2–order differential invariants) If a function ν =ν(z) has
the following properties:
1. The function ν is a scalar.
2. ν(z) is a polynomial of the metric gAB(z), the density ρ(z), their inverses, and z–derivatives
thereof in the point z.
3. ν is invariant under constant rescaling of the density ρ → λρ, where λ is a z–independent pa-
rameter.
4. ν scales as ν → λν under constant Weyl scaling gAB → λgAB, where λ is a z–independent pa-
rameter.
5. Each term in ν contains precisely two z–derivatives.
Then ν is of the form
ν = α νρ + β νρg
+ γ (ln
ρ
ρg
, ln
ρ
ρg
) , (3.4.9)
where α, β and γ are three arbitrary z–independent parameters.
Remarks: Conditions 1–5 are imposed, because the Laplacian (3.2.1) has these properties. Note that
if one collects the ρ–dependence into a function of ln ρ and its z–derivatives, the conditions 2 and 3
both exclude undifferentiated ln ρ–dependence (because ln ρ is not a finite polynomial in ρ and ρ−1,
and because ln ρ→ln ρ + ln λ is not invariant, respectively). So scalars like νρln(ρ/ρg) are excluded
from our considerations.
Sketched proof of Proposition 3.2: The first idea of the proof is to replace condition 1 with
a weaker condition
1′. The function ν is invariant under affine coordinate transformations zA → z′B = ΛB
AzA + λB.
Secondly, recall that every polynomial is a finite linear combinations of monomials. One can argue
that if ν(z) is a polynomial that satisfy condition 1′ plus conditions 2–5 of Proposition 3.2, then each
of its constituent monomials (that contributes nontrivially) must by themselves satisfy condition 1′
plus conditions 2–5. Thus one can limit the search (for a linear basis) to monomials. It follows from
13
lengthy but straightforward combinatorial arguments that a basis for the polynomials ν that satisfy
condition 1′ plus conditions 2–5 is:
ν(0)
ρ , ν(0)
ρg
, ν(1)
, ν(2)
, ν(3)
, ν(4)
, ν(5)
ρ , ν(5)
ρg
, ν(6)
ρ , ν(6)
ρg
, ν(7)
ρ , (3.4.10)
where ν
(0)
ρ , ν(1), ν(2), ν(3) were defined above, and
ν(4)
:= (−1)εA (
→
∂ℓ
∂zA
gAB
)gBC (gCD
←
∂r
∂zD
)(−1)εD , (3.4.11)
ν(5)
ρ := (−1)εA (
→
∂ℓ
∂zA
gAB
)(
→
∂ℓ
∂zB
ln ρ) , (3.4.12)
ν(6)
ρ := (ln ρ, ln ρ) , (3.4.13)
ν(7)
ρ := (ln ρ, ln ρg) . (3.4.14)
Thirdly, under an arbitrary infinitesimal coordinate transformation δzA = XA, one calculates
δν(4)
= 2(−1)εA (
→
∂ℓ
∂zA
gAB
)(
→
∂ℓ
∂zB
div1X)
+2gAB
(
→
∂ℓ
∂zB
→
∂ℓ
∂zA
XC
)gCD(gDE
←
∂r
∂zE
)(−1)εE , (3.4.15)
δν(5)
ρ = (ln ρ, div1X) − (−1)εA (
→
∂ℓ
∂zA
gAB
)(
→
∂ℓ
∂zB
div1X)
+gAB
(
→
∂ℓ
∂zB
→
∂ℓ
∂zA
XC
)(
→
∂ℓ
∂zC
ln ρ) , (3.4.16)
δν(6)
ρ = −2(ln ρ, div1X) , (3.4.17)
δν(7)
ρ = −(ln(ρgρ), div1X) . (3.4.18)
It is easy to check that the only linear combinations of the basis elements (3.4.10) that satisfy condition
1, are given by formula (3.4.9).
3.5 ∆ And ∆g
The Riemannian analogue ∆g of Khudaverdian’s ∆E operator [17, 18, 19, 20, 21, 22, 23] is defined as
∆g := ∆1 +
ν(1)
4
−
ν(2)
8
−
ν(3)
16
. (3.5.1)
We will prove below that the ∆g operator (3.5.1) takes semidensities to semidensities. It is obviously
manifestly independent of ρ. Next, we define a Riemannian analogue of the Grassmann–odd nilpotent
∆ operator in antisymplectic geometry [4]. The even ∆ operator, which takes scalar functions to
scalar functions, is defined for arbitrary ρ as
∆ := ∆ρ + νρ . (3.5.2)
This ∆ operator (3.5.2) is well–defined, because of Lemma 3.1. One may prove (by using methods
similar to the antisymplectic case [22, 4]), that the two operators ∆ and ∆g are related via a similarity–
transformation with
√
ρ,
∆g =
√
ρ∆
1
√
ρ
. (3.5.3)
14
Proof of eq. (3.5.3): Let σ denote an arbitrary argument for the ∆g operator. (The argument σ
is a semidensity, but we shall not use this fact.) Then, it follows from the explicit νρ formula (3.4.1)
that
(∆gσ) = (∆1σ) + (
ν(1)
4
−
ν(2)
8
−
ν(3)
16
)σ = (∆1σ) − (∆1
√
ρ)
σ
√
ρ
+ νρσ
=
√
ρ(∆1
σ
√
ρ
) + 2(
√
ρ,
σ
√
ρ
) + νρσ =
√
ρ(∆ρ
σ
√
ρ
) + νρσ =
√
ρ(∆
σ
√
ρ
) . (3.5.4)
Eq. (3.5.3) shows that the ∆g operator (3.5.1) takes semidensities to semidensities. The ∆ operator
(3.5.2) has, in turn, the remarkable property that the
√
ρ–conjugated operator
√
ρ∆ 1√
ρ is independent
of ρ. This is strikingly similar to what happens in the antisymplectic case, cf. Subsection 4.4. It is
interesting to investigate how unique this property is? Consider a primed operator
∆′
:= ∆ + ν = ∆ρ + νρ + ν , (3.5.5)
where ν is a most general zeroth–order term. (We will in this paper not consider the possibility of
changing second– and first–order parts of Laplace operators, i.e., we will only consider changes to the
zeroth–order term for simplicity.) It is easy to see from eqs. (3.5.3) and (3.5.5) that the corresponding
√
ρ–conjugated operator
√
ρ∆′ 1√
ρ is independent of ρ if and only if the shift term ν is ρ–independent.
On the other hand, by invoking Proposition 3.2, one sees that ν is ρ–independent if and only if ν = βνρg
is proportional to νρg
. So an operator of the form ∆′ = ∆ + βνρg
, for arbitrary coefficient β, is the
most general operator with this property. This is the minimal answer one could possibly have hoped
for, since a ρ–independence argument will never be able to detect the presence of a ρ–independent
shift term like βνρg
.
3.6 Levi–Civita Connection
A connection ∇(Γ) is called metric, if it preserves the metric g,
0 = (∇
(Γ)
A ˜g)BC = (
→
∂ℓ
∂zA
˜gBC) − ((−1)εAεB ΓBAC + (−1)εBεC (B ↔ C)) . (3.6.1)
Here we have lowered the Christoffel symbol with the metric
ΓABC := gADΓD
BC(−1)εC . (3.6.2)
The metric condition (3.6.1) reads in terms of the contravariant inverse metric
0 = (∇
(Γ)
A g)BC
≡ (
→
∂ℓ
∂zA
gBC
) + ΓA
B
DgDC
+ (−1)εBεC (B ↔ C) . (3.6.3)
The Levi–Civita connection is the unique connection ∇(Γ) that is both torsionfree T(Γ) =0 and metric
(3.6.1). The Levi–Civita formula for the lowered Christoffel symbol in terms of derivatives of the
metric reads
2ΓCAB = (−1)εAεC (
→
∂ℓ
∂zA
˜gCB) + (−1)(εA+εC)εB (
→
∂ℓ
∂zB
˜gCA) − (
→
∂ℓ
∂zC
˜gAB) . (3.6.4)
A density ρ is compatible (2.3.3) with the Levi–Civita Christoffel symbol (3.6.4) if and only if ρ is
proportional to the canonical density (3.1.10).
15
3.7 The Riemann Curvature
For a metric connection ∇(Γ), we prefer to work with a (0, 4) Riemann tensor (as opposed to a (1, 3)
tensor) by lowering the upper index with the metric (3.1.1). In terms of Christoffel symbols it is
easiest to work with expression (2.4.2):
RD,ABC := gDERE
ABC(−1)εC
= (−1)εAεD



→
∂ℓ
∂zA
ΓDBC + (−1)εE(εA+εD+1)+εC ΓEADΓE
BC



−(−1)εAεB (A ↔ B) . (3.7.1)
In the second equality of eq. (3.7.1) is used the metric condition (3.6.1). If the metric condition (3.6.1)
is used one more time on the first term in eq. (3.7.1), one derives the following skewsymmetry
RD,ABC = −(−1)(εA+εB)(εC +εD)+εC εD (C ↔ D) . (3.7.2)
This skewsymmetry becomes clearer if one instead starts from expression (2.4.6) and define
RAB,CD := RABC
E
˜gED = (−1)εD(εA+εB+εC)
RD,ABC . (3.7.3)
Then the skewsymmetry (3.7.2) simply translates into a skewsymmetry between the third and fourth
index:
RAB,CD = −(−1)εC εD (C ↔ D) . (3.7.4)
We note that the torsionfree condition has not been used so far in this Section 3.7. The first Bianchi
identity (2.4.7) reads (in the torsionfree case):
0 =
cycl. A,B,C
(−1)εAεC RAB,CD . (3.7.5)
The (A ↔ B) antisymmetry, the (C ↔ D) antisymmetry (3.7.4) and the first Bianchi identity (3.7.5)
imply that Riemann curvature tensor RAB,CD is symmetric with respect to an (AB ↔ CD) exchange
of two pairs of indices:
RAB,CD = (−1)(εA+εB)(εC +εD)
(AB ↔ CD) . (3.7.6)
This, in turn, implies that there is a version of the first Bianchi identity (3.7.5), where one sums
cyclically over the three last indices:
0 =
cycl. B,C,D
(−1)εBεD RAB,CD . (3.7.7)
It is interesting to compare Riemann tensors in the Riemannian case with the antisymplectic case.
In both cases, the (A ↔ B) antisymmetry and the Bianchi identity (3.7.5) hold, but the (C ↔ D)
antisymmetry (3.7.4) turns in the antisymplectic case into an (C ↔ D) symmetry (4.6.4), and there
is no antisymplectic analogue of the (AB ↔ CD) exchange symmetry (3.7.6), cf. Subsection 4.6.
3.8 Scalar Curvature
The scalar curvature is defined as
R := (−1)εB gBA
RAB = (−1)εA RABgBA
. (3.8.1)
16
Proposition 3.3 The Levi–Civita scalar curvature R is proportional to the scalar νρg
,
R = −4νρg
. (3.8.2)
Sketched proof of Proposition 3.3: Straightforward calculations shows that
R = −4ν(0)
ρg
− ν(1)
+ (−1)εA gAB
ΓB
D
C ΓC
DA , (3.8.3)
where
2(−1)εA gAB
ΓB
D
C ΓC
DA = −(−1)εA+εB ΓA
BC(gCB
←
∂r
∂zA
) = ν(2)
+
ν(3)
2
. (3.8.4)
As a corollary of Proposition 3.3 one gets that the νρ scalar (3.4.1) for arbitrary ρ is given by the
formula
νρ = ν(ρ; ρg, g) + νρg
=
ρg
ρ
(∆ρg
ρ
ρg
) −
R
4
. (3.8.5)
3.9 The ∆ Operator At ρ = ρg
When one restricts to ρ = ρg, the ∆ operator (3.5.2) reduces to the Laplace–Beltrami operator minus
a quarter of the Levi–Civita scalar curvature:
∆|ρ=ρg
= ∆ρg
+ νρg
= ∆ρg
−
R
4
. (3.9.1)
This is the even operator (1.0.3) already mentioned in the Introduction. But the important question
is: Does the zeroth–order term νρg
=−R/4 in the operator (3.9.1) have a property that distinguish it
from all the other zeroth–order terms? Yes, in the following sense:
1. Firstly, consider the most general ρ–independent operator of the form
∆ρg
+ ν , (3.9.2)
where ∆ρg
is the Laplace–Beltrami operator and ν is a general zeroth–order term. (Here it is
important that we only allow ρ–independent ν’s from the very beginning.)
2. Secondly, apply Proposition 3.2 to classify the possible zeroth–order terms ν. In detail, one
sees that ν = βνρg
is proportional to νρg
for some proportionality factor β. Hence the operator
(3.9.2) is actually
∆ρg
+ βνρg
. (3.9.3)
3. Thirdly, replace the canonical density ρg → ρ by an arbitrary density ρ. In other words, replace
the ρ–independent operator (3.9.3) with the corresponding ρ–dependent operator
∆′
:= ∆ρ + βνρ . (3.9.4)
More rigorously, one should consider an algebra homomorphism s : Ag → Aρ,g from the algebra
Ag of differential operators, that only depend on the metric g, to the algebra Aρ,g of differential
operators, that depend on both the density ρ and the metric g. The s homomorphism
17
should satisfy π ◦ s = IdAg
, where π : Aρ,g → Ag denotes the restriction map |ρ=ρg
and “◦” denotes
composition. Clearly such a procedure is in general highly ambiguous, but in the present
situation, where we are only interested in the ρ–extension of just two operators, namely the
second–order operator ∆ρg
and the zeroth–order operator νρg
, there is a preferred candidate for
the s homomorphism in this sector, i.e., ∆ρg
s
→ ∆ρ and νρg
s
→ νρ, respectively.
4. Fourthly, apply the
√
ρ–independence argument of Subsection 3.5. It follows that the
√
ρ–
conjugated ∆′ operator
√
ρ∆′ 1√
ρ becomes independent of ρ if and only if β =1. (In the antisymplectic
case ∆′ is also nilpotent if and only if β =1.) Thus we conclude that the coefficient β =1,
and hence the even ∆ operator (3.5.2) is singled out.
5. Fifthly, restrict to ρ = ρg. Hence one arrives at the preferred operator (3.9.1).
Needless to say, that the above argument depends crucially on the order of the above five steps. In
particular, if step 3 is performed before step 1 and 2, i.e., if one considers the most general ρ–dependent
zeroth–order term ν from the very beginning, the β coefficient in front of the zeroth–order term νρg
would remain arbitrary.
3.10 Particle In Curved Space
In this Subsection 3.10 we indicate how the ∆ operator (3.5.2) is related to quantization of a particle
in a curved Riemannian target space [7, 8, 9, 10, 11, 12, 13, 14, 15, 16] with a measure density ρ not
necessarily equal to the canonical density (3.1.10). The classical Hamiltonian action Scl is
Scl = dt pA ˙zA
− Hcl , Hcl =
1
2
pApBgBA
+ V , {zA, pB}P B = δA
B , (3.10.1)
where pA denote the momenta for the zA variables. We shall for simplicity not consider reparametrizations
of the time variable t. Moreover, we assume that the Riemannian metric gAB =gAB(z), the
density ρ=ρ(z), the potential V =V (z) and general coordinate transformations zA → z′B =fB(z) do
not depend explicitly on time t. The naive Hamiltonian operator ˆHρ is [7, 9, 10, 11]
ˆHρ − V (ˆz) =
1
2
ˆpr
A gAB
(ˆz) ˆpℓ
B =
1
2 ρ(ˆz)
ˆpA ρ(ˆz) gAB
(ˆz) ˆpB
(−1)εB
ρ(ˆz)
(3.10.2)
=
1
2
[ˆpA +
¯h
i
ln ρ(ˆz)
←
∂r
∂ˆzA
] gAB
(ˆz) [ˆpB(−1)εB −
¯h
i
→
∂ℓ
∂ˆzB
ln ρ(ˆz)] (3.10.3)
=
1
2
ˆpA gAB
(ˆz) ˆpB(−1)εB +
¯h2
2
ν(0)
ρ (ˆz) (3.10.4)
=
1
2
pApBgBA
(z)
∧
+
¯h2
2
ν(0)
ρ (ˆz) +
ν(1)(ˆz)
4
. (3.10.5)
The left, middle, and right momentum operators, denoted by ˆpℓ
A, ˆpA, and ˆpr
A, respectively, are related
as
(−1)εA
ρ(ˆz)
ˆpℓ
A ρ(ˆz) = ˆpA = ρ(ˆz) ˆpr
A
1
ρ(ˆz)
. (3.10.6)
The non–zero canonical equal–time commutator relations read
−[ˆpℓ
B, ˆzA
] = [ˆzA
, ˆpB] = [ˆzA
, ˆpr
B] = i¯hδA
B1 . (3.10.7)
18
The hat “∧” in eq. (3.10.5) denotes the corresponding Weyl–ordered operator. We mention for completeness
a temporal point–splitting operation “T” defined as [12]
T ˆF1(t) · · · ˆFn(t) =
1
n! π∈Sn
(−1)εF,π lim
t1, . . . , tn → t
tπ(1) > . . . > tπ(n)
ˆFπ(1)(tπ(1)) · · · ˆFπ(n)(tπ(n)) , (3.10.8)
where εF,π denotes the Grassmann sign factor arising from permuting
ˆF1(t1) · · · ˆFn(tn) −→ ˆFπ(1)(tπ(1)) · · · ˆFπ(n)(tπ(n)) . (3.10.9)
For most practical purposes, the temporal point–splitting “T” is the same as Weyl ordering “∧”. In
particular, Weyl–ordering “∧” and temporal point–splitting “T” yield the same two–loop quantum
correction:
pApBgBA(z)
∧
T ˆpA ˆpBgBA(ˆz)



