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Brownian motion, Stochastic calculus, supersymmetry, supermanifold, index theorem, morse theory, quantization
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\begin{document}
%
\begin{center}
{\large Supersymmetry and Brownian motion on supermanifolds}\\
\ \\
%
Alice Rogers\\
\ \\
Department of Mathematics \\
King's College \\
Strand, London WC2R 2LS \\
%
\end{center}
\vskip0.2in
\begin{center}
December 2001
\end{center}
\vskip0.2in
%
\begin{abstract}
An anticommuting analogue of Brownian motion, corresponding to
fermionic quantum mechanics, is developed, and combined with
classical Brownian motion to give a generalised Feynman-Kac-It\^o
formula for paths in geometric supermanifolds. This formula is
applied to give a rigorous version of the proofs of the
Atiyah-Singer index theorem based on supersymmetric quantum
mechanics. It is also shown how superpaths, parametrised by a
commuting and an anticommuting time variable, lead to a manifestly
supersymmetric approach to the index of the Dirac operator. After
a discussion of the BFV approach to the quantization of theories
with symmetry, it is shown how the quantization of the topological
particle leads to the supersymmetric model introduced by Witten in
his study of Morse theory.
\end{abstract}
%
\section{Introduction}
%
This survey concerns a battery of generalised probabilistic
techniques, originally motivated by path integration in fermionic
and supersymmetric quantum physics, which may be brought to bear
on some significant geometric operators.
A theory of Brownian motion in spaces with anticommuting
coordinates is developed, together with the corresponding
stochastic calculus. This is combined with conventional Brownian
motion to provide a mathematically rigorous version of the direct
and intuitive proofs of the Atiyah-Singer index theorem based on
supersymmetric quantum mechanics \cite{Alvare,FriWin}, and also of
the quantum tunneling calculations which are necessary to build
Witten's link between supersymmetry and Morse Theory
\cite{Witten82}.
The connection between anticommuting variables and geometry arises
via Clifford algebras. It seems to have first been observed in
the context of canonical anticommutation relations for fermi
fields (which are essentially Clifford algebra relations) that
such algebras have a simple representation in terms of
differential operators on spaces of functions of anticommuting
variables. It was thus in the physics literature that functions
of anticommuting variables were first considered, beginning with
the work of \cite{Martin} and ideas of Schwinger \cite{Schwin},
and extensively developed by Berezin \cite{Berezi1966} and by
DeWitt \cite{Dewitt1984}.
Anticommuting variables are not used to model physical quantities
directly; their use is motivated by the algebraic properties of
the function spaces of these variables. In application to physics,
results which are real or complex numbers emerge after what has
become known as Berezin integration (defined in equation
(\ref{BINTeq})) which essentially takes a trace. The approach to
fermions using anticommuting variables is particularly useful in
the context of supersymmetry symmetry because bose and fermi
degrees of freedom, which are related by symmetry transformations,
can then be handled in the same way. Similarly, when using
{\textsc{BRST}} techniques to consider theories with symmetry, the
use of anticommuting variables to handle ghost degrees of freedom
provides a unifying approach.
At this stage a comment on the prefix `super' is appropriate.
Originally used in the physical term {\em supersymmetry}, to
describe a symmetry which mingles bosonic and fermionic degrees of
freedom, the word has migrated into mathematics to describe a
generalisation or extension of a classical object to an object
which is in some sense $Z_2$-graded, with an even part usually
associated with the classical, commuting object and an odd part
associated with some anticommuting analogue.
A generic superspace is a space with coordinates some of which are
commuting and others anticommuting. There are two approaches to
superspace, a concrete one which generalises the actual space, and
an abstract one which generalises the algebra of functions on such
a space. Here generally the concrete approach will be used
because it allows simpler super analogues of the classical
concepts considered; however it should be born in mind that the
function algebras used in the concrete approach will normally
correspond to the generalised algebras in the more abstract
approach, so that the difference is more apparent than real.
The anticommuting analogue of probability theory developed here
does not mirror all aspects of classical probability theory, but
is largely restricted to features which relate directly to the
heat kernels of differential operators.
In section \ref{SSSMsec} of this survey superspace is described,
the concept of supersmooth function is defined together with the
related notions of differentiation and integration. The
construction of a particular class of supermanifold, obtained by
building odd dimensions onto a manifold using the data of a vector
bundle, is also given. These supermanifolds play a key role in
the geometric applications of the anticommuting probability
theory, because they carry natural classes of supersmooth
functions which correspond to spaces of forms and spinors on the
underlying manifold. Section \ref{SUSYQMsec} introduces fermionic
and supersymmetric quantum mechanics, explaining quantization in
terms of functions of commuting and anticommuting variables.
Systems are described whose Hamiltonians correspond to a number of
geometric Laplacians.
Section \ref{GPSsec} introduces the anticommuting analogue of
probability measure, and in particular fermionic Wiener measure.
In section \ref{SWMsec} fermionic Wiener measure is combined with
classical Wiener measure to give a super Wiener measure for paths
in superspace. The stochastic calculus for the corresponding super
Brownian paths is developed. Next, in section \ref{BMSMsec}, these
techniques are used to define Brownian paths on supermanifolds and
derive a Feynman-Kac-It\^o formula for certain Hamiltonians.
