PROBABILITY and GEOMETRY

Jean-Michel BISMUT

Departement de Math(~matique Universit~ Parts Sud

B~timent 425 91405 ORSAY

France

INTRODUCTION

I. SHORT TIME ASYMPTOTICS OF DIFFUSION BRIDGES AND HEAT KERNELS.

a) A finite dimensional analogue

b) Differential equations and stochastic differential equations.

c) Non singularity of @ .

d) Integration along the fiber in infinite dimensions.

e) A sketched proof.

f) A remark on the asymptotic expansions.

2. THE ATIYAH-SINGER INDEX THEOREM : A PROBABILISTIC PROOF.

a) A few definitions.

b) The Dirac operator.

c) The heat equation method.

d) A probabilistic construction of the heat equation semi-group.

e) An asymptotic representation of Tr s Pt(Xo,Xo) .

f) The asymptotics of Tr s Pt(Xo,Xo)

g) The Index Theorem of Atiyah-Singer.

3. LOCALIZATION FORMULAS IN EQUIVARIANT COHOMOLOGY AND THE INDEX THEOREM.

a) Assumptions and notations.

b) Localization formulas in equivariant cohomology.

c) A remark of Atiya/% and Witten.

d) The heat equation probabilistic proof asarigorous proof of localization.

e) Extension of the formalism of Atiyah and Witten to twisted Spin complexes.

f) Applications and extensions.

4. THE WITTEN COMPLEX AND THE MORSE INEQUALITIES.

a) Asstm~tions and notations.

b) The basic inequality.

c) The Morse inequalities : the non degenerate case.

REFERENCES.

The purpose of these notesis to give an introduction to a few

applications of probability theory in a variety of situations which

are connected with analysis or geometry.

In Section I, we show how to use stochastic differential equations

to obtain an explicit expression of the law of diffusion bridges by

integrating "along the fiber" the infinite dimensional Wiener measure.

The simplicity of the method - which is in principle based on an ele-

mentary application of the implicit function theorem in infinite dimen-

sions - limits its applicability to elliptic diffusions and certain few

hypoelliptic ones. However it at least offers a conceptual framework to

attack more compl~cate hypoelliptic situations. We here mostly follow

our work [17] which is based on the Malliavin calculus and uses large

deviation techniques. The main outcome is a compact formula which may

be used to calculate asymptotic expansions over bridges for any func-

tional of the considered bridge. For another approach to the same pro-

blem, we refer to Azencott [9], [i0].

In Section 2, and following our work [19], we describe a probabilis-

tic proof of the Index Theorem of Atiyah-Singer for Dirac operators.

Our presentation of the Index Theory is extremely brief. The proof is

based on the formula obtained in Section i. This formula is in fact

more precise than what is needed since it contains the whole asymptotic

expansion of the heat kernel. Also P. L~vy's well known area formulas

[57] play a key role in Index Theory (and also in fixed point theory).

We also refer to the proofs of Alvarez-Gaume [I], Friedan-Windey [31],

which are based on physical considerations, and to Getzler [35], [36]

and Berline-Vergne [15] for analytic proofs. In particular the second

proof of Getzler [36] gives the analytic counterpart of the probabilis-

tic proof. For fixed point Theory, we refer to [5], [8], [14], [19],

[21], [37].

In [4], Atiyah and Witten had exhibited a remarkable formal simila-

rity between the Index formula for the Dirac operator on the spin com-

plex and localization formulas in equivariant cohomology by Berline-

Vergne [13], [14], Duistermaat-Heckman [28]. In Section 3, and following

[25], we give a new proof of such formulas and show that the heat equa-

tion method, when interpreted probabilistically, is the corresponding

infinite dimensional rigorous extension. Still following [20], [25] we

also show that the functional integral constructs infinite dimensional

characteristic classes.

A precise understanding of the localization formulas of [13], [14], [28]

and an idea of Quillen [49] led us in [24] to the heat equation proof

of the Atiyah-Singer Index Theorem for families [8], which we also

briefly discuss.

In Section 4, we introduce the Witten complex, and describe Witten's

proof of the Morse inequalities [56]. Instead of s~ectral theory [56],

we still use the results of Section I, and follow our work [23].

References are only briefly indicated. Most of the time , we refer

to the original work (including ours) for more detailed references. This

is in particular true for Sections 1 and 2.

For the non probability oriented reader, the survey [18] might be

useful for a guided tour to Brownian motion.

I. Short time asymptotics of diffusign bridges and heat kernels.

In this section, we will aive a short survey of the results we ob-

tained in [17] on the short time asymptotics of diffusion bridges.

Two ideas will be exploited :

a) The first idea is the fact that the law of a smooth diffusion x.

can be obtained as the image of the Brownian measure P by solving

a stochastic differential equation.

b) The second idea is that if x I is the end point of the diffusion,

the conditional law of x . can be obtained by integration along the

fiber (xl=y) of the measure P considered as a differential form on

the infinite dimensional Wiener space.

In practice two instruments are used simultaneously :

a) The differential analysis on Wiener space, or Malliavin calculus,

combined with the theory of stochastic flows. This permits us essential-

ly to deal with Brownian motion almost as if it Was living in a finite

dimensional space. Also dependence of solutions on parameters can be con-

sidered in the usual C T sense, as envisioned in earlier work by

Blagoveshchenskii-Freidlin [26].

b) The formulation by Azencott of large deviation theory [9]. Indeed

Azencott established large deviation results as consequences of lar-

ge deviation on the Wiener space by studying the stochastic differen-

tial equation map.

Localization is obtained using Azencott's approach, and the diffe-

rential analysis is done using the Malliavin calculus. Of course the

necessary estimates are obtained in [].7] using the stochastic calculus

(martingale inequalities, Garcia-Rodemich-Rumsey's Theorem [33], etc..$

We refer to [17] for a much more systematic and precise exposition of

the ideas contained in this section. In particular, we want to under-

line strong connections with earlier work of De Wit~Morette-Maheswari-

Nelson [27] and also Elworthy-Truman [29].

The reader should keep in mind the following facts.

a) In the parametrix method for second order elliptic operators [32],

one is interested in the asymptotic expansion of the heat equation semi-

group Pt(x,y) Here we want much more, i.e. understand the structure

of the laws of the whole path.

b) In [17], a truly satisfactory answer has been given only in the

elliptic case. This is so because in the hypoelliptic situations

(H~rmander [39]) W÷Xl(W) in non singular only almost surely (this is

the essence of the Malliavin calculus [44]).

The non differentiability of Brownian motion plays here a key role.

Now the considered mapping may well become singular on differentiable

deterministic paths. In general for small time, the considered bridge

measures concentrate on singular points. This makes the problem of

understanding the asymptotic structure of the bridge very difficult

and interesting.

c) Rough estimates on the heat equation semi-group are needed, like the

estimate of Varadhan [52] for elliptic diffusions. Similar uniform esti-

mates are needed for non elliptic diffusions (see Gaveau [34],

L~andre [43]).

Other approches exist to attack the previously described problems.

a) In Molchanov [46], the asymptotic structure of the bridges in the

elliptic case was described in Gaussian a~proximation . However sto-

chastic differential equations were not used. Also Kifer [42] obtained

small time expansions of certain semi-groups usina a mixture of ana-

lytic and probabilistic techniques.

b) In [I0], Azencott obtained such expansions in the elliptic case

using the stochastic differential eauation describina the bridge itself.

