Chern-Gauss-Bonnet theorem via BV localization

Vyacheslav Lysov

London Institute for Mathematical Sciences,
Royal Institution, 21 Albemarle St, London W1S 4BS, UK

We present a new proof for the Chern-Gauss-Bonnet theorem. We represent the
Euler class integral as the partition function for zero-dimensional field
theory with on-shell supersymmetry. We rewrite the supersymmetric partition
function as a BV integral and deform the Lagrangian submanifold. The new
Lagrangian submanifold localizes the BV integral to the critical points of the
Morse function.

###### Contents

  1. 1 Introduction
  2. 2 Chern-Gauss-Bonnet theorem
  3. 3 Morse theory
  4. 4 Superspace integral representation for Euler class
  5. 5 Supersymmetric partition function
  6. 6 BV formalism
  7. 7 BV description of the curvature theory
  8. 8 BV localization
  9. 9 Proof of the Chern-Gauss-Bonnet theorem

##  1 Introduction

Chern-Gauss-Bonnet theorem [1] is an equality between the curvature tensor
integral and the Euler characteristics of the compact manifold. Among many
different proofs for this theorem, there are proofs that rely on
supersymmetry. The distinct feature of such proofs [2, 3] is that they serve
as a simple example for various connections between the differential geometry
and quantum field theory (QFT).

The key observation in [2, 3] was that the Hodge-Dirac operator emerges as an
action of supersymmeties on a Hilbert space of the $\mathcal{N}=2$
supersymmetric quantum mechanics (1-dimensional QFT). The index of the Hodge-
Dirac operator, which equals the Euler characteristic, can be expressed via
the twisted partition function in supersymmetric quantum mechanics. The
partition function is an integral over the loop space, known as the path
integral in quantum mechanics. The key feature of the $\mathcal{N}=2$
supersymmetric quantum mechanics is the four-fermion term with the Riemann
tensor coupling. The supersymmetry allows for a deformation of quantum
mechanics by a superpotential, given by a Morse function, without changing the
partition function. Hence, in the limit of a large deformation, the partition
function simplifies to the weighted sum over the critical point of the Morse
function.

The success of the quantum mechanical methods for the proof of Chern-Gauss-
Bonnet theorem inspired similar constructions for the zero-dimensional
supersymmetric QFTs, see [4, 5, 6, 7]. The crucial difference is that the path
integral for zero-dimensional supersymmetric QFT is just an ordinary finite-
dimensional integral over the supermanifold.

We describe a new proof of the Chern-Gauss-Bonnet theorem based on zero-
dimensional supersymmetric QFT. The main difference from existing proofs is
that we use an on-shell supersymmetric theory. The supersymmetry algebra for
such theory closes modulo equations of motion. The advantage of the on-shell
theory is that the corresponding QFT contains only variables that are relevant
to a problem. The even variables are coordinates on a manifold, while the odd
variables are responsible for the integral representation of the Pfaffian and
the differential form integration. The disadvantage is that we need to
generalize the supersymmetric localization to the case of on-shell
supersymmetry.

In our work [8] we used the Batalin-Vilkovisky (BV) formalism [9, 10] to
generalize the supersymmetric localization to a quadratic class on-shell
supersymmetric models. We show that the zero-dimensional supersymmetric QFT
for the Euler class integral belongs to the quadratic class and we can apply
the BV localization. The key feature of the BV localization is the ability to
deform the Lagrangian submanifold without changing the value of the integral.
We use the Lagrangian submanifold, constructed from the Morse function, to
localize the integral to the weighted sum over critical points. The sum,
according to the Morse theorem, equals the Euler characteristic.

Our proof provides a non-trivial consistency test for the novel BV
localization method proposed in [8]. Furthermore, there is a potential to
modify the proof so that the four fermion partition function will emerge as
the BV localization to a non-linear Lagrangian submanifold.

##  2 Chern-Gauss-Bonnet theorem

The Euler characteristic $\chi(X)$ for a smooth manifold $X$ is defined in
terms of its de Rham cohomology via

$\chi(X)=\sum_{k=0}^{\dim X}(-1)^{k}\dim H_{dR}^{k}(X).$ (2.1)

Let us recall the familiar theorem relating two-dimensional topology and
metric curvature.

######  Theorem 2.1.

(Gauss-Bonnet) Let $\Sigma$ be two-dimensional compact, closed manifold with
Riemann metric $g$, then the integral of the Ricci scalar $R$ is related to
the Euler characteristics $\chi(\Sigma)$ of the surface $\Sigma$ via

$\frac{1}{2\pi}\int_{\Sigma}\;R\sqrt{g}=\chi(\Sigma).$ (2.2)

Moreover, there is a generalization of the Gauss-Bonnet theorem for even-
dimensional smooth manifolds.

######  Theorem 2.2.

(Chern-Gauss-Bonnet) Let $X$ be compact closed $2n$-dimensional Riemann
manifold

$\chi(X)=\int_{X}e(\mathcal{R}),$ (2.3)

where $\mathcal{R}$ is the curvature form of (any) metric connection on the
tangent bundle.

The first proof of the theorem was given by Chern [1]. There are multiple
proofs [4, 5, 6, 7] relying on supersymmetry in physics inspired by the works
of Witten [2] and Alvarez-Gaume [3]. The supersymmetric proofs express the
Euler characteristics using the critical points for the superpotential, the
Morse function on $X$. Our proof is no exception, and we will also use Morse
theory results.

