# The Canonical BV Laplacian on Half-Densities

Alberto S. Cattaneo  Institut für Mathematik, Universität Zürich
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland<EMAIL_ADDRESS>
###### Abstract.

This is a didactical review on the canonical BV Laplacian on half-densities.

I acknowledge partial support of the SNF Grant No. 200020_192080, of the
Simons Collaboration on Global Categorical Symmetries, and of the COST Action
21109 (CaLISTA). This research was (partly) supported by the NCCR SwissMAP,
funded by the Swiss National Science Foundation.

> _Das Bekannte überhaupt ist darum, weil es bekannt ist, nicht erkannt_
>
> GWFH

###### Contents

  1. 1 Introduction
    1. 1.1 Overview
  2. 2 The standard BV Laplacian
  3. 3 The BV bracket
  4. 4 The BV Laplacian on functions
    1. 4.1 Digression: further properties of the BV Laplacian
  5. 5 The BV Laplacian on densities
    1. 5.1 Digression: Darboux expression of $\Delta_{\mu}^{(s)}$
  6. 6 Odd cotangent bundles
  7. 7 Conclusion of the argument
  8. A Some background details
    1. A.1 The Lie derivative
    2. A.2 Densities
  9. B Applications of the BV formalism
  10. C Historical remarks

##  1\. Introduction

The BV Laplacian on functions is a central ingredient of the
Batalin–Vilkovisky (BV) formalism. Its global definition is not canonical, yet
a remarkable fact observed by Khudaverdian [7, 8] is that its extension to
half-densities is so, and there are various ways to show it. In this review we
give a short, self-contained presentation which requires just a few simple
computations. It gives a special emphasis on the fact that half-densities have
a natural inner product.

This note has no pretence of originality. In Appendix C, we give a short
historical overview with the relevant references.

In Appendix A, we collect some technical background material. In Appendix B,
we recall why the BV formalism is important in applications.

###  1.1. Overview

We recall definitions and properties of BV Laplacians (on functions and
densities) on an oriented odd symplectic manifold with the goal of proving
that on half-densities it is canonically defined (whereas on other kinds of
densities, including functions, it requires the choice of a compatible
density). This goes as follows:

  1. (1)

On functions we define $\Delta_{\mu}f=\frac{1}{2}{\mathrm{div}}_{\mu}X_{f}$,
where $\mu$ is an even, nowhere vanishing111Throughout the paper, nowhere
vanishing means nonvanishing at every point of the body after the nilpotents
are set to zero. An even, nowhere vanishing density $\mu$ is the same as a
basis for the module of densities. The $k$-density $\mu^{k}$ is then a basis
for the module of $k$-densities. density, and $X_{f}$ denotes the hamiltonian
vector field of $f$.

  2. (2)

On half-densities we define
$\Delta^{(\frac{1}{2})}_{\mu}\sigma=\Delta_{\mu}(\sigma/\mu^{1/2})\mu^{1/2}$.

  3. (3)

We check that the leading term is invariant under symplectomorphisms and does
not depend on the choice of $\mu$.

  4. (4)

We check that $\Delta^{(\frac{1}{2})}_{\mu}$ is a symmetric operator. This
implies that in any Darboux chart only the zeroth-order term may depend on
$\mu$.

  5. (5)

We show that for a compatible density $\mu$, i.e., for $\mu$ satisfying
$\Delta_{\mu}^{2}=0$, the zeroth-order term of the previous point must vanish.
This shows that $\Delta^{(\frac{1}{2})}_{\mu}$ does not depend on the choice
of the compatible density $\mu$.

  6. (6)

We show, via the presentation of the given odd symplectic manifold as a
symplectomorphic odd cotangent bundle, that compatible densities always exist.

  7. (7)

We conclude that the BV Laplacian on half-densities, with the property of
squaring to zero, is canonically defined.

  8. (8)

As an aside, we also show how to define this canonical operator, denoted by
$\Delta$, directly on Darboux charts without any choice involved. Moreover, we
prove that the compatibility condition for a density $\mu$ can also be
expressed as $\Delta\mu^{\frac{1}{2}}=0$ and that the BV Laplacian on
functions can now be recovered as
$\Delta_{\mu}=\Delta(f\mu^{\frac{1}{2}})/\mu^{\frac{1}{2}}$.

We go through details, also checking signs carefully, to make this
presentation useful also as a reference. However, it should be observed that,
roughly, the above flow can be easily checked without going into sign
details—the only crucial point being (4), which requires some care.

######  Acknowledgment.

I thank A. Cabrera, D. Fiorenza, N. Moshayedi, M. Schiavina, and P. Ševera for
useful remarks. I am especially grateful to T. Voronov for several extremely
helpful exchanges.

##  2\. The standard BV Laplacian

On functions of odd variables $p_{1},\dots,p_{n}$ and even variables
$q^{1},\dots,q^{n}$, the standard BV Laplacian222This operator was introduced
to show that certain integrals are invariant under deformations of the
integration domain; see Appendix B. (a.k.a. BV Laplace operator or BV
operator) is defined as

$\triangle\coloneqq\partial_{i}\partial^{i},$

where we use Einstein’s summation convention and the shorthand notations

$\partial_{i}=\frac{\partial}{\partial
q^{i}}\quad\text{and}\quad\partial^{i}=\frac{\partial}{\partial p_{i}}.$

Another ingredient of the BV formalism is the BV bracket

(1) $(f,g)\coloneqq
f\overleftarrow{\partial{{}^{i}}}\,\overrightarrow{\partial{{}_{i}^{\phantom{i}}}}g-f\overleftarrow{\partial{{}_{i}^{\phantom{i}}}}\,\overrightarrow{\partial{{}^{i}}}g,$

where $f$ and $g$ are functions of homogeneous degree, and the arrows denote
the directions the derivatives are applied from. Explicitly,
$\overrightarrow{\partial{{}_{i}^{\phantom{i}}}}f=\partial_{i}f$,
$\overrightarrow{\partial{{}^{i}}}f=-\partial^{i}f$, and

$\begin{split}f\overleftarrow{\partial{{}_{i}^{\phantom{i}}}}&=\partial_{i}f,\\\
f\overleftarrow{\partial{{}^{i}}}&=-(-1)^{f}\partial^{i}f.\end{split}$

Here and in the following, when we put a function (or any other object) as an
exponent of $(-1)$ we mean its degree.333A more precise but more cumbersome
notation would be $(-1)^{|f|}$, where $|f|$ denotes the degree of $f$.

The standard BV Laplacian has the following properties which are easily proved
by direct computation:

(2a) $\displaystyle\triangle^{2}=0,$ (2b)
$\displaystyle\triangle(fg)=(\triangle f)g+(-1)^{f}f\triangle
g-(-1)^{f}(f,g),$ (2c) $\displaystyle\triangle\mathrm{e}^{S}=\left(\triangle
S-\frac{1}{2}(S,S)\right)\mathrm{e}^{S}.$

Here again $f$ and $g$ are functions of homogeneous degree, whereas the
function $S$ is assumed to be even. As an exercise, just using (2a) and (2b),
you can also show that

(3) $\triangle(f,g)=(\triangle f,g)-(-1)^{f}(f,\triangle g).$

The geometric interpretation of the content of this section is that the
$p_{i}$s and $q^{i}$s are Darboux coordinates for the odd symplectic form
$\omega={\mathrm{d}}p_{i}{\mathrm{d}}q^{i}$ with $(\ ,\ )$ its associated odd
Poisson bracket.444If the $q^{i}$s are coordinates on an open subset $U$ of
${\mathbb{R}}^{n}$, the functions of the $p$ and $q$ variables can be
identified with multivector fields on $U$. The BV bracket then gets identified
with the Schouten–Nijenhuis bracket. We will return to this, with a general
basis manifold, in Section 6; see footnote 15 on page 15. The main problem
with this, however, is that the standard BV Laplacian does not transform well
under symplectomorphisms, so it cannot be used as such to define an operator
on functions on an odd symplectic manifold. In the next sections, we will see
how to obviate this problem.

##  3\. The BV bracket

Let $(\mathcal{M},\omega)$ be an odd symplectic manifold. Given a function
$f$, we denote by $X_{f}$ its hamiltonian vector field defined via

$\iota_{X_{f}}\omega={\mathrm{d}}f.$

Given a second function $g$, we define the BV bracket as

$(f,g)\coloneqq(-1)^{f+1}X_{f}(g).$

This is an odd Poisson bracket, which in local Darboux coordinates agrees with
the one of the previous section:555This is a simple computation which however
requires checking signs. It is just needed to make sure that the conventions
we choose are compatible.