− ˆpA gAB
(ˆz) ˆpB(−1)εB =
1
4
[ˆpA, [ˆpB, gBA
(ˆz)]] = −
¯h2
4
ν(1)
(ˆz) . (3.10.10)
Note that Weyl–ordering “∧” and temporal point–splitting “T” are not covariant operations.
Now what should be the quantum Hamiltonian ˆH for the operator formalism? Obviously, one must
(among other things) demand that
1. ˆH is a scalar invariant.
2. ˆH is Hermitean.
3. ˆH has dimension of energy.
4. ˆH reduces to the classical Hamiltonian Hcl in the classical limit ¯h → 0.
The naive Hamiltonian operator (3.10.2) satisfies all these conditions 1–4. It is a scalar invariant, since
the momentum operators transform by definition under coordinate transformations zA → z′B = fB(z)
as
ˆp′ℓ
B = (
→
∂ℓ
∂fB(ˆz)
ˆzA
) ˆpℓ
A , (3.10.11)
ˆp′r
B = ˆpr
A (ˆzA
←
∂r
∂fB(ˆz)
) , (3.10.12)
ˆp′
B = (pA (zA
←
∂r
∂fB(z)
))∧
=
1
2
{ˆpA, ˆzA
←
∂r
∂fB(ˆz)
}+ . (3.10.13)
Note however that conditions 1–4 do not specify the quantum corrections to the quantum Hamiltonian
ˆH. For instance, one could add any multiple of ¯h2
νρ(ˆz) to ˆH without affecting conditions 1–4. Now
recall that every choice of ˆH in the operator formalism corresponds to a choice of action functional in
the path integral. One may fix the ambiguity by additionally demanding the following:
5. The operator formalism with the Hamiltonian operator ˆH should correspond to a Hamiltonian
path integral formulation where the path integral action is the pure classical action Scl with no
quantum corrections, i.e.,
zf | exp −
i
¯h
ˆH∆t |zi = zf , tf |zi, ti ∼
z(tf )=zf
z(ti)=zi
[dz][dp] exp
i
¯h
Scl[z, p] . (3.10.14)
19
The reader may wonder why we invoke the path integral formulation. The point is that, on one hand,
there is no unique way of telling what part of an operator should be considered as quantum corrections,
while on the other hand, there is a well–defined quantum part of an action functional, namely all the
terms of order O(¯h). The phase space path integral in quantum mechanics does not need to be
renormalized (unlike configuration space path integrals or quantum field theories), so it is consistent
to demand that the bare quantum corrections of the action functional vanish. Condition 5 determines
in principle the Hamiltonian operator ˆH to all orders in ¯h, but we shall in this paper truncate ˆH at
two–loop order, i.e., ignore possible higher–order quantum corrections of order O(¯h3
) for simplicity.
According to standard heuristic arguments, it follows from condition 5 that the quantum Hamiltonian
ˆH ∼ T(Hcl) (3.10.15)
is equal to the time–ordered classical Hamiltonian T(Hcl). However, time–ordering “T” is not a
geometrically well–defined operation, at least not if one uses the temporal point–splitting (3.10.8). It
should only be trusted modulo terms that contains single–derivatives of the metric [12]. In detail, the
time–ordered classical Hamiltonian T(Hcl) is given by the following non–covariant expression
T(Hcl) = ˆHρ −
¯h2
2
ν(0)
ρ (ˆz) +
ν(1)(ˆz)
4
, (3.10.16)
cf. eqs. (3.10.5) and (3.10.10). The combination
ν(0)
ρ +
ν(1)
4
= νρ +
ν(2)
8
+
ν(3)
16
= νρ +
(−1)εA
4
gAB
ΓB
D
C ΓC
DA (3.10.17)
is the νρ scalar (3.4.1) plus non–covariant terms that contain single–derivatives of the metric, cf. eq.
(3.8.3). Equations (3.10.15), (3.10.16) and (3.10.17) therefore strongly suggest that the full quantum
Hamiltonian ˆH is
ˆH = ˆHρ −
¯h2
2
νρ(ˆz) = T(Hcl) −
¯h2
16
ν(2)
(ˆz) +
ν(3)(ˆz)
2
, (3.10.18)
where we are neglecting possible quantum corrections of order O(¯h3
). The operator (3.10.18) satisfies
condition 1–5. For instance, it is a scalar invariant because of Lemma 3.1. We shall provide further
details concerning condition 5 in eq. (3.10.39) below. The preferred operator (3.10.18) also has an
extra feature:
6. The three operators
ˆHg = ρ(ˆz) ˆH
1
ρ(ˆz)
, ˆH , or
1
ρ(ˆz)
ˆH ρ(ˆz) (3.10.19)
are independent of ρ, if one declares that the left, middle, or right momentum operators ˆpℓ
A, ˆpA,
or ˆpr
A are independent of ρ, respectively.
We are now ready to relate the ∆ operator (3.5.2) to a particle in a curved space. The main point is
that the Hamiltonian (3.10.18) becomes ∆=∆ρ+νρ from eq. (3.5.2) if we identify
ˆzA
↔ zA
, ˆpℓ
A ↔
¯h
i
→
∂ℓ
∂zA
, (3.10.20)
ˆHρ − ˆV ↔ −
¯h2
2
∆ρ , ˆH − ˆV ↔ −
¯h2
2
∆ , ˆHg − ˆV ↔ −
¯h2
2
∆g . (3.10.21)
20
In detail, let |z, t ρ := |z, t / ρ(z) denote the instantaneous eigenstate ˆzA(t)|z, t ρ = zA|z, t ρ, and let
the eigenstate |z, t be the corresponding semidensity state with normalization dN z |z, t z, t| = 1
and Grassmann–parity ε (|z, t )=0. As a check, note that the formula (3.10.14) is covariant since it is
implicitly understood that the path integral contains one more momentum integration n
i=1 dp(t2i−1)
than coordinate integration n−1
i=1 dz(t2i) for any temporal discretization
ti ≡ t0 < t1 < . . . < t2n−1 < t2n ≡ tf . (3.10.22)
The momentum operators ˆpℓ
A, ˆpA, or ˆpr
A act on the eigenstates as follows:
ρ z, t|ˆpℓ
A(t) =
¯h
i
→
∂ℓ
∂zA ρ z, t| , z, t|ˆpA(t) =
¯h
i
z, t|
←
∂r
∂zA
, (3.10.23)
ˆpr
A(t)|z, t ρ = i¯h|z, t ρ
←
∂r
∂zA
, ˆpA(t)|z, t = i¯h|z, t
←
∂r
∂zA
. (3.10.24)
Therefore, the Hamiltonians ˆHρ, ˆH, and ˆHg translate into the Laplace operators ∆ρ, ∆, and ∆g:
ρ z, t| ˆHρ(t) − ˆV (t) = −
¯h2
2
∆ρ ρ z, t| , (3.10.25)
ρ z, t| ˆH(t) − ˆV (t) = −
¯h2
2
∆ ρ z, t| , (3.10.26)
z, t| ˆHg(t) − ˆV (t) = −
¯h2
2
∆g z, t| , (3.10.27)
cf. eqs. (3.2.1), (3.5.2) and (3.5.3), respectively. The time–evolution of states and operators are
governed by
ρ z, t| ˆH(t) = i¯h
d
dt ρ z, t| , z, t| ˆH(t) = i¯h
d
dt
z, t| , (3.10.28)
ˆH(t)|z, t ρ =
¯h
i
d
dt
|z, t ρ , ˆH(t)|z, t =
¯h
i
d
dt
|z, t , (3.10.29)
i¯h
d
dt
−
∂
∂t
ˆF(t) = [ ˆF(t), ˆH(t)] , (3.10.30)
where ˆF(t)=F(ˆz(t), ˆp(t), t) is an arbitrary operator that may depend explicitly on time t. We should
mention that semidensity states appear in geometric quantization [28].
Let us now calculate the left–hand side of eq. (3.10.14), i.e., the transition element zf |e−β ˆH|zi in the
operator formalism using expression (3.10.18) as the Hamiltonian [13]. Here we define
∆t := tf − ti , β :=
i
¯h
∆t , γ :=
i
¯h∆t
. (3.10.31)
It is better to change coordinates (zA
i ; zA
f ) → (zA
m; ∆zA) from the start–point zA
i and the endpoint zA
f
to the midpoint zA
m and the displacement ∆zA, where
zA
m :=
zA
f + zA
i
2
, ∆zA
:= zA
f − zA
i . (3.10.32)
In fact, we will suppress the subscript “m” since it is implicitly understood from now on that all
quantities are to be evaluated at the midpoint. We are going to rewrite all operators in terms of
symbols [29]. The Weyl symbol HW for the quantum Hamiltonian (3.10.18) reads
HW := ( ˆH)W = Hcl +
¯h2
16
ν(2)
+
ν(3)
2
, (3.10.33)
21
cf. eq. (3.10.18). Two Weyl symbols F and G are multiplied together via the Groenewold/Moyal ∗
product. It can be graphically represented as:
F ∗ G = F exp[→]G = FG + (F → G) +
1
2
(F →
→ G) + O(→3
) , (3.10.34)
2
i¯h
(F → G) = (F ← G) − (F → G) = {F, G}P B , (3.10.35)
← :=
←
∂r
∂zA
→
∂ℓ
∂pA
, → :=
←
∂r
∂pA
(−1)εA
→
∂ℓ
∂zA
. (3.10.36)
We will also need the zp–ordered and the pz–ordered symbols. They can be expressed in terms of the
Weyl symbol (·)W as
Fzf
p =
zf | ˆF|p
zf |p
= exp