##  3 Morse theory

For a smooth manifold $X$ we consider a smooth function $h:X\to\mathbb{R}$,
such that the critical points $\hbox{Crit}(h)=\\{p\in X\;|\;dh(p)=0\\}$ of $h$
is a finite set of points. Critical point $p\in\hbox{Crit}(h)$ is a non-
degenerate critical point if the Hessian matrix $\partial_{i}\partial_{j}h(p)$
has maximal rank. The non-degenerate critical points are isolated. A Morse
function is a smooth function with non-degenerate critical points only. By
Morse lemma, we can arrange the coordinates in the vicinity of a non-
degenerate critical point, such that the Hessian matrix takes a standard form
$\hbox{diag}(1,...,1,-1,...,-1)$. The number of negative entries is the Morse
index ${\gamma}(p)$ of the critical point $p$. We denote the set of critical
points with Morse index ${\gamma}(p)=k$ by $Crit_{k}(h)$.

######  Theorem 3.1.

(Morse) For a smooth function $h:X\to\mathbb{R}$ with non-degenerate critical
points the following holds

  * •

Morse inequality for de Rham cohomology

$\dim H_{dR}^{k}(X)\leq|Crit_{k}(h)|.$ (3.1)

  * •

Morse equality for Euler characteristic

$\begin{split}\chi(X)=\sum_{k=0}^{\dim X}(-1)^{k}|Crit_{k}(h)|.\end{split}$
(3.2)

##  4 Superspace integral representation for Euler class

The first step of the Chern-Gauss-Bonnet theorem’s proof is to represent the
Euler class as a supermanifold integral. Certain signs in the integral
representation will depend on the conventions for the integration over the odd
variables. We define a Berezin integral over multiple odd variables
$\theta_{1},\ldots,\theta_{m}$ via

$\int D\theta_{m}\cdots
D\theta_{1}\;f(\theta_{1},\ldots,\theta_{m})=\frac{\partial}{\partial\theta^{m}}\cdots\frac{\partial}{\partial\theta^{2}}\frac{\partial}{\partial\theta^{1}}f.$
(4.1)

The integration measure in the integral (4.1) is commonly referred to as the
coordinate Berezinian, which we denote as

$D^{m}\theta=D\theta_{m}\cdots D\theta_{1}.$ (4.2)

######  Lemma 4.1.

For an arbitrary Riemann metric $g$ on a smooth $2n$-dimensional compact
manifold $X$ the Euler class has a superspace integral representation

$\int_{X}e(\mathcal{R})=\frac{(-1)^{n}}{(2\pi)^{n}}\int_{T[1]X\oplus
T[1]X}d^{2n}x\;D^{2n}\psi D^{2n}\bar{\psi}\;(\det
g)^{-\frac{1}{2}}\;e^{-\frac{1}{4}R_{abcd}\psi^{a}\psi^{b}\bar{\psi}^{c}\bar{\psi}^{d}},$
(4.3)

where $\mathcal{R}_{ab}$ is the curvature 2-form, related to the Riemann
curvature tensor $R_{abcd}$ via

$\mathcal{R}_{ab}=\frac{1}{2}\sum_{cd}R_{abcd}\;dx^{c}\wedge dx^{d}.$ (4.4)

###### Proof.

The definition of the Euler class is

$e(\mathcal{R})=\frac{1}{(2\pi)^{n}}\hbox{Pf}(\mathcal{R}).$ (4.5)

The Pfaffian of the matrix has a Berezin integral (4.1) representation

$\hbox{Pf}(A)=\frac{(-1)^{n}}{n!2^{n}}\epsilon^{a_{1}b_{1}a_{2}b_{2}...a_{n}b_{n}}A_{a_{1}b_{1}}....A_{a_{n}b_{n}}=\int_{\mathbb{R}^{0|2n}}D^{2n}\bar{\psi}\;\;e^{-\frac{1}{2}A_{ab}\bar{\psi}^{a}\bar{\psi}^{b}}.$
(4.6)

Using the Pfaffian representation (4.6) the Euler class (4.5) is

$e(\mathcal{R})=\frac{(-1)^{n}}{(2\pi)^{n}}\int_{\mathbb{R}^{0|2n}}D^{2n}\bar{\psi}\;\;e^{-\frac{1}{2}\mathcal{R}_{ab}\bar{\psi}^{a}\bar{\psi}^{b}}.$
(4.7)

We can include the factor of $\sqrt{\det g}$ and change the integration over
$\mathbb{R}^{0|2n}$ in (4.7) to the integration over fibers of $T[1]X$, i.e.

$e(\mathcal{R})=\frac{(-1)^{n}}{(2\pi)^{n}}\int_{T[1]X/X}D^{2n}\bar{\psi}\;\;(\det
g)^{-\frac{1}{2}}\;\;e^{-\frac{1}{2}\mathcal{R}_{ab}\bar{\psi}^{a}\bar{\psi}^{b}}.$
(4.8)

The space of differential forms $\Omega^{\ast}(X)$ on $X$ can be identified
with the space of functions on a supermanifold $T[1]X$, while the integration
over $X$ of differential form $\omega(x,dx)$ is identical to the integration
of the corresponding function $\omega(x,\psi)$ over the supermanifold $T[1]X$
with the standard Berezinian, i.e.