######  Lemma 3.1.

In Darboux coordinates, $\omega={\mathrm{d}}p_{i}{\mathrm{d}}q^{i}$, the BV
bracket agrees with the standard one defined in (1).

###### Proof.

Writing $X=X^{i}\partial_{i}+X_{i}\partial^{i}$, we get666We use the
convention of total degree, i.e., internal degree plus parity of the form
degree. This means that ${\mathrm{d}}$ is odd and that $\iota_{X}$ has parity
opposite to that of $X$. Moreover, $X^{i}$ has the same parity as $X$, whereas
$X_{i}$ has opposite parity. Finally, $f$ and $X_{f}$ have opposite parity to
each other.

$\iota_{X}\omega={\mathrm{d}}p_{i}\,X^{i}+X_{i}\,{\mathrm{d}}q^{i}={\mathrm{d}}p_{i}\,X^{i}-(-1)^{X}{\mathrm{d}}q^{i}\,X_{i}.$

From

${\mathrm{d}}f={\mathrm{d}}p_{i}\,\partial^{i}f+{\mathrm{d}}q^{i}\,\partial_{i}f$

we then get

(4a) $\displaystyle(X_{f})^{i}$
$\displaystyle=\partial^{i}f=(-1)^{f+1}f\overleftarrow{\partial{{}^{i}}}$ and
(4b) $\displaystyle(X_{f})_{i}$
$\displaystyle=(-1)^{f}\partial_{i}f=(-1)^{f}f\overleftarrow{\partial{{}_{i}^{\phantom{i}}}},$

so $(-1)^{f+1}X_{f}(g)$ is given by (1). ∎

The BV bracket satisfies several properties.

######  Proposition 3.2.

For all functions $f$, $g$, and $h$, we have

$\displaystyle[X_{f},X_{g}]$ $\displaystyle=X_{(f,g)},$ $\displaystyle(f,gh)$
$\displaystyle=(f,g)\,h+(-1)^{(f+1)g}g\,(f,h),$ $\displaystyle(g,f)$
$\displaystyle=-(-1)^{(f+1)(g+1)}(f,g),$ $\displaystyle(f,(g,h))$
$\displaystyle=((f,g),h)+(-1)^{(f+1)(g+1)}(g,(f,h)).$

The last three properties say that $(\ ,\ )$ is an odd Poisson bracket,777In
particular this means that $(\ ,\ )$ is a super Lie bracket with respect to
the opposite parity of its arguments. whereas the first says the the map
$f\mapsto X_{f}$ is a morphism of super Lie algebras.

###### Proof.

We prove only the first identity. The others are also easily obtained from the
definitions (or recovered from the identical identities for the standard BV
bracket which one obtains in each Darboux chart). We have

$\displaystyle\iota_{[X_{f},X_{g}]}\omega=[\mathsf{L}_{X_{f}},\iota_{X_{g}}]\omega=\mathsf{L}_{X_{f}}\iota_{X_{g}}\omega=\mathsf{L}_{X_{f}}{\mathrm{d}}g$
$\displaystyle=(-1)^{f+1}{\mathrm{d}}\mathsf{L}_{X_{f}}g=(-1)^{f+1}{\mathrm{d}}X_{f}(g)={\mathrm{d}}(f,g)=\iota_{X_{(f,g)}}\omega,$

where $\mathsf{L}_{X_{f}}$ denotes the Lie derivative with respect to $X_{f}$.
∎

##  4\. The BV Laplacian on functions

Let $(\mathcal{M},\omega)$ be an odd symplectic manifold and $\mu$ an even,
nowhere vanishing density on $\mathcal{M}$, which we assume to be orientable.
We then define the $\mu$-dependent BV Laplacian as

(5) $\Delta_{\mu}f\coloneqq\frac{1}{2}{\mathrm{div}}_{\mu}X_{f}.$

Recall (or see Appendix A.2) that the divergence operator of a vector field
$X$ with respect to an even, nowhere vanishing density $\mu$ is defined via

${\mathrm{div}}_{\mu}X\,\mu=\mathsf{L}_{X}\mu,$

where $\mathsf{L}_{X}$ denotes the Lie derivative.

Again we have to make sure that this definition of the BV Laplacian agrees
with the standard one in the appropriate case:

######  Proposition 4.1.

In Darboux coordinates, $\omega={\mathrm{d}}p_{i}{\mathrm{d}}q^{i}$, with
standard density
$\mu_{\text{stand}}\coloneqq{\mathrm{d}}^{n}q\,{\mathrm{d}}^{n}p$, we have

$\Delta_{\mu_{\text{stand}}}=\triangle.$

###### Proof.

With the standard density the divergence of a vector field
$X=X^{i}\partial_{i}+X_{i}\partial^{i}$ is given by (14a), i.e.,

${\mathrm{div}}_{\mu_{\text{stand}}}X=\partial_{i}X^{i}-(-1)^{X}\partial^{i}X_{i}.$

If we now insert $X_{f}$ as calculated in (4), we get

${\mathrm{div}}_{\mu_{\text{stand}}}X_{f}=2\partial_{i}\partial^{i}f.$

∎

Next we check how the operator depends on the choice of density. Note that,
given two even, nowhere vanishing densities $\mu$ and $\widetilde{\mu}$, there
is a unique even, nowhere vanishing function $h$ such that
$\widetilde{\mu}=h\mu$.

######  Proposition 4.2.

We have

$\Delta_{h\mu}=\Delta_{\mu}+\frac{1}{2}\frac{1}{h}X_{h}.$

###### Proof.

The divergence operator depends on the choice of density as

${\mathrm{div}}_{h\mu}X_{f}={\mathrm{div}}_{\mu}X_{f}+\frac{1}{h}X_{f}(h)$

(see point (2) in Proposition A.4). However,
$X_{f}(h)=(-1)^{f+1}(f,h)=-(h,f)=X_{h}(f)$, since $h$ is even. ∎

We can always write $h=\pm\mathrm{e}^{S}$ with $S$ an even function. This way
the formula simplifies to

$\Delta_{h\mu}=\Delta_{\mu}+\frac{1}{2}X_{S},$

since $X_{\mathrm{e}^{S}}=\mathrm{e}^{S}X_{S}$. In particular, we have the

######  Lemma 4.3.

In Darboux coordinates with $\mu=\pm\mathrm{e}^{S}\mu_{\text{stand}}$, we have

(6) $\Delta_{\mu}=\triangle+\frac{1}{2}X_{S}=\triangle-\frac{1}{2}(S,\ ).$

This, together with Proposition 4.2, gives the

######  Proposition 4.4.

The BV Laplacian on functions is a second-order differential operator, and its
leading term is independent of the choice of density.

######  Remark 4.5.

Under the assumptions of Lemma 4.3, from (6), also using (3) and the Jacobi
identity, we get

$\Delta_{\mu}^{2}f=-\frac{1}{2}(F_{S},f)$

with

(7) $F_{S}\coloneqq\triangle S-\frac{1}{4}(S,S).$

We then see that $\Delta_{\mu}^{2}=0$ if and only if $F_{S}$ is constant. As
$F_{S}$ is odd, this happens if and only if $F_{S}=0$.888Note that this
condition can also be rephrased as $\triangle\mathrm{e}^{\frac{1}{2}S}=0$. The
appearance of the factor $\frac{1}{2}$ in the exponent will become clear when
we study half-densities (see Lemma 5.13). Therefore, in general the property
of equation (2a) does not extend to the global BV Laplacian.

######  Definition 4.6.

We say that an even, nowhere vanishing density $\mu$ is compatible with the
odd symplectic structure if $\Delta_{\mu}^{2}=0$.

######  Remark 4.7.

We will see that every odd symplectic manifold admits a compatible density.

######  Remark 4.8.

As we have seen in Remark 4.5, the property of equation (2a) does not extend
in general. One can try to obviate it by modifying the definition of the BV
operator. For example, in Darboux coordinates with
$\mu=\pm\mathrm{e}^{S}\mu_{\text{stand}}$, we could define

$\widetilde{\Delta}_{\mu}\coloneqq\Delta_{\mu}+F_{S}=\triangle+\frac{1}{2}X_{S}+\frac{1}{2}F_{S}$

with $F_{S}$ as in (7). It is an easy exercise (show first that
$\Delta_{\mu}F_{S}=0$ for every $S$) to show that
$\widetilde{\Delta}_{\mu}^{2}=0$ for every $\mu$. However, since
$\widetilde{\Delta}_{\mu}$ is not of the form in Lemma 4.3 (unless, of course,
$F_{S}=0$), it is not a BV Laplacian.

