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%Lower case a b c d e f g h i j k l m n o p q r s t u v w x y z
%Digits 0 1 2 3 4 5 6 7 8 9
%Exclamation ! Double quote " Hash (number) #
%Dollar $ Percent % Ampersand &
%Acute accent ' Left paren ( Right paren )
%Asterisk * Plus + Comma ,
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%Colon : Semicolon ; Less than <
%Equals = Greater than > Question mark ?
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%MACROS
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\parskip = 1.5ex plus .5ex minus .1ex
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\def\Box{\blacksquare}
\input amssym.def
\input amssym
%**************************************references
\newcount\refno
\def\Refname#1{\relax \global\advance\refno by 1
\xdef#1{{\noexpand\rm[\number\refno]}}#1}
%
\def\Ref#1{\lbrack #1\rbrack}
\def\Adref{\Ref{??}}
\def\Bib#1#2#3#4#5#6{%\frenchspacing
\item{[#1]}#2, ``#3'',{\bf #4} (#5), #6.}
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%Secno etc
\newcount\henry%this is section number
\newcount\figgle%this is equation number
\newcount\david%this is theorem, definition etc number
%
\def\Secno#1{\david=0 \figgle = 0 \global\advance \henry by 1
\vskip 0.2truein
{\noindent \bf\number\henry .\ \ #1} \vskip 0.1truein}
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%
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%
%**********************************************************sundries
%
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\def\frac#1#2{{#1\over #2}}
%
%********************************************************symbols
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\def\Ebold{\Bbbc{E}}
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\def\Gam{\gamma^5}
\def\Tra{{\rm trace}\,}
\def\Stra{{\rm supertrace}\,}
\def\Han{{n\over2}}
\def\Ham{{m\over2}}
\def\Sumam{\sum_{a=1}^m}
\def\Sumim{\sum_{i=1}^m}
\def\Sumrn{\sum_{r=1}^n}
\def\Sumu{\sum_{\mu\in M_m}}
\def\Sumkhn{\sum_{a=1}^{\Han}}
\def\Deltint#1#2#3{\exp i\Sumam #1^a (#2^a#3^a) }
\def\Comp{\Bbbc{C}}
\def\cinf{C^{\infty}}
%section2
\def\Cinfp{\cinf{}'(S(E),\Comp)}
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\def\Det#1{\delta_{\eta^{#1}}}
\def\Tdel#1#2{\theta^{#1}\delta_{\theta^{#2}}}
\def\Edet#1#2{\eta^{#1}\delta_{\eta^{#2}}}
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\theta^i\theta^k\Del j
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\def\Wbokminus{-R_i^j(x)\Tdel ij - \half R_{ki}{}^{j\ell}(x)
\theta^i\theta^k\Del j
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%brackets************
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\def\Lbd{\Biggl(}
\def\Rba{\bigr)}
\def\Rbb{\Bigr)}
\def\Rbc{\biggr)}
\def\Rbd{\Biggr)}
\def\Lsb{\Bigl[}
\def\Rsb{\Bigr]}
\def\Lsc{\biggl[}
\def\Rsc{\biggr]}
%*******************************section2
\def\Hdr{Hodge de Rham}
\def\Alp{{\textstyle{1\over\sqrt2}}}
\def\str{{\rm str}}
\def\Str{{\rm Str}}
\def\Mfun{\tau_{\alpha\beta}}
\def\Efun{g_{\alpha\beta}}
\def\Xiab{\Xi^{ab}}
%section3
\def\Sumim{\sum_{i=1}^m}
\def\Sumam{\sum_{a=1}^m}
\def\Sumip{\sum_{i=1}^p}
\def\xit#1#2{x^{#1}_{#2}}
\def\eiat#1#2#3{e^{#1}_{#2,#3}}
\def\Gamimk#1#2#3{\Gamma_{#2#3}^{#1}}
\def\Som{S(O(M),E)}
\def\Lscal{L_{Scal}}
%******************************section3
%\def\Aroof{\hat A}
\def\str{{\rm str\,}}
%******************************section4
\def\Str{{\rm Str}\,}
\def\str{{\rm str}\,}
\def\Asint{\Biggl[{\rm tr}\,
\exp\Lbc {-F\over2\pi}\Rbc \det \Lbc {i\Omega/2\pi\over\tanh
i\Omega/2\pi}\Rbc^{\half}\Biggr]}%Integrand for AS theorem
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i\rho^i \theta^j g_{ij}(x)}
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\def\Cuft#1#2#3#4{\tilde{F}_{#1#2\,#3}{}^{#4}}
\def\Hamt{{\tilde H}}
\def\Hamtz{{\tilde H}^0}
\def\Hamtone{{\tilde H}^1}
\def\Lap{\half(d+\delta)^2}
\def\Sflat{S(\Real^m\times\Comp^n)}
\def\Pri#1{#1'{}}
\def\Kert#1{\Til K_{#1}(x,\Pri x,\theta,\Pri{\theta},\eta,
\Pri{\eta})}
\def\Kertz#1{\Til K_{#1}^0(x,\Pri
x,\theta,\Pri{\theta},\eta,\Pri{\eta})}
\def\Kertzz#1{\Til K_{#1}^0(0,0,\theta,\Pri{\theta},\eta,
\Pri{\eta})}
\def\Fac#1{(2\pi #1)^{-m/2}}
\def\Strk{\str \Til{K}_t(0,0)}
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\def\Fs{F_s(x_s,\theta_s,\rho,\eta_s,\eta,\kappa)}
%\def\Step#1{{\noindent\bf Step #1}}
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\def\Etilo#1#2#3{\Che{e}^{#1}_{#2\,#3}}
\def\Xtilo#1#2{\Che{x}^{#1}_{#2}}
\def\Xitilo#1#2{\Che{\xi}^{#1}_{#2}}
\def\Rhotilo#1#2{\Che{\rho}^{#1}_{#2}}
\def\Pitilo#1#2{\Che{\pi}^{#1}_{#2}}
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\def\Etatiloo#1{\Che{\eta}_{#1}}
\def\Xtil#1{\Til{x}_{#1}}
\def\Xitil#1{\Til{\xi}_{#1}}
\def\Rhotil#1{\Til{\rho}_{#1}}
\def\Pitil#1{\Til{\pi}_{#1}}
%\def\Sigmatiloo#1{\Che{\sigma}_{#1}}
\def\Etatil#1{\Til{\eta}_{#1}}
\def\Etil#1#2#3{\Til{e}^{#1}_{#2\,#3}}
\def\Rtil#1#2#3#4{\Che{R}_{#1#2#3#4}}
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\def\Fnotil#1#2#3#4{{F}_{#1#2\,#3}{}^{#4}}
\def\Thetap#1{{\theta^
{#1}\over\sqrt t}}
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\def\Xhatt#1{\frac{\hat{x}^{#1}}{\textstyle3}}
\def\Tmt{t^{m\over2}}
\def\Fint{\Bint d^m\theta\,d^n\eta\,d^n\kappa \Intt ds\,\Fac s}
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ds\,
\Fac s}
\def\Phit{{\phi\over\sqrt {2\pi t}}}
\def\Phitt#1{{\phi^{#1}\over\sqrt {2\pi t}}}
\def\Fs{F_s(x,\xi,\Thetap{},\alpha,{\eta})}
\def\Evalp{\Eval{x=\Til x_{t-s},\xi = \Til{\xi}_{t-s}, \eta =
\Til{\eta}_{t-s}}}
\def\Evalo{\Eval{x=\Che x_{t-s},\xi = \Che{\xi}_{t-s}, \eta =
\Che{\eta}_{t-s}}}
\def\Evalf{\Eval{x=b_{t-s},\xi = \theta_{t-s}, \alpha =
{\eta}^0_{t-s}}}
\def\Etaz#1#2{\eta^{0\,#1}_{#2}}
\def\Kappaz#1#2{\kappa^{0}_{#1\,#2}}
\def\Thetaz#1#2{\theta^{#1}_{#2}}
\def\Rhoz#1#2{{\rho}^{#1}_{#2}}
\def\Bz#1#2{b^{#1}_{#2}}
\def\Integrand{\Lsb\exp \Lbb\int_{0}^{t-s} \Thetap i\Thetap j \Ftil
ijpq
\Etaz qu \Kappaz pu du + \Rtil ijkl \Thetaz iu\Thetaz ju\Rhoz ku
\Rhoz lu\cr
&+\half\Rtil i{}j{}\Thetaz iu\Rhoz ju du + \Thetaz iu\Rhoz ju \Bz ku
\Rtil ijkl d\Bz lu + \Thetaz iu\Rhoz ju \Bz ku \Rtil ijkl\Thetaz iu
\Rhoz ju \Bz ku \Rtil ijkl du \Rbb}
\def\Limt{\lim_{t\to0}}
\def\Hamf{\Til{H}^2}
\def\Hamz{\Til{H}^0}
\def\Etah#1{\hat{\eta}^{#1}}
\def\Psih#1{\hat{\chi}^{#1}}
\def\Curv#1#2{\Omega_{#1}{}^{#2}}
\def\Fer#1#2{\theta^{#1}_{#2}-i\pi^{#1}_{#2}}
%
\hyphenation{super-mani-fold}
\hyphenation{anti-commuting}
%END OF MACROS
\henry =0%section counter
%FRONT PAGE
{\nopagenumbers
\vglue 1truein
\centerline{STOCHASTIC CALCULUS IN
SUPERSPACE II: differential forms,}
\centerline{supermanifolds and the Atiyah-Singer
index
theorem\footnote*{Research supported by the Royal Society}}
\bigskip
\centerline{Alice Rogers}
\bigskip
\centerline{Department of Mathematics}
\centerline{King's College}
\centerline{Strand}
\centerline{London WC2R 2LS}
\centerline{E-mail F.A.Rogers @ UK.AC.KCL.CC.OAK}
\bigskip
\centerline{July 1992}
\vskip 1truein
\centerline{Abstract}
\medskip
Starting with vector bundles over manifolds, supermanifolds are
constructed whose function algebras correspond to twisted differential
forms. Stochastic calculus for bosonic and fermionic Brownian
paths is used to provide a geometric construction of Brownian paths on
these supermanifolds. A Feynman-Kac formula for the heat kernel of the
Laplace-Beltrami operator is then derived. This is used to provide a
simple, rigorous version of the supersymmetric proofs of the
Atiyah-Singer index theorem.