The concept of supersymmetry implies more than the mere presence
of anticommuting and commuting variables; an odd operator, known
as the supercharge, which squares to the Hamiltonian or Laplacian
of the theory is required. (In some cases the Hamiltonian is the
sum of squares of more than one supercharge.) Supersymmetric
quantum mechanical models are described in section
\ref{SUSYQMsec}, while geometric examples of supercharges are the
Hodge-De Rham operator and the Dirac operator which are considered
in later sections. Using the `super' stochastic machinery, these
operators can be analysed. In section \ref{ASIsec} the methods are
applied to make rigorous the supersymmetric proofs of the
Atiyah-Singer index theorem given by Alvarez-Gaum\'e \cite{Alvare}
and by Friedan and Windey \cite{FriWin}, while in section
\ref{MTTsec} it is used to verify the key instanton calculation in
Witten's linkage of Morse theory with supersymmetric quantum
mechanics. The Dirac operator is studied in section
\ref{DIRACsec}, where a further super construction, the Brownian
super path parametrised by both a commuting time $t$ and an
anticommuting time $\tau$, is introduced. Anticommuting variables
may also be used to handle the ghost degrees of freedom used in
the quantization of theories with symmetry. These ideas are
briefly described in section \ref{BRSTsec}, and the used in the
following section to show that quantization of the topological
particle leads to the supersymmetric model used by Witten to
analyse Morse theory.
Other work involving anticommuting analogues of probability measures
includes that of Applebaum and Hudson \cite{AppHud}, Barnet, Streater and
Wilde \cite{BarStrWil}, Haba and Kupsch
\cite{HabKup}. An alternative approach to considering paths in
superspace is to consider differential forms on loop space, as in the
work of Jones and L\'eandre \cite{JonLea}.
The work in this survey shows how an extended notion of path integral may
be applied to a variety of quantum mechanical systems. When considering
quantum field theory the standard approach is by functional integrals,
but an interesting alternative approach, involving Brownian motion over
loop groups, and over torus groups and higher dimensional objects, has
been considered in papers by Brzezniak and Elworthy
\cite{BrzElw}, by Brzezniak and L\'eandre
\cite{BrzLea}, and by L\'eandre \cite{Lea01}. Here too fermionic analogues
might be possible.
%
\section{Superspace and supermanifolds}\label{SSSMsec}
%
This section introduces the analysis and geometry of anticommuting
variables. As explained in the introduction, we use a concrete approach
to the concept of supermanifold. We begin with some algebraic notions,
leading to the key concept of supercommutative superalgebra.
\begin{Def}{\rm
A \Defi{super vector space} is a vector space $V$ together with a
direct sum decomposition
\begin{equation}%\label{}
V = V_{0} \oplus V_{1}.
\end{equation}
The subspaces $V_{0}$ and $V_{1}$ are referred to respectively as
the \Defi{even} and \Defi{odd} parts of $V$.
}\end{Def}
%
We will normally consider homogeneous elements, that is elements
$\Scap{X}$ which are either even or odd, with parity denoted by
$\Gp{\Scap{X}}$ so that $\Gp{\Scap{X}}=i$ if $\Scap{X}$ is in
$\Real_{S,i},i=0,1$. (Arithmetic of parity indices $i=0,1$ is always
modulo $2$.)
%
\begin{Def} {\rm
\begin{enumerate}
\item A \Defi{superalgebra} is a super vector space $A= A_0 \oplus A_1$
which is also an algebra which satisfies $A_i A_j \subset A_{i+j}$.
\item The superalgebra is \Defi{supercommutative} if, for all
homogeneous $\Scap{X}, \Scap{Y}$ in $A$,
$\Scap{X}\Scap{Y} = \Cfac{X}{Y} \Scap{Y}\Scap{X}$.
\end{enumerate}
}
\end{Def}
The effect of this definition is that in a superalgebra the
product of an even element with an even element and that of an odd
element with an odd element are both even while the product of an
odd element with an even element is odd; if the algebra is
supercommutative then odd elements anticommute, and the square of
an odd element is zero.
The basic supercommutative superalgebra used is the real
Grassmann algebra with generators
$\One, \beta_1, \beta_2, \dots$ and relations
\begin{equation}%\label{}
\One\beta_i = \beta_i \One= \beta_i, \qquad
\beta_i \beta_j = - \beta_j \beta_i.
\end{equation}
This algebra, which is denoted $\Rs$, is a superalgebra with
$\Rs := \Real_{S,0} \oplus \Real_{S,1}$
where $\Real_{S,0}$ consists of linear combinations of products
of even numbers of the anticommuting generators, while
$\Real_{S,1}$ is built similarly from odd products.
It is useful to introduce multi-index notation: for a positive
integer $n$, let $M_n$ denote the set of all multi-indices of the
form
$\mu := \mu_{1} \ldots \mu_{k}$
with $1 \leq \mu_{1} < \ldots < \mu_{k} \leq n$ together with the
empty multi-index {\small{$\emptyset$}}; also let $|\mu|$ denote
the length of the multi-index $\mu$, and for any suitable
$n$-component object $\xi^1, \dots, \xi^n$ define
$\xi^\emptyset := \One$ (the appropriate unit for the objects $\xi^i$) and
$\xi^\mu := \One \xi^{\mu^1} \dots \xi^{\mu_{|\mu|}}$.