−
i¯h
2
→
∂ℓ
∂pA
→
∂ℓ
∂zA
f


 FWf
= exp


(
∆zA
2
−
i¯h
2
→
∂ℓ
∂pA
)
→
∂ℓ
∂zA


 FW , (3.10.37)
Fpzi
=
p| ˆF|zi
p|zi
= exp



i¯h
2
→
∂ℓ
∂pA
→
∂ℓ
∂zA
i


 FWi
= exp


(
i¯h
2
→
∂ℓ
∂pA
−
∆zA
2
)
→
∂ℓ
∂zA


 FW . (3.10.38)
The transition element (or propagator) in the operator formalism now becomes
zf |e−β ˆH
|zi = dN
p zf |e− 1
2
β ˆH
|p p|e− 1
2
β ˆH
|zi = dN
p zf |p p|zi (e− 1
2
β ˆH
)zf
p (e− 1
2
β ˆH
)pzi
=
dN p
(2π¯h)N
e
i
¯h
pA∆zA
(e− 1
2
β ˆH
)W ∗ (e− 1
2
β ˆH
)W =
dN p
(2π¯h)N
e
i
¯h
pA∆zA
(e−β ˆH
)W
=
dN p
(2π¯h)N
e
i
¯h
pA∆zA
e−βHW
× 1 +
β2
4
(HW
→
→ HW ) +
β3
6
(HW → HW → HW ) + O(→4
)
= (2πi¯h∆t)− N
2 ρg e
1
2
γ∆zAgAB∆zB
e−βV
1 −
¯h2
β
16
ν(2)
+
ν(3)
2
+
¯h2
β2
8
(Hcl
→
← Hcl) − (Hcl
→
→ Hcl)
+
¯h2
β3
24
(Hcl → Hcl ← Hcl) + (Hcl ← Hcl → Hcl) − 2(Hcl → Hcl → Hcl)
+ O(→4
, ¯h3
)
p= ¯h
i
←
∂r
∂∆z
= (2πi¯h∆t)− N
2 ρg e−βV
1 −
¯h2
β
24
R +
¯h2
β
6
e− 1
2
βV
(∆ρg
e− 1
2
βV
) + O((∆z)2
, ¯h3
)
= (2πi¯h∆t)− N
2 ρ e− 1
2
βV
1 +
¯h2
β
6
→
∆ +O((∆z)2
, ¯h3
) e− 1
2
βV
ρ=ρg
. (3.10.39)
22
In the first equality of eq. (3.10.39) we summed over a complete set of momentum states |p , so that
it becomes possible to replace operators by symbols. The zp–ordered and the pz–ordered symbols
(3.10.37) and (3.10.38) were used in the second and third equality. We performed integration by part
of pA in the third equality. In the sixth equality, we replaced all non–Gaussian appearances of the
momenta pA by derivatives with respect to the displacement ∆zA, and performed the pA integration.
After the integration over pA, the terms downstairs in the seventh expression (that are either quadratic
or cubic in Hcl) read
(Hcl
→
← Hcl) ∼
1
β
ν(2)
+ O((∆z)2
) , (3.10.40)
(Hcl
→
→ Hcl) ∼ gAB
→
∂ℓ
∂zB
→
∂ℓ
∂zA
V −
1
2β
ln g −
1
2β
ν(3)
+ O((∆z)2
) , (3.10.41)
(Hcl → Hcl → Hcl) ∼
1
β2
ν(2)
+
(−1)εA
β
(
→
∂ℓ
∂zA
gAB
)
→
∂ℓ
∂zB
V −
1
2β
ln g
+O((∆z)2
) , (3.10.42)
(Hcl → Hcl ← Hcl) ∼
1
β2
ν(1)
−
ν(3)
2
+
1
β
gAB
→
∂ℓ
∂zB
→
∂ℓ
∂zA
V −
1
2β
ln g
+O((∆z)2
) , (3.10.43)
(Hcl ← Hcl → Hcl) ∼ V −
1
2β
ln g, V −
1
2β
ln g −
1
2β2
ν(3)
+ O((∆z)2
) . (3.10.44)
All the individual contributions of eqs. (3.10.40)–(3.10.44) have been collected in the eighth and ninth
expression of eq. (3.10.39). The eighth expression is the well-known covariant formula for the path
integral, i.e., the right–hand side of eq. (3.10.14). In the phase space path integral, the R/24 term
arises from the integration over quantum fluctuations [12]. In the ninth (and last) expression of
eq. (3.10.39), the R/24 term conspires with the Beltrami–Laplace operator to produce yet another
appearance of the ∆ operator (3.5.2).
3.11 First–Order SAB
Matrices
After considering quantization of a particle on a curved space in Subsection 3.10, we shall continue
with the investigation of Riemannian manifolds. We will assume for the remainder of the Riemannian
Sections 3 and 6 that the density ρ = ρg is equal to the canonical density (3.1.10).
Because of the presence of the metric tensor gAB, the symmetry of the general linear (= gl) Lie–algebra
(2.7.6) reduces to an orthogonal Lie–subalgebra. Its generators SAB
∓ read
SAB
∓ := CA
gBC
→
∂ℓ
∂CC
+ Y A
gBC
→
∂ℓ
∂Y C
∓ (−1)εAεB (A ↔ B) , (3.11.1)
ε(SAB
∓ ) = εA + εB , p(SAB
∓ ) = 0 , (3.11.2)
SA
∓C := SAB
∓ gBC(−1)εC . (3.11.3)
The SAB
∓ matrices are called first–order matrices, because they are first–order differential operators
in the CA and Y A variables. The SAB
− matrices satisfy an orthogonal Lie–algebra:
[SAB
∓ , SCD
∓ ] = (−1)εA(εB+εC)
gBC
SAD
− + SBC
− gAD
∓ (−1)εAεB (A ↔ B) , (3.11.4)
[SAB
∓ , SCD
± ] = (−1)εA(εB+εC)
gBC
SAD
+ − SBC
+ gAD
∓ (−1)εAεB (A ↔ B) . (3.11.5)
23
Note that the eqs. (3.11.4) and (3.11.5) remain invariant under a c–number shift
SAB
+ → S′AB
+ := SAB
+ + αgAB
1 , (3.11.6)
where α is a parameter.
3.12 ΓA
Matrices
The standard Dirac operator is only defined on a spin manifold, it depends on the vielbein, and
we shall describe it in Subsections 6.4–6.6. But first we shall introduce a poor man’s version of ΓA
matrices and the so–called Hodge–Dirac operator in the next Subsections 3.12–3.15. This construction
will work for a general Riemannian manifold, which is not necessarily a spin manifold.
The ΓA matrices can be defined via a Berezin–Fradkin operator representation [30, 31]
ΓA
λ ≡ ΓA
:= CA
+ λPA
, PA
:= gAB
→
∂ℓ
∂CB
, (3.12.1)
ε(ΓA
) = εA , p(ΓA
) = 1 (mod 2) . (3.12.2)
where λ is a Bosonic parameter with ε(λ)=0=p(λ), which is introduced to bring our presentation
of the Riemannian case in closer analogy with the antisymplectic case, see Subsection 4.9. One may
interpret λ as a Planck constant. The ΓA matrices satisfy a Clifford algebra
[ΓA
, ΓB
] = 2λgAB
1 . (3.12.3)
The ΓA matrices form a fundamental representation of the an orthogonal Lie–algebra (3.11.4):
[SAB
∓ , ΓC
] = ΓA
±λ gBC
∓ (−1)εAεB (A ↔ B) . (3.12.4)
If one commutes a metric connection ∇
(T)
A in the TA
B representation (2.7.4) with a ΓB matrix, one
gets
[∇
(T)
A , ΓB
] = −ΓA
B
C ΓC
. (3.12.5)
The minus sign on the right–hand side of eq. (3.12.5) can be explained as follows: The contravariant
flat ΓA matrices are passive bookkeeping devices that ultimately should be contracted with an active
covariant tensor field ηA. It is this implicitly written ηA that we are really differentiating. Thus there
should be a minus sign.
The ∇
(T)
A realization (2.7.4) can be identically rewritten into the following S± matrix realization
∇
(S)
A :=
→
∂ℓ
∂zA
−
1
2 ±
Γ±
A,BC SCB
± (−1)εB , (3.12.6)
i.e., ∇
(T)
A = ∇
(S)
A , where
Γ±
A,BC(−1)εC :=
1
2
(−1)εAεB ΓBAC ± (−1)εB εC (B ↔ C) . (3.12.7)
The Levi–Civita Γ±
A,BC connection reads:
Γ+
A,BC =
1
2
(
→
∂ℓ
∂zA
gBC) ,
Γ−
A,BC =
1
2
(˜gAB
←
∂r
∂zC
) + (−1)(εB +1)(εC +1)
(B ↔ C) . (3.12.8)
Note that both the SAB
− and the SAB
+ matrices are needed in the matrix realization (3.12.6).
24
3.13 C Versus Y
The SAB matrices (3.11.1) treat the CA and the Y A variables on complete equal footing, whereas the
ΓA matrices (6.4.1) contain only the C’s. Just from demanding that the ΓA matrices carry definite
Grassmann– and form–parity, such C ↔ Y symmetry breaking seems unavoidable. Further analysis of
the Riemannian case reveals that it is only possible to write a Berezin–Fradkin operator representation
(6.4.1) of the Clifford algebra (6.4.3) using the CA variables. (The CA variables are also preferred in the
antisymplectic case as well, see Subsection 4.B below.) One may ponder if there are situations where
the Y variables are useful instead? Yes. The democracy between C and Y gets restored in a bigger
framework that allows for both even and odd, Riemannian and symplectic manifolds, cf. Table 1.
For instance, the Y A variables are the only ones suitable for writing down a Berezin–Fradkin–like
representation
˜ΓA
:= Y A
+ λωAB
→
∂ℓ
∂Y B
, ε(˜ΓA
) = εA , p(˜ΓA
) = 0 , (3.13.1)
of the Heisenberg algebra
[˜ΓA
, ˜ΓB
] = 2λωAB
1 = −(−1)εAεB (A ↔ B) (3.13.2)
in even symplectic geometry [32, 33, 34]. (The Y A variables are also preferred in the odd Riemannian
case [25, 35, 36].)
Returning to the even Riemannian case, we will for simplicity only consider the CA variables from
now on, i.e., we shall from now on put the Y A variables to zero Y A → 0 everywhere, in particular
inside the TA
B matrices (2.7.5) and the SAB matrices (3.11.1).
3.14 Hodge ∗ Operation
One may formally define a Hodge ∗ operation on exterior forms η = η(z; C) ∈ Ω•0(M) as a fiberwise
Fourier transformation
(∗η)(z; C) :=
dN C′
ρ
e
i
¯h
C′∧C
η(z; C′
) , (3.14.1)
where we have introduced the shorthand notation
C′
∧ C := C′A
gAB ∧ CB
. (3.14.2)
The Hodge ∗ operation is an involution ∗2 ∼ Id. Note that the Hodge dual ∗η in general is a distri-
bution.
In detail, the Hodge ∗ operation is built out of two operations: Firstly, a fiberwise Fourier transform
Γ •
(T∗
M) ≡ Ω•0(M) ∋ η
F
→ π = Fη ∈ Γ •
(TM) , (3.14.3)
that takes exterior forms η=η(z; C) to multivectors
π = π(z; B) =
1
m!
πA1···Am
(z) Bℓ
Am
∧ · · · ∧ Bℓ
A1
, (3.14.4)
where Bℓ
A ≡(−1)εA Br
A and
Bℓ
A ∧ Bℓ
C = −(−1)εAεC Bℓ
C ∧ Bℓ
A , ε(Bℓ
A) = εA , p(Bℓ
A) = 1 . (3.14.5)
25
The Fourier transform F itself only depends on the density ρ:
(Fη)(z; B) :=
dN C
ρ
e
i
¯h
CA∧Bℓ
A η(z; C) . (3.14.6)
Secondly, a flat map
Γ(TM) ∋ X
♭
→ η = X♭
∈ Γ(T∗
M) , (3.14.7)
that takes vectors X =XABℓ
A to co–vectors η=ηACA. The Riemannian flat map ♭ is X♭
A = XBgBA,
or equivalently, in terms of basis elements,
Bℓ
A = gABCB
. (3.14.8)
Altogether, the Hodge ∗ operation can be written as
(∗η)(z; C) = (Fη)(z; B)
Bℓ
A=gABCB
. (3.14.9)
In contrast to the Riemannian case, there is no good way to construct an antisymplectic Hodge ∗
operation. This is because the antisymplectic flat map Bℓ
A = EABCB carries the opposite Grassmann–
parity ε(Bℓ
A) = εA + 1, cf. Subsection 4.1.
Proposition 3.4 The Hodge adjoint de Rham operator, also known as the Hodge codifferential, is:
∗d∗ ∼ δ := (−1)εA



1
ρ
→
∂ℓ
∂zA
ρ − (
→
∂ℓ
∂zA
gBC)CC
PB
(−1)εB


 PA
= (−1)εA



1
ρ
→
∂ℓ
∂zA
ρ −
1
2
(
→
∂ℓ
∂zA
gBC)SCB
+ (−1)εB


 PA
. (3.14.10)
Proof of Proposition 3.4:
(∗d∗η)(z, C) =
dN C′
ρ
e
i
¯h
C′∧C
C′A
→
∂ℓ
∂zA
dN C′′
ρ
e
i
¯h
C′′∧C′
η(z, C′′
)
= (−1)εA
dN C′
ρ



→
∂ℓ
∂zA
+
i
¯h
(
→
∂ℓ
∂zA
C ∧ C′
)



dN C′′
ρ
C′A
e
i
¯h
(C′′−C)∧C′
η(z, C′′
)
= −(−1)εA
i
¯h
dN C′′
ρ



→
∂ℓ
∂zA
− (
→
∂ℓ
∂zA
C ∧ P)



dN C′
ρ
PA
e
i
¯h
(C′′−C)∧C′
η(z, C′′
)
∼
(−1)εA
ρ



→
∂ℓ
∂zA
− (
→
∂ℓ
∂zA
C ∧ P)


 ρPA
η(z, C) . (3.14.11)
26
3.15 Hodge–Dirac Operator D(T)
= d + λδ
We shall for the remainder of Section 3 assume that the connection is the Levi–Civita connection.
Central for our discussion are the TA
B generators (2.7.5). They act on exterior forms η ∈ Ω•0(M),
i.e., functions η=η(z; C) of z and C. (Recall that the Y A variables are put to zero Y A → 0.)
The Dirac operator D(T) in the TA
B representation (2.7.4) is a ΓA matrix (3.12.1) times the covariant
derivative (2.7.4)
D(T)
:= ΓA
∇
(T)
A = CA
∇
(T)
A + λPA
∇
(T)
A = d + λδ , (3.15.1)
ε(D(T)
) = 0 , p(D(T)
) = 1 (mod 2) . (3.15.2)
The component of the Dirac operator to zeroth order in λ,
D(T)
λ=0
= CA
∇
(T)
A = CA



→
∂ℓ
∂zA
− ΓA
B
C CC
→
∂ℓ
∂CB


 = CA
→
∂ℓ
∂zA
= d , (3.15.3)
is just the exterior de Rham derivative d, because the connection is torsionfree. The component of
the Dirac operator to first order in λ,
[
→
∂ℓ
∂λ
, D(T)
] = PA
∇
(T)
A = [PA
, ∇
(T)
A ] + (−1)εA ∇
(T)
A PA
= ΓA
ACPC
+ (−1)εA