\vfil
PACS 02.40,02.50
\eject}
%SECTION 1
\Secno{Introduction}
In this paper superspace stochastic calculus is
used to define covariant Brownian paths on supermanifolds, and thus to
extend fermionic path integration to curved space. The construction is
used to study the Laplace-Beltrami operator on differential forms on a
Riemannian manifold.
The stochastic techniques employed in this paper are somewhat different
than those in other work where probabilistic methods are used to study
the Laplace-Beltrami operator on forms and related objects. The key
difference is the use of fermionic Brownian paths which are paths in a
space of anticommuting variables. This approach, which is described in
\Refname\GBM\ and the companion
paper \Refname\Scal, is designed to resemble as closely as possible both
standard Wiener paths and the
fermionic paths of the physics literature which were introduced by
Martin \Refname\Mar\ and have proved a powerful tool in heuristic
calculations. Rigorous fermionic path integration along these
lines has also been considered by Haba \Refname\Haba.
Superspace stochastic calculus considers Brownian paths in
superspace, a space parametrised by both commuting and anticommuting
variables. Such spaces do not directly model physical space, but are
useful mathematical constructs because the spaces of functions
naturally defined on them carry representations of fermionic
operators defined in physics, and also (in a geometric context) of
differential operators on forms on
manifolds and on cross-sections of spin bundles. One aim of the
current paper is to
show that these superspace paths, when handled in a rigorous manner,
lead to new and useful analytic techniques.
In the context of conventional probability theory (without anticommuting
variables) an extended study of probabilistic techniques applied to many
aspects of analysis on manifolds has been made by Elworthy
\Refname\Elwsf; this work includes a straightforward probabilistic proof
of the Gauss-Bonnet-Chern theorem. There
are a number of other works on applications of probabilistic methods
to index theory and localisation, such as the
work of Bismut \Refname\Bis, Jones and
Leandre \Refname\JLe, Leandre \Refname\Lea, Lott \Refname\Lot\ and
Watanabe \Refname\Wat. These other works use more technical
probabilistic
methods, such as Malliavin calculus, than those of this paper, where
analytic estimates based on comparisons of solutions of stochastic
differential equations are used. Also, in this paper heat kernels are
obtained by Duhamel's formula, rather than by the use of Brownian
bridges. Supermanifolds are also used in Getzler's proof of the index
theorem, which uses pseudo-differential operator methods to obtain the
necessary heat kernel asymptotics \Refname\Getzz. A detailed account of
these methods may be found in the recent book of Berline, Getzler and
Vergne \Refname\BerGetVer.
When considering fermions in curved space, and differential forms on
Riemannian manifolds, flat global superspace must be replaced by
what is
known as a supermanifold; essentially supermanifolds are extensions
of ordinary manifolds to include anticommuting coordinates. Section 2
defines some supermanifolds which can be constructed in a natural way
from a vector bundle over a Riemannian manifold. The construction does
not depend in any essential way on which of the various approaches to
supermanifolds existing in the literature is used. The supermanifolds
constructed allow superspace stochastic
teqniques to be applied to various physical and geometric problems.
Section 3 of this paper contains a geometric formulation of Brownian
paths on supermanifolds.
As in the classical treatment of Brownian paths on a manifold
\Refname\Elw \Refname\IkW, this is done by constructing stochastic
differential equations globally on the supermanifold. Next, using key
technical results from the companion paper \Scal, these paths
are
used to give a Feynman-Kac
formula for the Laplace-Beltrami operator for twisted differential
forms.
In the final section of the paper this formula is used to give
a rigorous version of the very
simple proofs of the index theorem using supersymmetry due to
Alvarez-Gaum\'e \Refname\AG\ and to Friedan and Windey \Refname\FW. In
\AG\ and \FW\ the path integral calculations are carried out
by physicist's methods which are not entirely rigorous, particularly
in curved space; the stochastic machinery developed in this paper
allows these steps to be made rigorous without spoiling the
underlying simplicity and elegance of the approach.
The proofs of the
index theorem given in \AG\ and \FW\ used formulae for the index of a
differential operator in terms of an evolution
operator $\exp -Ht$, where $H$ is the Hamiltonian of a supersymmetric
system; the first example of such a formula was given by MacKean and
Singer \Refname\MS, subsequently Witten showed that
properties of supersymmetric quantum mechanics could be used to derive
many analogous formulae \Refname\Wit. The evolution operator was then
expressed in terms of path
integrals. Using techniques which are not fully rigorous, it was then
shown that, for the purposes of calculating the index, the complicated
curved-space Hamiltonian $H$ could be replaced by a much simpler
flat-space supersymmetric magnetic oscillator Hamiltonian whose
evolution operator was then calculated exactly by standard methods. It
is the validity of this approximation which is established by the
stochastic calculus techniques developed in this paper. The approach
taken in this paper thus retains much
of the simplicity of the original supersymmetric proofs \AG,\FW.
%SECTION 2
\Secno{Supermanifolds and differential forms}
This section is purely geometrical,
constructing supermanifolds which provide the appropriate arena for the
Brownian paths of sections 3 and 4. The supermanifolds are constructed
using the data of a vector bundle over a conventional
manifold in such a way that the space of supersmooth functions on
this supermanifold is isomorphic to the space of twisted differential
forms on the manifold. In section 3 this construction
is used to transfer superspace Brownian motion to bundles of twisted
differential forms.
A supermanifold is a space which has some commuting and some
anticommuting coordinates.
There are a number of different approaches to supermanifolds in the
literature, which are broadly equivalent; the constructions in this
paper do not depend on the detailed aspects of any particular
approach. The important fact is that all local coordinates belong to
a graded commutative algebra, with even elements (which commute with
elements of either parity) represented by lower-case Latin letters
and odd elements (which commute with even elements but anticommute
with any odd element) denoted by Greek letters. The dimension of a
supermanifold is an ordered pair of integers which specifies the
number of even and odd local coordinates. Some facts about the
analysis of commuting and anticommuting variables are given in
section 2 of the companion paper \Scal. More detailed accounts may
be found in \Refname\Superman. As in the case of conventional manifolds,
a
supermanifold can be entirely characterised by its coordinate
transition functions, an approach that is used in this section. Details
of the reconstruction of a supermanifold from its transition functions
may be found in \Refname\Erice.
Let $M$ be a smooth, compact $m$-dimensional real manifold and let $E$
be a
smooth $n$-dimensional Hermitian vector bundle over $M$. Suppose that
$\{U_{\alpha}|\alpha \in \Lambda\}$ is an open cover of $M$ by sets
which are both coordinate neighbourhoods of $M$ and local
trivialisation neighbourhoods of $E$. For each $\alpha \in \Lambda$
let $\phi_{\alpha}:U_{\alpha}\rightarrow \Real^m$ be the coordinate
map on $U_\alpha$, and for each $\alpha,\beta\in\Lambda$ let
$h_{\alpha\beta}:U_{\alpha}\cap U_{\beta}\rightarrow U(n)$ be the
transition function of the bundle $E$ (so that for each $p\in
U_{\alpha}\cap U_{\beta}$\ $\Lba h_{\alpha\beta}{}^r{}_s(p)\Rba$ is a
unitary $n\times n$ matrix). Additionally let
$\{\tau_{\alpha\beta}|\alpha,\beta\in\Lambda\}$ be the coordinate
transition functions on $M$, that is
$\tau_{\alpha\beta}:\phi_{\beta}(U_{\alpha}\cap
U_{\beta})\rightarrow\phi_{\alpha}(U_{\alpha}\cap U_{\beta})$ with
$\tau_{\alpha\beta}=\phi_{\alpha}\circ\phi_{\beta}^{-1}$, and let
$\Lba m_{\alpha\beta}{}^i{}_k(x_{\beta})\Rba=\bigl(\partial
x^i_{\alpha}/\partial
x^k_{\beta}\bigr)$ be the
corresponding Jacobian matrix.