The set $M_{\infty}$ is defined in a similar way, but with no upper limit
on the indices. A typical element $A$ of $\Rs$ may then be expanded as
\begin{equation}\label{GENEXPeq}
\Scap{A} = \sum_{\lambda \in M_{\infty}} \Scap{A}_{\lambda}\beta_{\lambda}
\end{equation}
where each coefficient $\Scap{A}_{\mu}$ is a real number. For each
$\lambda \in M_{\infty}$ there is a \Defi{generator projection}
map
\begin{equation}\label{GENPROJeq}
P_{\lambda}:\Rs \to \Real ,
\quad
\Scap{A} \mapsto \Scap{A}_{\lambda},
\end{equation}
a particular case of this being the augmentation map
\begin{equation}\label{BODYeq}
\epsilon:\Rs \to \Real ,
\quad
\Scap{A} \mapsto \Scap{A}_{\emptyset}.
\end{equation}
%
Clearly $\Rs$ is an infinite-dimensional vector space. However our
use of $\Rs$ is purely algebraic, so that we do not need to equip
it with a norm, and the following notion of convergence is
sufficient:
\begin{Def}\label{CONVdef}
{ \rm
A sequence $\Scap{X}_k, k= 1, \dots$ of elements of $\Rs$ is said
converge to the limit $\Scap{X}$ in $\Rs$ if each coefficient
$\Scap{X}_{k,\lambda}$ in the expansion \break
$\Scap{X}_{k} = \sum_{\lambda \in M_{\infty}}
\Scap{X}_{k,\lambda} \beta_{\lambda}$ converges to the coefficient
$\Scap{X}_{\lambda}$ in the expansion \break
$\Scap{X} = \sum_{\lambda \in M_{\infty}}
\Scap{X}_{\lambda} \beta_{\lambda}$.
}\end{Def}
The Grassmann algebra $\Rs$ is used to build $(m,n)$-dimensional
superspace $\Rstt{m}{n}$ in the following way:
\begin{Def}%\label{}
{ \rm
$(m,n)$-dimensional \Defi{superspace} is the space
\begin{equation}%\label{}
\Rstt{m}{n} = \underbrace{\Rs{}_0\times\dots\times \Rs{}_0}_{m {\rm\ copies}}
\times \underbrace{\Rs{}_1\times\dots\times \Rs{}_1}_{n {\rm\ copies}}.
\end{equation}
}\end{Def}
A typical element of $\Rstt{m}{n}$ is written $(x; \xi)$ or
$(x^1, \dots, x^m; \xi^1, \dots, \xi^n)$, where the convention is
used that lower case Latin letters represent even objects and lower
case Greek letters represent odd objects, while small capitals are used
for objects of mixed or unspecified parity.
Consider first functions with domain the space $\Rstt{0}{n}$ of which a
typical element is
$\xi := (\xi^{1},\ldots,\xi^{n})$, that is, functions of purely
anticommuting variables. (For simplicity it will be assumed that
$n$ is an even number.) We will consider functions on this space
which are multinomials in the anticommuting variables. Using the
multi-index notation introduced above, a multinomial function
$\Scap{F}$ may then be expressed in the standard form
\begin{eqnarray}
\Scap{F}:\Rstt{0}{n} & \longrightarrow & \Rs \nonumber \\
(\xi^{1},\dots,\xi^{n}) & \mapsto &
\sum_{\mu \in M_{n}} \Scap{F}_{\mu} \xi^{\mu} \label{susmfn}
\end{eqnarray}
where the coefficients $\Scap{F}_{\mu}$ are real numbers. Such
functions will be known (anticipating the terminology for
functions of both odd and even variables) as \Defi{supersmooth}.
More general supersmooth functions, with the coefficients
$\Scap{F}_{\mu}$ taking values in $\Comp$, $\Rs$, or some other
algebra are also possible.
Differentiation of supersmooth functions of anticommuting
variables is defined by linearity together with the rule
\begin{equation}
\frac{\partial \xi^{\mu}}{\partial \xi^{j}} =
\left\{
\begin{array}{cl}
(-1)^{{k}-1} \xi^{\mu_{1}} \ldots \widehat{\xi^{{k}}} \ldots \xi^{\mu_{|\mu|}},
& \Mbox{ if $j=\mu_{{k}}$ for some ${k}$, $1 \leq {k} \leq |\mu|$,} \\
0 & \Mbox{otherwise,}
\end{array}
\right.
\label{graddiff}
\end{equation}
where the caret\ {}$\,{\widehat{}}\,${}\ indicates an omitted
factor.
Integration of functions of purely anticommuting variables is
defined algebraically by the Berezin rule
\cite{Martin,Berezi1966}:
\begin{equation}\label{BINTeq}
\Bint d^n\xi \, \Scap{F}(\xi) = \Scap{F}_{1 \ldots n},
\end{equation}
where $\Scap{F}(\xi)= \sum_{\mu \in M_{n}} \Scap{F}_{\mu}
\xi^{\mu}$ as in (\ref{susmfn}), so that $\Scap{F}_{1 \ldots n}$
is the coefficient of the highest order term.