→
∂ℓ
∂zA
− (−1)(εA+1)εB+εC ΓBAC CC
PB


 PA
= (−1)εA



1
ρg
→
∂ℓ
∂zA
ρg − (
→
∂ℓ
∂zA
gBC)CC
PB
(−1)εB


 PA (3.14.10)
= : δ , (3.15.4)
is the Hodge adjoint de Rham operator. Equations (3.15.3) and (3.15.4) prove the last equality in eq.
(3.15.1).
The Laplacian ∆
(T)
ρg
in the TA
B representation (2.7.4) is
∆(T)
ρg
:= (−1)εA ∇AgAB
∇
(T)
B = (−1)εA ∇
(T)
A gAB
∇
(T)
B + ΓA
AC gCB
∇
(T)
B
=
(−1)εA
ρg
∇
(T)
A ρggAB
∇
(T)
B . (3.15.5)
Theorem 3.5 (Weitzenb¨ock’s formula for exterior forms) The difference between the square of
the Dirac operator D(T) and the Laplacian ∆
(T)
ρg
in the TA
B representation (2.7.4) is
D(T)
D(T)
− λ∆(T)
ρg
= −
λ
4
SBA
− RAB,CD SDC
− (−1)εC +εD (3.15.6)
= −λCA
RAB PB
+
λ
2
CB
CA
RAB,CD PD
PC
(−1)εC +εD . (3.15.7)
Remarks: The square D(T)D(T) = λ(dδ + δd) is known as the form Laplacian. The Laplacian ∆
(T)
ρg
is
equal to the Bochner Laplacian.
27
Proof of Theorem 3.5: The square is a sum of three terms
D(T)
D(T)
=
1
2
[D(T)
, D(T)
] = I + II + III . (3.15.8)
The first term is
I :=
1
2
[ΓB
, ΓA
]∇
(T)
A ∇
(T)
B = λgBA
∇
(T)
A ∇
(T)
B . (3.15.9)
The second term is
II := ΓA
[∇
(T)
A , ΓB
]∇
(T)
B
(3.12.5)
= −ΓA
ΓA
B
C ΓC
∇
(T)
B = −(−1)εC ΓB
CA ΓA
ΓC
∇
(T)
B
= −(−1)εC λΓB
CA gAC
∇
(T)
B = λ
(−1)εA
ρg
(
→
∂ℓ
∂zA
ρggAB
)∇
(T)
B . (3.15.10)
Together, the first two terms I + II form the Laplace operator (3.15.5):
I + II = λ∆(T)
ρg
. (3.15.11)
The third term yields the curvature terms:
III := −
1
2
ΓB
ΓA
[∇
(T)
A , ∇
(T)
B ] =
1
2
ΓB
ΓA
RAB
D
C TC
D = −
1
4
ΓB
ΓA
RAB,CD SDC
− (−1)εC +εD
= −
1
4
CB
CA
+ λ(SBA
− + gBA
) + λ2
PB
PA
RAB,CD SDC
− (−1)εC +εD
= −
1
2
CB
CA
RAB,CD CC
PD
(−1)(εC +1)(εD+1)
−
λ
4
SBA
− RAB,CD SDC
− (−1)εC +εD
−
λ2
2
PB
PA
RAB,CD PC
CD
(−1)(εC +1)(εD+1)
= −
λ
4
SBA
− RAB,CD SDC
− (−1)εC +εD = −λCB
PA
RAB,CD CD
PC
(−1)εC +εD
= −λCB
RBA,CDgDA
PC
(−1)(εA+1)(εC +1)+εD + λCB
RBA,DC CD
PC
PA
(−1)εA+(εC +1)(εD+1)
= −λCB
RBAC
A
PC
(−1)(εA+1)(εC +1)
+ λCD
CB
RBA,DC PC
PA
(−1)εA(εD+1)+εC
= −λCB
RBC PC
+
λ
2
CB
CD
RDB,AC PC
PA
(−1)εA+εC . (3.15.12)
Here the first Bianchi identity (3.7.5) was used to cancel terms proportional to zeroth and second
order in λ.
3.A Appendix: Is There A Second–Order Formalism?
For the standard Dirac operator, which will be discussed in Subsections 6.4–6.6, it is natural to
replace the first–order sab
− matrices (6.3.1) with the second–order σab
− matrices (6.6.1). Therefore, it is
natural to speculate if it is possible to replace the first–order SAB
± matrices (3.11.1) with the following
second–order matrices:
ΣAB
∓ :=
1
4λ
ΓA
ΓB
∓ (−1)εAεB (A ↔ B) , (3.A.1)
ε(ΣAB
∓ ) = εA + εB , p(ΣAB
∓ ) = 0 . (3.A.2)
(The names first– and second–order refer to the number of CA–derivatives.) On one hand, the matrices
ΣAB
− =
1
4λ
{ΓA
, ΓB
}+ =
1
2λ
CA
CB
+
1
2
SAB
− +
λ
2
PA
PB
. (3.A.3)
28
yield precisely the same non–Abelian Lie–algebra (3.11.4) and fundamental representation (3.12.4) as
the SAB
− matrices. Moreover, the SAB
− matrices rotate the ΣAB
− matrices
[ΣAB
− , SCD
− ] = (−1)εA(εB+εC)
gBC
ΣAD
− + ΣBC
− gAD
− (−1)εAεB (A ↔ B) . (3.A.4)
However, the commutator of ΣAB
− and SCD
+ does not close,
[ΣAB
− , SCD
+ ] = (−1)εA(εB+εC)
gBC ˜ΣAD
− ˜ΣBC
gAD
− (−1)εAεB (A ↔ B) , (3.A.5)
where the tilde generators
˜ΣAB
:= −
1
2λ
CA
CB
+
1
2
SAB
+ +
λ
2
PA
PB
(3.A.6)
have no (A ↔ B) symmetry or antisymmetry. On the other hand, the matrices
ΣAB
+ :=
1
4λ
[ΓA
, ΓB
]
(3.12.3)
=
1
2
gAB
1 (3.A.7)
are proportional to the identity operator, and thus behave very differently from the non–Abelian SAB
+
matrices.
The problem with a substitution SAB
∓ → ΣAB
∓ is that the SAB
+ matrices appear in the matrix realization
(3.12.6). On one hand, the ΣAB
− representation (3.A.1) is not suitable, because it couples pathologically
to the non–vanishing SAB
+ sector, and, on the other hand, the ΣAB
+ matrices are Abelian, and therefore
pathological by themselves. Hence, it is doubtful if the substitution SAB
∓ → ΣAB
∓ makes any sense at
all. In any case, we shall dismiss the second–order ΣAB
∓ matrices (3.A.1) from now on.
4 Antisymplectic Geometry
4.1 Metric
Let there be given an antisymplectic metric, i.e., a closed two–form
E =
1
2
CA
EAB ∧ CB
= −
1
2
EAB CB
∧ CA
∈ Ω20(M) , (4.1.1)
of Grassmann–parity
ε(EAB) = εA + εB + 1 , ε(E) = 1 , p(EAB) = 0 , (4.1.2)
and with antisymmetry
EBA = −(−1)εAεB EAB . (4.1.3)
The closeness condition
dE = 0 (4.1.4)
reads in components
cycl. A,B,C
(−1)εAεC (
→
∂ℓ
∂zA
EBC) = 0 . (4.1.5)
The antisymplectic metric EAB is assumed to be non–degenerate, i.e., there exists an inverse contravariant
(2, 0) tensor field EAB such that
EAB EBC
= δC
A . (4.1.6)
29
The inverse EAB has Grassmann–parity
ε(EAB
) = εA + εB + 1 , (4.1.7)
and symmetry
EBA
= −(−1)(εA+1)(εB+1)
EAB
. (4.1.8)
The closeness condition (4.1.4) has no Riemannian analogue. It is the integrability condition for the
local existence of Darboux coordinates.
4.2 Odd Laplacian ∆ρ
The odd Laplacian ∆ρ, which takes scalar functions in scalar functions, is defined as
2∆ρ :=
(−1)εA
ρ
→
∂ℓ
∂zA
ρEAB
→
∂ℓ
∂zB
, ε(∆ρ) = 1 , p(∆ρ) = 0 . (4.2.1)
Note the factor of 2 in the odd Laplacian (4.2.1) as compared with the Riemannian case (3.2.1). It is
similar in nature to the factor of 2 in difference between eqs. (3.1.1) and (4.1.1). Both are introduced
to avoid factors of 2 in Darboux coordinates.
The antibracket (f, g) of two functions f =f(z) and g=g(z) can be defined via a double commutator
with the odd Laplacian, acting on the constant unit function 1,
(f, g) := (−1)εf [[
→
∆ρ, f], g]1 ≡ (−1)εf ∆ρ(fg) − (−1)εf (∆ρf)g − f(∆ρg) + (−1)εg fg(∆ρ1)
= (f
←
∂r
∂zA
)EAB
(
→
∂ℓ
∂zB
g) = −(−1)(εf +1)(εg+1)
(g, f) . (4.2.2)
The antibracket (4.2.2) satisfies a Jacobi identity,
cycl. f,g,h
(−1)(εf +1)(εh+1)
(f, (g, h)) = 0 , (4.2.3)
because of the closeness condition (4.1.4).
4.3 Odd Scalar νρ
A Grassmann–odd function νρ can be constructed from the antisymplectic metric E and an arbitrary
density ρ as
νρ := ν(0)
ρ +
ν(1)
8
−
ν(2)
24
, (4.3.1)
where
ν(0)
ρ :=
1
√
ρ
(∆1
√
ρ) , (4.3.2)
ν(1)
:= (−1)εA (
→
∂ℓ
∂zA
EAB
←
∂r
∂zB
)(−1)εB , (4.3.3)
ν(2)
:= −(−1)εB (
→
∂ℓ
∂zA
EBC)(zC
, (zB
, zA
))
= (−1)εAεD (
→
∂ℓ
∂zD
EAB
)EBC(ECD
←
∂r
∂zA
) . (4.3.4)
Here ∆1 is the odd Laplacian (4.2.1) with ρ = 1, and (·, ·) is the antibracket (4.2.2).
30
Lemma 4.1 The odd quantity νρ is a scalar, i.e., it does not depend on the coordinate system.
The proof of Lemma 4.1 is given in Ref. [22]. Below follows an antisymplectic version of Proposition 3.2.
Proposition 4.2 (Classification of 2–order differential invariants) If a function ν =ν(z) has
the following properties:
1. The function ν is a scalar.
2. ν(z) is a polynomial of the metric EAB(z), the density ρ(z), their inverses, and z–derivatives
thereof in the point z.
3. ν is invariant under constant rescaling of the density ρ → λρ, where λ is a z–independent pa-
rameter.
4. ν scales as ν → λν under constant Weyl scaling EAB → λEAB, where λ is a z–independent
parameter.
5. Each term in ν contains precisely two z–derivatives.
Then ν is proportional to the odd scalar νρ
ν = α νρ , (4.3.5)
where α is z–independent proportionality constant.
The proof of Proposition 4.2 is similar to the proof of Proposition 3.2.
4.4 ∆ And ∆E
Khudaverdian’s ∆E operator [17, 18, 19, 20, 21, 22, 23], which takes semidensities to semidensities, is
defined using arbitrary coordinates as
∆E := ∆1 +
ν(1)
8
−
ν(2)
24
. (4.4.1)
It is obviously manifestly independent of ρ. That it takes semidensities to semidensities will become
clear because of eq. (4.4.3) below. The Jacobi identity (4.2.3) precisely encodes the nilpotency of ∆E.
The Grassmann–odd nilpotent ∆ operator, which takes scalar functions to scalar functions, can be
defined as defined as
∆ := ∆ρ + νρ . (4.4.2)
In fact, every Grassmann–odd, nilpotent, second–order operator is of the form (4.4.2) up to a Grassmann–
odd constant [4]. We shall dismiss Grassmann–odd constants since they do not satisfy all the five
assumptions of Proposition 4.2. The ∆E operator and the ∆ operator are related via
√
ρ–conjugation
[22, 4]
∆E =
√
ρ∆
1
√
ρ
. (4.4.3)
The proof is almost identical to the corresponding Riemannian calculation (3.5.4).
31
Recall how the zeroth–order term is determined in the Riemannian case, where no nilpotency principle
was available, cf. Subsections 3.5 and 3.9. There we applied a ρ independence test. Could one do a
similar analysis in the antisymplectic case? Yes. In detail, consider an operator
∆′
:= ∆ + ν = ∆ρ + νρ + ν , (4.4.4)
where ν is a most general zeroth–order term. It is easy to see from eqs. (4.4.3) and (4.4.4) that the
corresponding
√
ρ–conjugated operator
√
ρ∆′ 1√
ρ is independent of ρ if and only if the shift term ν is
ρ–independent. From Proposition 4.2, one then concludes that ν = 0 has to be zero, i.e., the form of
the ∆ operator (4.4.2) can be uniquely reproduced from a ρ–independence test and knowledge about
possible scalar structures.
4.5 Antisymplectic Connection
A connection ∇(Γ) is called antisymplectic, if it preserves the antisymplectic metric E,
0 = (∇
(Γ)
A E)BC = (
→
∂ℓ
∂zA
EBC) − ((−1)εAεB ΓBAC − (−1)εB εC (B ↔ C)) . (4.5.1)
Here we have lowered the Christoffel symbol with the metric
ΓABC := EADΓD
BC(−1)εB . (4.5.2)
We should stress that there is not a unique choice of an antisymplectic, torsionfree, and ρ–compatible
connection ∇(Γ), i.e., a connection that satisfies eqs. (4.5.1), (2.2.3) and (2.3.3). On the other hand, it
can be demonstrated that such connections ∇(Γ) exist locally for N > 2, where N = dim(M) denotes
the dimension of the manifold M. (There are counterexamples for N =2 where ∇(Γ) need not exist.)
The mere existence of an antisymplectic and torsionfree connection ∇(Γ) implies that the two–form E
is closed (4.1.4), if we hadn’t already assumed it in the first place. (Curiously, while it is impossible
to impose closeness relations in Riemannian geometry, the closeness relations are almost impossible
to avoid in geometric structures defined by two–forms.) The antisymplectic condition (4.5.1) reads in
terms of the contravariant (inverse) metric
0 = (∇
(Γ)
A E)BC
≡ (
→
∂ℓ
∂zA
EBC
) + ΓA
B
DEDC
− (−1)(εB+1)(εC +1)
(B ↔ C) . (4.5.3)
4.6 The Riemann Curvature
For an antisymplectic connection ∇(Γ), we prefer to work with a (0, 4) Riemann tensor (as opposed to
a (1, 3) tensor) by lowering the upper index with the metric (4.1.1). In terms of Christoffel symbols it
is easiest to work with expression (2.4.2):
RD,ABC := EDF RF
ABC
= (−1)εA(εD+1)