The required supermanifold $S(E)$ is the $(m,m+n)$ dimensional
supermanifold built over $M$ with local coordinates
$(x_{\alpha}^1,\dots,x_{\alpha}^m,\theta_{\alpha}^1,\dots,
\theta_{\alpha}^m, \eta_{\alpha}^1,\dots,\eta_{\alpha}^n)$ and
transition functions
$$\eqalign{T_{\alpha\beta}&:(x_{\beta}^1,\dots,x_{\beta}^m,
\theta_{\beta}^1,
\dots, \theta_{\beta}^m, \eta_{\beta}^1,\dots,\eta_{\beta}^n)\cr
\mapsto \bigl(\tau_{\alpha\beta}^1(x_{\beta}),
\dots,\tau_{\alpha\beta}^m(x_{\beta}),
&m_{\alpha\beta}{}^1{}_k(x_{\beta})\theta_{\beta}^k
,\dots,m_{\alpha\beta}{}^m{}_k(x_{\beta})\theta_{\beta}^k,\cr
&h_{\alpha\beta}{}^1{}_r(\phi_{\beta}^{-1}(x_{\beta}))\eta_{\beta}^r,
\dots,h_{\alpha\beta}{}^n{}_r(\phi_{\beta}^{-
1}(x_{\beta}))\eta_{\beta}^r\bigr).\cr}
\Eqno$$
(Here and elsewhere the convention that repeated indices are to be
summed over their range is used.)
A useful space of functions is
%eggas a global version of the space $C^{\infty}{}{^\prime}
%egg(\Real^{m,m+n},\Comp)$ introduced in \Scal,
the space
$C^{\infty}{}{^\prime} (S(E))$ of
functions $f$ which locally take the form
$$ f(x,\theta,\eta) = \Sumu \Sumrn f_{\mu r} (x) \theta^{\mu} \eta^r
\Eqno
$$
where $\mu=\mu_1\dots\mu_k $ is a multi-index with
$1\le\mu_1<\dots<\mu_k\le m$,
$M_m$ is the set of all such multi-indices (including the empty one),
$\theta^{\mu}=\theta^{\mu_1}\dots\theta^{\mu_k}$ and each $f_{\mu
r}$ is in $ \cinf(\Real^m,\Comp)$. (It should be noted that these
functions are linear in the $\eta^r$ but may be multilinear in the
$\theta^i$; the nature of the transition functions (2.1) ensure that
such functions may be consistently defined.) Now it may be seen
that, again as a result of the choice of transition functions (2.1),
there is a globally defined map
$$
I:\Gamma(\Omega(M)\otimes E)\to \cinf{}'(S(E),\Comp)
$$
which may be obtained from the local prescription
$$\eqalign{
I(s)(x,\theta,\eta)&=\Sumu \Sumrn f_{\mu r} (x)
\theta^{\mu} \eta^r\cr
{\rm if\ \ \ \ } s(x)&=\Sumu \Sumrn f_{\mu r} (x) dx^{\mu} e^r\cr
}\Eqno$$
(where $\Omega(M)$ is the bundle of smooth forms on $M$ and
$(e^1,\dots,e^n)$ is the appropriate basis of the fibre of the bundle
$E$). Also,
this map is an isomorphism of vector spaces, and indeed of sheaves.
The crucial feature of this construction, which is itself quite
simple, is that it allows one to express the Hodge de Rham operator
on twisted differential forms on $M$ as a differential operator on
the extended space of functions $\Cinfp$. Fermionic path integration
techniques then allow one to use stochastic methods to analyse these
operators in the usual way.
Explicitly, suppose that $M$ is a Riemannian manifold with metric
$g$, and that a connection has been chosen on the bundle $E$.
Then, in local coordinates, if one
introduces the notation $\partial_i = {\partial/\partial x^i}$,
$\Del i = {\partial/\partial\theta^i}$ and $\Det r =
{\partial/\partial\eta^r}$, the Hodge de Rham operator $d +
\delta$ takes the form
$$
d + \delta = (\theta^i - g^{ij}\Del j) \partial_i
- g^{i\ell} \Chr ijk \theta^j\Del{\ell}\Del{k} +(\theta^i -
g^{ij}\Del j) \Con irs \eta^r\Det s.
\Eqno$$
where $\Chr ijk$ are the Christoffel symbols of the Riemannian
connection on $(M,g)$ and $\Con irs$ are the components of the
connection one form on $E$ pulled back to the coordinate
neighbourhood by the section corresponding to the local trivialisation.
Using the notation $\psi^i = \theta^i -
g^{ij}\partial/\partial\theta^j$, this takes the simpler form
$$
d + \delta = \psi^i\bigl( \partial_i
- \Chr ijk \theta^j\Del{k} - \Con irs \eta^r\Det s\bigr).
\Eqno$$
As will emerge in section 5, it is the heat kernel of the square of
this operator, the Laplace-Beltrami operator, which can be studied by
fermionic stochastic calculus, and is relevant to the proof of the
Atiyah-Singer index theorem. For these purpose it is useful to
establish the following lemma, which is a twisted version of the
Weitzenbock formula relating the Bochner Laplacian to the
Laplace-Beltrami operator. The proof uses the method of Cycon, Froese,
Kirsch and Simon \Refname\Sim, generalised to the twisted case. (A
normalisation factor of $\half$ is
included in the Laplace-Beltrami operator to conform with the standard
normalisation
of Brownian motion.)
\Theno{Lemma}{Let $L = \half(d+ \delta)^2$ be the Laplace-Beltrami
operator on the space of functions $\Cinfp$. Then
$$L = -\half\Lba B \Wbokminus\Rba \Eqno$$
where $R_{ki}{}^{mq} $ are the components of the curvature of
$(M,g)$, $F_{ijr}{}^s$ are the components of the curvature of the
connection on $E$, and $B$ is the twisted Bochner Laplacian,
$$B = g^{ij}(D_i D_j - \Chr ijk D_k)\Eqno$$
with
$$D_i = \partial_i - \Chr ijk \Tdel jk - \Con irs \Edet rs.
\Eqno$$}
\Proof $$\eqalign{
L &= \half\Lba\psi^iD_i\psi^jD_j\Rba \cr
&=\half\Lba\bigl(\half\{\psi^i,\psi^j\} +
\half[\psi^i,\psi^j]\bigr)D_iD_j +
\psi^i[D_i,\psi^j]D_j \Rba \cr}
\Eqno$$
(where $\{,\}$ denotes an anticommutator and $[,]$ denotes a
commutator).
Now
$$ \half \{\psi^i,\psi^j\} = -g^{ij} \Eqno$$
while
$$\eqalign{
\half[\psi^i,\psi^j]&D_iD_j\cr
&=-\half[\psi^i,\psi^j]
\bigl((\partial_i\Chr jk{\ell} - \Chr ik{\Ellp} \Chr j{\Ellp}m)
\Tdel k{\ell}
+(\partial_i \Con jrs - \Con irt \Con jts)\Edet rs\bigr)\cr
&=\Wbok.\cr
}\Eqno$$
Also
$$ \eqalign{
[D_i,\psi^j]&=-\partial_i g^{jk} \Del k - \Chr ikj \theta^k -
\Chr ik{\ell}g^{jk} \Del{\ell}\cr
&= -\Chr ikj\psi^k.\cr
} \Eqno$$
Thus
$$\eqalign{ \psi^i[D_i,\psi^j]D_j&= -\psi^i\psi^k\Chr ikj D_j\cr
&=g^{ik}\Chr ikj D_j.\cr
}\Eqno$$
Hence
$$\eqalign{
L&= -\half\Lba g^{ij}D_iD_j - g^{ij} \Chr ijk D_k\cr
&\quad \Wbokminus \Rba\cr
}\Eqno$$
as required.
One further supermanifold is required for the stochastic
constructions in the following section. This is a super extension
$\Som$ of the bundle of orthonormal frames on the $m$-dimensional
manifold $M$ (again twisted according to the Hermitian bundle $E$).
The supermanifold $\Som$ can be defined by its coordinate transition
functions; suppose that
$\{U_{\alpha}|\alpha \in \Lambda\}$ is again an open cover of $M$ by
sets
which are both coordinate neighbourhoods of $M$ and local
trivialisation neighbourhoods of $E$.
Let $x^i_{\alpha},b^u_{\alpha},i=1,\dots,m,u=1,\dots,\half
m(m-1)$ be local coordinates on $O(M)|_{U_{\alpha}}$. Then $\Som$ is
the
$(\half m(m+1),m+n)$-dimensional supermanifold with local coordinates
$x_{\alpha}^i,b^u_{\alpha},\theta^j_{\alpha},\eta^r_{\alpha}$. On
overlapping neighbourhoods $U_{\alpha}$ and $U_{\beta}$ the
coordinates $x^i,\theta^j,\eta^r$ have the transition functions
defined by equation (2.1), while the coordinates $b^u$ transform as
on the bundle of orthonormal frames $O(M)$.
It should be noted that anti-commuting variables are introduced in order
to provide function
algebras which have geometric and physical applications;
anticommuting variables do not directly model physical situations. An
important algebraic tool is the Berezin integral \Refname\Ber\ which
integrates out the anticommuting variables to give a real or complex
number. This is defined in the following manner: suppose that $f$ is
a polynomial function of $p$ anticommuting variables $\alpha^1,\dots,
\alpha^p$. Then
$$ f(\alpha) = k \alpha^1\alpha^2\dots\alpha^p + {\rm\ lower\ order
\ terms}
\Eqno$$
where $k$ is a real or complex number. The Berezin integral is then
defined by
$$
\Bint f(\alpha) = k.
\Eqno$$
The integral defined by this simple prescription has a number of
useful properties which will be exploited in later sections of this
paper.
%section 3
\Secno{Geometric Brownian paths on supermanifolds}
This section constructs Brownian paths on the supermanifolds
introduced in section 2. Before discussing these generalised Brownian
paths in curved superspace, a brief review of path integral
techniques for classical manifolds is given.