This integral can be used to define a Fourier transform with
useful properties. If $\Scap{F}$ is a supersmooth function on
$\Rstt{0}{n}$, then the Fourier transform $\Ft{\Scap{F}}$ of
$\Scap{F}$ is defined by
\begin{equation}%\label{}
\Ft{\Scap{F}} (\rho)
= \Bint \Df^n \xi \, \Scap{F}(\xi) \exp i \rho.\xi
\end{equation}
where $\rho.\xi = \sum_{i=1 \dots n} \rho^i \xi^i$. A simple calculation
establishes the Fourier inversion theorem
\begin{equation}\label{FITeq}
\Ft{\Ft{\Scap{F}}}= \Scap{F}.
\end{equation}
%
Any linear operator $K$ on the space of supersmooth functions of
purely anticommuting variables has integral kernel $\Ker{K}$
taking
$\Rstt{0}{n} \times \Rstt{0}{n} $ into $\Rs$ defined
by
\begin{equation}\label{IKeq}
K \Scap{F}(\xi) = \Bint d^n\theta \, \Ker{K}(\xi,\theta) \Scap{F}(\theta).
\end{equation}
Using the Fourier inversion theorem we see that
\begin{eqnarray}
\delta(\xi,\theta)
&=& \Bint \Df^n \rho \, \Expi{- \rho . (\xi - \theta)}\End
&=& \prod_{i=1}^{n} (\xi^i-\theta^i)
\label{DELFeq}
\end{eqnarray}
is the kernel of the identity operator, that is
\begin{equation}%\label{}
\int \Df^n \theta \delta(\xi,\theta) \Scap{F}(\theta)
= \Scap{F}(\xi).
\end{equation}
More generally, if $K$ is a differential operator acting on
supersmooth functions on $\Rstt{0}{n}$, then
\begin{equation}%\label{}
\Ker{K} (\xi, \theta) = K_{\xi} \delta (\xi, \theta)
\end{equation}
where the subscript $\xi$ indicates that derivatives are taken
with respect to this variable.
The link between anticommuting variables and Clifford algebras, which
underpins the various constructions in this paper, arises from the
operators
\begin{equation}\label{CLIFFORDOPSeq}
\psi^i = \xi^i + \Dbd{\xi^i}, \qquad i=1, \dots, n
\end{equation}
which satisfy the anticommutation relations
\begin{equation}\label{ANTICOMeq}
\Com{\psi^i}{\psi^j} = \psi^i \psi^j + \psi^j \psi^i = \delta^{ij}.
\end{equation}
(In the super algebra context, the commutator of two operators $A$ and
$B$ is defined by $\Com{A}{B} = AB - \Cfac{A}{B} B A$.)
If $\mu$ is a multi-index in $M_n$ then
\begin{equation}\label{KERNELeq}
\Ker{\psi^{\mu}}(\xi, \theta)
= \Bint \Df^n \rho (\xi+i \rho)^{\mu}
\Expi{-\rho . (\xi - \theta)},
\end{equation}
a fact which leads to the fermionic Feynman-Kac formulae exploited in
this paper.
In the case of operators acting on a graded algebra a useful and natural
quantity is the supertrace. If
$\Wi$ is the operator on the algebra in question which acts as $1$ on
even elements and $-1$ on odd elements, the supertrace is of an operator
$K$ is defined to be the trace of the operator $\Wi K$. It may readily be
shown that integration of the kernel of $K$ at coincident points gives
the supertrace, that is
\begin{equation} \label{STRFeq}
\Str K = \Bint d^n\xi \, \Ker{K}(\xi,\xi)
\end{equation}
while the standard trace may be obtained from the formula
\begin{equation} \label{TRFeq}
\Tr K = \Bint d^n\xi \, \Ker{K}(\xi,-\xi).
\end{equation}
More general supertraces also occur, where $\gamma$ is replaced by
some other involution.
In order to extend the notion of supersmooth to functions on the
more general superspace $\Rstt{m}{n}$, we must take note of the
fact that an even Grassmann variable is not simply a real or
complex variable. However the necessary class of functions can be
captured by defining supersmooth functions on $\Rstt{m}{0}$ as
extensions by Taylor expansion from smooth functions on $\Real^m$.
\begin{Def}%\label{}
{ \rm
The function $\Scap{F}: \Rstt{m}{0} \to \Rs$ is said to be
\Defi{supersmooth} if there exists a function $\tilde{\Scap{F}}:\Real^m \to
\Rs$ whose combination with each generator projection (c.f.
(\ref{GENPROJeq})) is smooth, such that
\begin{eqnarray}%\label{}
\Scap{F}(x^1, \dots, x^m)&=& \tilde{\Scap{F}}(\epsilon(x))
+ \sum_{i=1}^{n} (x^i- \epsilon(x^i)\One)
\frac{\partial \tilde{\Scap{F}}}{\partial x^i}(\epsilon(x)) \End
&& +\sum_{i,j=1} (x^i- \epsilon(x^i)\One)(x^j- \epsilon(x^j)\One)
\frac{\partial^2 \tilde{\Scap{F}}}{\partial x^i \partial x^j}
(\epsilon(x)) \dots \, .
\end{eqnarray}
(Although this Taylor series will be infinite, it gives well
defined coefficients for each $\beta_{\lambda}$ in the expansion
(\ref{GENEXPeq}), so that the value of $\Scap{F}$ is a
well-defined element of $\Rs$.)