(−1)εB
→
∂ℓ
∂zA
ΓDBC + (−1)εF (εA+εD)
ΓF ADΓF
BC



−(−1)εAεB (A ↔ B) . (4.6.1)
In the second equality of eq. (4.6.1) is used the antisymplectic condition (4.5.1). If the antisymplectic
condition (4.5.1) is used one more time on the first term in eq. (4.6.1), one derives the following
symmetry
RD,ABC = (−1)(εA+εB)(εC +εD)+εC εD (C ↔ D) . (4.6.2)
32
This symmetry becomes clearer if one instead starts from expression (2.4.6) and defines
RAB,CD := RABC
F
EF D = −(−1)εA+εB+(εA+εB+εC)εD RD,ABC . (4.6.3)
Then the symmetry (4.6.2) simply translates into a symmetry between the third and fourth index:
RAB,CD = (−1)εC εD (C ↔ D) . (4.6.4)
The Ricci 2–form is then
RAB =: RAB
C
C(−1)εC = RAB,CDEDC
(−1)εC . (4.6.5)
We note that the torsionfree condition has not been used so far in this Section 4.6. The first Bianchi
identity (2.4.7) reads (in the torsionfree case):
0 =
cycl. A,B,C
(−1)εAεC RAB,CD . (4.6.6)
4.7 Odd Scalar Curvature
The odd scalar curvature is defined as
R := EBA
RAB = RABEBA
. (4.7.1)
Proposition 4.3 For an arbitrary, antisymplectic, torsionfree, and ρ–compatible connections ∇Γ, the
scalar curvature R does only depend on E and ρ through the odd νρ scalar [4]
R = −8νρ . (4.7.2)
The proof of Proposition 4.3 is given in Ref. [4]. It is extended to degenerate anti–Poisson structures
in Ref. [23, 37]. In particular, one concludes that the odd scalar curvature R does not depend on
the connection used, and the odd ∆ operator (4.4.2) reduces to the odd ∆ operator (1.0.1) in the
Introduction.
Altogether, we have now established a link between the zeroth–order terms in the even and odd ∆
operators (1.0.3) and (1.0.1):
Riemannian zeroth order term Antisymplectic zeroth order term
−
R
4
= νρg
←→ 2νρ = −
R
4
. (4.7.3)
The left (resp. right) equality is due to Proposition 3.3 (resp. 4.3). Both zeroth–order terms are
characterized by the same ρ–independence test described in Subsections 3.9 and 4.4 (up to a subtlety on
how to switch back and forth between ρ–dependent and ρ–independent formalism in the Riemannian
case). It is no coincidence that the same coefficient minus–a–quarter appears on both sides of the
correspondence (after the odd ∆ operator has been multiplied with an appropriate factor 2). At
the mathematical level, this is basically because the zeroth–order terms are determined by the ν
(0)
ρ
building blocks alone, where the inverse metrics gAB and EAB enter in a similar manner, and only
linearly. For expressions that do not depend on the metric tensors gAB and EAB, and only have
an linear dependence of the inverse metrics gAB and EAB, respectively, one does not see the effects
that distinguish Riemannian and antisymplectic geometry, such as e.g., opposite Grassmann–parity,
closeness relations and the Jacobi identities.
33
4.8 First–Order SAB
Matrices
Because of the presence of the antisymplectic tensor EAB, the symmetry of the general linear (= gl)
Lie–algebra (2.7.6) reduces to an antisymplectic Lie–subalgebra. Its generators SAB
± read
SAB
± := CA
(−1)εB PB
∓ (−1)(εA+1)(εB +1)
(A ↔ B) , PA
:= EAB
→
∂ℓ
∂CB
, (4.8.1)
ε(SAB
± ) = εA + εB + 1 , p(SAB
± ) = 0 , (4.8.2)
SA
±C := SAB
± EBC(−1)εC . (4.8.3)
The SAB
+ matrices satisfy an antisymplectic Lie–algebra:
[SAB
± , SCD
± ] = (−1)εA(εB+εC+1)+εB EBC
SAD
+ − SBC
+ EAD
∓(−1)(εA+1)(εB +1)
(A ↔ B) , (4.8.4)
[SAB
± , SCD
∓ ] = (−1)εA(εB+εC+1)+εB EBC
SAD
− + SBC
− EAD
∓(−1)(εA+1)(εB +1)
(A ↔ B) . (4.8.5)
Note that the eqs. (4.8.4) and (4.8.5) remain invariant under a c–number shift
SAB
+ → S′AB
+ := SAB
+ + αEAB
1 , (4.8.6)
where α is a parameter.
4.9 ΓA
Matrices
Guided by the analysis of Appendix 4.B, we now define antisymplectic ΓA matrices via the following
Berezin–Fradkin operator representation [30, 31]
ΓA
θ ≡ ΓA
:= CA
+ (−1)εA θPA
= CA
− PA
θ , ε(ΓA
) = εA , p(ΓA
) = 1 (mod 2) ,
(4.9.1)
where θ is a nilpotent Fermionic parameter with ε(θ)=1 and p(θ)=0. The ΓA matrices satisfy a
Clifford–like algebra
[ΓA
, ΓB
] = 2(−1)εA θEAB
1 . (4.9.2)
The ΓA matrices form a fundamental representation of the antisymplectic Lie–algebra (4.8.4):
[SAB
± , ΓC
] = ΓA
±θ(−1)εB EBC
∓ (−1)(εA+1)(εB +1)
(A ↔ B) . (4.9.3)
If one commutes an antisymplectic connection ∇
(T)
A in the TA
B representation (2.7.4) with a ΓB
matrix, one gets
[∇
(T)
A , ΓB
] = −ΓA
B
C ΓC
. (4.9.4)
34
4.10 Dirac Operator D(T)
= d + θδ
We shall for the remainder of Section 4 assume that the connection is antisymplectic, torsionfree and
ρ–compatible.
The Dirac operator D(T) in the TA
B representation (2.7.4) is a ΓA matrix (4.9.1) times the covariant
derivative (2.7.4)
D(T)
:= ΓA
∇
(T)
A = d + θδ , ε(D(T)
) = 0 , p(D(T)
) = 1 (mod 2) . (4.10.1)
Unlike the Riemannian case of Subsection 3.15, the component δ of the Dirac operator to first order
in θ does not have an interpretation as a Hodge codifferential, since there is no antisymplectic Hodge
∗ operation. Even worse, it depends explicitly on the Christoffel symbols:
δ := (−1)εA PA
∇
(T)
A = (−1)εA [PA
, ∇
(T)
A ] + (−1)εA ∇
(T)
A PA
= ΓA
ACPC
+ (−1)εA



→
∂ℓ
∂zA
+ (−1)εAεB ΓBAC CC
PB


 PA
= (−1)εA



1
ρ
→
∂ℓ
∂zA
ρ + ΓABC CC
PB


 PA
. (4.10.2)
Nevertheless, there exists a close antisymplectic analogue of Weitzenb¨ock’s formula (3.15.7), cf. eq.
(4.10.5) below. The odd Laplacian ∆
(T)
ρ in the TA
B representation (2.7.4) is
2∆(T)
ρ := (−1)εA ∇AEAB
∇
(T)
B =
(−1)εA
ρ
∇
(T)
A ρEAB
∇
(T)
B . (4.10.3)
Theorem 4.4 (Antisymplectic Weitzenb¨ock type formula for exterior forms) The difference
between the square of the Dirac operator D(T) and twice the odd Laplacian ∆
(T)
ρ in the TA
B representation
is
D(T)
D(T)
− 2θ∆(T)
ρ =
θ
4
(−1)εB+εC SBA
− RAB,CD SDC
+ (4.10.4)
= −θCA
RAB PB
+
θ
2
CB
CA
RAB,CD PD
PC
(−1)εC . (4.10.5)
Proof of Theorem 4.4: The square is a sum of three terms
D(T)
D(T)
=
1
2
[D(T)
, D(T)
] = I + II + III . (4.10.6)
The first term is
I :=
1
2
[ΓB
, ΓA
]∇
(T)
A ∇
(T)
B = (−1)εB θEBA
∇
(T)
A ∇
(T)
B . (4.10.7)
The second term is
II := ΓA
[∇
(T)
A , ΓB
]∇
(T)
B
(4.9.4)
= −ΓA
ΓA
B
C ΓC
∇
(T)
B = −(−1)εC ΓB
CA ΓA
ΓC
∇
(T)
B
= −(−1)εB θΓB
CA EAC
∇
(T)
B = θ
(−1)εA
ρ
(
→
∂ℓ
∂zA
ρEAB
)∇
(T)
B . (4.10.8)
35
Together, the first two terms I + II form the odd Laplacian (4.10.3):
I + II = 2θ∆(T)
ρ . (4.10.9)
The third term yields the curvature terms:
III := −
1
2
ΓB
ΓA
[∇
(T)
A , ∇
(T)
B ] =
1
2
ΓB
ΓA
RAB
D
C TC
D =
1
4
ΓB
ΓA
RAB,CD SDC
+ (−1)εC
=
1
4
CB
CA
+ (−1)εB θ(SBA
− + EBA
) RAB,CD SDC
+ (−1)εC
=
1
2
CB
CA
RAB,CD CC
PD
(−1)εC εD +
θ
4
(−1)εB+εC SBA
− RAB,CD SDC
+
=
θ
4
(−1)εB+εC SBA
− RAB,CD SDC
+ = (−1)εA+εB θCB
PA
RAB,CD CD
PC
= −(−1)(εA+1)(εC +1)
θCB
RBA,CDEDA
PC
− θCB
RBA,DC CD
PC
PA
(−1)εA+εCεD
= −(−1)(εA+1)(εC +1)
θCB
RBAC
A
PC
+ θCD
CB
RBA,DC PC
PA
(−1)εA(εD+1)
= −θCB
RBC PC
+
θ
2
CB
CD
RDB,AC PC
PA
(−1)εA . (4.10.10)
Here the first Bianchi identity (4.6.6) was used one time in the θ–independent sector.
4.A Appendix: Is There A Second–Order Formalism?
There are no deformations of the first–order SAB
− matrices (4.8.1). The general second–order deformation
of the SAB
+ matrices (4.8.1) reads
ΣAB
+ := SAB
+ + αEAB
1 + βPA
PB
θ , (4.A.1)
where α and β are two parameters. The second–order ΣAB
+ matrices satisfy precisely the same antisymplectic
Lie–algebra (4.8.4) as the SAB
+ matrices. Moreover, the SAB
+ matrices rotate the ΣAB
+
matrices,
[ΣAB
+ , SCD
+ ] = (−1)εA(εB+εC+1)+εB EBC
ΣAD
+ − ΣBC
+ EAD
− (−1)(εA+1)(εB +1)
(A ↔ B) . (4.A.2)
The ΣAB
+ matrices interact with the ΓC and the SCD
− matrices as follows
[ΣAB
+ , ΓC
] = ΓA
(1+β)θ(−1)εB EBC
− (−1)(εA+1)(εB +1)
(A ↔ B) , (4.A.3)
[ΣAB
+ , SCD
− ] = (−1)εA(εB+εC+1)+εB EBC ˜ΣAD
+ ˜ΣBC
EAD
−(−1)(εA+1)(εB+1)
(A ↔ B) , (4.A.4)
where the generators
˜ΣAB
:= SAB
− + βPA
PB
θ (4.A.5)
have no (A ↔ B) symmetry or antisymmetry. According to eq. (4.A.3), one must choose the parameter
β = 0 to be zero, in order to ensure that the ΣAB
+ matrices rotates the ΓA matrices in the correct way.
One concludes that a consistent antisymplectic second–order formulation does not exist, regardless of
whether the pathological SAB
− sector decouples or not, and we shall abandon the subject. See also
comment in the Conclusions.
36
4.B Appendix: What Is An Antisymplectic Clifford Algebra?
In this Appendix 4.B, we shall motivate the definition (4.9.2) of an antisymplectic Clifford algebra
given in Subsection 4.9. Intuitively, one would probably assume that an antisymplectic Clifford algebra
should be
ΓA
⋆ ΓB
− (−1)(εA+1)(εB+1)
(A ↔ B)
?
= 2EAB
1 , (4.B.1)
where the “⋆” denotes a Fermionic multiplication, ε(⋆) = 1, cf. question 3 in the Introduction. We
will now expose some of the weaknesses of the proposal (4.B.1). (A question mark “?” on top of an
equality sign “=” indicates that a formula may be ultimately wrong.) It follows from eq. (1.0.6) that
the form degree of the ⋆ multiplication must vanish, p(⋆) = 0. Let us assume that the ⋆ multiplication
is invertible and commute with the ΓA matrices,
ΓA
⋆ − (−1)ε(ΓA)
⋆ΓA
≡ [ΓA
, ⋆]
?
= 0 . (4.B.2)
Then one can bring the Clifford algebra (4.B.1) on a Riemannian form,
ΓA
ΓB
+ (−1)εAεB (A ↔ B) = 2gAB
1 , (4.B.3)
where the Riemannian metric gAB is a product of ⋆−1 and the antisymplectic metric EAB,
gAB
:= (−1)ε(ΓA)
⋆−1
EAB
= (−1)εAεB (A ↔ B) , ε(gAB
) = εA + εB . (4.B.4)
The Riemannian structure (4.B.4) is non–commutative,
[gAB
, gCD
] = −2(−1)εB +εC ⋆−2
EAB
ECD
= 0 , (4.B.5)
since [⋆−1, ⋆−1] = 2⋆−2 = 0, and hence the metric (4.B.4) is not a classical Riemannian metric. We
would like to interpret the left–hand side of eq. (4.B.3) as a commutator [ΓA, ΓB], cf. definition (1.1.3).
This implies that the Grassmann– and form–parity of the ΓA matrices are
ε(ΓA
) = εA , p(ΓA
) = 1 (mod 2) . (4.B.6)
The only natural candidate for a Berezin–Fradkin operator representation [30, 31] is
ΓA
= CA
+ gAB
→
∂ℓ
∂CB
≡ CA
− PA
⋆−1
, ε(CA
) = εA , p(CA
) = 1 , (4.B.7)
where the CA variables commute with the ⋆ multiplication, [CA, ⋆] = 0, and they carry the same
Grassmann– and form–parities as the ΓA matrices. The PA derivatives are defined in eq. (4.8.1).
However, the Berezin–Fradkin operator representation (4.B.7) does not satisfy the Clifford algebra
(4.B.3) due to the non–commutative metric (4.B.5). The representation does also violate the commutation
relation (4.B.2). There appear extra terms on the respective right–hand sides,
[ΓA
, ⋆−1
] = −2 ⋆−2
PA
, (4.B.8)
[ΓA
, ΓB
] = 2gAB
1 − 2 ⋆−2
PA
PB
(−1)εB . (4.B.9)
The original antisymplectic Clifford algebra (4.B.1) looks even more complicated:
1
2
ΓA
⋆ ΓB
− (−1)(εA+1)(εB +1)
(A ↔ B) = SAB
+ − EAB
1 + PA
PB
⋆−1
. (4.B.10)
One idea would be to try to correct the Clifford algebra (4.B.9) by adding higher–order terms O(⋆−2)
to the Berezin–Fradkin operator representation (4.B.7), but unfortunately there is no obvious way
37
to do that. Another idea is to take the limit ⋆−1 → 0 in some appropriate way at the end of the
calculations. The approach that we shall pursuit in this paper is to take θ ≡ ⋆−1 as a fundamental
object, i.e., forgetting that it originally was an inverse of ⋆, and then assume that it is nilpotent
θ2 = ⋆−2 = 0. Then the ΓA matrices and the θ variable commute [ΓA, θ] = 0, the Riemannian metric
(4.B.4) becomes an ordinary commutative structure, and the Clifford algebra (4.B.3) is restored. The
price is that the Fermionic ⋆ multiplication (4.B.1), which ironically was our initial clue, does not
exist.
5 General Spin Theory
5.1 Spin Manifold
Let W be a vector space of the same dimension as the manifold M. Let the vectors (=points) in
W have coordinates wa of Grassmann–parity ε(wa) = εa (and form–degree p(wa) = 0). It is assumed
that the flat index “a” (denoted with a small roman letter) of the vector space W runs over the same
index–set as the curved index “A” (denoted with a capital roman letter) of the manifold M. In a slight
misuse of notation, let TW := M × W (resp. T∗W := M × W∗) denote the trivial vector bundle over
M with the vector space W (resp. dual vector space W∗) as fiber. Let ∂r
a and
→
dwa denote dual bases in
W and W∗, respectively, of Grassmann–parity ε(
→
dwa)=εa =ε(∂r
a). The form–parities p(
→
dwa)=p(∂r
a)
are either all 0 or all 1, depending on applications, whereas a 1–form dwa with no arrow “→” always
carries odd form–parity p(dwa)=1 (and Grassmann–parity ε(dwa)=εa).
Let us assume that M is a spin manifold, i.e., that there exists a bijective bundle map
e = ∂r
a ea
A
→
dzA
: Γ(TM) → Γ(TW) , (5.1.1)
e−1
= ∂r
A eA
a
→
dwa
: Γ(TW) → Γ(TM) . (5.1.2)
The intertwining tensor field ea
A is known as a vielbein. (There are topological obstructions for
the existence of a global vielbein. However, it would be out of scope to describe global notions for
supermanifolds here, such as, orientability and Stiefel–Whitney classes. The interesting topic of index
theorems for Dirac operators will for similar reasons be omitted in this paper.)
Note that the superdeterminant sdet(ea
A) = 0 of the vielbein transforms as a density under general
coordinate transformations. In general, the vielbein ea
A is called compatible with the measure density
ρ, if
ρ ∼ sdet(ea
A) (5.1.3)
is proportional to the vielbein superdeterminant sdet(ea
A) with a z–independent proportionality factor.
In this case, the notion of volume is unique (up to an overall rescaling).
5.2 Spin Connection ∇(ω)
= d + ω
A connection ∇(ω) = d + ω : Γ(TM) × Γ(TW) → Γ(TW) in the bundle TW is known as a spin
connection, where
∇
(ω)
A =
→
∂ℓ
∂zA
+ ∂r
b ωb
Ac
→
dwc
. (5.2.1)
The total connection ∇ = d + Γ + ω contains both a Christoffel symbol ΓB
AC, which acts on curved
indices, and a spin connection ωb
Ac, which acts on flat indices. We will always demand that the total
38
connection ∇ preserves the vielbein
0 = (∇Aeb
C) = (
→
∂ℓ
∂zA
eb
C) − (−1)εAεb eb
B ΓB
AC + ωA
b
c ec
C . (5.2.2)
This condition (5.2.2) fixes uniquely the spin connection as
ωb
Ac := Γb
Ac − fb
Ac , (5.2.3)
ωA
b
c := ΓA
b
c − fA
b
c = (−1)εAεb ωb
Ac , (5.2.4)
ωa
b
c := Γa
b
c − fa
b
c = (eT
)a
A
ωA
b
c , (5.2.5)
ωb
ac := Γb
ac − fb
ac = (−1)εaεb ωa
b
c , (5.2.6)
where
Γb
Ac := eb
B ΓB
AC eC
c , (5.2.7)
ΓA
b
c := (−1)εAεb Γb
Ac , (5.2.8)
Γa
b
c := (eT
)a
A
ΓA
b
c , (5.2.9)
Γb
ac := (−1)εaεb Γa
b
c , (5.2.10)
fA
b
c := (
→
∂ℓ
∂zA
eb
D)eD
c , (5.2.11)
fb
Ac := (−1)εAεb fA
b
c , (5.2.12)
fa
b
c := (eT
)a
A
fA
b
c , (5.2.13)
fb
ac := (−1)εaεb fa
b
c . (5.2.14)
Here the transposed vielbein is
(eT
)A
a
:= (−1)(εa+1)εA ea
A . (5.2.15)
The condition (5.2.2) implies in many cases that one can transfer concepts/objects back and forth
between TM and TW by simply multiplying with appropriate factors of the vielbein. Firstly, the
spin connection ∇
(ω)
A : Γ(TW) → Γ(TW) can in a certain sense be thought of as the connection
∇
(Γ)
A : Γ(TM) → Γ(TM) conjugated with the vielbein e : Γ(TM) → Γ(TW), i.e., roughly speaking
a product of three matrices,
e∇
(Γ)
A e−1
= ∂r
b eb
B
→
dzB
(
→
∂ℓ
∂zA
+ ∂r
D ΓD
AE
→
dzE
) ∂r
C eC
c
→
dwc
=
→
∂ℓ
∂zA
+ (−1)εAεD ∂r
b eb
D(
→
∂ℓ
∂zA
eD
c)
→
dwc
+∂r
b Γb
Ac
→
dwc (5.2.2)
= ∇
(ω)
A . (5.2.16)
Secondly, the torsion tensors T(ω)b
AC for the ∇(ω) connection is equal to the torsion tensor T(Γ)B
AC
for the ∇(Γ) connection up to a vielbein factor:
T(ω)a
BC = ea
A T(Γ)A
BC . (5.2.17)
This follows from
T(ω)
≡
1
2
dzA
∧ ∂r
b T(ω)b
AC dzC
:= [∇(ω) ∧, e] = [dzA
→
∂ℓ
∂zA
+ dzA
∂r
b ωb
Ad
→
dwd ∧, ∂r
c ec
C dzC
]
= dzA
∧ ∂r
b