Clearly it is not straightforward to transfer Brownian motion to the
setting of a general manifold, because Brownian motion, while
invariant under rigid Euclidean transformations, will not survive
more general coordinate transformations. Various analogues of
Brownian motion on Riemannian manifolds have been considered, two
of which will be described here because they are suitable for
generalisation to superspace. The first method is straightforward in
principle - one simply replaces Wiener measure with a measure whose
finite-dimensional marginal distributions are based on the heat
kernel of the Laplacian
of the manifold, an approach which leads almost trivially to a
Feynman-Kac formula
for a Hamiltonian which is the sum of the Laplacian and arbitrary
first order and scalar terms. This approach has been used in
conjunction with fermionic Brownian motion in \Refname\GBC\ to analyse
the
Hodge-de Rham operator on a Riemann manifold, and hence prove the
Gauss-Bonnet-Chern formula; however its usefulness is limited
by its dependence on information about the heat kernel of the
Laplacian, which is itself a highly non-trivial object.
A second approach to Brownian paths on manifolds, described in \Elw\ and
\IkW, is to use paths which are solutions
of stochastic differential equations, the stochastic differential
equations being defined in a manner which is globally valid. The second
order correction term in the \Ito\
formula (equation (3.1) of \Scal) prevents one from using vector
fields and other tensorial objects in the obvious way, a difficulty
which
is overcome by modifying the \Ito integral so that a compensating
term is included in the transformation rule. The most elegant way of
doing this
is to introduce the symmetric product or Stratonovich integral, which
is defined in the following way.
\Defno{Definition} Suppose that $X_s$ and $Y_s$ are Stochastic integrals
with
$$\eqalign{
dX_s &= f_{a,s} db^a_s + f_{0,s} ds,\cr
dY_s &= g_{a,s} db^a_s + g_{0,s} ds.\cr
}\Eqno$$
Then $$
Y_s\circ dX_s =_{def} Y_s(f_{a,s} db^a_s + f_{0,s} ds)+ \half f_{a,s}
g_{a,s} ds.
\Eqno$$
(Here $b^a_s,a=1,\dots,m$ denotes $m$-dimensional Brownian motion, and
$db^a_s$ denotes an \Ito\ differential.) As before, a repeated index
is summed over its range.
Now suppose that, for $a=0,1,\dots,m$, $A_a$ are vector fields on a
$p$-dimensional manifold $N$. In a local coordinate system
$(x^1,\dots,x^p)$ where $A_a = A^i_a(x)\partial/\partial
x^i$ consider the stochastic differential equation
$$
dx^i_s = A_a^i(x_s)\circ db_s^a + A^i_0(x_s) ds.
\Eqno$$
By applying the \Ito\ calculus one finds that the equation
(\number\henry.3)
is
form invariant under change of coordinate $\tilde x_s = \tilde
x(x_s)$,
where $\tilde x$ is a change of coordinate function. The solution to
such an equation exists globally, as can be shown by a careful
patching argument \Elw,\IkW. Such equations
enable one to define a notion of stochastic flow on a manifold. They
also enable one to construct functions on $N \times \Real^+$ which
satisfy the differential equation
$$\eqalign{
{\partial f\over\partial t} &= \half A_aA_af\cr
{\rm with\ \ \ } f(x,0) &= h(x)\cr
}\Eqno$$
where $h$ is a smooth function on $N$ and $A_a$ has suitable
properties, as in the following theorem.
\Theno{Theorem}{
Suppose that $x_t$ is a solution to (\number\henry.3) with initial
condition $x_0 =
x\in M$. Then $f(x,t) = \Ebold(h(x_t))$ satisfies the differential
equation (3.4).} %check
\Oproof
Using the \Ito\ formula, and omitting details of the
patching of solutions over different coordinate neighbourhoods,
$$\eqalign{
\Ebold\Lba h(x_t)\Rba &- h(x)\cr
&= \Ebold\Lbb \Intt \partial_i h(x_s) dx^i_s
+\Intt\half A^i_a(x_s) A^j_a(x_s) \partial_i\partial_j h(x_s) ds\Rbb.\cr
}\Eqno$$
Now
$$
dx^i_s = A^i_a(x_s) db^a + \half A^j_a(x_s)\Lba \partial_j A^i_a
(x_s)\Rba ds.
\Eqno $$
Thus, using the fact that \Ito\ increments $db_s^a$ have zero
expectation and are independent of $b_u$ when $u \leq s$, one finds that
$$\eqalign{
\Ebold\Lba h(x_t)\Rba &- h(x)\cr
&= \Ebold\Lbb \Intt \half A^j_a(x_s)\Lba \partial_j A^i_a(x_s)\Rba
\partial_ih(x_s)
+ \half A^i_a (x_s)A^j_a(x_s) \partial_i\partial_j h(x_s) ds\Rbb \cr
&=\Intt \half A_a A_a h(x_s) ds.\cr
}\Ename\ELW $$
Hence, for suitable $A_a$,
$$
\Ebold\Lba h(x_t)\Rba = e^{\half(A_aA_a)t} h(x)
\Eqno $$
and thus $f(x,t)=\Ebold\Lba h(x_t) \Rba$ solves (\number\henry.3).
(The $x$ dependence of $f(x,t)$ comes from the initial condition
satisfied by $x_t$.) \Endproof
In the case of a Riemannian manifold $(M,g)$, when one wishes to
construct
paths which will help in the study of the Laplacian and related
operators, one uses $O(M)$, the bundle of orthonormal frames on $M$,
as the manifold $N$, and considers the $m$ canonical horizontal
vector
fields on this bundle as the vector fields $A_a$. Suppose that
$(p,e_a)$ is a point in $O(M)$, that is, $p$ is a point in $M$ and
$e_a,a=1,\dots,m$ is an orthonormal basis of the tangent space at
$p$. Then, if $x^i$ are local coordinates at $p$ and
$$
e_a = e_a^i{\partial\over\partial x^i},
\Eqno$$
the $m$ canonical horizontal vector fields on $O(M)$ are
$$
V_a = e_a^i{\partial\over\partial x^i} - e^i_a e^j_b \Chr ijk (x)
{\partial \over \partial e^k_b},
\Eqno$$
where the functions $\Chr imk$ are the Christoffel symbols for the
Riemannian connection.
In this case, following the general form of
(\number\henry.3), one obtains the stochastic differential equations
$$\eqalign{
d\xit is &= \eiat ias \circ db^a_s\cr
d\eiat ias &= - \eiat jas \eiat kbs \Chr jki \bigl(x_s\bigr)\circ
db^b_s .\cr
}\Eqno$$
To see the connection with the Laplacian, suppose that
$x_s^i,e^i_{a,s}, (a,i=1,\dots,m)$ are solutions to
(\number\henry.11)
with initial condition (in a particular coordinate system) $x_0 =
x\in M, e^i_{a,0} = e^i_a$, where $e_a = e^i_a
{\partial\over\partial x^i} $ is an orthonormal frame at $x$. Then,
again omitting details of the patching of solutions on different
coordinate charts, if $f$ is a smooth function on $O(M)$,
$$
\Ebold\Lba f(x_t, e_t)\Rba -f(x,e) = \Intt \half V_a V_a f(x_s,e_s)
ds.
\Eqno$$
In the particular case where $f$ depends on $x$ only, (that is, $f =
g
\circ \pi$ where $\pi:O(M) \to M$ is the projection map, and $g \in
\cinf(M)$),
$$
\Ebold\Lba g(x_t)\Rba - g(x) = \Intt -\Lscal\, g(x_s) ds
\Eqno$$
where $\Lscal = -\half \Lba g^{ij} \partial_i\partial_j + g^{ij}
\Chr ijk\partial_k\Rba$ is
the scalar Laplacian. This follows since
$$ \half V_aV_a f(x) = -\Lscal\, f(x)
\Eqno$$
and one can show that $e^i_{a,s} e^j_{a,s} = g^{ij}(x_s)$ almost
surely. Hence one finds that $f(x,t) = \Ebold \Lba g(x_t)\Rba$
satisfies
$$
{\partial f\over\partial t} = -\Lscal f.
\Eqno$$
The two
approaches to Brownian motion on Riemann manifolds are closely
related, because the $x$ components of the process which
solves the stochastic differential equations (\number\henry.11) have
finite dimensional distributions
built from the heat kernel of the Laplacian.
Turning now to
supermanifolds, a construction analogous to this second method will
be described. While more general supermanifolds are possible,
attention in this paper will be restricted to those of the type
constructed in section 2,
because of their use in
geometry and supersymmetric physics; the general approach is
applicable to other situations.
The Brownian paths in superspace which will now be defined will lead
to a Feynman-Kac formula for the Laplace-Beltrami operator
$L=(d+\delta)^2$ acting on $\Cinfp$ introduced in section 2. Letting
$(\theta^a_t,\rho^a_t) a=1,\dots,m$ denote $m$-dimensional fermionic
Brownian paths \Scal\ and letting $(x^i,e^i_a,\theta^i,\eta^r)$ be
coordinates of a point in the extended bundle of orthognal frames
$\Som$ introduced in section 2, consider the $m + m^2 + n$ stochastic
differential equations
$$\eqalign{
\xit it &= x^i+ \Intt\eiat ias \circ db^a_s\cr
\eiat iat &= e^i_a + \Intt-e^{\ell}_{a,s}\Chr k{\ell}i \bigl(x_s\bigr)
\eiat kbs \circ db^b_s
\cr
\xi^i_t&= \theta^i +\theta^a_t \eiat iat + \Intt \Lbb-
\xi^j_s \Chr jki(x_s) \eiat kbs \circ db^b_s\cr
&\qquad\qquad-\theta^a_t de^i_{a,s} +\Quai \xi^j_s
R^i{}_{jkl}(x_s)\xi^k_s \pi^{\ell}_s ds\Rbb \cr
\eta_t^p &= \eta^p + \Intt \Lbb -\eiat jas \eta^q_s \Con jqp(x_s) \circ
db^a_s\cr
&\qquad+{\textstyle{1\over4}} \eta^q_s
\Lba \xi^i_s+i\pi^i_s \Rba\Lba \xi^j_s+i\pi^j_s \Rba
F_{ij\,q}{}^p(x_s) ds\Rbb, \cr
}\Ename\SDE$$
where
$$
\pi^{\ell}_s = e^{\ell}_{a,s} \rho^a_s.