}\end{Def}
A supersmooth function on the general superspace $\Rstt{m}{n}$
can now be defined.
\begin{Def}\label{SSFUNCdef}
{ \rm
A function $\Scap{F}: \Rstt{m}{n} \to \Rs$ is said to be supersmooth
if there exist supersmooth functions $\Scap{F}_{\mu}, \mu \in M_n$ of
$\Rstt{m}{0}$ into $\Rs$ such that
\begin{equation}%\label{}
\Scap{F}(x, \xi) = \sum_{\mu\in M_{n}} \Scap{F}_{\mu} (x) \xi^{\mu}
\end{equation}
for all $(x,\xi)$ in $\Rstt{m}{n}$.
}\end{Def}
Integration of supersmooth functions is defined by a combination of
Berezin integration and conventional Riemann integration.
\begin{Def} \label{SUPERINTEGRALdef} {\rm
If $\Scap{F}:\Rstt{m}{n} \to \Rs$ is supersmooth and $V \subset \Real^m$,
then the integral of $\Scap{F}$ over $V$ is defined to be
\begin{equation}\label{SUPERINTEGRALeq}
\Sint{V} d^mx \, d^n \xi \, \Scap{F}(x, \xi)
= \int_{V} d^mx \, \left( \Bint d^n\xi \, \Scap{F}(x, \xi) \right).
\end{equation}
}
\end{Def}
Using the Berezinian, which is the superdeterminant of the matrix of
partial derivatives, a change of variable rule may be obtained which is
valid for functions of compact support.
Up to this point we have used the prefix super merely to indicate the
presence of an anticommuting extension of some classical commuting
object. The concept of supersymmetry involves the further feature that
the Hamiltonian (or, geometrically, the Laplacian) of the system is the
square of an odd operator known as the supercharge, or the sum of squares
of several supercharges. Many examples of this are given in the following
section on supersymmetric quantum mechanics. The corresponding time
evolution may also have a square root, which will be defined by
introducing an odd parameter $\tau$ in conjunction with the usual time
parameter $t$, and defining some rather special objects on the
$(1,1)$-dimensional superspace parametrised by $(t;\tau)$. The starting
point is the superderivative
$D\Subtt = \Dbd{\tau} + \tau \, \Dbd t$ acting on functions
$\Fsc(t; \tau)$ on the superspace $\Rstt11$. Since $(\Dbd{\tau})^2=0$,
this has the property that
\begin{equation}%\label{}
(D\Subtt)^2 = \Dbd{t},
\end{equation}
so that $D\Subtt$ is a square root of the generator of time
translations. It is also possible to introduce a notion of
integration between even and odd limits of a function $\Fsc$ on
$\Rstt11$ in the following way:
\begin{equation}\label{SUPERINTLIMeq}
\Intzztt \Dss \Fsc(s;\sigma) =
\Bint d\sigma \int_0^{t+\sigma\tau} ds \Fsc(s;\sigma).
\end{equation}
It may then be shown by direct calculation that this integral
provides a square root of the fundamental theorem of calculus in
the sense that that
\begin{equation}\label{SFUNDTHECALCeq}
\Intzztt \Dss D\Subss \Fsc(s;\sigma)
= \Fsc(t;\tau) - \Fsc(0;0).
\end{equation}
If we now introduce a superpath $X:\Rstt11 \to \Rstt{p}{q}$
and let
\begin{equation}
DX^i= \Dss D\Subss X^i(s, \sigma), i=1, \dots p+q,
\end{equation}
then by using the chain rule for derivatives together with
(\ref{SFUNDTHECALCeq}) the integral along $X$ of the gradient of a
function $\Scap{G}$ on $\Rstt{p}{q}$ can be expressed as the
difference between the values of $\Scap{G}$ at the endpoints of
the super path:
\begin{equation}\label{INTALONGSUPERPATHeq}
\Intzztt DX^i \Ds{i} \Scap{G}(X(s;\sigma)) =
\Scap{G}(X(t;\tau)) - \Scap{G}(X(0;0)).
\end{equation}
(The usual summation convention is applied for repeated indices.) In
section \ref{DIRACsec} a stochastic version of this result is established
and applied to the Dirac operator.
In many applications of anticommuting variables the simple superspace
$\Rstt{m}{n}$ must be replaced by a more general supermanifold. In this
article it will be sufficient to consider supermanifolds constructed in
a standard way from the data of a smooth vector bundle $E$ over a smooth
manifold $M$. (A theorem of Batchelor
\cite{Batche1979} shows that all smooth supermanifolds may be
obtained in this way.) The idea of the construction is to patch
together local pieces of the supermanifold using the change of
coordinate functions of the manifold and the transition functions
of the bundle. A careful definition would need an excursion into
supermanifold theory \cite{GTSM}; here it will suffice to define
change of coordinate functions, a full description of the patching
construction may be found in \cite{ERICE}.