If $\Psi\colon\mathcal{M}\to\mathcal{N}$ is a diffeomorphism, we have the
pushforward

$\Psi_{*}\colon\operatorname{End}(C^{\infty}(\mathcal{M}))\to\operatorname{End}(C^{\infty}(\mathcal{N}))$

defined by $\Psi_{*}P=\Psi_{*}\circ P\circ\Psi_{*}^{-1}$, where on the right-
hand side $\Psi_{*}$ denotes the pushforward of functions. The BV Laplacian
transforms nicely under symplectomorphisms.

######  Proposition 4.9.

For every symplectomorphism $\Psi$,

$\Psi_{*}\Delta_{\mu}=\Delta_{\Psi_{*}\mu}.$

###### Proof.

The divergence operator changes under a diffeomorphism as

$\Psi_{*}{\mathrm{div}}_{\mu}X_{f}={\mathrm{div}}_{\Psi_{*}\mu}\Psi_{*}X_{f}$

(see point (1) in Proposition A.4). If $\Psi$ is a symplectomorphism, we also
have $\Psi_{*}X_{f}=X_{\Psi_{*}f}$. ∎

######  Remark 4.10.

In conjunction with Proposition 4.4, this implies that under a
symplectomorphism $\Psi\colon\mathcal{M}\to\mathcal{M}$, the leading term of
the BV Laplacian on functions is invariant under symplectomorphisms.

######  Remark 4.11.

The proposition also implies that a symplectomorphism sends compatible
densities to compatible densities.

###  4.1. Digression: further properties of the BV Laplacian

Even though we are not going to use this in the following, it is good to know
that the properties stated in equations (2b) and (2c) also hold for the global
BV Laplacian.

######  Proposition 4.12.

For every even, nowhere vanishing density $\mu$, every functions $f$ and $g$,
and every even function $S$, we have

(8a)
$\displaystyle\Delta_{\mu}(fg)=(\Delta_{\mu}f)g+(-1)^{f}f\Delta_{\mu}g-(-1)^{f}(f,g),$
(8b)
$\displaystyle\Delta_{\mu}\mathrm{e}^{S}=\left(\Delta_{\mu}S-\frac{1}{2}(S,S)\right)\mathrm{e}^{S}.$

###### Proof.

The identities may be proved by observing, thanks to (6), that in every
Darboux chart the BV Laplacian differs from $\triangle$ by a vector field.

They may also be proved directly using properties of the divergence operator
(see point (3) in Proposition A.4) and of hamiltonian vector fields. To
illustrate this, we prove the second identity. Since
$X_{\mathrm{e}^{S}}=\mathrm{e}^{S}X_{S}$, we have

${\mathrm{div}}_{\mu}X_{\mathrm{e}^{S}}={\mathrm{div}}_{\mu}(\mathrm{e}^{S}X_{S})=\mathrm{e}^{S}{\mathrm{div}}_{\mu}X_{S}+X_{S}(\mathrm{e}^{S}),$

which proves the identity once we observe that
$X_{S}(\mathrm{e}^{S})=-\mathrm{e}^{S}(S,S)$. ∎

######  Remark 4.13.

As we have seen in Remark 4.8, it is possible to define an operator
$\widetilde{\Delta}_{\mu}$ that squares to zero, no matter what $\mu$ is, so
as to satisfy the extension of the property of equation (2a). However, since
this is obtained by adding to $\Delta_{\mu}$ a multiplication operator, we see
that now (8a) is no longer satisfied (unless, of course, $F_{S}=0$).

##  5\. The BV Laplacian on densities

A differential operator $P$ defined on functions on a (super)manifold
$\mathcal{M}$ can be extended to sections of a trivial line bundle $L$ over
$\mathcal{M}$ once an even, nowhere vanishing section $\lambda$ of $L$ has
been chosen. For $\sigma\in\Gamma(L)$ there is a uniquely determined function
$f$ such that $\sigma=f\lambda$, and we set

$P^{(\lambda)}\sigma\coloneqq P(f)\lambda.$

######  Lemma 5.1.

The leading term of the differential operator $P^{(\lambda)}$ does not depend
on the choice of $\lambda$.

###### Proof.

If $\widetilde{\lambda}$ is also a nowhere vanishing section, then there is a
uniquely determined nowhere vanishing function $h$ with
$\widetilde{\lambda}=h\lambda$. For $\sigma=f\widetilde{\lambda}=fh\lambda$,
we have

$P^{(\widetilde{\lambda})}\sigma=P(f)\widetilde{\lambda}=P(f)h\lambda.$

On the other hand, we have

$P^{(\lambda)}\sigma=P(fh)\lambda=((P_{\text{leading}}f)\,h+\cdots)\lambda,$

where the dots denote terms where less than the maximum number of derivatives
hit $f$. ∎

We now want to extend the BV Laplacian $\Delta_{\mu}$ to $s$-densities (see
Appendix A.2 for a review). Since we already have an even, nowhere vanishing
section, $\mu$, of the density bundle, we can use it to get our reference
section, $\mu^{s}$, of the $s$-density bundle.999Since we are interested also
in nonintegral $s$, in particular $s=\frac{1}{2}$, it is essential to require
that $\mathcal{M}$ is oriented. That is,

######  Definition 5.2.

For an $s$-density $\sigma$, which we can uniquely write as $\sigma=f\mu^{s}$,
we set101010 We avoid the more pedantic notation $\Delta_{\mu}^{(\mu^{s})}$ on
the left-hand side.

$\Delta_{\mu}^{(s)}\sigma\coloneqq\Delta_{\mu}(f)\mu^{s}.$

Note that $\Delta_{\mu}^{(0)}$ is the same as $\Delta_{\mu}$.

######  Remark 5.3.

An immediate consequence of this definition is that $\Delta_{\mu}^{(s)}$
squares to zero on $s$-densities if and only if $\Delta_{\mu}$ does so on
functions (i.e., $\mu$ is compatible with the odd symplectic structure).

Proposition 4.4 and Lemma 5.1 immediately imply

######  Proposition 5.4.

The BV Laplacian on $s$-densities is a second-order differential operator, and
its leading term is independent of the choice of reference density.

Proposition 4.9 immediately implies

######  Proposition 5.5.

For every symplectomorphism $\Psi$ and for every $s$,

$\Psi_{*}\Delta^{(s)}_{\mu}=\Delta^{(s)}_{\Psi_{*}\mu}.$

######  Remark 5.6.

In particular, this implies that, for every $s$, under a symplectomorphism
$\Psi\colon\mathcal{M}\to\mathcal{M}$, the leading term of the BV Laplacian on
$s$-densities is invariant under symplectomorphisms.

Note that the product of an $s$-density $\sigma$ and a $(1-s)$-density $\tau$
yields a density, which can be integrated (if it has compact support). Given a
differential operator $P$ on $s$-densities, we define its transpose $P^{t}$ as
the differential operator on $(1-s)$-densities that satisfies

$\int_{\mathcal{M}}P\sigma\,\tau=(-1)^{P\,\sigma}\int_{\mathcal{M}}\sigma\,P^{t}\tau$

for all $s$-densities $\sigma$ with compact support (the upper index
“$P\,\sigma$” denotes the product of the degrees; see footnote 3 on page 3).
The case $s=\frac{1}{2}$ is special, since transposition becomes an
endomorphism on the space of differential operators.

######  Proposition 5.7.

We have

$(\Delta_{\mu}^{(s)})^{t}=\Delta_{\mu}^{(1-s)}$

In particular, $\Delta_{\mu}^{(\frac{1}{2})}$ is symmetric.