(−1)εAεb
→
∂ℓ
∂zA
eb
C + ωb
Ac ec
C


 dzC (5.2.2)
= dzA
∧ ∂r
b eb
B ΓB
AC dzC
39
=
1
2
dzA
∧ ∂r
b eb
B T(Γ)B
AC dzC
. (5.2.18)
In particular, the two connections ∇(Γ) and ∇(ω) are torsionfree at the same time.
Thirdly, if the ∇
(Γ)
A connection and the vielbein ea
A are both compatible with the density ρ, cf. eqs.
(2.3.3) and (5.1.3), then the spin connection ∇
(ω)
A becomes traceless,
ωA
b
b(−1)εb
(5.2.2)
= 0 . (5.2.19)
Fourthly, the two Riemann curvature tensor R(Γ) and R(ω) are related, see next Subsection 5.3. Fifthly,
the two connections ∇(Γ) and ∇(ω) respect an additional structure, such as a Riemannian (resp. an
antisymplectic) structure at the same time, cf. Subsection 6.1 (resp. Subsection 7.1).
5.3 Spin Curvature
The spin curvature R(ω) is defined as (half) the commutator of the ∇(ω) connection (5.2.1),
R(ω)
=
1
2
[∇(ω) ∧, ∇(ω)
] = −
1
2
dzB
∧ dzA
⊗ [∇
(ω)
A , ∇
(ω)
B ]
= −
1
2
dzB
∧ dzA
⊗ ∂r
d R(ω)d
ABc
→
dwc
, (5.3.1)
R(ω)d
ABc =
→
dwd
[∇
(ω)
A , ∇
(ω)
B ]∂r
c
= (−1)εd
εA (
→
∂ℓ
∂zA
ωd
Bc) + ωd
Ae ωe
Bc − (−1)εAεB (A ↔ B) . (5.3.2)
The two types of Riemann curvature tensors R(Γ) and R(ω) are equal up to conjugation with vielbein
factors
R(ω)d
ABc = ed
D R(Γ)D
ABC eC
c , (5.3.3)
basically because curvature is a commutator of connections,
e ∂r
D R(Γ)D
ABC
→
dzC
e−1
= e[∇
(Γ)
A , ∇
(Γ)
B ]e−1 (5.2.16)
= [∇
(ω)
A , ∇
(ω)
B ]
= ∂r
d R(ω)d
ABc
→
dwc
. (5.3.4)
5.4 Covariant Tensors with Flat Indices
Let
Ωmn(W) := Γ m
(T∗
W) ⊗ n
(T∗
W) (5.4.1)
be the vector space of (0, m+n)–tensors ηa1···amb1···bn (z) that are antisymmetric with respect to the
first m indices a1 . . . am, and symmetric with respect to the last n indices b1 . . . bn. As usual, it is
practical to introduce a coordinate–free notation
η(z; c; y) =
1
m!n!
cam
∧ · · · ∧ ca1
ηa1···amb1···bn
(z) ⊗ ybn
∨ · · · ∨ yb1
. (5.4.2)
Here the variables ya are symmetric counterparts to the one–form basis ca ≡ dwa.
ca ∧ cb = −(−1)εaεb cb ∧ ca , ε(ca) = εa , p(ca) = 1 ,
ya ∨ yb = (−1)εaεb yb ∨ ya , ε(ya) = εa , p(ya) = 0 .
(5.4.3)
40
The covariant derivative can be realized on covariant tensors η ∈ Ωmn(W) by a linear differential
operator
∇
(t)
A :=
→
∂ℓ
∂zA
− ωA
b
c tc
b , (5.4.4)
where
ta
b := ca
→
∂ℓ
∂cb
+ ya
→
∂ℓ
∂yb
(5.4.5)
are generators of the Lie–algebra gl(W), which reflects infinitesimal change of frame/basis in W, cf.
eq. (2.7.6). The relation with the ∇
(T)
A realization (2.7.4) is
∇
(T)
A η(z; eb
BCB
; ec
CY C
) = (∇
(t)
A η)(z; c; y)
cb
= eb
BCB
yc
= ec
C Y C
, (5.4.6)
because of condition (5.2.2), where η=η(z; c; y)∈Ω••(W) is a flat covariant tensor. The relationship
(5.4.6) between the ∇(T) and the ∇(t) realizations, where one puts cb = eb
BCB and yc = ec
CY C, is of
course just a particular case of the more general correspondence (5.2.16) between the ∇(Γ) and the
∇(ω) connections.
5.5 Local Gauge Transformations
Consider for simplicity a flat one–form η = ηa(z)ca ∈ Ω10(W). The covariant derivative reads
(∇Aη)c = (
→
∂ℓ
∂zA
ηc) − ηb ωb
Ac . (5.5.1)
Under a local gauge transformation
ηa = η′
b Λb
a , c′a
= ca
, (5.5.2)
where the group element Λa
b =Λa
b(z) is z–dependent, the spin connection ωb
Ac obeys the well–known
affine transformation law for gauge potentials,
Λb
a ωa
Ac = (−1)εAεb (
→
∂ℓ
∂zA
Λb
c) + ω′b
Ad Λd
c , (5.5.3)
so that the covariant derivative transforms covariantly,
(∇Aη)a = (∇Aη′
)b Λb
a . (5.5.4)
6 Riemannian Spin Geometry
6.1 Spin Geometry
Assume that the vector space W is endowed with a constant Riemannian metric
g(0)
= ya
g
(0)
ab ∨ yb
∈ Ω02(W) , (6.1.1)
called the flat metric. It has Grassmann–parity
ε(g
(0)
ab ) = εa + εb , ε(g(0)
) = 0 , p(g
(0)
AB) = 0 , (6.1.2)
41
and symmetry
g
(0)
ba = −(−1)(εa+1)(εb
+1)
g
(0)
ab . (6.1.3)
Furthermore, assume that the vielbein ea
A intertwines between the curved gAB metric and the flat
g
(0)
ab metric:
gAB = (eT
)A
a
g
(0)
ab eb
B . (6.1.4)
As a consequence, the canonical Riemannian density (3.1.10) is compatible with the vielbein, i.e., it
is proportional to the vielbein superdeterminant,
ρg := sdet(gAB) = sdet(g
(0)
ab ) sdet(ea
A) ∼ sdet(ea
A) , (6.1.5)
cf. eq. (5.1.3). A spin connection ∇(ω) is called metric, if it preserves the flat metric,
0 = −∇
(ω)
A g
(0)
bc = ωA,bc − (−1)(εb+1)(εc+1)
ωA,cb , (6.1.6)
i.e., the lowered ωA,bc symbol should be skewsymmetric in the flat indices. Here we have lowered the
ωA,bc symbol with the flat metric
ωA,bc := (−1)εAεb ωbAc(−1)εc , ωbAc(−1)εc := g
(0)
bd ωd
Ac . (6.1.7)
In particular, the two connections ∇(Γ) and ∇(ω) are metric at the same time, as a consequence of
the correspondence (5.2.2) and (6.1.4). Note that we shall from now on put the ya variables to zero
ya → 0 everywhere, in analogy with the Y a variables of Subsection 3.13.
6.2 Levi–Civita Spin Connection
The Levi–Civita spin connection ∇(ω) is by definition the unique spin connection that corresponds
to the Levi–Civita connection ∇(Γ) via the identifications (5.2.2) and (6.1.4). It is both torsionfree
T(ω) =0 and preserves the metric (6.1.6). The Levi–Civita formula for the spin connection in terms of
the vielbein reads
−2ωbac = (−1)εaεb fa[bc] + (−1)(εa+εb
)εc fc[ba] + fb[ac] , (6.2.1)
where
fbac := g
(0)
bd fd
ac(−1)εc , ωbac := g
(0)
bd ωd
ac(−1)εc , (6.2.2)
and where fa[bc] := fabc − (−1)εb
εc facb, cf. eqs. (5.2.11)–(5.2.14).
6.3 First–Order sab
Matrices
Because of the presence of the flat metric gab
(0), the symmetry of the general linear Lie–algebra gl(W)
reduces to an orthogonal Lie–subalgebra o(W). Its generators sab
∓ read
sab
∓ := ca
pb
∓ (−1)εaεb (a ↔ b) , pa
:= gab
(0)
→
∂ℓ
∂cb
, (6.3.1)
ε(sab
∓ ) = εa + εb , p(sab
∓ ) = 0 , (6.3.2)
sa
∓c := sab
∓ g
(0)
bc (−1)εc . (6.3.3)
42
The transposed operator of a differential operator that depend on the flat ca–variables is now defined to
imitate integration by part. (This becomes important in Lemma 6.4 below.) Explicitly, the transposed
fundamental operators are
1T
= 1 , (ca
)T
= ca
, (pa
)T
= −pa
. (6.3.4)
Therefore the transposed sab
∓ matrices read
(sab
− )T
= −sab
− , (sab
+ )T
= 2gab
(0)1 − sab
+ . (6.3.5)
The ∇
(t)
A realization (5.4.4) can be identically rewritten into the following sab matrix realization
∇
(s)
A :=
→
∂ℓ
∂zA
−
1
2
ωA,bc scb
−(−1)εb =
→
∂ℓ
∂zA
−
1
2
ωA
b
c sc
−b , (6.3.6)
i.e., ∇
(t)
A = ∇
(s)
A for a metric spin connection. One gets a projection onto the sab
− matrices (rather
than the sab
+ matrices), because a metric spin connection ωA,bc is antisymmetric, cf. eq. (6.1.6). Note
that in the sab representation — not only the connection (6.3.6) — but also the curvature — carries
a minus–a–half normalization:
[∇
(s)
A , ∇
(s)
B ] = −
1
2
RAB
d
c sc
−d . (6.3.7)
This can be explained as follows: The minus sign is caused by that the sab representation acts on
covariant tensors (as opposed to contravariant tensors), and the factor 1
2 because the ta
b generator
(5.4.5) becomes 1
2 sa
−b after the metric symmetrization.
The sab
− matrices satisfy an o(W) Lie–algebra:
[sab
∓ , scd
∓ ] = (−1)εa(εb
+εc)
gbc
(0) sad
− + sbc
− gad
(0) ∓ (−1)εaεb (a ↔ b) . (6.3.8)
6.4 γa
Matrices And Clifford Algebras
The flat γa matrices can be defined via a Berezin–Fradkin operator representation [30, 31]
γa
λ ≡ γa
:= ca
+ λpa
, ε(γa
) = εa , p(γa
) = 1 (mod 2) . (6.4.1)
The transposed γa matrices correspond to a change in the parameter λ ↔ −λ:
(γa
)T
:= ca
− λpa
= γa
−λ . (6.4.2)
The γa matrices satisfy a Clifford algebra
[γa
, γb
] = 2λgab
(0)1 . (6.4.3)
The γa matrices commute with the transposed (γb)T matrices
[γa
, (γb
)T
] = 0 . (6.4.4)
Let V be the vector space
V := span ca
⊕ span pa
= span γa
⊕ span (γa
)T
, (6.4.5)
and let
T(V ) :=
∞
m=0
V ⊗m
= (span 1) ⊕ V ⊕ V ⊗V ⊕ V ⊗V ⊗V ⊕ . . . (6.4.6)
43
be the corresponding tensor algebra. Let I(V ) be the two–sided ideal generated by
[ca ⊗, cb
] , [pa ⊗, cb
] − gab
1 , [pa ⊗, pb
] , (6.4.7)
or equivalently, the two–sided ideal generated by
[γa ⊗, γb
] − 2gab
1 , [γa ⊗, (γb
)T
] , [(γa
)T ⊗, (γb
)T
] + 2gab
1 . (6.4.8)
Then the Heisenberg algebra, or equivalently, the Clifford algebra Cl(V ) is isomorphic to the quotient
Cl(V ) ∼= T(V )/I(V ) . (6.4.9)
Each element of Cl(V ) is a differential operator in the ca–variables, and may be Wick/normal–ordered
in a unique way, so that all the c–derivatives (the p’s) stands to the right of all the c’s. This is also
known as cp–ordering.
There is another important description of the Clifford algebra Cl(V ) as a tensor product of two
(mutually commutative) Clifford algebras
Cl(V ) ∼= Cl(γ) ⊗ Cl(γT
) , (6.4.10)
where
Cl(γ) =
∞
m=0
span γa1
γa2
· · · γam ∼= T(γ)/I(γ) , (6.4.11)
Cl(γT
) =
∞
m=0
span (γa1
)T
(γa2
)T
· · · (γam
)T ∼= T(γT
)/I(γT
) . (6.4.12)
Since the γ matrices commute with the transposed γT matrices, it is possible to unshuffle an arbitrary
element in Cl(V ) into a γγT –ordered form, i.e., so that all the γ matrices stand to the left of all the
γT matrices. For instance, the γγT –ordered form of the γa and the (γa)T matrices are
γa
= γa
⊗ 1 ,
(γa
)T
= 1 ⊗ (γa
)T
, (6.4.13)
respectively. For more complicated expressions, the γγT –ordered form will in general not be unique,
since e.g., the γ matrices do not commute among themselves. Nevertheless, the γγT –ordering bears
some resemblance with, e.g., the method of holomorphic and antiholomorphic blocks in conformal field
theory.
The γa matrices form a fundamental representation of the o(W) Lie–algebra (6.3.8):
[sab
∓ , γc
] = γa
±λ gbc
(0) ∓ (−1)εaεb (a ↔ b) . (6.4.14)
As a consequence, if one commutes a metric spin connection (6.3.6) with a flat γa matrix, one gets
[∇
(s)
A , γb
] = −ωA
b
c γc
. (6.4.15)
A curved γA matrix is now defined as a flat γa matrix dressed with the inverse vielbein in the obvious
way:
γA
:= eA
a γa
= γa
(eT
)a
A
, ε(γA
) = εA , p(γA
) = 1 (mod 2) . (6.4.16)
(Similar straightforward rules applies to other objects when switching between flat and curved indices.)
If one commutes a metric spin connection (6.3.6) with a curved γA matrix, one gets
[∇
(s)
A , γB
] = −ΓA
B
C γC
, (6.4.17)