$\int_{X}\omega(x,dx)=\int_{T[1]X}d^{2n}xD^{2n}\psi\;\;\omega(x,\psi).$ (4.9)

Using an integral representation (4.8) and a superspace representation (4.9)
for the differential forms integration we arrive at equality (4.3) of the
lemma, hence the proof is complete. ∎

######  Remark 4.2.

There are different signs, used in the literature for the Chern-Gauss-Bonnet
theorem, due to different conventions for the Berezin integration (4.1) and
Pfaffian definition (4.6). The consistency of signs can be checked using
$X=\mathbb{S}^{2n}$, the $2n$-dimensional round sphere of radius $\ell$. The
Euler characteristic is $\chi(\mathbb{S}^{2n})=2$.

The $\mathbb{S}^{2n}$ is a maximally symmetric space, so the Riemann curvature
tensor simplifies to

$R_{abcd}=\frac{R}{2n(2n-1)}(g_{ac}g_{bd}-g_{bc}g_{ad})=\frac{1}{\ell^{2}}(g_{ac}g_{bd}-g_{bc}g_{ad}),$
(4.10)

where $g_{ab}$ is the metric components and $R=\frac{2n(2n-1)}{\ell^{2}}$ is
the Ricci curvature.

The exponent in (4.3) simplifies to

$-\frac{1}{4}R_{abcd}\psi^{a}\psi^{b}\bar{\psi}^{c}\bar{\psi}^{d}=\frac{1}{2\ell^{2}}(g_{ab}\psi^{a}\bar{\psi}^{b})^{2}.$
(4.11)

The integral (4.3) evaluates into

$\begin{split}\chi(\mathbb{S}^{2n})&=\frac{(-1)^{n}}{(2\pi)^{n}}\int_{T[1]\mathbb{S}^{2n}\oplus
T[1]\mathbb{S}^{2n}}d^{2n}x\;D^{2n}\psi D^{2n}\bar{\psi}\;(\det
g)^{-\frac{1}{2}}\;e^{-\frac{1}{4}R_{abcd}\psi^{a}\psi^{b}\bar{\psi}^{c}\bar{\psi}^{d}}\\\
&=\frac{(-1)^{n}}{(2\pi)^{n}}\int_{\mathbb{S}^{2n}}d^{2n}x(\det
g)^{-\frac{1}{2}}\;\frac{(-1)^{n}(2n)!}{\ell^{2n}2^{n}n!}\cdot\det
g=\frac{(2n)!}{\ell^{2n}(4\pi)^{n}n!}\cdot\hbox{Vol}(\mathbb{S}^{2n})\\\
&=\frac{(2n)!}{\ell^{2n}(4\pi)^{n}n!}\cdot\frac{(4\ell^{2}\pi)^{n}(n-1)!}{(2n-1)!}=\frac{2n}{n}=2.\end{split}$
(4.12)

##  5 Supersymmetric partition function

The curvature term in (4.3) is the feature of $\mathcal{N}=2$ supersymmetric
quantum mechanical sigma-models [2, 3] on curved manifolds. Moreover, the
curvature term is the only term that remains if we reduce the dimension of the
source space from one to zero. Hence, we expect it to be invariant under the
zero-dimensional version of the supersymmetry transformations. Furthermore, we
can interpret the integral (4.3) for the Euler class as the supersymmetric
partition function.

######  Lemma 5.1.

The superspace integral (4.3) is the partition function for an $\mathcal{N}=2$
on-shell supersymmetric theory, i.e.

$\int_{X}e(\mathcal{R})=\frac{(-1)^{n}}{(2\pi)^{n}}\int_{M}\mu_{M}\;\;e^{-S}$
(5.1)

on a supermanifold $M=T[1]X\oplus T[1]X$ with coordinates
$x^{i},\psi^{i},\bar{\psi}^{i}$, classical action

$S[x,\psi,\bar{\psi}]=\frac{1}{4}R_{abcd}\psi^{a}\psi^{b}\bar{\psi}^{c}\bar{\psi}^{d}$
(5.2)

and the integration measure $\mu_{M}=(\det
g)^{-\frac{1}{2}}\;\;d^{2n}xD^{2n}\psi D^{2n}\bar{\psi}$.

###### Proof.

The $\mathcal{N}=2$ supersymmetric partition function is a superspace integral
of an exponential function of the classical action, such that the action $S$
and the integration measure $\mu_{M}$ are invariant under the action of the
two odd vector fields ${Q}$ and $\bar{Q}$. Moreover, the vector fields are
nilpotent and anticommute on the critical set of $S$.

We use the supersymmetric quantum mechanics results [2, 3] as a motivation for
the supersymmetry vector fields

$Q=\bar{\psi}^{j}\frac{\partial}{\partial
x^{j}}+\psi^{k}\bar{\psi}^{l}\Gamma_{kl}^{j}\frac{\partial}{\partial\psi^{j}},\;\;\bar{Q}=\psi^{j}\frac{\partial}{\partial
x^{j}}-\psi^{k}\bar{\psi}^{l}\Gamma_{kl}^{j}\frac{\partial}{\partial\bar{\psi}^{j}}.$
(5.3)

The action (5.2) is invariant under the supersymmetry vector fields, i.e.