Suppose that $M$ has dimension $m$ and $E$ has dimension $n$; the
supermanifold $\Sm{M}{E}$ is then $(m,n)$-dimensional. If
$\{ U_{\alpha} | \alpha \in \Lambda \}$ is an open cover
of $M$ by sets which are both coordinate neighbourhoods of $M$ and
local trivialization neighbourhoods of $E$, then for each $\alpha$
in $\Lambda$ there are local coordinates
$\Srow{x_{\alpha}}{m}{\xi_{\alpha}}{n}$ for the supermanifold;
change of coordinates on any overlap between charts is defined in
terms of the coordinate maps $\phi_{\alpha}:U_{\alpha} \to
\Real^m$ of $M$ and the vector bundle transition functions
$g_{\alpha\beta}:U_{\alpha} \cap U_{\beta} \to \Mbox{GL}(n | \Real)$
by
\begin{eqnarray}
x^i_{\beta}(x_{\alpha};\xi_{\alpha})&=&
\phi^i_{\beta}(x_{\alpha}) \qquad i=1, \dots, m \End
\xi^j_{\beta}(x_{\alpha};\xi_{\alpha})&=&
\sum_{k=1}^n g\,{}_{\alpha\beta}{}^j{}_k(x_{\alpha}) \xi_{\alpha}^k
\qquad j=1, \dots, n.
\end{eqnarray}
A particular example of this construction is the supermanifold
$\Sm{M}{TM}$ obtained from the tangent bundle of a manifold $m$.
Explicitly, if $M$ has dimension $m$, then $\Sm{M}{TM}$ has dimension
$(m,m)$, and local coordinates $\Srow{x_{\alpha}}{m}{\xi_{\alpha}}{m}$
which change according to the rule
\begin{eqnarray}
x^i_{\beta}(x_{\alpha};\xi_{\alpha})&=&
\phi^i_{\beta}(x_{\alpha}) \qquad i=1, \dots, m \End
\xi^j_{\beta}(x_{\alpha};\xi_{\alpha})&=&
\sum_{k=1}^{m} \Dbdf{x_{\beta}^j}{x_{\alpha}^k}(x_{\alpha})\, \xi_{\alpha}^k
\qquad j=1, \dots, m.
\end{eqnarray}
Supersmooth functions on this supermanifold are then naturally identified
with forms on $M$; in local coordinates this identification may be
expressed as
\begin{equation}\label{FORMFUNCTIONeq}
\Sum{\mu \in M_m}{} f_{\mu}(x) \xi^{\mu} \leftrightarrow
\Sum{\mu \in M_m}{} f_{\mu}(x) dx^{\mu},
\end{equation}
so that Berezin integration on the supermanifold corresponds to the
standard integration of top forms on the manifold and the exterior
derivative takes the form $d = \xi^{i}\Dbd{x^i}$.
For geometric applications of superspace path integration, the
significant supermanifold is
$\Som$, the underlying manifold being the orthonormal frame bundle $O(M)$
of a Riemannian manifold
$(M,g)$, and the vector bundle over $O(M)$ being the bundle induced by
projection of
$O(M)$ onto $M$ of the product of the tangent bundle of $M$ and a bundle
$E$ over $M$. There is a natural definition of Brownian motion on this
supermanifold, based on the Brownian motion on manifolds defined by
Elworthy \cite{Elwort} and by Ikeda and Watanabe \cite{IkeWat}, whose
construction is described in section \ref{SWMsec}.
A further supermanifold with geometric applications is the supermanifold
$\Sm{M}{T(M) \otimes E}$, where $E$ is an $n$-dimensional Hermitian vector
bundle over $m$; if one takes local coordinates
$(x^1, \dots,x^m; \xi^1, \dots, \xi^m, \eta^1, \dots, \eta^n)$, then
supersmooth functions which are linear in the $\eta$ variables correspond
to forms on $M$ twisted by $E$. Given a metric $g$ on $M$ and a
connection $A$ for $E$, the Hodge-de Rham operator takes the form
\begin{equation}\label{HODGEDERHAMeq}
d + \delta =
\psi^{i} \left( \Dbd{x^i} - \Chri{i}{j}{k} \xi^j \Dbd{\xi^k}
- A_{i \, r}{}^s \eta^r \Dbd{\eta^s}\right)
\end{equation}
where $\Chri{i}{j}{k}$ are the Christoffel symbols for the metric
$g$ and the Clifford algebra operators $\psi^i$
are now adapted to curved space, taking the form
$\psi^i = \xi^i + g^{ij}(x) \Dbd{\xi^j}$. The corresponding
Weitzenbock formula is then
\begin{equation}\label{WFlem}
-2(d + \delta)^2 =
B - R_i^j(x) \xi^i \Dbd{\xi^j} -
\Half R_{ki}{}^{jl}(x) \xi^{i}\xi^{k}\Dbd{\xi^{j}}\Dbd{\xi^{k}}
+ \Frac14 [\psi^i, \psi^j] {F}_{ij \, r}{}^s(x) \eta^r
\Dbd{\eta^s}
\end{equation}
where $R_{ki}{}^{jl}$ are the components of the curvature of $g$, $F_{ij
\, r}{}^{s}$ are the components of the curvature of $A$ and $B$ is the
twisted Bochner Laplacian
\begin{equation}\label{BOCNERLAPLACeq}
B= g^{ij}\left( D_{i} D_{j} - \Chri{i}{j}{k} D_{k} \right)
\end{equation}
with $D_i = \Dbd{x^i}- \Chri{i}{j}{k} \xi^j \Dbd{\xi^k}
- A_{i \, r}{}^{s} \eta^r \Dbd{\eta^s}$.