###### Proof.

Write $\sigma=f\mu^{s}$ and $\tau=g\mu^{1-s}$. Then

$\Delta_{\mu}^{(s)}\sigma\,\tau=\Delta_{\mu}f\,g\,\mu.$

But

$\displaystyle
2\Delta_{\mu}f\,g={\mathrm{div}}_{\mu}X_{f}\,g=(-1)^{(f+1)g}g\,{\mathrm{div}}_{\mu}X_{f}$
$\displaystyle=(-1)^{(f+1)g}({\mathrm{div}}_{\mu}(gX_{f})-(-1)^{(f+1)g}X_{f}(g))$
$\displaystyle=(-1)^{(f+1)g}{\mathrm{div}}_{\mu}(gX_{f})+(-1)^{f}(f,g),$

where we have used part (3) of Proposition A.4. Similarly,

$\sigma\,\Delta_{\mu}^{(1-s)}\tau=f\,\Delta_{\mu}g\,\mu,$

and

$\displaystyle
2f\,\Delta_{\mu}g=f\,{\mathrm{div}}_{\mu}X_{g}={\mathrm{div}}_{\mu}(fX_{g})-(-1)^{f(g+1)}X_{g}(f)$
$\displaystyle={\mathrm{div}}_{\mu}(fX_{g})+(-1)^{fg+f+g}(g,f)={\mathrm{div}}_{\mu}(fX_{g})+(f,g).$

Therefore,

$2\Delta_{\mu}f\,g-(-1)^{f}2f\,\Delta_{\mu}g=(-1)^{(f+1)g}{\mathrm{div}}_{\mu}(gX_{f})-(-1)^{f}{\mathrm{div}}_{\mu}(fX_{g}),$

which implies, thanks to part (4) of Proposition A.4, that111111Note the
different sign in (8a), which instead yields
$\int_{\mathcal{M}}(\Delta_{\mu}^{(s)}\sigma\,\tau+(-1)^{\sigma}\sigma\,\Delta_{\mu}^{(1-s)}\tau)=\int_{\mathcal{M}}(\Delta_{\mu}f\,g+(-1)^{f}f\,\Delta_{\mu}g)\,\mu=(-1)^{f}\int_{\mathcal{M}}(f,g)\,\mu.$
Subtracting the two identities yields the interesting formula
$\int_{\mathcal{M}}(f,g)\,\mu=2\int_{\mathcal{M}}f\,\Delta_{\mu}g\,\mu,$ which
in particular shows that the integral of the BV bracket of two functions (at
least one of which has compact support) against the density $\mu$ vanishes if
one of the two functions is in the kernel of $\Delta_{\mu}$.

$\int_{\mathcal{M}}(\Delta_{\mu}^{(s)}\sigma\,\tau-(-1)^{\sigma}\sigma\,\Delta_{\mu}^{(1-s)}\tau)=0.$

∎

######  Remark 5.8.

By inspection in the proof one sees that it is essential that we have used the
same density $\mu$ to define the BV Laplacian on functions and to extend it to
$s$-densities.

We now come to the fundamental consequence of this proposition.

######  Lemma 5.9.

In each Darboux chart, there is an odd function $G_{\mu}$, depending on $\mu$,
such that

$\Delta_{\mu}^{(\frac{1}{2})}\sigma=(\triangle
f+G_{\mu}f)\,\mu_{\text{stand}}^{\frac{1}{2}},$

where we have written $\sigma=f\,\mu_{\text{stand}}^{\frac{1}{2}}$.

###### Proof.

Since $\Delta_{\mu}^{(\frac{1}{2})}$ is a second-order differential operator
and its leading term is independent of $\mu$, we have

$\Delta_{\mu}^{(\frac{1}{2})}\sigma=(\triangle
f+Y_{\mu}(f)+G_{\mu}\,f)\,\mu_{\text{stand}}^{\frac{1}{2}},$

for some vector field $Y_{\mu}$ and some function $G_{\mu}$. Since $\triangle$
has constant coefficients, it is symmetric. Moroever, since $Y_{\mu}$ is a
vector field, we have $Y_{\mu}^{t}=-Y_{\mu}+$some function. Since
$\Delta_{\mu}^{(\frac{1}{2})}$ is symmetric, we get $Y_{\mu}=0$. ∎

This implies the following

######  Corollary 5.10.

For every compatible $\mu$, the BV Laplacian on half-densities has a canonical
form in every Darboux chart:

$\Delta_{\mu}^{(\frac{1}{2})}\sigma=\triangle
f\,\mu_{\text{stand}}^{\frac{1}{2}},$

where we have written $\sigma=f\,\mu_{\text{stand}}^{\frac{1}{2}}$. In
particular, $\Delta_{\mu}^{(\frac{1}{2})}$ is independent of the choice of the
compatible density $\mu$.

###### Proof.

Using $\triangle^{2}=0$, $G_{\mu}^{2}=0$, and (2b), we get

$0=(\Delta_{\mu}^{(\frac{1}{2})})^{2}\sigma=(\triangle
G_{\mu}\,f+(G_{\mu},f))\mu_{\text{stand}}^{\frac{1}{2}}.$

Therefore, for every function $f$, we have

$\triangle G_{\mu}\,f+(G_{\mu},f)=0.$

In particular, for $f=1$, we get $\triangle G_{\mu}=0$, so the condition
simplifies to $(G_{\mu},f)=0$ for every $f$. This implies that $G_{\mu}$ is
constant, but then it must vanish because it is odd.121212This argument fails
if one wants to work in families, as in this case odd constants are allowed. ∎

In every Darboux chart, we can consider the BV Laplacian
$\Delta_{\mu_{\text{stand}}}^{(\frac{1}{2})}$, which sends
$f\,\mu_{\text{stand}}^{\frac{1}{2}}$ to $\triangle
f\,\mu_{\text{stand}}^{\frac{1}{2}}$. If we move from this Darboux chart to
another one via the transition map $\phi$, the operator goes to
$\phi_{*}\Delta_{\mu_{\text{stand}}}^{(\frac{1}{2})}$, which, by Proposition
5.5, is $\Delta_{\phi_{*}\mu_{\text{stand}}}^{(\frac{1}{2})}$. However, since
$\mu_{\text{stand}}$ is a compatible density (and therefore, by Remark 4.11,
also $\phi_{*}\mu_{\text{stand}}$ is a compatible density), by the corollary
this is again $\Delta_{\mu_{\text{stand}}}^{(\frac{1}{2})}$ (in the new
chart). Therefore, we have the

######  Theorem/Definition 5.11.

There is a canonical BV operator $\Delta$ acting on half-densities, defined as
$\Delta_{\mu_{\text{stand}}}^{(\frac{1}{2})}$ in every Darboux chart.

Note that, by its very definition, the canonical BV operator is invariant
under symplectomorphisms and satisfies $\Delta^{2}=0$.

Corollary 5.10 may now be reformulated as

######  Theorem 5.12.

For every compatible $\mu$, the BV Laplacian on half-densities is equal to the
canonical BV operator:

$\Delta_{\mu}^{(\frac{1}{2})}=\Delta.$

By this theorem we have that the canonical BV operator is a BV Laplacian on
half-densities whenever the odd symplectic manifold admits compatible
densities,131313Unlike in the even symplectic case, where a canonical choice
of density is provided by the top exterior power of the symplectic form, there
is no canonical density in the odd case. which we have not proved yet.

On the other hand, what is remarkable is that $\Delta$ is canonically defined
(by Theorem/Definition 5.11) without choosing any density (in particular it
would exist even if there were no compatible densities).

The canonical BV operator gives another way of defining the BV Laplacian on
functions. Namely, given an even, nowhere vanishing density $\mu$, we define a
differential operator $\widehat{\Delta}_{\mu}$ on functions via

(9) $\widehat{\Delta}_{\mu}f=\Delta(f\mu^{\frac{1}{2}})\,\mu^{-\frac{1}{2}}.$

Note that $\widehat{\Delta}_{\mu}^{2}=0$ for every $\mu$. However, it is not a
BV Laplacian in general. We actually have the

######  Lemma 5.13.

For every $\mu$,

$\widehat{\Delta}_{\mu}=\widetilde{\Delta}_{\mu}$

in every Darboux chart, where $\widetilde{\Delta}_{\mu}$ is the operator
defined in Remark 4.8.

###### Proof.

Writing $\mu=\mathrm{e}^{S}\mu_{\text{stand}}$, we have

$\widehat{\Delta}_{\mu}f\,\mathrm{e}^{\frac{S}{2}}\mu_{\text{stand}}^{\frac{1}{2}}=\Delta(f\mathrm{e}^{\frac{S}{2}}\mu_{\text{stand}}^{\frac{1}{2}})=\triangle(f\mathrm{e}^{\frac{S}{2}})\,\mu_{\text{stand}}^{\frac{1}{2}}.$

One can easily compute

$\triangle(f\mathrm{e}^{\frac{S}{2}})=\left(\triangle
f-\frac{1}{2}(S,f)+\frac{1}{2}F_{S}\right)\mathrm{e}^{\frac{S}{2}}=\widetilde{\Delta}_{\mu}f\,\mathrm{e}^{\frac{S}{2}},$

with $F_{S}$ as in (7). ∎

This yields the following

######  Corollary 5.14.

Let $\mu$ be an even, nowhere vanishing density. Then

  1. (1)

$\widetilde{\Delta}_{\mu}$ is globally defined;

  2. (2)

$\mu$ is compatible if and only if $\widehat{\Delta}_{\mu}=\Delta_{\mu}$;

  3. (3)

$\mu$ is compatible if and only if $\Delta\mu^{\frac{1}{2}}=0$.