$Q(S)=\bar{Q}(S)=0.$ (5.4)

Indeed, the vector field’s action evaluates into

$\begin{split}Q(S)&=\left(\bar{\psi}^{j}\frac{\partial}{\partial
x^{j}}+\psi^{k}\bar{\psi}^{l}\Gamma_{kl}^{j}\frac{\partial}{\partial\psi^{j}}\right)R_{abcd}\psi^{a}\psi^{b}\bar{\psi}^{c}\bar{\psi}^{d}\\\
&=\partial_{j}R_{cdab}\bar{\psi}^{j}\bar{\psi}^{c}\bar{\psi}^{d}\psi^{a}\psi^{b}+2\Gamma_{aj}^{e}R_{cdbe}\bar{\psi}^{j}\bar{\psi}^{c}\bar{\psi}^{d}\psi^{a}\psi^{b}=\nabla_{[j}R_{ab]cd}\bar{\psi}^{j}\bar{\psi}^{a}\bar{\psi}^{b}\psi^{c}\psi^{d}=0.\end{split}$
(5.5)

We used the symmetry properties of the Riemann tensor
$R_{abcd}=-R_{bacd}=-R_{cdab}$ and the Bianchi identity
$\nabla_{[j}R_{ab]cd}=0$ in the last equality. The vanishing of $\bar{Q}(S)$
is proven in a similar way.

The invariance of the integration measure $\mu_{M}$ is equivalent to the
divergence-free condition for the supersymmetry vector fields, i.e.

$\hbox{div}_{\mu_{M}}Q=\hbox{div}_{\mu_{M}}\bar{Q}=0.$ (5.6)

The divergence of the vector field $v$ with respect to the measure
$\mu=\rho\cdot\mu_{c}$ is given by

$\hbox{div}_{\mu_{M}}v=\hbox{div}_{\mu_{c}}v+v(\ln\rho),$ (5.7)

For the vector field

$v=v^{a}\frac{\partial}{\partial
x^{a}}+u^{a}\frac{\partial}{\partial\psi^{a}}+\bar{u}^{a}\frac{\partial}{\partial\bar{\psi}^{a}}$
(5.8)

the divergence in the standard measure $\mu_{c}=d^{2n}xD^{2n}\psi
D^{2n}\bar{\psi}$ is

$\hbox{div}_{\mu_{c}}v=\partial_{x^{a}}v^{a}-(-1)^{|v|}\partial_{\psi^{a}}u^{a}-(-1)^{|v|}\partial_{\bar{\psi}^{a}}\bar{u}^{a}.$
(5.9)

For the vector fields (5.3) the divergence evaluates into

$\begin{split}\hbox{div}_{\mu_{X}}Q&=\hbox{div}_{\mu_{c}}Q+Q(\ln(\det
g)^{-\frac{1}{2}})=\frac{1}{2}\bar{\psi}^{a}\partial_{a}\ln\det
g-\frac{1}{2}\bar{\psi}^{a}\partial_{a}\ln\det g=0,\\\
\hbox{div}_{\mu_{X}}\bar{Q}&=\hbox{div}_{\mu_{c}}\bar{Q}+\bar{Q}(\ln(\det
g)^{-\frac{1}{2}})=\frac{1}{2}\psi^{a}\partial_{a}\ln\det
g-\frac{1}{2}\psi^{a}\partial_{a}\ln\det g=0.\end{split}$ (5.10)

The algebra of supersymmetry vector fields (5.3) is

$\begin{split}[Q,Q\\}&=-\psi^{b}\bar{\psi}^{c}\bar{\psi}^{d}R_{~{}bcd}^{a}\partial_{\psi^{a}}=-2g^{ab}\;\partial_{\psi^{b}}S\;\partial_{\psi^{a}},\\\
[\bar{Q},\bar{Q}\\}&=-\psi^{c}\psi^{d}\bar{\psi}^{b}R_{~{}bcd}^{a}\partial_{\bar{\psi}^{a}}=-2g^{ab}\;\partial_{\bar{\psi}^{b}}S\;\partial_{\bar{\psi}^{a}},\\\
[\bar{Q},Q\\}&=\psi^{c}\psi^{d}\bar{\psi}^{b}R_{~{}bcd}^{a}\partial_{\psi^{a}}+\bar{\psi}^{c}\bar{\psi}^{d}\psi^{b}R_{~{}bcd}^{a}\partial_{\bar{\psi}^{a}}=-g^{ab}\;\partial_{\psi^{b}}S\;\partial_{\bar{\psi}^{a}}-g^{ab}\;\partial_{\bar{\psi}^{b}}S\;\partial_{\psi^{a}}.\end{split}$
(5.11)

The right-hand side of the graded commutators in (5.11) vanish on the critical
set $M_{crit}=\\{\phi\in M\;|\;dS(\phi)=0\\}$ hence the vector fields become
nilpotent and anticommuting. We checked all required properties of the
supersymmetric partition function, so the proof is complete.

∎

In physics terminology the supersymmetry algebra (5.11) is the on-shell $d=0$
$\mathcal{N}=2$ supersymmetry algebra. The number $\mathcal{N}$ refers to the
number of independent supersymmetries, i.e. the dimension of the corresponding
odd part of the superalgebra. The term on-shell indicates that the nontrivial
commutators of supercharges are proportional to the equations of motion for
the action (5.2), i.e.