cf. eqs. (5.2.4) and (6.4.15). The result (6.4.17) can be summarized as saying that the total connection
∇ = d + Γ + ω commutes with the γA matrices: [∇A, γB] = 0.
44
6.5 Dirac Operator D(s)
For a general discussion of Dirac operators, see e.g., Ref. [38]. We shall for the remainder of the
Section 6 assume that the connection is the Levi–Civita connection.
Central for our discussion are the sab matrices (6.3.1). They act on flat exterior forms η ∈ Ω•0(W),
i.e., functions η=η(z; c) of the zA and ca variables.
The Dirac operator D(s) in the sab representation (6.3.6) is a γA matrix (6.4.16) times a covariant
derivative (6.3.6)
D(s)
:= γA
∇
(s)
A , ε(D(s)
) = 0 , p(D(s)
) = 1 (mod 2) . (6.5.1)
The Laplace operator ∆
(s)
ρg
in the sab representation (6.3.6) is
∆(s)
ρg
:= (−1)εA ∇AgAB
∇
(s)
B = (−1)εA ∇
(s)
A gAB
∇
(s)
B + ΓA
AC gCB
∇
(s)
B
=
(−1)εA
ρg
∇
(s)
A ρggAB
∇
(s)
B . (6.5.2)
Theorem 6.1 (cp–ordered Weitzenb¨ock formula for flat exterior forms) The difference between
the square of the Dirac operator D(s) and the Laplace operator ∆
(s)
ρg
in the sab representation (6.3.6)
is
D(s)
D(s)
− λ∆(s)
ρg
= −
λ
4
sBA
− RAB,CD sDC
− (−1)εC +εD (6.5.3)
= −λcA
RAB pB
+
λ
2
cB
cA
RAB,CD pD
pC
(−1)εC +εD . (6.5.4)
Proof of Theorem 6.1: Almost identical to the proof of Theorem 3.5 because of eq. (5.3.3).
6.6 Second–Order σab
Matrices
We now replace the first–order sab
∓ matrices (6.3.1) with second–order matrices:
σab
∓ (λ) ≡ σab
∓ :=
1
4λ
γa
γb
∓ (−1)εaεb (a ↔ b) = σab
∓ ⊗ 1 , (6.6.1)
ε(σab
∓ ) = εa + εb , p(σab
∓ ) = 0 , (6.6.2)
σa
∓c := σab
∓ g
(0)
bc (−1)εc . (6.6.3)
(The names first– and second–order refer to the number of ca–derivatives.) The transposed σab
∓
matrices read
(σab
∓ )T
= ±
1
4λ
(γa
)T
(γb
)T
∓ (−1)εaεb (a ↔ b) = ∓σab
∓ (−λ) = 1 ⊗ (σab
∓ )T
. (6.6.4)
In the last expression of eqs. (6.6.1) and (6.6.4) we wrote the σab
∓ and the (σab
∓ )T matrices on a
γγT –ordered form. In particular, the σab
∓ matrices decouple completely from the (σab
∓ )T matrices,
[σab
∓ , (σcd
∓ )T
] = 0 , [σab
∓ , (σcd
± )T
] = 0 . (6.6.5)
45
On one hand, the matrices
σab
− =
1
4λ
{γa
, γb
}+ =
1
2λ
ca
cb
+
1
2
sab
− +
λ
2
pa
pb
(6.6.6)
satisfy precisely the same non–Abelian o(W) Lie–algebra (6.3.8) and fundamental representation
(6.4.14) as the sab
− matrices. On the other hand, the matrices
σab
+ :=
1
4λ
[γa
, γb
]
(6.4.3)
=
1
2
gab
(0)1 (6.6.7)
are proportional to the identity operator, and thus Abelian.
The sab
− matrices can be expressed in terms of the σab
− matrices and their transposed,
sab
− = σab
− + σab
− (−λ) = σab
− ⊗ 1 − 1 ⊗ (σab
− )T
, (6.6.8)
as a consequence of eq. (6.6.6). In contrast, the sab
+ matrices can not be expressed in terms of the σab
∓
matrices and their transposed.
The first–order ∇
(s)
A realization (6.3.6) can be identically rewritten into the following second–order
σσT matrix realization
∇
(σσT )
A :=
→
∂ℓ
∂zA
−
1
2
ωA,bc σcb
− ⊗ 1 − 1 ⊗ (σcb
− )T
(−1)εb =
→
∂ℓ
∂zA
−
1
2
ωA
b
c σc
−b ⊗ 1 − 1 ⊗ (σc
−b)T
,
(6.6.9)
i.e., ∇
(t)
A = ∇
(s)
A = ∇
(σσT )
A for a metric spin connection. In contrast, the first–order ∇
(S)
A realization
(3.12.6) does in general not have a second–order formulation for a metric connection, even if the
manifold is a spin manifold, cf. Appendix 3.A. This is despite the fact that the first–order realizations
∇
(S)
A and ∇
(s)
A are closely related via condition (5.2.2),
∇
(S)
A η(z; eb
BCB
) = (∇
(s)
A η)(z; c)
cb=eb
BCB
, (6.6.10)
where η=η(z; c; y)∈Ω•0(W) is a flat exterior form. Here the SAB
∓ and sab
∓ matrices act by adjoint
action on the CC and cc variables as
[SAB
∓ , CC
] = CA
gBC
∓ (−1)εAεB (A ↔ B) , [sab
∓ , cc
] = ca
gbc
(0) ∓ (−1)εaεb (a ↔ b) , (6.6.11)
cf. eqs. (3.11.1) and (6.3.1), respectively. The crucial difference is that the ∇
(S)
A realization (3.12.6)
contains a non–trivial S+ sector, while the ∇
(s)
A realization (6.3.6) has no s+ sector. This has its root
in the fact that the flat metric condition (6.1.6) is an algebraic condition, while the curved metric
condition (3.6.1) is a differential condition. (Curiously, it is just opposite for the torsionfree conditions:
the curved torsionfree condition is an algebraic condition, while the flat torsionfree condition is a
differential condition, cf. eqs. (2.2.2) and (5.2.18).)
6.7 Lichnerowicz’ Formula
It is convenient to define a totally symmetrized combination of three γa matrices as
γa1a2a3 :=
1
3! π∈S3
(−1)επ,a γ
aπ(1) γ
aπ(2) γ
aπ(3) , (6.7.1)
46
where (−1)επ,a is the sign factor that arises when one does a π–permutation of three supercommuting
objects with the same Grassmann– and form–parity as the γa matrices, say, the c’s
ca1 ∧ ca2 ∧ ca3 = (−1)επ,a c
aπ(1) ∧ c
aπ(2) ∧ c
aπ(3) , (6.7.2)
cf. (5.4.3). The symmetrized γabc matrix can be reduced with the help of the Clifford relation (6.4.3)
as
γabc
= γa
γb
γc
− λgab
(0) γc
+ (−1)εbεc λgac
(0) γb
− γa
λgbc
(0) . (6.7.3)
Theorem 6.2 (γγT –ordered Lichnerowicz’ formula [6]) The square of the Dirac operator D(σσT )
in the σσT representation (6.6.1) is
D(σσT )
D(σσT )
= λ∆(σσT )
ρg
−
λ
4
R +
λ
2
σBA
− RAB,CD ⊗ (σDC
− )T
(−1)εC +εD . (6.7.4)
Proof of Theorem 6.2: One derives that the square of the Dirac operator D(σσT ) is the Laplacian
∆
(σσT )
ρg
plus a curvature term, by proceeding along the lines of the proof of Theorem 3.5:
D(σσT )
D(σσT )
=
1
2
[D(σσT )
, D(σσT )
] = λ∆(σσT )
ρg
−
1
2
γB
γA
[∇
(σσT )
A , ∇
(σσT )
B ] . (6.7.5)
When one γγT –decomposes the curvature term, it splits in two parts:
−
1
2
γB
γA
[∇
(σσT )
A , ∇
(σσT )
B ] =
1
4
γB
γA
RAB
d
c σc
−d ⊗ 1 − 1 ⊗ (σc
−d)T
= III + IIIT
, (6.7.6)
where
IIIT
:=
λ
2
σBA
− RAB,CD ⊗ (σDC
− )T
(−1)εC +εD , (6.7.7)
and
III := −
1
4
γB
γA
RAB,CD σDC
− (−1)εC +εD = −
1
8λ
γB
γA
RAB,CD γD
γC
(−1)εC +εD
=
1
8λ
(−1)(εA+εB)εC γB
γA
γC
RAB,CD γD
(−1)εD
(6.7.3)
=
1
8λ
γCBA
+ γC
λgBA
− λgCB
γA
+ (−1)εAεB λgCA
γB
RAB,CD γD
(−1)εD
= −
1
4
gCB
γA
RAB,CD γD
(−1)εD =
1
4
(−1)(εA+εB)(εD+1)
RABD
B
γA
γD
(−1)εD
= −
1
4
RDA γA
γD
(−1)εD = −
λ
4
RDA gAD
(−1)εD = −
λ
4
R . (6.7.8)
Here the first Bianchi identity (3.7.5) was used one time.
6.8 Clifford Representations
The spinor representations S and ST can be defined as Fock spaces
S := Cl(γ)|0 =
∞
m=0
span ca1
ca2
· · · cam
|0 , pa
|0 = 0 , (6.8.1)
ST
:= Cl(γT
)|0T
=
∞
m=0
span pa1
pa2
· · · pam
|0T
, ca
|0T
= 0 . (6.8.2)
47
The constraints pa|0 =0 (resp. ca|0T =0) are consistent, because the pa’s (resp. the ca’s) commute.
The representation (6.8.1) and (6.8.2) are of course just two possibilities out of infinitely many equivalent
choices of Fock space representations. A different class of vacua |1 and |1T are defined via
σab
− |1 = 0 , (σab
− )T
|1T
= 0 . (6.8.3)
They both represent the singlet/trivial representation of the orthogonal Lie–group O(W). Again, the
constraints (6.8.3) for the vacua are consistent, since the σab
− (resp. the (σab
− )T ) matrices form Lie–
algebras. All the above constraints are examples of first–class constraints. More generally, assume
that |Ω and |ΩT are two arbitrary consistent vacua (that are not necessarily related). Let V and VT
denote the corresponding vector spaces
V := Cl(γ)|Ω , VT
:= Cl(γT
)|ΩT
. (6.8.4)
The Clifford algebra Cl(V ) ∼= Cl(γ) ⊗ Cl(γT ) is defined to act on the tensor product V ⊗ VT via a
γγT –ordered form, i.e., the γa matrices act on the first factor V and the transposed (γa)T matrices
act on the second factor VT . In detail, if |v ∈ V and |vT ∈ VT are two (not necessarily related)
states, then
γa
.(|v ⊗ |vT
) := (γa
|v ) ⊗ |vT
, (6.8.5)
(γa
)T
.(|v ⊗ |vT
) := (−1)ε(γa)·ε(v)
|v ⊗ (γa
)T
|vT
. (6.8.6)
By definition, V is a Clifford bundle, while VT is a dual/contragredient Clifford bundle.
A Lie–algebra element x ∈ so(W) is of the form
x =
1
2
(−1)εa xabsba
− =
1
2
xa
bsb
−a =
1
2
xa
b σb
−a ⊗ 1 − 1 ⊗ (σb
−a)T
, (6.8.7)
where
xab = (−1)(εa+1)(εb
+1)
(a ↔ b) , xa
c := gab
(0)xbc . (6.8.8)
A γγT –ordered form of a generic special orthogonal Lie–group element g=ex ∈ SO(W) is
exp
1
2
xa
bsb
−a = exp
1
2
xa
bσb
−a ⊗ exp −
1
2
xc
d(σd
−c)T
. (6.8.9)
In this way the vector space VT becomes a dual/contragredient representation of the special orthogonal
Lie–group SO(W), hence the name.
6.9 Intertwining Operator
Consider the intertwining operator
s := dN
θ eθaγa
⊗ eθb
(γb)T
, (6.9.1)
where θa are integration variables with Grassmann–parity ε(θa)=εa and form–parity p(θa) = 1 (mod 2).
Lemma 6.3 The intertwining operator s is invariant under the adjoint action exse−x =s of the special
orthogonal Lie–group SO(W). Equivalently, the intertwining operator s commute with the so(W) Lie–
algebra generators [sab
− , s]=0.
48
Proof of Lemma 6.3: The adjoint action rotates the γa matrices,
exp
1
2
xc
dσd
−c γa
exp −
1
2
xe
f σf
−e = (ex
)a
bγb
,
exp −
1
2
xc
d(σd
−c)T
(γa
)T
exp
1
2
xe
f (σf
−e)T
= (ex
)a
b(γb
)T
, (6.9.2)
where
(ex
)a
b := δa
b + xa
b +
1
2!
xa
cxc
b +
1
3!
xa
cxc
dxd
b +
1
4!
xa
cxc
dxd
exe
b + . . . . (6.9.3)
Hence one may change integration variables θa → θ′
b = θa(ex)a
b in the integral (6.9.1). The Jacobian
vanishes for special orthogonal transformations
ln sdet(ex
)a
b = (−1)εa xa
a = (−1)εa gab
(0)xba = 0 . (6.9.4)
Lemma 6.4 The corresponding intertwining state
||s := s.(|Ω ⊗ |ΩT
) = dN
θ eθaγa
|Ω ⊗ eθb
(γb)T
|ΩT
(6.9.5)
is invariant under the action of the special orthogonal Lie–group SO(W). Equivalently, the so(W)
Lie–algebra generators sab
− annihilate the intertwining state sab
− ||s =0.
Proof of Lemma 6.4:
ex
||s = dN
θ eθaγa
exp
1
4λ
(−1)εc xcdγd
γc
|Ω ⊗ eθb(γb)T
exp −
1
4λ
(−1)εe xef (γf
)T
(γe
)T
|ΩT
= dN
θ exp
1
4λ
(−1)εc xcd˜γd
˜γc
eθaγa
|Ω ⊗ exp −
1
4λ
(−1)εe xef (˜γf
)T
(˜γe
)T
eθb
(γb)T
|ΩT
= ||s , (6.9.6)
where we have introduced (a kind of) Fourier transformed γ matrices
˜γa
:=
→
∂ℓ
∂θa
+ gab
(0)θb , (˜γa
)T
:= −
→
∂ℓ
∂θa
+ gab
(0)θb , (6.9.7)
which satisfy
˜γa
exp θbγb
= exp θbγb
γa
, −(˜γa
)T
exp θb(γb
)T
= exp θb(γb
)T
(γa
)T
. (6.9.8)
In the last equality of eq. (6.9.6), we performed integration by part, which turns ˜γa into (˜γa)T , and
vice–versa.