###### Proof.

The first two properties are now trivial. We prove the third. Setting $f=1$ in
(9) yields
$\Delta(\mu^{\frac{1}{2}})=\widehat{\Delta}_{\mu}1\,\mu^{\frac{1}{2}}$. Lemma
5.13 then implies that $\Delta(\mu^{\frac{1}{2}})=0$ if and only if in every
Darboux chart $F_{S}=\widetilde{\Delta}_{\mu}1=0$. By Remark 4.5 this happens
if and only if $\mu$ is compatible. ∎

###  5.1. Digression: Darboux expression of $\Delta_{\mu}^{(s)}$

In a Darboux chart we have $\mu=\mathrm{e}^{S}\mu_{\text{stand}}$. If
$\sigma=f\,\mu_{\text{stand}}^{s}$, we then have
$\sigma=\mathrm{e}^{-sS}f\,\mu^{s}$, so

$\Delta_{\mu}^{(s)}\sigma=\Delta_{\mu}(\mathrm{e}^{-sS}f)\,\mu^{s}.$

By using (6) and the properties of $\triangle$ in (2), we get

$\Delta_{\mu}^{(s)}\sigma=\left(\triangle
f+\left(s-\frac{1}{2}\right)(S,f)-\left(s\triangle
S+\frac{1}{2}s(s-1)(S,S)\right)f\right)\mu_{\text{stand}}^{s}.$

One can now explictly check that
$(\Delta_{\mu}^{(s)})^{t}=\Delta_{\mu}^{(1-s)}$. Moreover, one explicitly sees
that the first-order term vanishes for every $S$ if and only if
$s=\frac{1}{2}$. Finally, for $\mu$ compatible, i.e., $\Delta
S=\frac{1}{4}(S,S)$, the zeroth-order term becomes $\frac{1}{4}(2s-1)s(S,S)$,
which vanishes for $s=\frac{1}{2}$ or $s=0$.

This formula, for $s=\frac{1}{2}$, yields a different way to prove Corollary
5.10 and therefore Theorem/Definition 5.11 and Theorem 5.12. However, proving
this formula is a bit more laborious than proving Proposition 5.7.

##  6\. Odd cotangent bundles

We now focus on an odd cotangent bundle $\Pi T^{*}M$,141414Here $\Pi$ denotes
the change-of-parity functor. It simply means that the fiber coordinates are
now regarded as odd. where $M$ is an ordinary manifold. We will see that it is
easy to show the existence of compatible densities in this case. Since every
odd symplectic manifold is symplectomorphic to the odd cotangent bundle of its
body [13], this implies that every odd symplectic manifold has a compatible
density and therefore that the canonical BV operator is indeed a BV Laplacian
on half-densities.

The first observation is that functions on $\Pi T^{*}M$ are the same as
multivector fields on $M$: $C^{\infty}(\Pi
T^{*}M)=\Gamma(\Lambda^{\bullet}TM)$ as super algebras.151515The interested
reader might also appreciate that $(C^{\infty}(\Pi T^{*}M),(\ ,\ ))$ and
$(\Gamma(\Lambda^{\bullet}TM),[\ ,\ ]_{\text{SN}})$ are naturally isomorphic
odd Poisson algebras, where $(\ ,\ )$ is the canonical BV bracket, associated
to the canonical symplectic form, and $[\ ,\ ]_{\text{SN}}$ is the
Schouten–Nijenhuis bracket.

The second observation is about the density bundle on $\Pi T^{*}M$. If
$\phi_{\alpha\beta}$ denotes a transition function on $M$, then the
corresponding transition function on $T^{*}M$ has a fiber part given by
$(({\mathrm{d}}\phi_{\alpha\beta})^{*})^{-1}$. Therefore, the Berezinian of
the corresponding transition function for $\Pi T^{*}M$ is
$({\mathrm{d}}\phi_{\alpha\beta})^{2}$. This means that densities on $\Pi
T^{*}M$ transform like $2$-densities on $M$. Since we assume $M$ to be
oriented, we also have that half-densities on $\Pi T^{*}M$ transform like top
forms on $M$. More precisely, we have a canonical isomorphism

$\Gamma(\operatorname{Dens}^{\frac{1}{2}}(\Pi
T^{*}M))\simeq\Gamma(\Lambda^{\bullet}TM)\otimes_{C^{\infty}(M)}\Omega^{n}(M),$

where $n$ is the dimension of $M$. The next observation is that we have a
canonical isomorphism (of $C^{\infty}(M)$-modules)

$\phi\colon\begin{array}[t]{ccc}\Gamma(\Lambda^{\bullet}TM)\otimes_{C^{\infty}(M)}\Omega^{n}(M)&\to&\Omega^{n-\bullet}(M)\\\
X\otimes v&\mapsto&\iota_{X}v\end{array}$

We can use this isomorphism to define an operator on half-densities:

$\mathcal{D}\coloneqq\phi^{-1}\circ{\mathrm{d}}\circ\phi,$

where ${\mathrm{d}}$ is the de Rham differential. One immediately sees that
$\mathcal{D}^{2}=0$ and that $\mathcal{D}v=0$ for every top form $v$ on $M$.

######  Proposition 6.1.

The canonical operator $\mathcal{D}$ is the same as the canonical BV operator
$\Delta$ on half-densities.

###### Proof.

We just have to check how $\mathcal{D}$ acts in Darboux coordinates. Let
$q^{1},\dots,q^{n}$ denote coordinates on a chart of $M$ and let
$p_{1},\dots,p_{n}$ be the corresponding odd fiber coordinates. By linearity
it is enough to consider a half-density $\sigma=g\,\mu_{\text{stand}}$ with
$g(p,q)$ of the form $f(q)\,p_{i_{1}}\cdots\,p_{i_{k}}$, with pairwise
distinct indices $i_{j}$. In the identification described above this
corresponds to
$\sigma=f(q)\,\partial_{i_{1}}\wedge\dots\wedge\partial_{i_{k}}\otimes{\mathrm{d}}^{n}q$,
so

$\phi(\sigma)=f(q)\,\iota_{\partial_{i_{1}}}\cdots\,\iota_{\partial_{i_{k}}}{\mathrm{d}}^{n}q.$

We then have

${\mathrm{d}}\phi(\sigma)={\mathrm{d}}f(q)\,\iota_{\partial_{i_{1}}}\cdots\,\iota_{\partial_{i_{k}}}{\mathrm{d}}^{n}q=\partial_{i}f(q)\,{\mathrm{d}}q^{i}\wedge\iota_{\partial_{i_{1}}}\cdots\,\iota_{\partial_{i_{k}}}{\mathrm{d}}^{n}q,$

since the form
$\iota_{\partial_{i_{1}}}\cdots\,\iota_{\partial_{i_{k}}}{\mathrm{d}}^{n}q$
has a constant coefficient, so it is closed. If the index $i$ is different
from all the $i_{j}$s, the corresponding term vanishes, since
${\mathrm{d}}q^{i}$ is already contained in ${\mathrm{d}}^{n}q$. Otherwise, we
have

$\begin{split}{\mathrm{d}}q^{i_{j}}\wedge\iota_{\partial_{i_{1}}}\cdots\,\iota_{\partial_{i_{k}}}{\mathrm{d}}^{n}q&=(-1)^{j-1}{\mathrm{d}}q^{i_{j}}\wedge\iota_{\partial_{i_{j}}}\iota_{\partial_{i_{1}}}\cdots\,\widehat{\iota}_{\partial_{i_{j}}}\cdots\,\iota_{\partial_{i_{k}}}{\mathrm{d}}^{n}q\\\
&=(-1)^{j-1}\iota_{\partial_{i_{1}}}\cdots\,\widehat{\iota}_{\partial_{i_{j}}}\cdots\,\iota_{\partial_{i_{k}}}{\mathrm{d}}^{n}q,\end{split}$

where the caret denotes omission. On the other hand,

$\partial_{i_{j}}(p_{i_{1}}\cdots\,p_{i_{k}})=(-1)^{j-1}p_{i_{1}}\cdots\,\widehat{p}_{i_{j}}\cdots\,p_{i_{k}}.$

Therefore,

$\phi(\triangle
g\,\mu_{\text{stand}})={\mathrm{d}}\phi(g\,\mu_{\text{stand}}),$

which shows that $\Delta=\mathcal{D}$. ∎

We then have $\Delta v=0$ for every top form $v$ on $M$ (regarded as a half-
density on $\Pi T^{*}M$). As a notation, we write $\mu_{v}$ for the even,
nowhere vanishing density $v^{2}$ on $\Pi T^{*}M$ associated to a volume form
$v$ on $M$. We then have

$\Delta\mu_{v}^{\frac{1}{2}}=0,$

which shows, by point (3) of Corollary 5.14, that $\mu_{v}$ is a compatible
density. We then have the

######  Proposition 6.2.

An odd cotangent bundle possesses compatible densities.