\Eqno$$
The existence of local solutions to such stochastic differential
equations
was established in theorem 5.2 of \Scal, while the usual
patching techniques allow a global solution to be constructed, since
they transform covariantly under change of coordinates.
In order to establish a Feynman-Kac formula for the Laplace-Beltrami
operator $L= (d+ \delta)^2$, vector fields $W_a (a=1,...,m)$ on
$\Som$ must be defined which are the canonical horizontal vector
fields on $\Som$ regarded as a bundle over $M$ with connection
$(\Gamma,A)$. In a local coordinate
system $(x^i,e^i_a,\theta^i,\eta^p)$ on $\Som$ these vector fields
take the form
$$\eqalign{
W_a &= e^i_a{\partial\over\partial x_i} - e^j_a e^k_b \Chr jki
{\partial\over\partial e^i_b}
- e^j_a\theta^k \Chr jki {\partial\over\partial\theta^i} -
e^j_a\eta^r \Con jrs {\partial\over\partial\eta^s}.\cr
}\Eqno$$
The key property of the vector fields $W_a$ is that, when acting on
functions on $\Som$ which are independent of the $e^i_a$ (that is, on
functions of the form $f = g\circ\pi$ where $\pi$ is the canonical
projection of $\Som$ onto $S(E)$) it is related to the Laplace-Beltrami
operator $L = \half(d + \delta)^2$ by
$$ -\half\Lba W_aW_a \Wbokminus\Rba = L,
\Eqno$$
as may easily be seen from Lemma (2.1).
The following Feynman-Kac formula may then be established quite
directly using theorem (\number\henry.2).
\Theno{Theorem}{Suppose that $(x^i_t,\eiat iat,\eta^r_t)$ satisfy
\SDE. Then
$$
\exp(-Lt)g(x,\theta,\eta) = \Ebold \Lba g(x_t,\xi_t,\eta_t)\Rba
\Ename\FK$$
where $L = \half(d + \delta)^2$ is the Laplace-Beltrami operator
acting on $\Cinfp$.}
\Proof Using \Elw\ and the superspace \Ito\ formula
(theorem 3.5 of \Scal),
$$\eqalign{
\Ebold\Lba &g(x_t,\xi_t,\eta_t)\Rba - g(x,\theta,\eta)\cr
&=\Ebold\Lbb\Intt \half W_aW_ag(x_s,\xi_s,\eta_s) - \Quai
R^i{}_{jk\ell}(x_s)\xi^k_s
\pi^{\ell}_s \xi^j_s\Del{i} g(x_s,\xi_s,\eta_s)\cr
&+ \Qua(\xi^i_s+i\pi^i_s)(\xi^j_s+i\pi^j_s)F_{ij\,p}{}^q\eta^p_t\Det q
g(x_s,\xi_s,\eta_s) ds\Rbb\cr
&\quad=\Ebold\Lbb\Intt \half\Lba W_aW_a -R_i^j(x)\Tdel ij \cr
&- \half R_{ki}{}^{j\ell}(x) \theta^i\theta^k\Del j \Del {\ell} +
{1\over4}[\psi^i,\psi^j]F_{ijr}{}^s(x) \Edet rs \Rba g(x_s,\xi_s,\eta_s)
ds\Rbb,\cr
}\Eqno$$
using properties of fermionic paths (\Scal\ equation (2.15)). Hence, if
$$\eqalign{
f(x,\theta,\eta,t) &=\Ebold \Lba g(x_t,\xi_t,\eta_t)\Rba,\cr
f(x,\theta,\eta,t) - f(x,\theta,\eta,0) &= \Intt -
Lf(x,\theta,\eta,s) ds,\cr
}\Eqno$$
and the result follows.
\Endproof
%SECTION 4
\Secno{The Atiyah-Singer Index theorem}
The supermanifold stochastic techniques developed in
the previous section will now be used to establish the Atiyah-Singer
index
theorem. As in the paper by Atiyah, Bott and Patodi \Refname\ABP, the
method
used is to establish a stronger local version of the theorem in the
particular case of the twisted Hirzebruch signature complex, by
studying the heat kernel of the Laplacian.
The full theorem may then be inferred by the
$K$-theoretic arguments presented in \ABP.
The starting point is the formula of McKean and Singer \MS\ and Witten
\Wit\
expressing the index of the complex in terms of the heat kernel of
the Laplacian. As before, suppose that $(M,g)$ is a
compact Riemannian manifold of dimension $m=2k$, and that $E$ is an
$n$-dimensional hermitian vector bundle over $M$. The formula of McKean
and Singer then states that
$$
{\rm Index\ }(d+\delta) = Str \exp(-Lt).\Ename\MKS
$$
Here $d+\delta$ is, as before, the \Hdr\ operator and $L=
\half(d+\delta)^2$ is the Laplace-Beltrami operator,
while Str denotes the supertrace. With the identification of the space
of twisted forms on $M$ with the space of functions $\Cinfp$ on the
supermanifold $S(E)$ set
up in section 2, the supertrace can be defined in the following
manner. First, the standard involution $\tau$ may be defined on $\Cinfp$
by the formula
$$\eqalign{
\tau\Lbb\Sumu&\Sumrn f_{\mu r}(x) \theta^{\mu}\eta^r \Rbb \cr
&= \Sumu\Sumrn\Bint d^m\rho {1\over\sqrt{{\rm det}(g_{ij}(x))}}\exp
i\rho^i \theta^j g_{ij}(x)\,\,f_{\mu r}(x) \rho^{\mu}\eta^r.\cr
}\Eqno$$
(In this expression $\Row{\rho}m$ are anticommuting variables and
the integral is the Berezin integral defined in section 2. The
definition is
independent of the choice of local coordinates.) The
supertrace is now defined for a suitable operator $O$ on $\Cinfp$ by
the formula
$$
\Str O = {\rm Tr}\, \tau O
\Eqno$$
where ${\rm Tr}$ denotes the conventional trace. It emerges in the
proof of McKean and Singer's formula that the right hand side of
\MKS\
is in fact independent of the real
parameter $t$.
The Atiyah-Singer index theorem for the twisted Hirzebruch signature
complex takes the following form.
\Theno{Theorem}{With the notation of section 2,
$$
{\rm Index}\,(d+\delta) = \int_M \Asint,
\Eqno$$
where $F$ is the curvature 2-form of a connection on the bundle $E$,
$\Omega$ is the Riemann curvature 2-form of $(M,g)$ and the
square brackets indicate projection onto the $m$-form component of
the integrand.}
Combining this with the McKean and Singer formula
\MKS\
one sees that an equivalent result is
$$
\Str\exp(-Ht) = \int_M \Asint.
\Eqno$$
(Again this result is valid for all $t$.)
The following theorem is a stronger, local version of this result.
\Theno{Theorem}{With the notation of Theorem \number\henry.1,
if $p\in M$,
$$
\lim_{t\to 0}\,\, \str \exp(-Ht)(p,p)\,{\rm dvol} = \Asint\Bigg|_p,
\Ename\ASI$$
where $\str$ denotes the $2^mn \times 2^mn$ matrix supertrace, as
opposed to the full operator supertrace $\Str$, so that $\str \exp(-
Ht)(p,q)$ is then the kernel of the operator on $C^{\infty}(M)$
obtained by this partial supertrace.}
The strategy for proving this theorem is to use the Feynman-Kac
formula (Theorem (3.3))
to analyse the operator $\exp(-Ht)$ and then, using Duhamel's formula
to extract information about the kernel (as in Getzler
\Refname\Get), show that in the limit as
$t$ tends to zero only the required terms survive. The proof is thus
carried out in several steps.
\Step1{An expression for the matrix supertrace in terms of Berezin
integrals}
%
As a peliminary, the matrix supertrace of an operator will be
expressed in terms of a Berezin integral of its kernel. Suppose that
$G(p)$ denotes the $2^p$-dimensional space of polynomial functions of
$p$ anticommuting variables; then a linear operator $O$ on this space
has a kernel
$O(\xi,\zeta)=O(\xi_1,\dots,\xi_p,\zeta_1,\dots,\zeta_p)$ which is a
function of $2p$ anticommuting variables, and satisfies
$$
O\,f (\xi) = \Bint d^p \zeta\, O(\xi,\zeta) f(\zeta)
\Eqno$$
for all functions $f$ in $G(p)$.
It can be shown by direct calculation that
$${\rm tr}\,O = \Bint d^p \xi\, O(\xi,-\xi).