As is well-known, such equations are singular and necessitatea careful

analysis.

Also note that in Fefferman [30], Sanchez-Calle [51]} much progress

has been done in the understandina of the time behavior of the heat

equation semi-grou D associated with ~ 2 1Xi under H~rmander's assumptions

[39], in relation with the geometry of the vector fields XI...X m

This might also lead ultimately to a better understandinq of the bridges

themselves.

In the next sections we will apply the techniques of Section 1 only

in comparatively simple situations where only a few terms of the small

time expansions are needed. However the fact of having a global descrip-

tion of the law of the bridge will help us considerably, specially in

proving the Index Theorem.

This section is organized as follows.

In a), a finite dimensional analogue is described. In b) , certain

stochastic differential equations are introduced. In c) , the non sinau-

larity of a certain mapDing is considered. In d) , an integration alona

the fiber procedure in infinite dimensions isgiven, which leads to the

asymptotic representation of certain bridaes. In e) , a proof of the

results of d) is sketched. In f), asymptotic expansions are considered.

In the whole text, d will denote S tratonovitch differentials, and

6 It8 differentials [18].

a) A finite dimensional analoque.

Let } be a C ~ mapping from R n into R m . We endow R n with lwl 2

the Gaussian measure dP(w) =e 2 dw Let d~(y) be the imaoe mea- (/2~) n

sure by ~ By definition for any f6C~(R m)

(i.i) f nf[¢(w) ]dP(w) = ~Rmf(Y) dP (y) R

Now assume that m<n , and that % is a submersion, i.e. that the

rank of %'(w) is m at every w .

Set

K = ~-l{y} y

If Ky is non empty, it is a n -m dimensional submanifold of R n.

Also dP~(y)_ ~ is of the form

(1.2) dP¢(y) = m(y)dy

In the sense of classical differential aeometry, m(y)dy is the in- ]wl 2

tegral along the fiber K of e 2 dw which we write in the form : n t

Y (~2~)

lw[ 2

(1.3) m(y)dy = f e 2 dw K (g2~) n Y

In particular if under a C ~ change of coordinates in R nw÷~(w)=x ,

%'(x)= ~(9-1(x)) is a projection

(1.4) x = (y,y') ÷ }' (x) : y

if g(x)dx is the law dP evaluated in the new coordinates, then

(z .5) re(y) : ~Rn_mg(y,y')dy'

However in general (1.4) is only valid locally. Another may of re-

writing (1.3) is to use an explicit factorization of dx . Namely let

'~ R m R n d~Y(w) be the area element in K Let ~ (w) : ÷ be the Y

adjoint of ~' (w) Then (1.3) can be explicitly rewritten as

(1.6)

lwl 2 d~Y(w)

m(y) = I e 2 .......... , ,~ K (/2~) n [det(~ # ) (w) ]1/2 Y

(1.3) -(1.6) contains in fact more information than m(y)

obvious that

It is

(1.7) dP y(w) -

lwl 2 e 2 dc y (w)

m(y) (/2~) n[det¢'#'*(w) ]1/2

is the law of w conditional on %(w)= y • This shows that

lw I 2

h (w) e 2 da y (w)

1.8) IRnf[ ¢(w) ] h(w) dP(w)= ~R mf(y) dV[~- K (\/2~)n det [~'#'*(w)] I/~

Y

We now fix y6R m K Y

Figure l

We assume that 16R n is the unique element of minimal Euclidean norm

!n K Let V be a "small" neighborhood of % in R n Y

Set

(1.9) H I = TIKy ; H 2 = HI~

Clearly 16H 2

As shown in Fiqure i, we can Darametrize locally K by H I For 2 2 y

wl6Hl small enough, there is a unique v (with v small enough)

such that

(I.i0) (I +wl,v2)6K Y

2 1 v depends smoothly on w

1 We now find that in the local coordinates w

(i.il)

Also

Set

(1.12)

dP 1 (w I )

ll+v2 (wl) 12 2

m (y) dpy (w) e (/2~) m det[ ~ (l+wl+v2(wl))]

~v 2

iwll 2

2 e dw 1

(/2 ~ ) n-m

det[ ~--i(l+wl+v2)] = det[~' (l+wl+v 2) ..... ~'*(~) ] ~V 2 [det(%'%'*(1) ]1/2

dP I (w I) =

lwll

e 2 dw ] (/2~) n-m

is the Gaussian measure on H I We find that

(1.13) h (w) m (Y) dPY (w) = F

h[l+wl+v 2 (w I) ]e

SIwl I<~E (/2~) m

I ~+v 2 (w i) 12

2 [det(~'¢'*) (I) ]I/2dPl (wl)

det[%' (l+wl+v2 (wl)) %'* (~) ]

Now replace ¢(w) by

mt(Y) , dP[ (W)

ded smooth h

(i.i4)

%t(w) = }(/tw)

will denote the corresponding objects. Then for a boun-

(eventually depending on t )

fgh(w) mt(Y)dP~(w)=flw I i<~//th(~+/twl+v2"('/twl))(/2~t) m

I ~+v 2 (/tw i) 12 - 2t i/2

e [det#'#'~(,l) ] dP] (w I)

[det~' (~+/twl+v 2(/tw I)) ~'~(l) ]

Mow (1.14) is interesting for the followino reasons :

• Assume that as t++0 , mt(Y) dp~(w)_ localizes on 1 f

se that for any k6N

in the sen-

_L~L 2

(1.15) mt(Y)P~(Cv) = e 2t o(t k)

{we may eventually replace o(t k) by e -×/t with

r.h.s, of (1.14) is then an accurate evaluation of

X > 0 ) The

up to e 2t o(tk).

~h (w) m t (y) dP y (w)

• If (1.15) is verified, we have a very precise evaluation of mr(Y) ,

but also a precise understandina of P~ as t++0 . In particular, we

may Taylor expand mt(Y) in the form

2t

(i.16) mt(Y) =~(/2~t) n[ao(Y)+al(Y)t+...]

and more qenerally, we may expand

(1.17) ~h t(w) m t(y)dPy t(w) •

However all the information is contained in the compact formula (1.14).

• In a neighborhood of 1 , all the P~ are described by means of

the unique probability space (H I , dPl(wl)) This will be of utmost

importance in an infinite dimensional context.

In general (1.15) is an easy estimate only if K Y

wise some work has to be done to prove (1.15).

is compact. Other-

b) Diffezential eauations and stochastic differential equations.

The basic idea which we will develoo is that in certain infinite di-

mensional situations, (1.14) still makes sense.

Set

(1.18) H = L2([0,1] ; R k)

Let X 1 .,.,X k be a family of vector fields on R m whose components

belong to C~(R m) .

10

Fix x 6R m . For h6H , consider the differential equation o

(1.19) k

dx h = E X. (xh)hids l

1

h x (0) = x

o

(from now on, the summation signs will be omitted).

Set

h (1.20) }(h) = x 1

We are in a situation analogue to what is done in Section i, except

that R n is renlaced by H

] k Let P be the Wiener measure on C([0,1] ; R k) wt=(w~...w t)

denotes the corresDondina Brownian motion.