The algebra bundle (6.4.9) of differential operators in the ca–variables, or equivalently polynomials in
γ and γT , is isomorphic to the bispinor bundle S ⊗ ST . The bundle isomorphism is
Cl(V ) ∼= Cl(γ) ⊗ Cl(γT
) ∋ F
∼=
→ F||s ∈ S ⊗ ST ∼= End(S) . (6.9.9)
The bispinor bundle S ⊗ ST ∼= End(S) is, in turn, isomorphic (as vector bundles) to the bundle
of endomorphisms: S → S. Let us also mention that the Weyl symbol
∼=
→ operator isomorphism
•(V )
∼=
→ Cl(V ) from the exterior algebra ( •(V ); ∗), equipped with the Groenewold/Moyal ∗ product,
to the Heisenberg algebra (Cl(V ); ◦), is known as the Chevalley isomorphism in the context of
Clifford algebras.
49
6.10 Schr¨odinger–Lichnerowicz’ Formula
We will be interested in how the Dirac operator acts on a Clifford bundle V ⊗ |1T ∼= V and a tensor
Clifford bundle V ⊗ VT .
Theorem 6.5 (Schr¨odinger–Lichnerowicz’ formula [5, 6]) On a Clifford bundle V ⊗ |1T ∼= V,
the square of the Dirac operator D(σ) is equal to the Laplacian ∆
(σ)
ρg
minus a quarter of the scalar
curvature R,
D(σ)
D(σ)
= λ∆(σ)
ρg
−
λ
4
R . (6.10.1)
Proof of Theorem 6.5: This is a Corollary to Lichnerowicz’ formula (6.7.4).
The Schr¨odinger–Lichnerowicz’ formula (6.10.1) corresponds to naively substituting the first–order
matrices sab
− → σab
− in the ∇(s) realization (6.3.6) with the second–order matrices σab
− . The analysis in
Subsections 6.6 and 6.8 shows in detail why this replacement is geometrically sound and in fact very
natural.
Theorem 6.6 The square of the Dirac operator D(σσT ) on a tensor Clifford bundle V ⊗ VT becomes
equal to the Laplace–Beltrami operator ∆ρg
when it is projected on the singlet representation ||s ,
D(σσT )
D(σσT )
f||s = λ(∆ρg
f)||s , (6.10.2)
where f =f(z) is an arbitrary scalar function.
Proof of Theorem 6.6: This is a Corollary to the Weitzenb¨ock formula (6.5.3).
7 Antisymplectic Spin Geometry
7.1 Spin Geometry
Assume that the vector space W is endowed with a constant antisymplectic metric
E(0)
=
1
2
ca
E
(0)
ab ∧ cb
= −
1
2
E
(0)
ab cb
∧ ca
∈ Ω20(W) , (7.1.1)
called the flat metric. It has Grassmann–parity
ε(E
(0)
ab ) = εa + εb + 1 , ε(E(0)
) = 1 , p(E
(0)
AB) = 0 , (7.1.2)
and symmetry
E
(0)
ba = −(−1)εaεb E
(0)
ab . (7.1.3)
Furthermore, assume that the vielbein ea
A intertwines between the curved EAB metric and the flat
E
(0)
ab metric:
EAB = (eT
)A
a
E
(0)
ab eb
B . (7.1.4)
50
A spin connection ∇(ω) is called antisymplectic, if it preserves the flat metric,
0 = −∇
(ω)
A E
(0)
bc = ωA,bc − (−1)εb
εcωA,cb , (7.1.5)
i.e., the lowered ωA,bc symbol should be symmetric in the flat indices. Here we have lowered the ωA,bc
symbol with the flat metric
ωA,bc := (−1)εAεb ωbAc , ωbAc := E
(0)
bd ωd
Ac(−1)εA . (7.1.6)
In particular, the two connections ∇(Γ) and ∇(ω) are antisymplectic at the same time, as a consequence
of the correspondence (5.2.2) and (7.1.4).
7.2 First–Order sab
Matrices
Because of the presence of the flat metric Eab
(0), the symmetry of the general linear Lie–algebra gl(W)
reduces to an antisymplectic Lie–subalgebra. Its generators sab
± read
sab
± := ca
(−1)εb pb
∓ (−1)(εa+1)(εb
+1)
(a ↔ b) , pa
:= Eab
(0)
→
∂ℓ
∂cb
, (7.2.1)
ε(sab
± ) = εa + εb + 1 , p(sab
± ) = 0 , (7.2.2)
sa
±c := sab
± E
(0)
bc (−1)εc . (7.2.3)
The ∇
(t)
A realization (5.4.4) can be identically rewritten into the following sab matrix realization
∇
(s)
A :=
→
∂ℓ
∂zA
+
1
2
ωA,bc scb
+(−1)εb =
→
∂ℓ
∂zA
−
1
2
ωA
b
c sc
+b , (7.2.4)
i.e., ∇
(t)
A = ∇
(s)
A for an antisymplectic spin connection. One gets a projection onto the sab
+ matrices
(rather than the sab
− matrices), because an antisymplectic spin connection ωA,bc is symmetric, cf. eq.
(7.1.5).
The sab
+ matrices satisfy an antisymplectic Lie–algebra:
[sab
± , scd
± ] = (−1)εa(εb+εc+1)+εb Ebc
(0) sad
+ − sbc
+ Ead
(0) ∓ (−1)(εa+1)(εb+1)
(a ↔ b) . (7.2.5)
7.3 γa
Matrices
The flat γa matrices can be defined via a Berezin–Fradkin operator representation [30, 31]
γa
θ ≡ γa
:= ca
+(−1)εa θpa
= ca
−pa
θ , ε(γa
) = εa , p(γa
) = 1 (mod 2) . (7.3.1)
The γa matrices satisfy a Clifford–like algebra
[γa
, γb
] = 2(−1)εa θEab
(0)1 . (7.3.2)
The γa matrices form a fundamental representation of the antisymplectic Lie–algebra (7.2.5):
[sab
± , γc
] = γa
±θ(−1)εb Ebc
(0) ∓ (−1)(εa+1)(εb+1)
(a ↔ b) . (7.3.3)
51
As a consequence, if one commutes an antisymplectic spin connection (7.2.4) with a flat γa matrix,
one gets
[∇
(s)
A , γb
] = −ωA
b
c γc
. (7.3.4)
Similarly, if one commutes an antisymplectic spin connection (7.2.4) with a curved γA matrices, one
gets
[∇
(s)
A , γB
] = −ΓA
B
C γC
, (7.3.5)
cf. eqs. (5.2.4) and (7.3.4).
7.4 Dirac Operator D(s)
We shall for the remainder of Section 7 assume that the connection is antisymplectic, torsionfree and
ρ–compatible.
The Dirac operator D(s) in the sab representation (7.2.4) is a γA matrix (7.3.1) times a covariant
derivative (7.2.4)
D(s)
:= γA
∇
(s)
A , ε(D(s)
) = 0 , p(D(s)
) = 1 (mod 2) . (7.4.1)
The odd Laplacian ∆
(s)
ρ in the s representation (7.2.4) is
2∆(s)
ρ := (−1)εA ∇AEAB
∇
(s)
B =
(−1)εA
ρ
∇
(s)
A ρEAB
∇
(s)
B . (7.4.2)
Theorem 7.1 (Antisymplectic Weitzenb¨ock type formula for flat exterior forms) The difference
between the square of the Dirac operator D(s) and twice the odd Laplacian ∆
(s)
ρ in the sab
representation (7.2.4) is
D(s)
D(s)
− 2θ∆(s)
ρ =
θ
4
(−1)εB+εC sBA
− RAB,CD sDC
+ (7.4.3)
= −θcA
RAB pB
+
θ
2
cB
cA
RAB,CD pD
pC
(−1)εC . (7.4.4)
Proof of Theorem 7.1: Almost identical to the proof of Theorem 4.4 because of eq. (5.3.3).
7.A Appendix: Shifted s′ab
+ Matrices
We have already seen in Appendix 4.A that there are no consistent antisymplectic second–order
deformations of the sab
+ matrices. The only remaining deformation is a c–number shift,
s′ab
+ := sab
+ + αEab
(0)1 , (7.A.1)
s′a
+b := sa
+b + α(−1)εa δa
b 1 , (7.A.2)
with a parameter α, cf. eq. (4.8.6). These shifted s′ab
+ matrices satisfy the same Lie–algebra (7.2.5)
and fundamental representation (7.3.3) as the sab
+ matrices. Moreover, the shift does not affect the sab
representation (7.2.4) of the spin connection, because of tracefree condition (5.2.19). Similarly, the
curvature
[∇
(s)
A , ∇
(s)
B ] = −
1
2
RAB
d
c sc
+d . (7.A.3)
is unaffected, since the shift–term is proportional to the Ricci two–form RAB =0, which is zero. Thus
we conclude that the c–number shift sab
+ → s′ab
+ has no effects at all on the construction.
52
8 Conclusions
The main objective of the paper is to gain knowledge about the deepest and most profound geometric
levels of the field–antifield formalism [1, 2, 3]. It is imperative to better understand the geometric
meaning of the odd scalar curvature R, which sits as a zeroth–order term in the odd ∆ operator (1.0.1),
and which descends to the quantum master equation ∆ exp[ i
¯h W] = 0 as a two–loop contribution:
(W, W) = 2i¯h∆ρW − ¯h2 R
4
. (8.0.1)
We have in this paper investigated the hypothesis that the zeroth–order term −R/4 of (twice) the odd
∆ operator (1.0.1) is related to the zeroth–order term −R/4 in the Schr¨odinger–Lichnerowicz formula
(6.10.1). We have so far been unable to give a closed argument that such relationship exists. In fact,
Theorem 6.6 indicates that there is no relation, as explained in the Introduction. Some of the main
results of the paper are the following.
• We have classified scalar invariants of suitable dimensions that depend on the density ρ and the
metric, cf. Proposition 3.2 and Proposition 4.2.
• We have identified (via a ρ–independence argument) a Riemannian counterpart (3.9.1) of the
antisymplectic ∆ operator (1.0.1), that takes scalars to scalars, and we have, in terms of formulas,
traced the minus–a–quarter coefficient in front of R from the Riemannian to the antisymplectic
side, cf. Subsection 4.7.
• We have tied the Riemannian ∆ operator (3.5.2) to the quantum Hamiltonian ˆH for a particle
moving in a curved Riemannian space, cf. Subsection 3.10.
• We have derived the Laplace-Beltrami operator ∆ρg
by projecting the square of the bispinor
Dirac operator D(σσT ) to a singlet state ||s , cf. Theorem 6.6.
• We have found a first–order formalism for antisymplectic spinors and proved two Weitzenb¨ock–
type identities (Theorem 4.4 and Theorem 7.1) that are in exact one–to–one correspondence
with their Riemannian siblings (Theorem 3.5 and Theorem 6.1).
However, there is in our approach no antisymplectic analogue of the Riemannian second–order formalism
and the Schr¨odinger–Lichnerowicz formula (6.10.1). A bit oversimplified, this is because the
canonical choice for antisymplectic second–order ΣAB
± matrices is
ΣAB
±
?
=
1
4
ΓA
⋆ΓB
∓(−1)(εA+1)(εB +1)
(A ↔ B) , ε(ΣAB
± ) = εA+εB +1 , p(ΣAB
± ) = 0 , (8.0.2)
where “⋆” is a Fermionic multiplication, ε(⋆) = 1. This choice (8.0.2) meet all the requirements of
Grassmann–parity and symmetry, and is a direct analogue of the Riemannian second–order ΣAB
± matrices
(3.A.1). Unfortunately, such ⋆ multiplication does not admit a Berezin–Fradkin representation
of the ΓA matrices, cf. Appendix 4.B. We instead introduced a Fermionic nilpotent parameter θ,
which may formally be identified with the inverse ⋆−1, and which serves as a Fermionic analogue
of the “Planck constant” λ from the Riemannian case. Then the ⋆ multiplication itself should be
identified with the inverse θ−1, which is an ill–defined quantity, and hence the above formula (8.0.2)
for the ΣAB
± matrices does not make sense. Note however that the nilpotent θ parameter breaks the
non–degeneracy of the Clifford algebra (4.9.2).
53
Acknowledgement: We would like to thank Poul Henrik Damgaard, the Niels Bohr Institute and
the Niels Bohr International Academy for warm hospitality. I.A.B. would also like to thank Michal
Lenc and the Masaryk University for warm hospitality. The work of I.A.B. and K.B. is supported by
the Ministry of Education of the Czech Republic under the project MSM 0021622409. The work of
I.A.B. is supported by grants RFBR 08–01–00737, RFBR 08–02–01118 and LSS–1615.2008.2.
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