\Eqno$$
Thus if $O$ is an operator on $\Cinfp$ one has the local coordinate
expression
$$\eqalign{
\str O(x,y)&={\rm tr}\,\tau O(x,y)\cr
&= \Bint d^m\rho d^m\theta d^n\eta\,O(x,y,\rho,-\theta,\eta,-
\eta) \Tauk.\cr
}\Ename\STR$$
\Step2{The construction of a locally equivalent metric and connection
on the super
extension of $\Real^m \times \Comp$}
%
Theorem (4.2)
is a local result; in order to prove this
theorem at an arbitary but fixed point $p\in M$ it is in fact
sufficient to replace the manifold $M$ by
$\Real^m$ and
the bundle $E$ over $M$ by the trivial bundle $\Real^m\times\Comp^n$
over $\Real^m$ with metric and connection satisfying certain conditions,
and to prove
the result for this simpler situation. The construction of a suitable
metric and connection will now be given. Suppose that $W$ is an open
subset of $M$ containing $p$ which has compact closure and is both a
coordinate neighbourhood
of $M$ and a local trivialisation neighbourhood of the bundle $E$,
and that $U$ is also an open subset of $M$ containing $p$ with
$\overline U \subset W$. Also let $\phi:W \to\Real^m$ be a system of
normal coordinates on $W$ based at $p$ which satisfy $\det
g_{ij}(x)=1$ at all points of $U$ (and, of course, $\phi(p)=0$) \Sim.
Additionally a
local trivialisation of the bundle $E$ is chosen such that
$A_{i\,r}{}^s (0) = 0$.
Then the required metric $\Til g$ on $\Real^m$ is a metric satisfying
$$\eqalign{
\Til g_{ij}(\Row xm)&=g_{ij}(\Phiv(\Row xm)) {\rm\ \ when\
}x\in\phi(U)\cr
\Til g_{ij}(\Row xm)&= \delta_{ij} {\ \ when\ }x\notin \phi(W)\cr
{\rm and\ \ } \det \Til g_{ij} = 1 {\rm\ \ throughout\ }\Real^m.\cr
}\Eqno$$
Also, the required connection is a connection satisfying
$$\eqalign{
\Til A_{i\,r}{}^{s}(\Row xm)&=A_{i\,r}{}^{s}(\Phiv(\Row xm)) {\rm\ \
when\ }x\in \phi(U),\cr
\Til A_{i\,r}{}^{s}(\Row xm) &= 0{\rm \ \ when\ }x\notin \phi(W).\cr
}\Eqno$$
Some simple consequences of this definition are that,
if $\Cur ijkl$ denotes the Riemann curvature of $(M,g)$ and
$\Cuf ijrs$ denotes the curvature of the connection $A$ on the bundle
$E$, while tildes denote the corresponding quantities on $\Real^m$
and $\Real^m \times \Comp^n$,
$$\eqalign{
\Curt ijkl(\Row xm) &= \Cur ijkl(\Phiv(\Row xm)) \cr
{\rm and}\qquad\Cuft ijrs(\Row xm) &= \Cuf ijrs(\Phiv(\Row xm)) \cr
}\Eqno$$
on $\phi(U)$. Also one has the standard Taylor expansions in normal
coordinates \ABP
$$\eqalign{
\Til{g} _{ij} (x) &= \delta_{ij} - \Third x^k x^{\ell}
\Til{R}_{ki\ell j}(0) + \dots, \cr
\Chrt ijk (x) &= \Third x^{\ell} (\Curt ljik(0) + \Curt lijk(0)) +
\dots, \cr
\Cont irs(x) &= -\half x^j \Cuft ijrs(0) + \dots\,.\cr
}\Eqno$$
Cutting and pasting arguments, for instance as presented by Cycon,
Froese, Kirsch and Simon in \Sim, show
that (if $\Hamt$ denotes the Laplacian $\Lap$ on $\Sflat$ with metric
$\Til{g}$ and connection
$\Til A$)
$$
\lim_{t\to0}( \str\exp(-Ht) (p,p)- \str\exp(-\Hamt t(0,0)) =0.
\Ename\CPA$$
Thus in the rest of this section it will be sufficient to
consider $\Hamt$ on $\Sflat$ in place of $H$ on $S(E)$.
\vfil\eject
\Step3{The use of the Feynman-Kac formula to give an explicit
expression for the required matrix supertrace}
%
Letting $\Hamtz$ be the flat Laplacian
$$\Hamtz = -\half \partial_i\partial_i
\Eqno$$
on $\Sflat$, and$\Kert t$ and $\Kertz t$ denote the heat kernels
$\exp-\Hamt t(x,\Pri{x},\theta,\Pri{\theta},\eta,\Pri{\eta})$ and
$\exp-\Hamtz t(x,\Pri{x},\theta,\Pri{\theta},\eta,\Pri{\eta})$,
Duhamel's formula \Get\ states that
$$\eqalign{
&\Kert t - \Kertz t\cr
=\Intt &ds \, e^{-(t-s)\Hamt}\Hamdif \Kertz s
\cr}\Eqno$$
where all differential operators act with respect to the variables
$x,\theta$ and $\eta$. Now
$$\eqalign{
&\quad\Kertz s \cr
&= \Bint d^m\rho d^n \kappa \Fac{s} \exp-{(x-
\Pri{x})^2\over2s} \exp-i\rho(\theta-\Pri{\theta})\exp-i\kappa(\eta-
\Pri{\eta})\cr
}\Eqno$$
and direct calculation shows that $\str\Til{K}^0_t(x,\Pri{x})= 0$.
Thus, using \STR,
$$
\eqalign{
&\Strk\cr
= \Bint d^m\theta &d^m\Pri{\theta} d^n \eta \Intt ds
\Lbc \Lba e^{-(t-s)\Hamt}\Hamdif\Kertzz{s}\Rba \Eval{\Pri{\eta} = -
\eta}\cr
&\qquad\qquad\times\exp -i\theta\Pri{\theta}\Rbc.\cr
}\Eqno$$
Now, using the Feynman-Kac formula \FK,
$$\eqalign{
&e^{-(t-
s)\Hamt}\Hamdif\Ktilz_s(0,0,\theta,\Pri{\theta},\eta,\Pri{\eta})\cr
&\quad=\Ebold\Bint d^m\rho d^n\kappa \Fac{s}F_s(\Xtil{t-s},
\Xitil{t-s},\rho,\Etatil{t-s},\kappa) \exp -i\rho\Pri{\theta}
\exp i\kappa\Pri{\eta}\cr
}\Eqno$$
where
$$
F_s(x,\theta,\rho,\eta,\kappa) =
\Hamdif\Lba \exp-{x^2\over2s}\exp-i\rho\theta \exp -
i\kappa\eta\Rba
\Eqno$$
(with differential operators again acting with respect to $x$,
$\theta$ and $\eta$) and $\Xtil s$, $\Xitil s$ and $\Etatil s$ satisfy
the
stochastic
differential equation \SDE\
(in normal coordinates on $\Sflat$)
with initial conditions
$\Xtil0 = 0$, $\Til{e}^i_{a\,0}=\delta^i_a$, $\Xitil{0} = \theta$ and
$\Etatil0= \eta$.
Hence
$$ \eqalign{
&\qquad\Strk \cr
&=\Ebold\Lbb\Bint d^m\theta d^n\Pri{\theta} d^m\rho d^n\eta
d^n \kappa
\cr &
\Intt ds \Fac{s}F_s(x_{t-s},\xi_{t-s},\rho,\eta_{t-s},\kappa) \exp -
i\theta\Pri{\theta} \exp i\rho\Pri{\theta}\exp-i\kappa\eta\Rbb.\cr
}\Eqno$$
Thus, integrating out $\Pri{\theta}$ and $\rho$, one obtains
$$\eqalign{
\Strk &= \cr
\Ebold \Bint &d^m\theta d^n\eta d^n \kappa \Intt ds \Fac{s}
F_s(x_{t-s},\xi_{t-s},\theta,\eta_{t-s},\kappa,)\exp -i\kappa\eta.\cr
}\Eqno$$
\Step4{The replacement of the Hamiltonian $\Til{H}$ by an equivalent
Hamiltonian $\Che{H}$ of simpler form.}
%
The next step is to construct a simpler Hamiltonian $\Che H$ on
$\Sflat$with
heat kernel $\Che K_t(x,\Pri x,\theta,\Pri{\theta},\eta,\Pri{\eta})$
such that
$$\lim_{t\to 0} \str\Che K_t(0,0) = \lim_{t\to 0}\str\Til K_t(0,0),
\Eqno$$
so that the required supertrace can be calculated. The modified
Hamiltonian $\Che H$ is obtained by considering the following
stochastic differential equations (which are a simplification of \SDE).
$$\eqalign{
\Xtilo it &= b^i_t \cr
\Xitilo it &= \theta^i + \theta^a_t\delta^i_a + \Intt \Lbb\Third\Xitilo
js\Xtilo {\ell}s (\Cur {\ell}kji +
\Cur{\ell}jki) db^k_s \cr
&+ \Third\Xitilo js R^i_j ds -\Quai \Xitilo js
\Xitilo ks \Pitilo{\ell}s R_{jk}{}^i{}_l\,ds\Rbb\cr
\Etatilo pt &= \eta^p + \Intt \Qua\Etatilo qs
(\Fer is)(\Fer js)\Cuf ijqp ds.\cr
}\Ename\SDEM
$$
with
$$
\Pitilo ls = \rho^a_s\delta^{\ell}_a.
\Eqno$$
(Here $R_{ijkl} = \Rtil ijkl(0)$ and $\Cuf ijpq = \Ftil ijpq(0)$,
with indices raised and lowered by $\Til{g}^1_{ij}(0)=\delta_{ij}$.)