For t >0 , consider the stochastic differential equation

(1.21) dx = X. (x) ./tdw i 1

x(0) = x o

where the differential dw is the Stratonovitch differential of w

[18]. Set

(1.22) ¢(/tdw) = x 1

The notations (1.20)-(i~22) are compatible.

Indeed :

a) P should be thought of as the Gaussian cylindrical measure on H dw

Of course a.s. ~s ~H , i.e.

p (H) : 0 .

b) As is well-known, by a result of Wona-Zakai,Strooc~-Varadhan [50], n

if w is replaced by its piecewise linear interpolation w on

dyadic time intervals [k , k+l[ , as n÷+~ x ~/t~n conver~es~ to the 2n 2 n

solution x of (1.21) in probability.

11

In particular

(1.23) ~(/t~ n) ÷ ~(/tdw) in probability .

More generally, we will take for granted all the results on stochas-

tic flows [16J, [17], [18], which quarantee that a.s., differentia-

tion with respect to parameters is possible on stochastic differential

equations.

c) Non singularity of

Let us now try to reproduce what has

the correspondence

been done in a). We will study

h ÷ ~(h)

h If x is the solution of (1.19), set

1.24) x h = ~h(x o)

h* ~ h* Let ~t be the derivative t. It is easy to verify that <0 t (x o)

is invertible. By proceeding as in [17, Chapter i], we find that for

v6H

h* rl, h*-i . (Xo)Vi ~' (h)v = q)l ]otq)s Xi) ds •

(%'~' ) (h) is the matrix

(1.25) < h* h*-I

_h h*rl, h*-i Xi ) (Xo) X i(xO),p >ds p +t P'~Pl ]O~PS ~I ~°s "

The non singularity of ¢ is equivalent to the invertibility of C~

h* C h is exactly the Modulo the irrelevant invertible ~i '

Malliavin covariance matrix of the nroblem [17, Chapter i].

As is well-known by the Malliavin calculus [44], if XI,...,X m

verify the assumptions of H~rmander [39] (i.e. if X 1 ... X m and their

Lie brackets span R m at each x 6Rm), then C dw is a.s. invertible.

This fact, and estimates on the L norm of II (cdW)-lll are the key P

steps in the probabilistic proof of H~rmander's theorem. The key point

is that w is not differentiable.

12

However recall that if we go back to the picture of Figure i, and

think in terms of large deviations, as t+%0 , in principle dP~ (w)

concentrates ona 16H , on which in general, even under H6rmander's

assumption, C 1 is not invertible.

m For a given y6R , if K is defined by

Y

(1.26) K = {h6H ; ¢(h) : y}. Y

K is generally a singular subset of H . Y

Conditions under which C h is invertible are given in [i7, Chapter i]

They include

a) The elliptic case : This is the case where XI,... X m span R n

everywhere. In this case ~ is obviously a submersion.

b) Certain few hypoelliptic situations which are modelled on the three

dimensional Heisenberg group (XI, X 2 being the first two generators

of its Lie algebra).

Except these two cases, there is little hope that the method outlined

in a) could work.

d) Integration along the fiber in infinite dimensions.

We now concentrate on the elliptic case.

For more general situations see [17] and Leandre [43]. Remember that

our purpose is to understand fully the small time asymptotics of certain

diffusion bridges (which live on the sets K for y6R m) Y

To simplify, we do the assumption that k=m . We then assume that

Xl(Xo) ... Xm(X O) span R m

Also, we will only interest ourselves in the disintegratlon or the

diffusion measure on K , i.e. on the law of diffusion loops which

start and end at Xo6R mxO For the general case see [17].

Here

K x = {hCH ; ~(h) = x ° } . o

13

Obviously the element of minimal norm in K x o

is l=0 .

We now use the picture of Figure i.

Using (1.24), it is clear that H I

exactly

(which coincides with TK x ) o

is

(1.27) H 1 = {v6H ; f~ vds = 0}

Its orthogonal H 2 is given by

(i .28) H 2 = {v6H ; v : cst }.

The cylindrical gaussian measure P1 on H I is the Brownian bridge

measure. Under dPl(wl),wl(0<s<l) will be a Brownian bridge in R TM j 1 ] dw ]

with w ° = w I = 0 ds should be thought as living in H 1 (although

it doesn't !).

Our next problem is to make sense of Figure i.

Namely we should make sense of the following equation

(1.29) dx = X. (x) [~t dw l'i +v 2'i ds] 1

x(o) = x O

1 where in (1.29), for a given w

be chosen uniquely so that

2 , for t small enough, v should

(1.30) X I = x O

2 1 But then obviously v will anticipate on x,w and so we get out

of the usual probabilistic folklore ! Also, as far as estimates are

concerned, it is well-known that if w I is described as a semi-martin-

gale, its drift is singular at t:l

So let us explain how to solve these difficulties.

i) . Take a usual Brownian motion w' Set

1 ! W = W -- SW I s s 1

1 W is then a Brownian bridge.

14

2). Replace equation (1.29) by

v

(1.31) dx = X. (x) [~t dw i + (v2,i_\/tw l,i) ds] 1

x(o) = x o

We will have to make sense of (1.31) and satisfy (1.30).

We now use stochastic flows [16, Chapter I -III]. Consider the equa-

tion

'i (1.32) dy = Xi(Y)~t dw

y(o) = Yo '

and its associated stochastic flow of C T diffeomorphisms ~s(~t dw')

so that in (1.32) , ys=~s~t dw,Yo)l

Consider the differential equation

(1.33) dz

z(o)

'i = (~s-iXi) (z) (v2'i-~t w 'l)ds

= x o

Of course (1.33) is solvable as a differential equation, if the

flow ~ does not grow too much at infinity. This is the case if

Xl,... X m have compact support.

So in the sequel, we assume that XI,...X m have compact support. This is

no restriction since we are essentially interested in what happens in a

neighborhood of x O

Since w' is now an usual Brownian motion, estimations on (1.31)

(1.33) are fairly standard [16, Chapter I-III].

SO as a definition of a solution of (1.31), we set

(1.34) x s = ~s(~t dw', z s)

15

Again this definition is sound from the point of view of the approxima-

tion of stochastic flows with the flows of differential equations, as

shown by the approximation result of [16,Theorems I.i.2 and 1.2.1].

A trivial application of the ordinary implicit function theorem

(with w' as a parameter) shows that indeed for t small enough, a

unique v 2 of small norm exists such that (1.30) is verified. The 2 w' 1 reader may wander wether v depends on or only on w . Approxi-

mation or the theory of enlargement of filtrations [40] shows that in

fact v 2 is a function of w I We will write v2(~t dw I)

In order to make sense formally of formula (1.14) in this situation,

we must show how to define the determinants appearing in (1.14).

This is very easy. As shown by (1.33), x. defined by (1.31) is C ~ in 2

v For t small enough, the analogue of the determinant in (1.14)

does not vanish and its inverse is uniformly bounded.

It is not entirely trivial to find how to replace the now meaningless

lwll~e//t. We will instead introduce a function G(t,dw I) which has the

following properties [17, Chapter 4].

• 0 ~< G ~< 1

• If G # 0 , v2(~t dw I) is well-defined.