%
(A proof of this result, generalising the proof given in \cite{Simetal},
may be found in \cite{SCSONE}.)
The Dirac operator for a spin manifold $M$ of even dimension $m$
may also be represented as a differential operator on a
supermanifold. In this case the supermanifold is constructed from
the vector bundle $E_{O(M)}$ associated to the bundle of
orthonormal frames of $M$ via the vector representation of
$SO(m)$. The supermanifold $\Spinsm$ has local coordinates
$(x^i;\eta^a)$, $a,i=1, \dots, m$, using the convention that
coordinate indices are from the middle of the alphabet while
orthonormal frame indices are from the beginning of the alphabet.
Defining operators $\psi^a = \eta^a + \Dbd{\eta^a}$ we see that
$\psi^a \psi^b + \psi^b \psi^a = 2 \delta^{ab}$, so that the
supersmooth functions over a point of $M$ define a
$2^m$-dimensional representation of the Clifford algebra on
$\Real^m$. The Dirac representation can obtained by considering
supersmooth functions with values in $\Rs \otimes \Comp$, and
restricting to the $2^{m/2}$-dimensional subspace of functions
which satisfy the $n$ conditions
\begin{equation}
\Psib^{2r-1}\Psib^{2r} \,f= i f, \quad r=1, \dots, m/2,
\label{DCeq}\end{equation}
with $\Psib^a = \eta^a - \Dbd{\eta^a}$.
Functions satisfying these conditions will be referred to as Dirac
functions. It is also useful to define the projection operator $P$
which projects arbitrary functions onto Dirac functions by setting
\begin{equation}
P= P_1 \dots P_{n/2}
\end{equation}
where, for each $r=1,\dots,n/2$, $P_r$ is the operator which
satisfies
\begin{eqnarray}
\lefteqn{P_r\left(g(\eta^1,\dots,\eta^{2r-2})(a +b\eta^{2r-1} + c\eta^{2r} +
d\eta^{2r-1} \eta^{2r}) h(\eta^{2r+1},\dots, \eta^n)\right)=} \End
&& g(\eta^1,\dots,\eta^{2r-2})\left(\Frac{a-id}{2}(1-i\eta^{2r-1}\eta^{2r})
+ \Frac{b+ic}{2}(\eta^{2r-1} -i\eta^{2r})\right)
h(\eta^{2r+1},\dots, \eta^n).\End
\end{eqnarray}
%
\section{Fermionic and supersymmetric quantum\hfil\break mechanics}
\label{SUSYQMsec}
%
In this section we describe the quantum mechanical models whose heuristic
path integral quantization is constructed rigorously in this survey. We
begin with a purely fermionic model, and investigate various
supersymmetric models, starting in flat space and then moving to curved
space where we develop the models used section \ref{ASIsec} for the
supersymmetric proof of the index theorem and in section \ref{MTTsec}
for the study of Morse theory.
In the canonical quantization of $n$-dimensional particle mechanics, the
classical observables $p_i$ (momentum) and $x^i$ (position) are replaced
by quantum operators which satisfy the canonical commutation relations
$[x^i,p_j]=i \delta^i_j$. The standard representation is the Schr\"odinger
representation, with $x^i, i=1, \dots,m$ realised as the multiplication
operator on $\Ltr{m}$, and $p_i= -i\Dbd{x^i}$. For fermionic operators
$\psi^i$ the canonical anticommutation relations can be represented as in
an analogous manner on functions of anticommuting variables $\xi^i$ by
setting $\psi^i = \xi^i + \Dbd{\xi^i}$, so that
$\Com{\psi^i}{\psi^j} = \delta^{ij}$.
When fermions and bosons are both present, wave functions are functions
$\Scap{F}(x,\xi)$, with time evolution determined as usual by the Schr\"odinger
equation $ i \Dbdf{\Scap{F}}{t} = H \Scap{F}$ (or, in Euclidean time,
$ \Dbdf{\Scap{F}}{t} = - H \Scap{F}$), where units are
used in which Plank's constant $\hbar$ is equal to $1$. In flat space
the standard Hamiltonians take the form
$H = \frac{1}{2m} p_i p_i + V(x, \xi)$. The free Hamiltonian, on which the super
Wiener measure developed in sections \ref{GPSsec} and \ref{SWMsec} is
based, is
$H_{0}= \frac{1}{2m} p_i p_i$; the fermionic contribution to the free
Hamiltonian is zero. (Generally we will use $m=1$.)
The defining feature of a supersymmetric theory is that the Hamiltonian
$H$ has the form $H= \Half \Com{Q}{Q} = Q^2$ (or, more generally,
$H=\Half \sum \Com{Q_i}{Q_i} = \sum Q_i^2$) where $Q$ (or $Q_i$) is an
odd `supercharge'. The Lagrangian for such a theory is symmetric under a
group of transformations which includes transformations of fermions into
bosons and vice versa. The simplest example of a supersymmetric
Hamiltonian is the free Hamiltonian $H_{0} = Q_{0}^2$ with
$Q_0= \psi^i p_i$. In sections
\ref{ASIsec}, \ref{DIRACsec} and \ref{MTTsec} geometric examples of
supersymmetric Hamiltonians are used.