Then, if $f \in C^{\infty}{}'(\Sflat)$,
$$\Ebold\Lba f(\Xtiloo t,\Xitiloo t,\Etatiloo t)\Rba - f(0,\theta,\eta)
=\Ebold \Lba\Intt -\Che{H} f(\Xtiloo s,\Xitiloo s,\Etatiloo s) ds\Rba
\Ename\FKone$$
where
$$\eqalign{
&\Che{H} = -\Lbb\half \partial_i\partial_i -\Qua R_{ij}{}^{kn'}
\Thetah i
\Thetah j \Del {n'}\Del k + \Third R^k_j\Thetah j \Del k\cr
&-\half R^k_j \Thetah j \Del k + \Qua\psi^i\psi^j \Etah q
\Ftil ijpq \Det q
-\Third\Thetah j \Xhat{\ell} (R_{\ell}{}^k{}_j{}^i+R_{\ell j}{}^{ki})
\Del i\partial_{k}\cr
& -{\textstyle{1\over18}} \Thetah {n'}\Thetah {m'}
\Xhat k \Xhat l (R_{kpn'}{}^i + R_{kn'p}{}^i)
(R_{\ell}{}^p{}_{m'}{}^j + R_{\ell m'}^{pj})
\Del i \Del j \Rbb.\cr
}\Eqno$$
(Here the physicist's convention of denoting operators with hats is
adopted to avoid ambiguities.)
Now, using Duhamel's formula again, and setting
$$G_s(x,\xi,\theta,\eta,\kappa) = \exp-\Lba {x^2\over2s}+i\theta\xi +
i\kappa\eta\Rba,
\Eqno$$
one finds that
$$\eqalign{
&\quad \Til K_t(0,0)-\Che K_t(0,0)\cr
&= \Ebold \Bint d^m\theta\,d^n\eta\,d^n\kappa \Intt ds\,\Fac s
\Lsb\cr
&\quad(\Til H - \Hamtz)G_s(x,\xi,\theta,\eta,\kappa)\Evalp\cr
&\qquad-(\Che H - \Hamtz)G_s(x,\xi,\theta,\eta,\kappa)\Evalo\Rsb.\cr
}\Eqno$$
At this stage it is useful to introduce the scaled variable $\phi =
\sqrt{2\pi t}\theta$; then, following the rules
of Berezin integration \Ber, $d\theta = \sqrt{2\pi t} d\phi$ and
thus
$$\eqalign{
&\quad \Til K_t(0,0)-\Che K_t(0,0)\cr
&= A(t) \qquad + \qquad B(t)\cr
&{\rm where\ \ \ } A(t) \cr
&= \Ebold\Fintt \Lsb\cr
&(\Til H - \Che H)G_s(x,\xi,\Phit,\eta,\kappa)\Evalo\Rsb\cr
&{\rm and\ \ \ \ \ } B\cr
& = \Ebold\Fintt\Lsb\cr
&(\Til H -\Hamtz) G_s(x,\xi,\Phit,\eta,\kappa)\Evalp\cr
&-(\Til H - \Hamtz)G_s(x,\xi,\Phit,\eta,\kappa)\Evalo\Rsb.\cr
}\Eqno$$
Now, using flat space Brownian motion techniques \Refname\Sima\ together
with
fermionic Brownian motion techniques \GBM, one can show that for
any
suitably regular function $f$ on $\Sflat$,
$$\eqalign{
&\quad\Ebold f(\Xtilo{}{t-s},\Xitilo{}{t-s},\Etatilo{}{t-s})\cr
&= \exp -\Che H(t-s) f(x,\Phit,\eta){\rm\ \ (by\ \FKone)}\cr
&=\Ebold\Lsb\exp \Lbb\int_{0}^{t-s} -\Quai(\Fer iu)(\Fer ju)\Ftil ijpq
\Etaz pu \Kappaz qu du -\Qua \Rnotil ijkl \Thetaz iu\Thetaz ju\Rhoz ku
\Rhoz lu\,du\cr
&+\Thirdi\Rnotil i{}j{}\Thetaz iu\Rhoz ju du -\Thirdi\Thetaz ju\Rhoz ku
\Bz {\ell}u
(R_{\ell kj}{}^i + R_{\ell j k}{}^i) d\Bz ku \Rbb\, f(\Bz{}{t-
s},\Phit+\Thetaz{}{t-s},\eta +
\Etaz {}{t-s})\Rsb.\cr
}\Ename\FKflat$$
(Here
$\Thetaz iu,\Rhoz ju,i,j = 1,\dots,m$ are fermionic Brownian paths,
while $\Etaz pu,\Kappaz qu$ are a further set of fermionic Brownian
paths, introduced to handle the twisted or gauge group part of the
operator.)
Thus $A(t)$ can be estimated using $\Bz{}u\sim\sqrt
u,\,\theta_u\sim1,\rho_u\sim1,\Etaz{}u\sim1,\Kappaz u{}\sim 1$.
(The fermionic part of these estimates follows from theorem
(3.3)
of \Scal.) This estimation shows that $A(t)\to 0$ as $t\to 0$.
Now the stochastic differential equations \SDE\ (in the
normal
coordinate system) and \SDEM\ are closely related, and one
might
expect that for small $s$ their solutions would be similar. In fact,
using the explicit construction of solutions developed in \Scal, one
can show (by induction) that
$\Til x_u-\Che{x}_u\sim \sqrt{u^3},\,\Til{\xi}_u-\Che{\xi}_u
\sim u$ and $\Til{\eta}_u-\Che{\eta}_u\sim u$ and $\Til{e}_u -
\Til{e}^1_u \sim u$. This enables one to show that $B(t) \to 0$ as
$t\to 0$. Thus
$$
\lim_{t\to0}\,\str \exp-\Til{H}t (0,0)
=\lim_{t\to0}\, \str \exp-\Che{H}t (0,0).
\Ename\ONE$$
\Step5{Evaluating the supertrace}
%
The final step in the proof of the Atiyah-Singer index theorem is to
evaluate $\lim_{t\to0}\str \exp-\Che{H}t (0,0)$ using flat space path
integral techniques (both classical \Sim\ and fermionic \GBM).
Now, once again using Duhamel's formula, and also using \FKflat,
$$\eqalign{
&\quad\str \exp - \Che{H}t(0,0)\cr
&= \Ebold\Lsc\Fintt d^m\theta'\Lsb\cr
&\quad\exp-\Che{H}(t-s)(\Che{H}-\Til{H}^0)G_s
(0,\Phit{},\theta',\eta,-\eta)\exp-i\kappa\eta\exp-
i\Phit\theta'\Rsb\Rsc\cr
&=\Ebold\Lsc\Fint \Lsb\cr
&\exp \Lbb\int_{0}^{t-s} \Qua(\Fer iu)(\Fer ju)
\Fnotil ijpq(i\Etaz qu \Kappaz pu - \delta^q_p) du \cr
&\,\,- \Qua\Rnotil ijkl \Thetaz iu\Thetaz ju\Rhoz ku
\Rhoz lu+\Thirdi\Rnotil i{}j{}\Thetaz iu\Rhoz ju du
\quad + \Thirdi\Thetaz ju\Rhoz ku \Bz {\ell}u
(R_{\ell kj}{}^i + R_{\ell j k}{}^i) d\Bz ku \Rbb\cr
&\quad\,\,(\Che{H}-\Hamz)G_s(x,\xi,\Phit{},\alpha,\kappa) \Evalf
\exp i\kappa\eta\Rsb.\cr
}\Eqno$$
Also, again using the estimates for flat space Brownian paths, it can
be seen that
$$\eqalign{
&\Limt \str \exp-\Che{H}t(0,0)\cr
&= \Ebold\Lsc\Limt \Fintt \Lsb\cr
&\quad(\Hamf-\Hamz)G_s(x,\xi,\Phit{},\alpha,\kappa)\Evalf \exp-
i\kappa\eta\Rsb\Rsc,\cr
}\Eqno$$
where
$$\eqalign{
&\qquad\Hamf\cr
& = -\Lbb\half\partial_i\partial_i -\Third \Phitt i\Phitt j
(R_{\ell}{}^k{}_{ji} + R_{\ell j}{}^k{}_i)
\Xhat l \partial_k \cr
&-{\textstyle{1\over18}}\Xhat k\Xhat{\ell}\Phitt {n'}\Phitt {m'}\Phitt
i\Phitt j
(R_{kpn'i} + R_{kn'pi})(R_{\ell}{}^p{}_{n'j} + R_{\ell
m'}{}^{p}{}_{j})\cr
&+ {1\over4}R_{ijk\ell}\Phitt i\Phitt j\Psih k\Psih{\ell}
+\Phitt i\Phitt j \Cuf ijpq \Etah p\Det q\Rbb. \cr
}\Eqno$$
(with $\Psih i=\theta^i + \partial/\partial\theta^i$).
Thus
$$\Limt \str \exp-\Che{H}t(0,0)=\Limt \str \exp-\Til{H}^2t(0,0).
\Ename\TWO$$
Now $\Hamf$ decouples into operators acting separately on the $x$
variables, the $\theta$ variables and the $\eta$ variables.