$\frac{\partial
S}{\partial\psi^{j}}=\frac{1}{2}R_{jbcd}\psi^{b}\bar{\psi}^{c}\bar{\psi}^{d},\;\;\;\frac{\partial
S}{\partial\bar{\psi}^{j}}=\frac{1}{2}R_{abjd}\psi^{a}\psi^{b}\bar{\psi}^{d}.$
(5.12)

Therefore, as long as equations of motions are satisfied, the on-shell
supersymmetry algebra is the same as the off-shell one.

$[Q,Q\\}=[\bar{Q},Q\\}=[\bar{Q},\bar{Q}\\}=0.$ (5.13)

##  6 BV formalism

For a supermanifold $M$ there is an odd symplectic manifold
$\mathcal{M}=T^{\ast}[1]M$ with the fiber coordinates known as the antifeilds.
In particular, for $M=T[1]X\oplus T[1]X$ with coordinates $x,\psi,\bar{\psi}$
the corresponding antifields are $x^{\ast},\psi^{\ast},\bar{\psi}^{\ast}$. The
canonical symplectic form on $\mathcal{M}$ is

$\omega=dx^{i}\wedge dx^{\ast}_{i}-d\psi^{i}\wedge
d\psi^{\ast}_{i}-d\bar{\psi}^{i}\wedge d\psi^{\ast}_{i}.$ (6.1)

The BV bracket for two functions $f,g$ with parities $|f|,|g|$ is

$\begin{split}\\{f,g\\}&=\partial_{x^{i}}f\;\partial_{x^{\ast}_{i}}g+(-1)^{|f|}\partial_{x^{\ast}_{i}}f\;\partial_{x^{i}}g-\partial_{\psi^{\ast}_{i}}f\;\partial_{\psi^{i}}g-(-1)^{|f|}\partial_{\psi^{i}}f\;\partial_{\psi^{\ast}_{i}}g\\\
&\qquad-\partial_{\bar{\psi}^{\ast}_{i}}f\;\partial_{\bar{\psi}^{i}}g-(-1)^{|f|}\partial_{\bar{\psi}^{i}}f\;\partial_{\bar{\psi}^{\ast}_{i}}g.\end{split}$
(6.2)

A Berezinian

$\mu=\rho(x,\psi,\bar{\psi})\;d^{2n}xD^{2n}\psi
D^{2n}\bar{\psi}D^{2n}x^{\ast}d^{2n}\psi^{\ast}d^{2n}\bar{\psi}$ (6.3)

on $\mathcal{M}$ is a compatible Berezinian on $(\mathcal{M},\omega)$, while
the corresponding BV Laplacian is

$\Delta_{\mu}=\frac{\partial}{\partial x^{i}}\frac{\partial}{\partial
x^{\ast}_{i}}-\frac{\partial}{\partial\psi^{i}}\frac{\partial}{\partial\psi^{\ast}_{i}}-\frac{\partial}{\partial\bar{\psi}^{i}}\frac{\partial}{\partial\bar{\psi}^{\ast}_{i}}+\frac{1}{2}\\{\ln\rho,\cdot\\}.$
(6.4)

######  Definition 6.1.

For an odd symplectic space with compatible Berezinian
$(\mathcal{M},\omega,\mu)$ an even function $\mathcal{S}\in
C^{\infty}(\mathcal{M})$ is a solution to the quantum master equation (QME) if
it satisfies

$\frac{1}{2}\\{\mathcal{S},\mathcal{S}\\}=\Delta_{\mu}\mathcal{S}=0.$ (6.5)

In physics literature, a solution to QME is often called a BV action. Given a
QME solution $\mathcal{S}$ there is a $\Delta_{\mu}$-closed exponential
function $f=e^{-\mathcal{S}}$.

######  Remark 6.2.

Our definition (6.5) for the quantum master equation differs from the
commonly-used definition

$\frac{1}{2}\\{\mathcal{S},\mathcal{S}\\}=\hbar\Delta_{\mu}\mathcal{S}$ (6.6)

in the BV literature [9, 10, 11, 12]. Our solution (6.5) to the QME is the
$\hbar$-independent solution of (6.6). An additional restriction on a solution
ensures that the exponent $f=e^{-\frac{1}{\hbar}\mathcal{S}}$ is not a formal
series expansion in $\hbar$, but rather a well-defined function.

######  Definition 6.3.

A Lagrangian submanifold $\mathcal{L}$ of an odd symplectic manifold
$(\mathcal{M},\omega)$ is

  * •

isotropic, i.e. $\omega|_{\mathcal{L}}=0$;

  * •

not a proper submanifold of another isotropic submanifold of $\mathcal{M}$.

For an odd function $V$ on $M$, there is an associate Lagrangian submanifold

${\cal L}_{V}=\hbox{graph}(dV)\subset T^{\ast}[1]M.$ (6.7)

For a supermanifold $\mathcal{N}$ there is a canonical map sending Berezinians
$\mu$ on $T^{\ast}[1]\mathcal{N}$ to Berezinians $\sqrt{\mu|_{\mathcal{N}}}$
on $\mathcal{N}$. Locally, for local coordinates $X^{a}$ on $\mathcal{N}$,
coordinates $(X^{a},\Xi_{a})$ on $T^{\ast}[1]\mathcal{N}$ and a Berezinian
$\mu=\rho(X,\Xi)d^{n}Xd^{n}\Xi$ the corresponding Berezinian in $\mathcal{N}$
is $\sqrt{\mu|_{\mathcal{N}}}=\sqrt{\rho(X,0)}d^{n}X$.