This implies, by Theorem 5.12, that $\Delta=\Delta^{(\frac{1}{2})}_{\mu_{v}}$.
Given a volume form $v$, we can now use (9) to get $\Delta_{\mu_{v}}$ on
functions. It is an easy exercise to see that

$\Delta_{\mu_{v}}=\phi_{v}^{-1}\circ{\mathrm{d}}\circ\phi_{v},$

where $\phi_{v}$ is the ($v$-dependent!) isomorphism

$\phi_{v}\colon\begin{array}[t]{ccc}C^{\infty}(\Pi
T^{*}M)=\Gamma(\Lambda^{\bullet}TM)&\to&\Omega^{n-\bullet}(M)\\\
X&\mapsto&\iota_{X}v\end{array}$

Note that, for a vector field $X$, we have
$\Delta_{\mu_{v}}X={\mathrm{div}}_{v}X$. For this reason, $\Delta_{\mu_{v}}$
is interpreted as the extension to multivector fields of the divergence
operator with respect to $v$.

##  7\. Conclusion of the argument

It is a theorem (see [13]) that every odd symplectic manifold $\mathcal{M}$
is, noncanonically, isomorphic to the odd cotangent bundle $\Pi T^{*}M$ of its
body $M$.161616This is a global Darboux theorem. If one is familiar with the
proof of Darboux’s theorem via Moser’s trick, the reason why this now works
globally is that it turns out that one has to integrate a vector field in the
odd directions, which has no problem of definition domain.

On $\Pi T^{*}M$ we have compatible densities (Proposition 6.2), e.g.,
$\mu_{v}$ for a volume form $v$ on $M$. If $\Psi\colon\Pi
T^{*}M\to\mathcal{M}$ is a symplectomorphism, which exists by the above
mentioned theorem, then $\Psi_{*}\mu_{v}$ is a compatible density on
$\mathcal{M}$ (Remark 4.11).

We therefore have—also using Theorem 5.12, equation (9), and point (2) of
Corollary 5.14—our final result:

######  Theorem 7.1.

Every odd symplectic manifold possesses compatible densities. For each
compatible density $\mu$, we have

$\Delta_{\mu}^{(\frac{1}{2})}=\Delta,$

with $\Delta$ the canonical BV operator on half-densities, and the BV
Laplacian $\Delta_{\mu}$ on functions may be recovered from

$\Delta_{\mu}f\,\mu^{\frac{1}{2}}=\Delta(f\mu^{\frac{1}{2}}).$

##  Appendix A Some background details

###  A.1. The Lie derivative

If $\Xi$ is an object (e.g., function, vector field, density) for which a
notion of pushforward $\Psi_{*}$ under a diffeomorphism $\Psi$ exists, we
define the Lie derivative with respect to a vector field $X$ as

$\mathsf{L}_{X}\Xi\coloneqq\left.\frac{\partial}{\partial
t}\right\rvert_{t=0}(\Phi^{X}_{-t})_{*}\Xi.$

For an even vector field $X$, the diffeomorphism $\Phi^{X}_{t}$ is the flow of
$X$ at time $t$. If $X$ is odd, the variable $t$ is also odd (so the
evaluation at zero in the formula is redundant), and we define the morphism
$\Phi^{X}\colon\Pi{\mathbb{R}}\times\mathcal{M}\to\mathcal{M}$ via171717This
defines the flow of $X$ at odd time $t$ if and only if $[X,X]=0$. Otherwise,
the flow property $\Phi^{X}_{t+s}=\Phi^{X}_{t}\circ\Phi^{X}_{s}$ is not
satisfied, so in general $\Phi^{X}$ is not a flow.

$(\Phi^{X})^{*}\colon\begin{array}[t]{ccc}C^{\infty}(\mathcal{M})&\to&C^{\infty}(\Pi{\mathbb{R}}\times\mathcal{M})\\\
f&\mapsto&f+tX(f)\end{array}$

where $t$ is the odd coordinate on $\Pi{\mathbb{R}}$.181818Here $\Pi$ denotes
the change-of-parity functor: $\Pi{\mathbb{R}}$ is the superdomain with one
odd coordinate. Note that in local coordinates $z^{\mu}$, with
$X=X^{\mu}{\small\frac{\partial}{\partial z^{\mu}}}$, we have in both cases

(10)
$(\Phi^{X})^{\mu}\coloneqq(\Phi^{X})^{*}z^{\mu}=z^{\mu}+tX^{\mu}+O(t^{2}),$

where of course $O(t^{2})=0$ in the odd case. Also note that in both cases we
have

$\Psi_{*}(\Phi^{X})_{*}=(\Phi^{\Psi_{*}X})_{*}\Psi_{*}$

for every diffeomorphism $\Psi$.191919In the even case, one just observes how
solutions of ODEs are mapped under diffeomorphisms. In the odd case, it is
just an immediate application of the chain rule. This implies that

$\mathsf{L}_{\Psi_{*}X}\Psi_{*}\Xi=\Psi_{*}\mathsf{L}_{X}\Xi.$

If two objects $\Xi_{1}$ and $\Xi_{2}$ as above can be multiplied, we
immediately get

(11)
$\mathsf{L}_{X}(\Xi_{1}\Xi_{2})=\mathsf{L}_{X}\Xi_{1}\,\Xi_{2}+(-1)^{X\Xi_{1}}\Xi_{1}\,\mathsf{L}_{X}\Xi_{2}.$

On a function $f$ or on a vector field $Y$, the Lie derivative is readily
computed as

$\mathsf{L}_{X}f=X(f)\quad\text{and}\quad\mathsf{L}_{X}Y=[X,Y].$

###  A.2. Densities

A density is an object that transforms in such a way that its integration may
be defined. For this we have to recall that, if $\Psi$ is a diffeomorphism
between two superdomains, then the change-of-variables formula for berezinian
integration involves the Berezinian $\operatorname{Ber}(\Psi)$ of the jacobian
matrix of $\Psi$ times the sign of the jacobian determinant of the reduction
to the body of $\Psi$. Since we assume throughout that the body of our
supermanifold is oriented, we will not have this factor. Therefore, given an
oriented atlas for $\mathcal{M}$ that has transition functions
$\phi_{\alpha\beta}$, we define the density bundle
$\operatorname{Dens}\mathcal{M}$ as the line bundle with transition functions
$\operatorname{Ber}(\phi_{\alpha\beta})^{-1}$. More generally, for any real
number $s$, we define the line bundle $\operatorname{Dens}^{s}\mathcal{M}$ of
$s$-densities with transition functions
$\operatorname{Ber}(\phi_{\alpha\beta})^{-s}$. Note that $1$-densities are the
same as densities, and $0$-densities are the same as functions.

With this definition the integral of a compactly supported202020The support is
defined in terms of the coefficient body functions. Namely, an object
(function, vector field, density,…) is compactly supported when in its
expansion in odd variables all the coefficients are compactly supported
functions on the body. density $\mu$ over $\mathcal{M}$ is defined as usual
(by choosing a partition of unity subordinated to the atlas and by berezinian
integration in each chart).

The pushforward of $s$-densities under diffeomorphisms is also defined. If we
go to charts, the pushforward is given by the pushforward of the representing
function times the Berezinian of the transformation to the power $-s$.

Armed with the pushforward, we can define the Lie derivative of an
$s$-density. In particular, we need the following

######  Lemma A.1.

In local coordinates $p_{1},\dots,p_{m}$ and $q^{1},\dots,q^{n}$ (odd and
even, respectively), the Lie derivative of the standard density
$\mu_{\text{stand}}={\mathrm{d}}^{n}q\,{\mathrm{d}}^{m}p$ is

$\mathsf{L}_{X}{\mu_{\text{stand}}}=(\partial_{i}X^{i}-(-1)^{X}\partial^{i}X_{i})\,{\mu_{\text{stand}}},$

where we used the expansion $X=X^{i}\partial_{i}+X_{i}\partial^{i}$.