Explicitly,
$$
\Hamf = \Hamf_x + \Hamf_{\theta} + \Hamf_{\eta}
\Eqno$$
where (after some use of the symmetry properties of $R_{ijkl}$),
$$\eqalign{
\Hamf_x&=-\Lbb\half \partial_i\partial_i +\half \Xhat{\ell}
\Phitt j\Phitt i
\Cur ji{\ell}k \partial_k\cr
&\qquad+{\textstyle{1\over8}}\Xhat k\Xhat{\ell}{\phi^i\phi^j
\phi^{n'}\phi^{m'}\over (2\pi t)^2}
R_{n'ikp} \Cur {m'}j{\ell}p\Rbb,\cr
\Hamf_{\theta} &= -{1\over4}R_{ijk\ell}{\phi^i\phi^j\over2\pi t}
\Psih k\Psih {\ell}\quad{\rm and}\cr
\Hamf_{\eta}&=-\Phitt i\Phitt j \Cuf ijpq \Etah
p\Det q. \cr
}\Eqno$$
Now $\exp-\Hamf_x t(0,0)$ can be evaluated using the result given by
Simon \Sim\ for $\Real^2$ that, if
$$
L= -\half\partial_i\partial_i + {iB\over{\textstyle2}}(x^1\partial_2-
x^2\partial_1) + {\textstyle{1\over8}}B^2((x^1)^2+(x^2)^2),
\Eqno$$
then
$$
\exp-Lt(0,0) = {B\over 4\pi\sinh(\half Bt)}.
\Eqno$$
Thus, if $\Curv kl = \half\phi^i\phi^j \Cur ijkl$ is regarded as an
$m\times m$ matrix, skew-diagonalised as
$$\Lba\Omega_k{}^l\Rba = \left(\matrix
{0 &\Omega_1 &\ldots&0 &0\cr
-\Omega_1 &0&\ldots&0 &0\cr
\vdots&\vdots&\ddots&\vdots&\vdots\cr
0&0&\ldots&0&\Omega_{\half m}\cr
0&0&\ldots&-\Omega_{\half m}&0\cr}\right),
\Eqno$$
$$\exp-\Hamf_x t(0,0)
= \prod_{k=1}^{m/2} {i\Omega_k\over2\pi t}\,
{1\over\sinh({i\Omega_k\over2\pi})}.
\Eqno$$
Also, using fermion paths \GBM\ or direct calculation,
$$\eqalign{
&\exp-t\Hamf_{\theta}(\theta,\theta') \cr
&= \Bint d^m\rho \Lbb\exp -
i\rho(\theta-\theta')\cr
& \qquad\prod_{k=1}^{m/2} \Lba \cosh {i\Omega_k\over2\pi}
+ (\theta^{2k-1}
+i\rho^{2k-1})(\theta^{2k}+i\rho^{2k}) \sinh
{i\Omega_k\over2\pi}\Rba\Rbb.\cr
}\Eqno$$
Thus
$$\eqalign{
\str \exp-\Hamf t(0,0)
&=\Bint d^m\phi \prod_{k=1}^{m/2} {i\Omega_k\over2\pi}\,
{1\over\sinh({i\Omega_k\over2\pi})} \cosh {i\Omega_k\over4\pi}
{\rm tr} (\exp -\phi^i\phi^j {F_{ij}\over2\pi}).\cr
}\Eqno$$
Hence, using \CPA, \ONE\ and \TWO,
$$\str \exp -H(p,p)\, {\rm dvol}
=\Biggl[{\rm tr}\,
\exp\Lbc {-F\over2\pi}\Rbc \det \Lbc {i\Omega/2\pi\over\tanh
i\Omega/2\pi}\Rbc^{\half}\Biggr]\Eval p
\Eqno$$
as required.
%REFERENCES
\medskip
\medskip
\centerline{\bf References}
\baselineskip = 11pt
\lineskip = 1.5pt
\lineskiplimit = 3pt
%
\parskip = .2ex plus .5ex minus .1ex
\parindent 3em
\def\Bib#1#2#3#4#5#6#7{\item{#1}#2, ``#3'', #4 {\bf #5} (#6),
#7.\hfil\break}
\def\Bibk#1#2{\item{#1}#2.\hfil\break}
%1 refnumber, 2 author, 3 title, 4 journal, 5 vlume, 6 year, 7 pages.
%**************************************************************
\frenchspacing
%
%SECTION 1
%
\Bib\GBM{A. Rogers}{Fermionic path integration and Grassmann
Brownian motion}{Comm. Math. Phys.}{113}{1987}{353-368}
%
\Bib\Scal{A. Rogers}{Stochastic Calculus in Superspace I:
Supersymmetric Hamiltonians}{J. Phys. A}{25}{1992}{447-468}
%
\Bib\Mar{J. Martin}{The Feynman principle for a Fermi system}{Proc.
Roy. Soc.}{A251}{1959}{543-549}
%
\Bibk\Haba{Z. Haba, ``Supersymmetric Brownian Motion'', Bielefeld
University Preprint 1986}
%
\Bibk\Elwsf{K.D. Elworthy, ``Geometric Aspects of diffusions in
manifolds'' in {\sl \'Ecole d'\'Et\'e de Probabilit\'es de Saint-Flour
XV-XVII, 1985-1987, ed P.L. Hennequin}, Lecture Notes in Mathematics
{\bf 1362}, Springer (1988)}
%
\Bib\Bis{J.-M. Bismut}{The Atiyah-Singer theorems; A probabilistic
approach. I. The index theorem}{J. Funct. Anal.}{57}{1984}{56-99}
%
\Bibk\JLe{J.D.S. Jones and R. Leandre, ``$L^p$-Chen forms on loop
spaces'',
in {\sl Stochastic Analysis, eds M.T. Barlow and N.H. Bingham},
Cambridge University Press (1991)}
%
\Bibk\Lea{R. Leandre, Sur le theoreme d'Atiyah-Singer, preprint}
%
\Bib\Lot{J. Lott}{Supersymmetric Path Integrals}{Comm. Math.
Phys.}{100}{1987}{605-629}
%
\Bibk\Wat{S. Watanabe, ``Short time asymptotic problems in Wiener
functional integration theory. Applications to heat kernels and index
theorems'', in {\sl Stochastic Analysis and related topics II, eds H.
Korezlioglu and A.S. Ustunel}, Springer Lecture Notes in Mathematics
{\bf 1444} (1988)}
%
\Bib\Getzz{E. Getzler}{Pseudodifferential Operators on Supermanifolds
and the Atiyah-Singer Index Theorem}{Comm. Math. Phys.}{92}{1983}{163-
178}
%
\Bibk\BerGetVer{N. Berline, E. Getzler and M. Vergne, {\sl Heat Kernels
and Dirac Operators} Springer, 1992}
%
\Bibk\Elw{K.D. Elworthy, {\sl Stochastic differential equations on
manifolds},
London Mathematical Society Lecture notes in Mathematics, Cambridge
University Press, (1982)}
%
\Bibk\IkW{N. Ikeda and S. Watanabe, {\sl Stochastic differential
equations and diffusion processes}, North-Holland, Amsterdam (1981)}
%
\Bib\AG{L.Alvarez-Gaum\'e}{Supersymmetry and the
Atiyah-Singer index theorem}{Comm. Math. Phys.}{90}{1983}{161-173}
%
\Bib\FW{D. Friedan and P. Windey}{Supersymmetric derivation of the
Atiyah-Singer index theorem and the chiral anomaly}{Nuc. Phys.}
{B235}{1984}{395-416}
%
\Bib\MS{H.P. McKean and I.M. Singer}{Curvature and Eigenvalues of the
Laplacian}{J. Diff. Geo.}{1}{1967}{43-69}
%
\Bib\Wit{E. Witten}{Constraints on supersymmetry breaking}{Nucl.
Phys.}{B202}{1982}{253}
%
%SECTION 2
%
\Bibk\Superman{C. Bartocci, U. Bruzzo and D. Hern\'andez-Ruip\'erez,
{\sl The Geometry of Supermanifolds}, Kluwer (1991)}
\item{}M. Batchelor, ``Graded manifolds and supermanifolds'' in {\sl
Mathematical Aspects of Superspace, eds. H.-J. Siefert, C.J.S. Clarke
and A. Rosenblum} Reidel (1983).
\item{}A. Rogers, ``A global theory of supermanifolds'',
Journ. Math. Phys. {\bf 21} (1980) 1352-1365.
%
\Bibk\Erice{A. Rogers, ``Integration and global aspects of
supermanifolds'',
in {\sl Topological properties and global structure of spacetime, eds
P. Bergmann and V. De Sabbata}, Plenum, New York (1985)}
%
\Bibk\Sim{H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon, {\sl
Schr\"odinger operators with applications to quantum mechanics and
global geometry}, Springer (1987)}
%
\Bibk\Ber{F.A. Berezin, {\sl The method of second quantization},
Academic Press New York (1966)}
%SECTION 3
%
\Bib\GBC{A. Rogers}{A superspace path-integral proof of the
Gauss-Bonnet-Chern theorem}{Journ. Geom. Phys.}{4}{1987}{417-437}
%
%SECTION 4
\Bib\ABP{M. Atiyah, R. Bott and V.K. Patodi}{On the heat equation and
the index theorem}{Inventiones Math.}{19}{1973}{279-330}
%
\Bib\Get{E. Getzler}{A short proof of the Atiyah-Singer index
theorem}{Toplogy}{25}{1986}{111-117}
%
\Bibk\Sima{B. Simon, {\sl Functional integration and quantum
mechanics}, New York, San Francisco, London: Academic Press (1979)}
%
\item{}
\bye