######  Definition 6.4.

For odd symplectic manifold with compatible Berezinian
$(\mathcal{M},\omega,\mu)$ a BV-integral of a function $f\in
C^{\infty}(\mathcal{M})$, such that $\Delta_{\mu}f=0$, over Lagrangian
submanifold $\mathcal{L}$ is an integral

$\int_{\mathcal{L}}f\;\sqrt{\mu|_{\mathcal{L}}}.$ (6.8)

######  Theorem 6.5.

(Batalin-Vilkovisky-Schwarz [11]) For an odd-symplectic manifold with
compatible Berezinian $(\mathcal{M},\omega,\mu)$ and compact body.

  1. 1.

For any $g\in C^{\infty}(\mathcal{M})$ and Lagrangian submanifold
$\mathcal{L}\subset\mathcal{M}$ the BV integral

$\int_{\mathcal{L}}\Delta_{\mu}g\;\sqrt{\mu|_{\mathcal{L}}}=0.$ (6.9)

  2. 2.

For a pair of Lagrangian submanifolds $\mathcal{L}$ and $\mathcal{L}^{\prime}$
with homologous bodies in the body of $\mathcal{M}$ and a function, such that
$\Delta_{\mu}f=0$ the following holds

$\int_{\mathcal{L}}f\;\sqrt{\mu|_{\mathcal{L}}}=\int_{\mathcal{L}^{\prime}}f\;\sqrt{\mu|_{\mathcal{L}^{\prime}}}.$
(6.10)

##  7 BV description of the curvature theory

According to results of [8] a supersymmetric partition function (5.1) defines
an approximate solution to the QME

$\mathcal{S}(\phi,\phi^{\ast})=S(\phi)+\mathcal{Q}+\Pi^{(2)}.$ (7.1)

The first term in (7.1) is the classical action (5.2) for the supersymmetric
system. The linear term in antifields, $\mathcal{Q}$, is the generator of
supersymmetries (5.3)

$\mathcal{Q}=({\varepsilon}\bar{\psi}^{j}+\bar{{\varepsilon}}\psi^{j})x_{j}^{\ast}+\psi^{k}\bar{\psi}^{l}\Gamma_{kl}^{j}({\varepsilon}\psi^{\ast}_{j}-\bar{{\varepsilon}}\bar{\psi}^{\ast}_{j}).$
(7.2)

The quadratic term in antifields $\Pi^{(2)}$ is a BV bivector, given by the
commutation relations (5.11)

$\Pi^{(2)}=\frac{1}{2}g^{kj}({\varepsilon}\psi^{\ast}_{j}-\bar{{\varepsilon}}\bar{\psi}^{\ast}_{j})({\varepsilon}\psi^{\ast}_{k}-\bar{{\varepsilon}}\bar{\psi}^{\ast}_{k}).$
(7.3)

Note that the bivector is not a constant bivector as in simple examples of
[8]. However it satisfies the necessary properties, so the QME solution (7.1)
is a quadratic type theory according to the refined classification of [8].

######  Proposition 7.1.

The BV action

$\begin{split}\mathcal{S}&=\frac{1}{4}R_{abcd}\psi^{a}\psi^{b}\bar{\psi}^{c}\bar{\psi}^{d}+({\varepsilon}\bar{\psi}^{j}+\bar{{\varepsilon}}\psi^{j})x_{j}^{\ast}+\psi^{k}\bar{\psi}^{l}\Gamma_{kl}^{j}({\varepsilon}\psi^{\ast}_{j}-\bar{{\varepsilon}}\bar{\psi}^{\ast}_{j})\\\
&\qquad\qquad\qquad+\frac{1}{2}g^{kj}({\varepsilon}\psi^{\ast}_{j}-\bar{{\varepsilon}}\bar{\psi}^{\ast}_{j})({\varepsilon}\psi^{\ast}_{k}-\bar{{\varepsilon}}\bar{\psi}^{\ast}_{k})\end{split}$
(7.4)

solves the quantum master equation on an odd symplectic space
$\mathcal{M}=T^{\ast}[1]M$ with canonical symplectic form (6.1) and a
compatible Berezinian

$\mu=\frac{1}{\det g}d^{2n}xD^{2n}\psi
D^{2n}\bar{\psi}D^{2n}x^{\ast}d^{2n}\psi^{\ast}d^{2n}\bar{\psi}^{\ast}.$ (7.5)

###### Proof.

The statement of proposition is that the approximate solution (7.1) does not
aquire higher order terms in antifields. Hence, we need to check the QME for
orders $\mathcal{O}(\phi^{\ast})^{2}$, $\mathcal{O}(\phi^{\ast})^{3}$ and the
triviality of BV Laplacian for the bivector (7.3).