%
\section{Grassmann probability spaces and \hfil\break fermionic Wiener
measure}
\label{GPSsec}
%
In this section we begin by defining a notion of Grassmann probability
space and random variable, based on the standard finite-dimensional
Berezin integral (\ref{BINTeq}) for functions of anticommuting variables.
The Berezin integral is essentially algebraic; it is neither the limit of
a sum nor an antiderivative, and lacks the positivity properties
necessary for the use of standard measure theoretic techniques. The
approach to defining an analogue of probability measure taken here, which
was first developed in \cite{GBM}, is to use finite dimensional marginal
distributions and Kolmogorov consistency conditions.
A particular example, Grassmann Wiener space, together with the
corresponding fermionic Brownian motion, is then constructed and its
relationship to the heat kernel of fermionic Hamiltonians is
demonstrated.
\begin{Def}%\label{}
{ \rm
An $n$\Defi{-Grassmann probability space of weight }$w$, where
$w$ is an element of $\Rs$, consists of
\begin{enumerate}
\item a set $A$;
\item for each finite subset $B$ of $A$ a supersmooth function
$\Scap{F}_{B}: (\Rstt{0}{n})^{B} \to \Rs$ such that if
$B=\{b_1, \dots,b_N \}$ and $B'=\{b_1, \dots,b_{N-1} \}$ then
\begin{enumerate}
\item
\begin{equation}%\label{}
\Bint \Df^{n} \theta_{b_1} \dots \Df^{n} \theta_{b_N} \,
\Scap{F}_{B}(\theta_{b_1}, \dots \theta_{b_N}) = w,
\end{equation}
\item
\begin{equation}%\label{}
\Bint \Df^{n} \theta_{b_N} \, \Scap{F}_{B}(\theta_{b_1}, \dots, \theta_{b_N}) =
\Scap{F}_{B'}(\theta_{b_1}, \dots, \theta_{b_{N-1}}).
\end{equation}
\end{enumerate}
\end{enumerate}
Such a space will be denoted
$\left( \Rstt{(0,n)}{A}, \{\Scap{F}_{B} \} \right)$.
}\end{Def}
Having built Kolmogorov consistency conditions into the
definition, a notion of Grassmann random variable can be defined
by a limiting process. The definition of such random variables is
rather cumbersome because the generalised measure we are using
does not have the usual positivity properties.
\begin{Def}\label{GRVdef}
{ \rm
Suppose that $\Fclass$ is a class of functions on
$\Rstt{r}{s}$. Then an $(r,s)$-dimens\-ion\-al Grassmann
random variable of class $\Fclass$ on a
Grassmann probability space
$\left( \Rstt{(0,n)}{A}, \{\Scap{F}_{B} \} \right)$
consists of
\begin{enumerate}
\item a sequence $B_1= \{ b_{1,1}, \dots, b_{1, \Num{B_1}}\},
B_2 = \{ b_{2,1}, \dots, b_{2, \Num{B_2}}\},
\dots$ of finite subsets of $A$,
where $\Num{B_k}$ denotes the number of elements in the set $B_k$;
\item a sequence $\Scap{G}_1,\Scap{G}_2, \dots$ of supersmooth
functions, with $\Scap{G}_k : (\Rstt{0}{n})^{B_k} \to \Rstt{r}{s}$, such
that for each function $\Scap{H}$ in $\Fclass$ the sequence
\begin{eqnarray}%\label{}
\lefteqn{I_{k}\Scap{H} = } \End
&& \Bint \Df^{n} \theta_{b_{k,1}} \dots \Df^{n} \theta_{b_{k,\Num{B_k}}} \,
\Scap{F}_{B}(\theta_{b_{k,1}}, \dots \theta_{b_{k,\Num{B_k}}}) \End
&& \qquad \times \Scap{H}( \Scap{G}_{k}(\theta_{b_{k,1}}, \dots \theta_{b_{k,\Num{B_k}}}))
\end{eqnarray}
tends to a limit in $\Rs$ as $k \to \infty$. (The notion of
convergence in $\Rs$ is given in Definition \ref{CONVdef}.)
\end{enumerate}
The limit of this sequence is the \Defi{Grassmann expectation} of $\Scap{F}(\Scap{G})$,
and is denoted $\Exg{\Scap{F}(\Scap{G})}$.
}\end{Def}
%
We now construct fermionic Wiener measure, which is the key
construction underpinning the probabilistic methods described in this
survey. The finite dimensional marginal distributions which determine
this measure are built from the heat kernel of the free fermionic
Hamiltonian. Since this Hamiltonian is zero, the measure is in fact built
from Grassmann delta functions (\ref{DELFeq}). In order to obtain a
Feynman-Kac formula which can handle differential operators of quite a
general class, the measure is over paths in phase space, that is, the
Fourier transform variables are not integrated out before defining the
measure.
\begin{Def}\label{FWMdef}
{ \rm
\begin{enumerate}
\item Let $A$ be the interval $[0, \infty)$.
Then $n$-dimensional fermionic Wiener space is
defined to be the $2n$-Grassmann probability space
$(\Rstt{0}{2n})^{A}, \{ \Scap{F}_{B} \})$ such that, given
$B= \{ t_1, \dots, t_N \} \subset A$ with
$0 \leq t_1 < t_2 < \dots