• For any k 6 N

P[G(t,dw I) ~ i] = ~ (t k)

The explicit definition of G is directly connected with the applica-

bility of the implicit function theorem.

we now state in this situation the main result of [17, Section 4].

Theorem I.i. For any bounded function h on ~([0,i] ; R m) the follo-

wing equality holds

16

i X (1.35) mt(x o) @[0,1];Rm)h(w)G(t,dwl)dPt°(w)

= I~[0,1];Rm ) h(~tw I + Sv2)G(9, dwl)

(~2wt) m exp

v 2 ] 2

2t

dP] (w I )

det ~__i (~tw I +v2(gtdwl)) ~v 2

Also for any k6N , and uniformly on compact subsets of R m

X (1.36) mt(x o) Pt°(G(t,dw I) ~i) :o(t k)

Remark I. (1.35) tells us in particular that in the region G(t,dwl)=], x o the description of the measure mt(xo)dP t (w) which we gave using the

implicit function theorem is exact.

Recall that in the probabilistic litterature [40],a Brownlan bridge

is usually described in terms of a u-process (in the sense of Doob).

Namely let Pt(x ,y) be the heat kernel associated with the operator o

n x2 " t z t i

2 x e . Then under Pt°,x. is described by the stochastic differential

equation

(i 37) dx X. (x) (~tdwi+ tXipt(l-sl (x'x°) ds

. = ..... )

l p t (l-s) ( x ' Xo)

x ( O ) = x O

However note that the drift is singular at s:l [17, Section 2].

The description of the bridge given in (1.35) eliminates the singularity~

Also we now will use the bridge to ohtafn Informations in particular on

Pt " On the contrary (1.37) uses Pt to construct the bridge. However

note that Azencott [10]was able to use (1.37) to also obtain asympto-

tic expansions.

17

e) A sketched proof.

We now briefly sketch the proof of Theorem i.I.

The basic idea is to split the Brownian measure P on H I , H 2 •

Clearly under the measure

dP (w I )

IV2{2 2 exp - ~ d v

(%/2 ~ t) n

if w is given by

1 v2S (1.38) W s = w s +

w is a standard Brownian motion. We now consider the stochastic diffe-

rential equation .

(1.39) dx = X. (x) (~tdwl'i+ v2'ids) l

x(0) = x o

which we solve as indicated in (1.29) - (1.34).

Approximation theory or enlargement of filtrations show that if w

given by (1.38) and x by (1.39) then

is

x I = #(~tdw)

Now Iv2[ 2

2 t dv 2 (1.40) fg(Xl)h(x)dP(w) = fg(Xl)h(x)dP I (wl)e

(~2~t) m

1 The idea is now to change variables in (1.39), i.e. for a given w ,

to make the change of variables v 2 This makes appear the deter- ÷x 1

minant Z~ By showing that a smooth disintegration of (1.40) is pos-

er 2 sible we obtain (1.35).

The most difficult point is obviously (1.36), which is the analogue

of (1.15). In [17], to prove (1.36), we use large deviation results on

18

diffusions defined by stochastic differential equations by

• using large deviation results on Brownian motion

• establishing simple "continuity" properties of stochas-

tic differential equations.

Also we use the uniform estimates on the heat kernel for elliptic

diffusions.

d2(Xo Yo ) d2(xo'Yo)-× }< %(Xo,Y o) ( ~ exm{- ' }

(1.41) exp{- 2t ~ 2t

of Molchanov [46], Azencott and al. []2].

TO establish large deviation results for bridges, we divide [0,i] in

the intervals [0,1/2] and [1/2,1] and use standard large deviation

theory for free diffusions as well as (1.41) and Varadhan's technique

[52], [53].

x A key step is to prove that under Pt O , the conditions of the im-

plicit f u n c t i o n t h e o r e m a r e v e r i f i e d u p t o o ( t k) . By p r o c e e d i n g a s

previously indicated, this is equivalent to establishing a large devia-

tion r e s u l t o n s t a n d a r d s t o c h a s t i c f l o w s .

This is done in [17] by an indirect technique wh±ch uses (1.37) and

large deviat±on estimates on

grad Pt

Pt

More recently, Kusuoka [59] has shown us how to establish large devia-

tion results for stochastic flows, which are as good as the correspon-

ding results for usual stochastic differential equations.

The idea of Kusuoka is as follows :

a) On a single trajectory of a flow ~s(~tdw,x o) , standard large de-

viations give bounds like e-X/t(×>0)

b) Take ~>0 arbitrarily small. If we now choose e ~/t points x o on

a uniform grid inabounded region (with ~<X ) , the bound will become

19

e-(×- n) / t

n -~/t c) Any point in a bounded region is at a distance at most e of a

point in the grid. Standard estimates on the module of continuity of a

flow [16, Chapter I] show that the remainin~ probability which should

be estimated is like C~ e -6/t (with ~ arbitrary large).

f) ~_remark on the asymptotic expansions.

In [17, Chapter 4], we show that (].35) can be expanded in powers of

t , thus obtainin~ the Minakshishundaram-Pleijel expansion of

mt(x O) =Pt(Xo,Xo) when h = ]

This is done by expanding IV212

2t det }~

8v 2

in the variable ~t using

iterated stochastic integrals.

However note that classically, in finite dimensions (see H~rmander

[58]), expansions of Laplace type integrals are obtained usfng a cerv

tain second order differential operator associated with the Laplace

integral.

Here the situation is infinite dimensional. So the corresponding

differential operator is infinite dimensional.

2. The Atiyah-Singer Index Theorem : a__probabilistic Eroof.

In this section, we give a probabilistic proof of the Atiyah-Singer

Index Theorem for Dirac operators, based on the asymptotic representa-

tion of Section i. We here follow our paper [19].

In a), we give a few definition . In b), we introduce the Dirac

operator D. In c) , the heat emuation method is briefly described. In

d), the heat equation semi-group associated with D 2 is constructed

probabilistically. In e) , an asymptotic representation of the super-

trace of the heat kernel is miven. In f) , the asymptotics of the

20

supertrace is explicitly found in terms of a Brownian bridge. In g),

the local Index formula is calculated using the well-known area formula

of P. L@vy [57]. This formula is proved again by means of an infinite

determinant. This will be of utmost importance in the next section in

connection with Atiyah [4].

This section includes neither motivation nor applications to the

Index Theorem. We refer to Atiyah-Bott [5], Atiyah-Singer [8], Atiyah

[3], in which the backround material is developed . The heat equation

method was introduced in Mc Kean-Singer [45], Patodi [47], Gilkey [37],

Atiyah-Bott-Patodi [7]. Physicist's proofs of the Index Theorem based

on supersymmetry were given by Alvarez-Gaume [i] Friedan-Windey [31],

Getzler gave a rigorous proof in [35] using pseudodifferential operators

techniques. Berline-Vergne [15] gave another proof using heat equation

in the bundle of frames.

The second proof of Getzler [36] is more directly related to our

proof in [19] . We refer to [36] for more details.

a) A few definitions.

n = 2£ is an integer. R n is the Euclidean space endowed

with an orthogonal oriented base el,.., e n

Recall that the Clifford algebra c(R n) is the algebra generated

over R by 1 , el,.., e n and the commutation relations.