The $\mathcal{O}(\phi^{\ast})^{2}$ order QME

$\begin{split}\\{\mathcal{Q},\Pi^{(2)}\\}&=\\{({\varepsilon}\bar{\psi}^{j}+\bar{{\varepsilon}}\psi^{j})x_{j}^{\ast},\Pi^{(2)}\\}+\\{\psi^{k}\bar{\psi}^{l}\Gamma_{kl}^{j}({\varepsilon}\psi^{\ast}_{j}-\bar{{\varepsilon}}\bar{\psi}^{\ast}_{j}),\Pi^{(2)}\\}\\\
&=-\frac{1}{2}\left(\partial_{j}g^{kl}+2g^{kc}\Gamma_{cj}^{l}\right)({\varepsilon}\bar{\psi}^{j}+\bar{{\varepsilon}}\psi^{j})({\varepsilon}\psi^{\ast}_{l}-\bar{{\varepsilon}}\bar{\psi}^{\ast}_{l})({\varepsilon}\psi^{\ast}_{k}-\bar{{\varepsilon}}\bar{\psi}^{\ast}_{k})=0.\end{split}$
(7.6)

The last equality follows from the symmetrization over $kl$ for the identity

$\begin{split}\partial_{j}g^{kl}+2g^{kc}\Gamma_{cj}^{l}=g^{ka}g^{lb}(\partial_{a}g_{bj}-\partial_{b}g_{aj})\end{split}$
(7.7)

The $\mathcal{O}(\phi^{\ast})^{3}$ order QME

$\\{\Pi^{(2)},\Pi^{(2)}\\}=0$ (7.8)

is rather trivial, since the bivector has only components along the fermionic
antifields $\psi^{\ast},\bar{\psi}^{\ast}$, while the coefficients, for these
components depend on $x$ only.

The BV Laplacian for bivector

$\Delta_{\mu}\Pi^{(2)}=\Delta_{\mu_{c}}\Pi^{(2)}+\frac{1}{2}\\{\ln\det
g(x),\Pi^{(2)}\\}=0$ (7.9)

is almost trivial since the determinant of metric $\det g$ has only
$x$-dependence while the $\Pi^{(2)}$ is independent on $x^{\ast}$. ∎

##  8 BV localization

The supersymmetric partition function (5.1) has a BV integral (6.8)
representation. In particular, it equals to the QME solution (7.4) integrated
over trivial Lagrangian submanifold ${\cal L}_{0}$, the zero section of the
$T^{\ast}[1]M$ bundle

$\frac{1}{(2\pi)^{n}}\int_{M}\mu_{M}\;\;e^{-S}=\frac{1}{(2\pi)^{n}}\int_{{\cal
L}_{0}}\sqrt{\mu}\;\;e^{-\mathcal{S}}.$ (8.1)

For a Morse function $W:X\to\mathbb{R}$, we define a Lagrangian submanifold
${\cal L}_{V}$, associated to with odd function $V=t\psi^{i}\partial_{i}W$. In
our coordinates

$\mathcal{L}_{V}=\left\\{x^{\ast}_{i}=t\psi^{j}\partial_{i}\partial_{j}W(x),\;\;\psi^{\ast}_{i}=t\partial_{i}W(x),\;\;\bar{\psi}^{\ast}_{i}=0\right\\}\subset
T^{\ast}[1]M.$ (8.2)

######  Lemma 8.1.

The $t\to\infty$ limit of the BV integral for the QME solution (7.4) with
${\varepsilon}=1,\;\bar{{\varepsilon}}=0$ over the Lagrangian submanifold
(8.2), is expressed via Morse indices of $W$, i.e.

$I_{V}=\frac{(-1)^{n}}{(2\pi)^{n}}\int_{{\cal
L}_{V}}\sqrt{\mu}\;\;e^{-\mathcal{S}}=\sum_{p\in Crit(W)}(-1)^{{\gamma}(p)}.$
(8.3)

###### Proof.

The restriction of the QME solution (7.4) to the Lagrangian submanifold (8.2)
evaluates into

$\mathcal{S}\Big{|}_{\mathcal{L}_{V}}=\frac{1}{4}R_{abcd}\psi^{a}\psi^{b}\bar{\psi}^{c}\bar{\psi}^{d}+t\partial_{j}\partial_{k}W\psi^{k}\bar{\psi}^{j}+t\psi^{k}\bar{\psi}^{l}\Gamma_{kl}^{j}\partial_{j}W+\frac{1}{2}t^{2}g^{kj}\partial_{j}W\partial_{k}W.$
(8.4)

The BV integral in the $t\to\infty$ simplifies since we can drop the curvature
term in (8.4) comparing to the $t$-enhanced quadratic fermionic terms

$I_{V}=\frac{(-1)^{n}}{(2\pi)^{n}}\int_{M}d^{2n}xD^{2n}\psi
D^{2n}\bar{\psi}\;\;(\det
g)^{-\frac{1}{2}}\exp\left(-t\nabla_{j}\partial_{k}W\psi^{k}\bar{\psi}^{j}-\frac{1}{2}t^{2}g^{kj}\partial_{j}W\partial_{k}W\right).$

The integral over fermions is Gaussian, so we can perform it exactly

$\int
D^{2n}\psi\;D^{2n}\bar{\psi}\;\;\exp\left(-t\nabla_{j}\partial_{k}W\psi^{k}\bar{\psi}^{j}\right)=(-1)^{n}t^{2n}\det(\nabla_{j}\partial_{k}W).$
(8.5)

An expression $g^{kj}\partial_{j}W\partial_{k}W$ is non-negative on $X$ and
takes zero values at critical points of $W$. Hence the integral over $X$ for
large $t$ localizes to the neighborhood of critical points of $W$, i.e.