###### Proof.

Using (10), we see that212121By $\delta^{i}_{j}$ we mean the Kronecker delta,
which is equal to $1$ for $i=j$ and to $0$ otherwise.

$\begin{split}\operatorname{Ber}(\Phi^{X}_{-t})&=\operatorname{Ber}\begin{pmatrix}\delta_{i}^{j}-\partial_{i}(tX^{j})&0\\\
0&\delta^{i}_{j}-\partial^{i}(tX_{j})\end{pmatrix}+O(t^{2})\\\
&=-\partial_{i}(tX^{i})-\partial^{i}(tX_{i})+O(t^{2})\\\
&=-t\partial_{i}(X^{i})-(-1)^{X}t\partial^{i}(X_{i})+O(t^{2}),\end{split}$

using that the paritiy of $t$ is the same as that of $X$. Applying the
definitions of Lie derivative and of pushforward of a density yields the
result. ∎

Applying (11) to the case of densities, we get

(12) $\mathsf{L}_{X}(h\mu)=X(h)\,\mu+(-1)^{Xh}h\,\mathsf{L}_{X}\mu$

for every function $h$. In particular, we get the general formula for the Lie
derivative in local coordinates:

$\begin{split}\mathsf{L}_{X}\mu&=(X(h)+(-1)^{Xh}h(\partial_{i}X^{i}-(-1)^{X}\partial^{i}X_{i}))\,{\mu_{\text{stand}}}\\\
&=(X(h)+(\partial_{i}X^{i}-(-1)^{X}\partial^{i}X_{i})h)\,{\mu_{\text{stand}}}\\\
&=X(h)\mu_{\text{stand}}+(\partial_{i}X^{i}-(-1)^{X}\partial^{i}X_{i})\mu,\end{split}$

with $\mu=h\mu_{\text{stand}}$. This implies the following

######  Lemma A.2.

For every function $f$, vector field $X$, and density $\mu$, we have

$\mathsf{L}_{fX}\mu=f\mathsf{L}_{X}\mu+(-1)^{fX}X(f)\mu.$

###### Proof.

It is enough to check the formula in local coordinates. We have

$\begin{split}\mathsf{L}_{fX}\mu&=fX(h)\mu_{\text{stand}}+(\partial_{i}(fX^{i})-(-1)^{f+X}\partial^{i}(fX_{i}))\mu\\\
&=fX(h)\mu_{\text{stand}}+f\,(\partial_{i}X^{i}-(-1)^{X}\partial^{i}X_{i})\mu+(\partial_{i}fX^{i}-(-1)^{f+X}\partial^{i}fX_{i})\mu\\\
&=f\mathsf{L}_{X}\mu+((-1)^{Xf}X^{i}\partial_{i}f-(-1)^{f+X}(-1)^{(f+1)(X+1)}X_{i}\partial^{i}f)\mu\\\
&=f\mathsf{L}_{X}\mu+(-1)^{fX}X(f)\mu.\end{split}$

∎

In the case of densities, the pushforward is related to the change-of-
variables formula. Namely, if $\Psi\colon\mathcal{M}\to\mathcal{N}$ is a
diffeomorphism (preserving the orientation of the bodies), then

(13) $\int_{\mathcal{N}}\Psi_{*}\mu=\int_{\mathcal{M}}\mu$

for every compactly supported density $\mu$. This leads to

######  Lemma A.3.

For every vector field $X$ and density $\mu$, one of which is compactly
supported, we have

$\int_{\mathcal{M}}\mathsf{L}_{X}\mu=0.$

###### Proof.

First consider the case when $\mu$ is compactly supported. In this case,
$(\Phi^{X}_{-t})_{*}\mu$ and hence $\mathsf{L}_{X}\mu$ are also compactly
supported. We then have

$\int_{\mathcal{M}}\mathsf{L}_{X}\mu=\int_{\mathcal{M}}\left.\frac{\partial}{\partial
t}\right\rvert_{t=0}(\Phi^{X}_{-t})_{*}\mu=\left.\frac{\partial}{\partial
t}\right\rvert_{t=0}\int_{\mathcal{M}}(\Phi^{X}_{-t})_{*}\mu=\left.\frac{\partial}{\partial
t}\right\rvert_{t=0}\int_{\mathcal{M}}\mu=0,$

where we have also used (13).

If on the other hand $\mu$ is not compactly supported, but $X$ is, we replace
$\mu$ with $\widetilde{\mu}=\rho\mu$, where $\rho$ is a compactly supported
bump function which is identically equal to $1$ on the support of $X$. Since
$\mathsf{L}_{X}\mu=\mathsf{L}_{X}\widetilde{\mu}$, we get the result by the
previous case. ∎

If we have an even, nowhere vanishing density $\mu$, then every other density
can be written as $f\mu$ for a uniquely determined function $f$. Therefore, we
can define the divergence operator via the following formula:

${\mathrm{div}}_{\mu}X\,\mu=\mathsf{L}_{X}\mu.$

The properties of the Lie derivative immediately imply properties for the
divergence operator:

######  Proposition A.4.

Let $\mu$ be an even, nowhere vanishing density $\mu$ and $X$ a vector field.
Then

  1. (1)

for every diffeomorphism $\Psi$,

$\Psi_{*}{\mathrm{div}}_{\mu}X={\mathrm{div}}_{\Psi_{*}\mu}\Psi_{*}X;$

  2. (2)

for every even, nowhere vanishing function $h$,

${\mathrm{div}}_{h\mu}X={\mathrm{div}}_{\mu}X+\frac{1}{h}X(h);$

  3. (3)

for every function $f$,

${\mathrm{div}}_{\mu}(fX)=f{\mathrm{div}}_{\mu}X+(-1)^{fX}X(f);$

  4. (4)

under the additional assumption that $X$ is compactly supported,

$\int_{\mathcal{M}}{\mathrm{div}}_{\mu}X\,\mu=0.$

Moreover, from the local coordinate expression for the Lie derivative, we get

(14a) $\displaystyle{\mathrm{div}}_{\mu_{\text{stand}}}X$
$\displaystyle=\partial_{i}X^{i}-(-1)^{X}\partial^{i}X_{i},$ and, rearranging
the terms, (14b) $\displaystyle{\mathrm{div}}_{\mu}X$
$\displaystyle=\frac{1}{h}(\partial_{i}(hX^{i})-(-1)^{X}\partial^{i}(hX_{i})),$

for $\mu=h\mu_{\text{stand}}$.

##  Appendix B Applications of the BV formalism

We recall here the main reason why the BV formalism was introduced: to study
the invariance of certain integrals.

We start considering the case of Section 2, where we have Darboux coordinates
$p_{1},\dots,p_{n}$ and $q^{1},\dots,q^{n}$. For the sake of the argument, we
rearrange each pair $(p_{i},q^{i})$ into a new pair $(x_{i},y^{i})$, with
$x_{i}=p_{i}$ and $y^{i}=q^{i}$ for some $i$s and $x_{i}=q^{i}$ and
$y^{i}=p_{i}$ for the other $i$s. Note that $x_{i}$ has opposite parity to
$y^{i}$ (but we do not insist on $y^{i}$ being even). We now use the shorthand
notation

$\partial_{i}=\frac{\partial}{\partial
y^{i}}\quad\text{and}\quad\partial^{i}=\frac{\partial}{\partial x_{i}}.$

Note that the BV Laplacian still reads $\triangle=\partial_{i}\partial^{i}$.

If $\psi$ is an odd function of the variables $y$, it makes sense, in terms of
parity, to set $x_{i}=(-1)^{x_{i}}\partial_{i}\psi$ (we will explain the
reason for the signs in a moment). For a function $f$ in the $(x,y)$
variables, we define

$\int_{\mathcal{L}_{\psi}}f\coloneqq\int
f|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}\;{\mathrm{d}}^{n}y,$

where $|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}$ means that we set each $x$
variable to the corresponding right-hand-side expression. The reason for the
notation is that $\mathcal{L}_{\psi}$ is a Lagrangian submanifold determined
by $\psi$. In fact,
$\omega={\mathrm{d}}x_{i}{\mathrm{d}}y^{i}={\mathrm{d}}(x_{i}{\mathrm{d}}y^{i})$
and

$(x_{i}{\mathrm{d}}y^{i})|_{x_{i}=(-1)^{x_{i}}\partial_{i}\psi}={\mathrm{d}}\psi.$

The first result is the BV lemma.