(2.1) e. e + e. e =-2~.. i 3 3 i ±3

c(R n) is spanned by the products eil...eip (with i I <i2... <ip) ,

As a representation of S0(n) , c(R n) is isomorphic (as a vector space)

to the exterior algebra A(R n) . c(R n) has a natural grading (which cor-

responds to the grading of A(Rn)). In particular c(R n) is Z 2 gra-

ded, and so has even and odd elements.

c(Rn) @RC identifies to End S , where S is a 2 £ dimensional

Hermitian space of spinors. Set

(2.2) T = i el...e n

Then

21

(2.3) T 2 = i

Set

S = {s6S ; Ts=s} ; S = {s6S ; ~s =-s}. +

Then

S =S+@S_

£-i S+,S_ are orthogonal subspaces of S , of dimension 2 The ele-

ments of ceVen(R n) (resp c°dd(Rn)) commute (resp. anticommute) with

Definition 2.1. If A6c(R n) , the supertrace TrsA is defined by

(2.4) Tr A :~ Tr[TA] s

If A6c (R n) ,

(2.5)

and so

(2.6)

as an element of

A = [E F!

G H

End ( S+@S_ )

Tr A --: Tr E - Tr H . s

we may write

It may be easily shown (see Atiyah-Bott [5 , p. 484])that if

il< .... <ip,p<n

(2.7) Tr e .... e = 0 s ]'i Ip

and that

(2.8) Tr s e I ... e n = (-2i)

Spin(n) denotes the double cover of S0(n) Since for n > 3 ,

~l(S0(n)) = Z 2 , for n > 3 , Spin(n) is the double cover of

S0(n)

Spin(n) can be embedded in ceVen(R n)

embedding.

We partly describe this

Let ~ be the set of (]%,n) real antisymmetric matrices. ~ is

the Lie algebra of S0(n) and Spin(n)

22

If A = (a3)66L, we identify A with the element of ceVen(R n)

1 ~j ei e (2.9) k (A) = ~ i j

The key facts are that k is a Lie algebra homomorphism and that if

f6R n is considered as an element of c(R n)

(2.10) [k (A) ,f] : Af .

in the r.h.s, of (2.10), Af is still considered as an element of

c (R n)

Spin(n) acts unitarily and irreducibly on S+ and S_

Set ~ be the projection Spin(n) ÷ SO (n)

If x6Spin(n) , for f6R n , xfx-16R n and also

-i (2.11) ~(X) f = xfX

If A6a , we also identify A with the 2-form X,y÷< Y,AY> , i.e.

with

i a i dx i A dx j (2.12) ~ 3

A ̂ k denotes the k th power of A in the exterior algebra A(R n)

Definition 2.2. The Pfaffian Pf(A) of A6a is defined by the rela-

tion

A Ai (2.13) Z--T = mfi dJ A ... Adx n

Note the relation

2 det A = (PfA)

The easiest way to see the deep relation of the Clifford algebra with

the heat kernel is the following [19, Theorem 1.5]

Theorem 2.3. Let t ÷ x t be a continuous mapping from R into

Spin(n) , which is C 1 at t = 0 , such that x ° is the identity.

23

If A6a is defined by

dx (2.14) A : ~TLt =o

then

(2.15) Trs[X t ]

lim - i£PfA • t++0 t n/2

Proof : For t small enough, we may write

(2.16) x t = exp[t A+o(t) ]

where o(t) is of length 2 in the Clifford algebra. Now

(tA+o(t))n/2 (2.17) exp(tA+o(t)) = I+ti+°(tl+ ~o(t n/2)

1 ! " " " (n/2) !

Using (2.7)-(2.9) , it is clear that the first n/2-1

length < n - 2 , i.e. give a 0 contribution to Tr s

elements have

So we find that the 1.h.s. of (2.].5) is given by

An/2 (2.18) Trs (n/2) !

However in (2.18) in the Clifford product A n/2 , terms containing two

identical factors like e I e 2 e I ... give a 0 contribution since their

length is < n . In the r.h.s, of (2.18) we can replace Clifford multi-

plication by exterior multiplication. Using (2.8), (2.18), (2.15)

follows. []

b) The Dirac operator.

M is a connected compact Riemannian manifold of dimension n=2Z

Let N be the S0(n) bundle of oriented orthonormal frames in TM .

We assume that M is spin, i.e. N lifts as a spin bundle N' ,

with

so that

N' ÷ N + M o

induces the covering projection Spin(n) ÷ S0(n)

24

The topological obstruction to the spin structure is the second

Stir±el- Whitney class w2(M)

Let F,F+ be the vector bundles

(2.19) F = N'Xspin(n) S

F+= _ N 'Xspin (n) S+

F , F+ are unitary bundles over M . TM acts by Clifford multipli-

cation over F and exchanges F+ and F_

The Levi-Civita connection on N lifts naturally to N' So F, F+

are endowed with a unitary connection. Let V be the covariant diffe-

rentiation operator.

R denotes the curvature tensor of TM . Using (2.9), we can iden-

tify R with the curvature tensor of F+

denotes a unitary bundle over M . We endow ~ with an unitary

connection. We still note V the covariant differentiation operator

on ~ .

TM acts on F®~ by Clifford multiplication on F, i.e. e6TM is

identified with e@l .

F(F+@~) denotes the C ~ sections of F±@~

There is a natural L 2 scalar product on F(F±®<) given by

< h,h' >=f < h,h' >(x) dx M

Definition 2.4. If el,.., e n is an orthonormal base of TM , the

Dirac operator D acting on the C a sections of F ~ ~ is given by

(2.20) n

D = Z e. v ! e.

1 l

D interchanges F(F+O<) and F(F_@~) D is formally self-adjoint

on F(F®<) Its principal symbol is ~-i~ , where GET M (identified

with TM ) is a Clifford multiplication operator. Since

25

D is an elliptic operator

Let D+ be the restriction of D to F(F+@~)

By standard results in index theory (see Treves [62, Chapter 3 ] )

Ker D+ , Ker D_ are finite dimensional.

Definition 2.5. The index of D+ is defined by

(2.21) Ind D+ = dim Ker D+ - dim Ker D_

This is the definition of the analytical index of D+ The purpo-

se of the Atiyah-Singer Index Theorem is to prove the equality of the

analytical index with a topological index determined by the principal

symbol of D .

c) The heat equation method.

If (el(x) ...el(x)) is a locally defined smooth section of N ,

us recall that the horizontal Laplacian A H acting on F(F ® ~) is

given by n ~H ~[(V )2

1 e i (x) - V e i (x) ] = re. (x) l

Also let K be the scalar curvature of M .

let

Theorem 2.6.

We first recall Lichnerowicz's formula [60].

If el,.., e n is an orthonormal base of TM I

given by

D 2 is

(2.22) D 2 H K 1

= - A +~+~ E e i ej ® L(ei,ej)

Proof : See [60] and [19, Theorem 1.9].

Remark i. The remarkable cancellations which explainLichnerowicz's for-

mula should not be thought of as "accidental". The proof of the Index

Theorem has already started. Indeed as we shall see, in Quillen's forma-

lism [49], D should be thought of as a generalized connection, and

D 2 as its curvature.