$I_{V}=\frac{(-1)^{2n}}{(2\pi)^{n}}\sum_{p\in\hbox{Crit}(W)}\int_{U_{p}}d^{2n}x\;(\det
g)^{-\frac{1}{2}}\;\exp\left(-\frac{1}{2}t^{2}g^{kj}\partial_{j}W\partial_{k}W\right)\;t^{2n}\det(\nabla_{j}\partial_{k}W).$
(8.6)

The integral (8.6) is invariant under the coordinate transformations on $X$.
By the Morse lemma, we can choose the local coordinates, such that near the
critical point $p$ Morse function $W$ takes the form

$W=W(p)-\frac{1}{2}\sum_{j=1}^{{\gamma}(p)}(y^{j})^{2}+\frac{1}{2}\sum_{j={\gamma}(p)+1}^{2n}(y^{j})^{2},$
(8.7)

where ${\gamma}(p)$ is the Morse index of the critical point $p$. The
determinant in Morse coordinates

$\det(\nabla_{j}\partial_{k}W)(p)=(-1)^{{\gamma}(p)}.$ (8.8)

We perform a variable change $z^{i}=ty^{i}$ so that the argument of the
exponent near the critical point $p$ simplifies

$t^{2}g^{kj}(x)\frac{\partial W}{\partial x^{j}}\frac{\partial W}{\partial
x^{k}}=t^{2}\tilde{g}^{kj}(y)y_{j}y_{k}=t^{2}\hat{g}^{kj}(t^{-1}z)z_{j}z_{k}=t^{2}\hat{g}^{kj}(0)z_{j}z_{k}+\mathcal{O}(t),$
(8.9)

where $\tilde{g}$ is the metric in $y$ coordinates and $\hat{g}$ is the metric
in $z$-coordinates. The integral (8.6) near the critical point $p$ for large
$t$ evaluates into

$\begin{split}I_{V}&=\sum_{p\in\hbox{Crit}(W)}\frac{(-1)^{{\gamma}(p)}t^{2n}}{(2\pi)^{n}}\int_{\mathbb{R}^{2n}}d^{2n}z\;(\det\hat{g})^{-\frac{1}{2}}\;\exp\left(-\frac{1}{2}t^{2}\hat{g}^{kj}(0)z_{j}z_{k}+\mathcal{O}(t)\right)\\\
&=\sum_{p\in\hbox{Crit}(W)}\frac{(-1)^{{\gamma}(p)}t^{2n}}{(2\pi)^{n}}\left(\frac{2\pi}{t^{2}}\right)^{n}(\det\hat{g}(0))^{-\frac{1}{2}}\frac{1}{\sqrt{\det\hat{g}^{-1}}(0)}(1+\mathcal{O}(t^{-1}))\\\
&=\sum_{p\in\hbox{Crit}(W)}(-1)^{{\gamma}(p)}+\mathcal{O}(t^{-1}).\end{split}$

The $t\to\infty$ limit removes the $\mathcal{O}(t^{-1})$ corrections and we
arrive into (8.3), so the proof is complete. ∎

######  Remark 8.2.

We can modify the lemma 8.1 if we use the vector field $v$ with isolated
critical points instead of the Morse vector field $g^{-1}(dW,\cdot)$. The
corresponding Lagrangian submanifold is associated with the odd function
$V=\psi^{j}g_{jk}v^{k}(x)$. The result of the BV integration in (8.3) over
this Lagrangian submanifold equals the sum over zeros of the vector field,
weighted by the corresponding indices. The sum is identical to the sum in the
Poincare-Hopf theorem, representing the Euler characteristics of $X$.

##  9 Proof of the Chern-Gauss-Bonnet theorem

The first step of the proof is to use the lemma 4.1 and turn the Euler class
integral over $X$ into the supermanifold integral over $M=T[1]X\oplus T[1]X$.
The next step is to use the lemma 5.1 to represent the supermanifold integral
as a supersymmetric partition function with on-shell supersymmetry. The third
step is to rewrite (8.1) the supersymmetric partition function as a BV
integral over the trivial Lagrangian submanifold for the BV action (7.4). In
particular, the proposition 7.1 implies that the on-shell supersymmetric
theory with action (5.2) is a quadratic type of BV supersymmetric theory in
refined classification from [8].

The key result in BV formalism is that the BV integral for the solution of the
quantum master equation is invariant under the deformations of the Lagrangian
submanifold, formalized by the theorem 6.5. The lemma 8.1 describes the BV
integral over limiting Lagrangian submanifold, defined by the Morse function
$W$ on $X$ and expresses the final result in terms of Morse index for the
function $W$. The Morse theorem 3.1 implies a relation between the weighted
sum in (8.3) and the Euler characteristic of $X$ and completes the proof.

## Acknowledgments

V.L. is grateful to Yasha Neiman, Andrey Losev, and Pavel Mnev for the useful
discussions on the material of this paper. The work of V.L. is supported by
the Arnold Fellowship at the London Institute for Mathematical Sciences.

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