######  Lemma B.1.

Suppose that $f=\triangle g$ and $g$ is integrable. Then
$\int_{\mathcal{L}_{\psi}}f=0$ for every $\psi$.

###### Proof.

We have

$\begin{split}\sum_{i}\partial_{i}(\partial^{i}g)|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}&=\sum_{i}(\partial_{i}\partial^{i}g)|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}\\\
&\phantom{=}\
+\sum_{ik}(\partial_{i}((-1)^{x_{k}}\partial_{k}\psi)\,\partial^{k}\partial^{i}g)|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}.\end{split}$

Since
$\sum_{ik}(-1)^{x_{k}}(\partial_{i}\partial_{k}\psi\,\partial^{k}\partial^{i}g)=0$,
as one can easily see by exchanging the indices, we get

$(\triangle
g)|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}=\sum_{i}\partial_{i}(\partial^{i}g)|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi},$

so $\int_{\mathcal{L}_{\psi}}f$ vanishes. ∎

This leads to the fundamental BV theorem.

######  Theorem B.2.

Let $\psi_{t}$ be a family of odd functions in the $y$ variables depending
smoothly on the even parameter $t$. If $f$ is integrable, on every
$\mathcal{L}_{\psi_{t}}$, and $\triangle f=0$, then

$I_{t}\coloneqq\int_{\mathcal{L}_{\psi_{t}}}f$

is constant.

###### Proof.

We have

$\dot{I}_{t}=\int\sum_{i}((-1)^{x_{i}}\partial_{i}\dot{\psi}\,\partial^{i}f)|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}\;{\mathrm{d}}^{n}y,$

where the dot denotes derivative with respect to $t$. From

$\triangle(\dot{\psi}\,f)=\sum_{i}\partial_{i}\partial^{i}(\dot{\psi}\,f)=\sum_{i}(-1)^{x_{i}}\partial_{i}(\dot{\psi}\,\partial^{i}f)=\sum_{i}(-1)^{x_{i}}\partial_{i}\dot{\psi}\,\partial^{i}f-\dot{\psi}\,\triangle
f,$

we get

$\dot{I}_{t}=\int\triangle(\dot{\psi}\,f)|_{x_{j}=(-1)^{x_{j}}\partial_{j}\psi}\;{\mathrm{d}}^{n}y=\int_{\mathcal{L}_{\psi_{t}}}\triangle(\dot{\psi}\,f)=0,$

where we have used the previous lemma. ∎

The BV theorem is used in quantum gauge theories,222222In this application,
the odd symplectic manifold is actually infinite dimensional, so $\triangle$
is not defined. What one does is either to proceed formally or to regularize
the theory, e.g., by replacing the space of fields with a finite-dimensional
approximation. where the choice of $\psi$ (called the gauge-fixing fermion)
corresponds to the choice of a gauge fixing and the invariance under
deformations of this choice yields the independence of the theory from the
gauge fixing, assuming of course that $\triangle f=0$.

Using the results of Sections 4 and 5, one can extend the BV lemma and the BV
theorem globally. The main observation is that the restriction of a half-
density on an odd symplectic manifold to a Lagrangian submanifold defines a
density there [13].

######  Theorem B.3.

Let $\mathcal{M}$ be an odd symplectic manifold and $\sigma$ a half-density.
The following hold:

  1. (1)

If $\sigma=\Delta\tau$ for some $\tau$, then

$\int_{\mathcal{L}}\sigma=0$

for every a Lagrangian submanifold $\mathcal{L}$ of $\mathcal{M}$ on which
$\sigma$ is integrable.

  2. (2)

If $\Delta\sigma=0$ and $\mathcal{L}_{t}$ is a smooth family of Lagrangian
submanifolds of $\mathcal{M}$ on which $\sigma$ is integrable, then

$I_{t}\coloneqq\int_{\mathcal{L}_{t}}\sigma$

is constant.

In the case of an odd cotangent bundle $\Pi T^{*}M$, one can show that every
Lagrangian submanifold is a smooth deformation of an odd conormal bundle $\Pi
N^{*}C$, where $C$ is a submanifold of $M$. Moreover, one can show that

$\int_{\Pi N^{*}C}\sigma=\int_{C}\phi(\sigma)$

for every half-density $\sigma$. By Stokes’ theorem and the characterization
of the canonical BV operator as in Proposition 6.1 (i.e.,
$\phi\circ\Delta={\mathrm{d}}\circ\phi$), one gets that

$\int_{\Pi N^{*}C_{1}}\sigma=\int_{\Pi N^{*}C_{2}}\sigma$

if $C_{1}$ and $C_{2}$ are homologous and $\Delta\sigma=0$.

By Schwarz’ theorem one can then generalize the global BV theorem a bit.

######  Theorem B.4.

Let $\mathcal{M}$ be an odd symplectic manifold and $\sigma$ a half-density
satisfying $\Delta\sigma=0$. Then

$\int_{\mathcal{L}_{1}}\sigma=\int_{\mathcal{L}_{2}}\sigma$

whenever $\mathcal{L}_{2}$ can be obtained from $\mathcal{L}_{1}$ by a
combination of smooth deformations and homologous changes of the body,
assuming that $\sigma$ is integrable on each intermediate step.

##  Appendix C Historical remarks

The BV formalism was developed by Batalin and Vilkovisky [1, 2] as a
generalization of the BRST formalism [3, 15], which in turn put gauge fixing
and the Faddeev–Popov determinant [5] into a cohomological setting. They
constructud the BV Laplacian (as in our Section 2) and proved their
fundamental theorem (in our notes, Theorem B.2). In addition, they showed
that, under suitable assumptions, a physical action can be extended to a BV
action $S$ satisfying the classical master equation $(S,S)=0$.

In [16] Witten recognized the relation between the BV operator on functions on
an odd cotangent bundle and the divergence operator on multivector fields on
the body (see the last paragraph of Section 6). The relation between the de
Rham differential and the divergence operator on multivector fields on an
ordinary manifold was already known to Soviet mathematicians in the ’80s (see,
e.g., [12]). Also note that Bernshtein and Leites [4] had already introduced
in 1977 the notion of “integral forms” on supermanifolds—where this notion
differs from that of differential forms—and defined the exterior differential
for them as the divergence operator.

Khudaverdian [6] was the first to give the global definition of the BV
Laplacian on functions, see our equation (5).

Khudaverdian was also the first to observe that an odd symplectic manifold
always admits global Darboux coordinates. This was used by Schwarz in [13],
where he also observed how to globalize and extend the BV theorem (in our
notes, Theorem B.3).

Finally, in [7, 8],232323Note that O. M. Khudaverdian and H. M. Khudaverdian
are just different spellings of the same name. Khudaverdian showed the
existence of a canonical BV Laplacian on half-densities (in our notes, this
corresponds to Theorem/Definition 5.11).

In a series of papers (among others, [9, 10, 11]), Khudaverdian and Voronov
further clarified the properties of the canonical BV Laplacian, essentially
covering all the constructions we present in these notes (and more). In [9]
they extended the construction of the odd Laplacian on half-densities to odd
Poisson manifolds. In [10] they discussed the principal and subprincipal
symbols of the BV Laplacians and their transposed operators (see our Section
5). In [11] they defined the canonical BV Laplacian on half-densities on an
odd cotangent bundle in terms of the de Rham differential of the corresponding
differential forms and then showed that this operator is invariant under all
symplectomorphisms—not just those coming from the base.

It is worth mentioning that Ševera [14] found a completely different
construction for the canonical BV Laplacian. The main observation is that the
complex of differential forms on $\mathcal{M}$ has two commuting coboundary
operators: the de Rham differential ${\mathrm{d}}$ and the operator
$\delta\coloneqq\omega\wedge\ $. It turns out that the associated spectral
sequence lives up to the $E_{2}$-term. More precisely, Ševera shows that
$E_{1}=H_{\delta}(\mathcal{M})$ is canonically isomorphic to
$\operatorname{Dens}^{\frac{1}{2}}(\mathcal{M})$ and that the induced
coboundary operator ${\mathrm{d}}_{1}$ vanishes, which implies that
$E_{2}=E_{1}=\operatorname{Dens}^{\frac{1}{2}}(\mathcal{M})$. Finally, Ševera
proves that the canonically induced coboundary operator ${\mathrm{d}}_{2}$ is
the canonical BV Laplacian and that all higher coboundary operators vanish.

## References

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