26

By standard elliptic theory, for t > 0 ,

kernel Pt(x,y) For (x,y)£M×M , Pt(x,y )

into (F+@~) x

tD 2 2

e is given by a smooth

maps linearly (F+®~)y

We can still define the supertrace of H6 End(F® ~) by an obvious

extension of (2.4), i.e. by setting

Tr H = Tr[ (,~®I)H] s

In particular TrsPt(x,x) is well-defined.

The first step in the heat equation method is the following (~c Kean-

Singer [45], Atiyah-Bott-Patodi [7]).

Theorem 2.7 . For any t > 0

(2.23) Ind D+ = S Trs[Pt(x,x)]dx M

Proof : The operators D D+ and D+D_ have a discrete spectrum. For I

I > 0 , let H± be the corresponding finite dimensional eigenspace in

F (F+®[)_ Since I > 0 , it is easy to find that D+ is an isomor-

phism from H I H 1 + on - The r.h.s, is obviously given by

I - dim H I) (2.24) Ind D+ + Z e-lt(dim H+ _

I > 0

Since dim H 1 = dim H I (2 23) is proved. D + - ! •

d) A probabili@tic construction of the heat equation semi-group.

Let X I,. .. X m be the standard horizontal vector fields on N .

is t h e c o n n e c t i o n f o r m on N , X. i s d e f i n e d by 1

If

(2.25) ~(X i) = 0

X* u-In* i = ei

(here el,.., e n is the canonical base of R n)

Take x 6M , u 6N For t > 0 , O O X

o rential e q u a t i o n

consider the stochastic diffe-

27

(2.26) h

du t = Z X$(u t) .~t dw 1 1

1

u(o) = u o

Thenvas is well known by the construction of Malliavin [61], Eells-

t n u t Elworthy [63], if Xs = s ' Xs/t is a standard Brownian motion on

÷ u w is the development M starting at x ° . The curve in T x M 8s o s

of x s in T x M , and is a standard EuClidean Brownian motion in TxoM.

In fact note t~at the generator of the diffusion (2.26) is the

operator t£ , where £ is given by

i n *2 £ = ~ E X i

1

A H

2

It is well-known that £ projects equivariantly on M as theLaplacian

Incidently note that £ is not elliptic on N . However since we

will consider the mapping w ÷x I , the submersion property used in sec-

tion i is preserved. So what will be done is an adaptation of what we

did in Section 1 For more details see [17, Chapter 4].

In particular UsU~ I_ is the parallel transport operator from TXoM

on T M . This parallel transport makes sense because using appro- x s

ximations [16, Chapter i], it is a limit of standard parallel transport

operators along piecewise C curves.

More generally, let T °'t be the parallel transportation operator s t

from fibers over x into fibers over x , and set o s

s,t ~ o,t]-I T O = LT s

To construct the semi-group e

Feynman-Kac formula.

tD 2

2 we use a matrix version of the

el,.., e n is an orthonormal oriented base of TxoM •

ys,t L (r O't e , T O't o xt s 1 s

~ s , t L {T ° ' t e i , ' r ° ' t o t s s

x s

ej) denotes the element of End

ej )~O'ts

X O

28

Definition 2.8. U t denotes the process in End(F@~) x s o

differential equation

defined by the

(2.27) t tl dU = U [-: t ~ e. e~ ® s,t L

t i<j i 3 o x s

u t = I Q

( O,t ei , o,t e ) ]ds s s j

We now have

Theorem 2.9.

(2.28)

If h6F(F®~) , the following identity holds.

tD 2 ........ 2 i K(Xst)dS}u t l,t h(~c~) ]

e h(x o) =E[exp{-t ~ 8 o o

Proof : It~'s formula (see [16,IX, Theorems 1.2-1.3]) shows that

s

s,t h(Xts ) = h(Xo)+ f ~ v,t ~H h(Xv)d v (2.29) <O O o

s

+ f Tv't . h(x~ 6w i o q~°,te

o v 1

Using (2.29), it is then easy to apply ItS's formula [18]

process

s K (x~)dv (2.30) exp{-t f 8 } Ut Ts ,t h(x t)

s o s o

to the

Taking expectations (which makes disappear the terms containing

we get (2.28) D

6w ),

e) An asymptotic representation of Tr s Pt(Xo, x o)

We now disintegrate (2.28) the way we did in Section I.

H I , H 2 are still given by (1.27), (1.28).

Pl is still defined as after (1.28). (1.29) is replaced by

(2.31) du t = X. (ut)(~t dw I' +v2'lds) 1

ut(0) = u o

Set

29

(2.32) t t

X = Z i] S S

2 V is adjusted so that

t X 1 = X O

Using (1.35), and (2.28), we see that for any k6N

1K(x~)ds

(2.33) Trs[Pt(Xo , Xo) ] = ~ 1 Trs[U~ ~l't]exp{-t~o • 8 ] (~2 ~t) n o

Iv 2 (~/tdw I I 2

exp - 2t

det $$ (~tdwl+v 2)

~v 2

G(t,dwl)dPl(wl)+o(t k) ,

where o(t k) is uniform on M .

We are now left with the task of studying the asymptotics as t++0

of (2.33).

f) The asymptoticS of TrsPt(x ° , x o) .

In (2.33) , as t++0

" G(t,dw I) ÷ i

• exp{-t~l O K(xt)ds --7----} ~ i

Also Figure i shows that at t =0

2 ~ v 2 (2.34) v = ~ = 0

so that

(2.35) Iv212

÷ 1 exp 2t

We now show the critical fact that

(2.36)

t l,t Tr s U 1 T o

(g2~t) n

has a limit.

boundedly

boundedly .

boundedly •

30

Indeed if rol't is considered as acting on TXoM ,

nition of the curvature R [19, Section 4] shows that

the very dell-

1 l,t I t w I) + o(t) o = -~I R x (u od~u °

O O

If L =0 , in view of Theorem 2.3, the sequence of events should

Indeed, if ~,t-~ is now considered as acting on F+,x ° , be obvious.

we have

1 l,t = exp{-~f R x (UodW 1 wl)+o(t) }

T O IU O o o

where o(t) is taken in a (identified with the antisymmetric elements

of End T M ) X o

We e x p a n d 1 , t as i n ( 2 . 1 7 ) , i . e . , i f 1 , t i s now c o n s i d e r e d as O O

acting on (F®~) X

O

t i 1 (2.37) Tol't = [i+( -~S R x (Uodwl,u ° w )+o(t))

O O

t 1 n/2+o (-~ S~Rxo(Uo dwl,u Ow )+o(t))

+...+ (n/2) !

t n/2 )

]0

[I + 0(t) ]

t in the form Similarly, we expand U 1

(2.38) I t t t t

Ult = i + ~O __~l Z.<j e. e.®l 3 Ts'O L x t(TO' el' T°'

s

e )ds 3

t o +S [- ~- Z e. e. @ ~s'tL t(x 't e i T°'te 0~< s~<s' ~< 1 i<j i 3 o x ' s j) ]

S

t ' tLxst ( ek~° ~ ez)] ds ds' [-~ Z e k e~ ® s , To,t t k<Z o , s'

+...+ o(t n/2)

we calculate Trs[U t T l't] and use (2.7) (2 8) proceeding Now if S O I • ,

as in (2.16) - (2.18), we see that the limit of (2.36) is simply obtained

by cancelling the factor t and replacing the Clifford products e i ej

31

by the Grassmann products (-2i)dxlAdx 3

Let q be the Riemannian orientation form of

is now a differential form of degree n

to,~'6 @A(M) we write o

to ~ tot

M. Trs[Pt(Xo,Xo)]q(x o)

n over M . If

if ~ and w' have the same component in the maximal degree n .

We finally obtain [19, Theorem 3.15].

Theorem 2.10. The following relation holds

(2.39) 1

lira Trs[Pt(Xo,Xo) ]q (xo)nSexDA{ -i t++0 4~ %RXo (Uo dwl'uo wl) }

L X

dP 1 (w l)iTr exp[-2.~T ,]

Remark 2.

valued in

equations

In [19] we proceeded as follows. Let y be a Brownian motion

a independent of w I Consider the stochastic differential

(2.40) dV l't = V l't s s

VI, t = I F o x

o

t Z e e dy j (-~ i<j i j i )

dV 2't = v2't[ E T s't L (T°'te , T O't ej)@¥3 i] s s i < j o x t s i

S

V 2 o = IS x

o

Using the stochastic calculus, it is elementary to prove that

(2.41) U~ E Y- l,t 2, n(n~l)t 2 ] = IV1 @Vl t]exp[ 16 ...........

We can then directly use Theorem 2.3 to calculate

(2.42) ~ l,t

lim Trs[V 'tTl't] ®Tr[V ,t r ]. t++0 o o

32

In fact if V 2 is the solution of

(2.43) dV~ = V2[Ls x (e i , ej)~7~]l O

V 2 = I O

(2.42) is given by

i R(Uo dwl,uo wl) Uo 71 uol 2 (2.44) (-i)n/2 Pf[ S 4~ + 2~ ]Tr V 1

O

Using (2.13), we can replace Pf[...] by e A['''] and proceed as

before to obtain (2.39).

This avoids the explicit expansions (2.37) - (2.38). In Section 3,

we will see the interpretation of (2.41).

For a related proof, see Azencott [ii].

g) The Index Theorem of Atiyah-Singer

In (2.39), Tr exp(~) is a representative of the Chern character

ch ~ of

We now evaluate the first term in the r.h.s, of (2.39).

Definition 2.11. A denotes the ad 0(n) invariant analytic function on

a such that if C6a has diagonal entries

0 X,

[ l] then --X. 0 ,

1

xi/2 (2.45) A~) =

1 shy

In the sequel A(~) should be understood as the analytic function " R A evaluated on ~ where all the products are replaced by exterior pro-

ducts.

We now have

Theorem 2.12.

(2.46)

The following identity holds :

1 Sexp A{-i = 4~ ]O R(Uodwl, uowl) } dPl(wl ) ~( R2~ )

33

Proof : Since R is the Levi-Civita curvature tensor it is known that

that 1 1 1

(2.47) f <X,f R(UodWl,UoWl)y>=- f <R(X,Y)Uo dwl,uowl > o o o

We should now evaluate

(2.48) 1

• dwl,uowl > } f e x p ^ { ~ f < R u o o

However now R can be replaced by any antisymmetric C6a Also

since Brownian bridge is rotation invariant, we can as well assume

that C has diagonal entries

o x. [ l] Using a well known formula of P. L6vy [57], we then know -x i o

that (2.48) is given by

~i/4~ 1

sh x./4~ l

The proof is finished. []

We finally obtain the Index Theorem of Atiyah-Singer [8],[7].

Theorem 2.13. Ind D+ is given by

(2.49) Ind D+ =f i(~) ch ~ . M

Remark 3. In [57], Yor shows that

1 ,2 _ w I 2 1 (2.50) E exp-iy[f w I'I dw I ' dw I' ] = o

2 1 y (E[exp-% f lwl'll2ds]) 2 =

shy o

the final equality being classical (for the simple proof by Williams,

see Yor [57]).

Also (2.50) can be directly calculated using the stochastic calcu-

lus on the Brownian bridge as in [17, Theorem 4.17].

We now explain another derivation of (2.50). Consider the symmetric

kernel K(s,t) on

(2.51)

34

2 [0,i] given by

K I'I =K 2'2 = 0

Kl'2(s,t) = y s < t

-y s > t

K2' 1 (s,t) = KI'2 (t,s)

We now use the notations H I , H 2 in (1.27), (1.28) with m = 2

If f = (fl,f2)CH I , then

f ]2 <K(s't) fs'ft > dsd t =2y[f (flf2-f2f})ds dt] [0,i o<s~t<l s t s t

In particular

(2.52) 1

S <K(s,t) ~wls,~wl t > =yf (ws l'Idwl'2-wl'2dwl'l)s s s o<~s~<t<~l o

In the r.h.s, of (2.52) d can be replaced by ~.

Let P1 be the projection operator from H On H I

~IK is a Hilbert-Schmidt operator on H I Let {lk } be its

eigenvalues, {<ok}k6N a corresponding base of orthonormal eigenvectors

in H 1 Since

(2.53) P1 K = Z I k <ok(S) @<0k(t)

we find that

(2.54) I k i

f < = I > ]2-1] o~<s~<t~<l o s

1

Since { f <~k(S) ,6w~ >} are mutually independent Gaussian variables, o

we finally find

(2.55) E [ exp-iyflwl'idwl'2-wl'2dw I'I] = 1 o {det2[I + i K ]}1/2

35

where

(2.56) det2(I +iK) = Z[I +iXk]e-ilk

The Ik can be easily explicitly calculated and are given by

(2.57) ~k = £1k6Z. ,

and each eigenspace is of dimension 2 .

So we find that

2 (2.58) [det2(I +iK) ] I/2 = ~ [i + --Y---]

kCZ+, k2~ 2

Of course this proves again the well khown result

Y (2.59) s'ty,~ =

y2 [1+ ]

k6Z+. k2~ 2

As we shall see, in the formalism of Atiyah-Witten [4], replacing

A(~) by the corresponding infinite product has a very interesting aeo-

metric interpretation.

3. Localization formulas in equivariant cohomology and the Index Theorem.

In [4] Atiyah and Witten made a very interesting remark on the path

integral representation of the Index of the Dirac operator for the Spin

complex. They showed that if one applies formall Z in infinite dimensions

a localization formula of Berline-Vergne [13], [14]] Duistermaat-Heckman

[28], the path integral should be equal on a priori grounds to the in-

tegral of a certain characteristic class over M , which turns out to

be exactly equal to A(R/2~) . The formal aoplication of the localiza-

tion formula gives then the right answer for the Index.

In [5], Atiyah and Bott discussed various aspects of localization

formulas, which also follow from purely algebraic arguments, and sug-

gest that a proof of such formulas should be given in infinite dimen-

sions.

In [20], we extended the Atiyah-Witten formalism to the case of a

36

twisted spin complex. We also shew that the Lefschetz fixed point for-

mulas of Atiyah-Bott [5] , Atiyah-Singer [8] could be given such a for-

mal interpretation. This formalism also helped us to find a heat equa-

tion proof of the infinitesimal Lefschetz formulas [21], [22].

Now the proofs of Berline-Vergne [13], Duistermaat-Heckman [28]

are difficult to extend in infinite dimensions, since they make an ex-

plicit use of Stokes formula on small balls. As is well known, it is

difficult to integrate on balls in infinite dimensions.

In [25], we took the opposite direction. Since the heat equation,

when interpreted probabilistically, produced the required localization

in infinite dimensions, could it be that such a proof could now be e__xx-

tended in finite dimensions so as to give another proof of the localiza-

tion formulas of [13] -[14]- [28]? We gave in [25] such a proof, which

reproduces in detail the various technical steps of the proof of the

Index Theorem given in Section 2. Later on [24], [25], inspired by the

finite dimensional baby proof, we could give two heat equation proofs

of the Index Theorem for families.

In this section, we want to introduce the reader to some of these

problems. In a), we give the main assumptions and notations. In b), we

give a brief sketch of our proof [25] of the localization formulas of

Berline-Vergne [13], [14], Duistermaat-Heckman [28]. In c), we summari-

ze the remarks of Atiyah and Witten [4]. In d), we show that the proof

of the Index Theorem in Section 2, and the proof of the localization

formula in b), are strictly parallel. In e), following [20], we show

how to extend the formalism of Atiyah and Witten [4] to twisted spin

complexes. In f), we briefly summarize various applications and exten-

sions. In particular, we briefly explain the superconnection formalism

of Quillen [49], and our heat equation proofs of the Index Theorem for

families [24].

a) Assumptions and notations.

M is a C ~ connected compact oriented Riemannian manifold of

dimension n .

X denotes a Killing vector field on M , i.e. X generates a

group of isometries of M .

87

v denotes the covariant differentiation operator for the Levi-

Civita connection of TM . R is the curvature tensor of TM . Set

(3.1) JX = v.X

Then JX is an antisymmetric (i,i) tensor. Also since V is X inva-

riant, we know [13] that

(3.2) VyJ x +R(X,Y) = 0 .

A(M) denotes the set of C ~ sections of the exterior algebra of

T M . An element u of A(M) can be written as

(3.3) ~ = ~o +'''+Un

where ~j is a j-form on M . If S

mension j , by definition

(3.4) f u : f ~j S S

Set

(3.5) F = {X :0}

is a submanifold of M of di-

Since N is orthogonal to TF which is stable by Jx ' N is stable

by JX " Also by (3.2) JX is parallel on F . JX is non degenerate

on N , which has even dimension. The 2-form < Y,Jx Z > being non de-

generate on N , N is naturally oriented. The orientation of N makes

Pf JX >0 . TF is then also oriented, so that TF@N is oriented like

TM .

The exterior differentiation operator d increases the degree of

~i by 1 , the interior multiplication operator i X decreases it by 1 .

Following Berline-Vergne [13], Witten [56], we now set the following

It is well-known that F is a finite union of totally geodesic con-

nected submanifolds of M . Let N be the normal bundle of F in M.

Since F is totally geodesic, the Levi-Civita connection of TM produ-

ces an Euclidean connection on N R N denotes the restriction of

R to N .

38

definition.

Definition 3.1. ~6A(M) will be said to be X equivariantly closed

(X e.c.) if

(3.6) (d + ix)~ = 0

(3.6) shows that if u = u0 + "'" + ~n '

(3.7) dnj + i x uj+2 = 0

then

b) Localization formulas in equivariant cohomology.

We now will give a new proof of the localization formulas of Berline-

Vergne [13], [14], Duistermaat-Heckman [28] (also see Atiyah-Bott [6]).

It will be more detailed than necessary since our purpose is to com-

pare it with the proof of the Index Theorem.

In what follows we use the notation JX + RN

P f [ - = - / T - - ]

in the following sense.

Recall that on N Pf(Jx ) > 0 . We can then expend analytically

JX + R N ] - l i n R N , P f [ ~ the variable replacing products

by exterior products. We then get a finite sum of differential 1

forms, is an even element in A(F) . JX + RN

P f [ ~ ]

We now have the formula of Berline-Vergne [13]-[14], Duistermaat-

Heckman [28].

Theorem 3.2. If ~ is X e.c., then

( 3 . 8 ) f ~ = f M F JX + RN

P f [ ~ ]

Proof : This proof is taken from [25]. Let X' be the i- form dual to

X.

We first claim that for any s

39

(3.9) ~ u = f exp-[s(d +ix)X']~ M M

Indeed (3.9) holds at 0 . Moreover.

(3.i0) ~-~ f exp {-s(d+ix)X'} ~ =- f(d + Jx)X' exp {-s(d+ix)X'}u M

Now if L X is the Lie derivative operator associated with X

(3.II) (d+ix)2 = L X .

Since X is Killing, LxX' = 0 . It follows that

(3.12) (d+ix) [ex p {- s(d+ix)X'} ~] = 0 .

So the r.h.s, of (3.10) is equal to

(3.13) -~(d+ix) [X' exp {- s(d+ix)X'} u ]

Now if v£A(M),~dv=0 . Also since ixV has degree <~n-i ,~ ixV=0 . M M

(3.13) is then clearly 0 and (3.9) holds.

We get for any t >0

dX'+Lxl 2}~. (3.14) f ~ : S exp {- 2t

M M

As t++0, (3.14) clearly localizes on F . By making the change

of coordinates y =~ty' in N , we get that as t++0 , if V is a

small neighborhood of F , (3.14) is very close to

dim N

(3.15) t 2 S exp{ - dX' (x,Vt,y) +

2t N

XI2 (x,w/ty) } ~ (x,~ty)

Now

Ixl 2 (x,gty) IJxYl 2 (3.16) 2t ~ ---7 + o(gt)

Using (3.2) and a non entirely trivial argument on differential

forms [25], we find that if RTF(Jxy,y) is the restriction of the

antisymmetric matrix R(Jxy,y) to TF , (3.15) converges to

40

RTF(oXy,y) IJxyl 2 (3.17) ~ [f exp{Jx - 2 ~ } ] ~

F N

Now the symmetries of the Levi-Civita curvature tensor R show

that if Y,Z6TF

(3.18) < y,RTF(Jxy,y)Z >= < RN(y,Z)y,Jx y >.

(3.17) is then equal to

< RN( - -)Y,Jx y> 1 xY~ ]Z (3.19) f [~ exp X- 2 2

F N

where R N is considered as a 2 form on F with values in EndN . The

gaussian integral (3.19) is readily evaluated to be equal to

J X + RN ~ 1 . Pf[---~-~--- The proof is finished. D

c) A remark of Atiyah and Witten.

We go back to the assumptions of Section 2.

We first consider the case of the spin complex i.e. ~ is the tri-

vial line bundle (with L = 0)

Take ~6S0(n) , 86Spin(n) such that ~8|= ~ .

Let 8 1 , . . . e ~ b e t h e a n g l e s a s s o c i a t e d w i t h ~ . T h e n i t f o l l o w s f r o m

( 2 . 7 ) , ( 2 . 8 ) a n d [ 5 , p . 4 8 4 ] , [ 1 9 , P r o p o s i t i o n 1 . 2 ] t h a t

n 0 . ~ = + ~ 2 s i n @ . ( 3 . 2 0 ) 1 T r s _

1

We now follow Atiyah [4]. Let M ~ be the space of smooth loops in

M s E S 1 = R / Z ~ x s T x ~ i s i d e n t i f i e d w i t h t h e s p a c e o f s m o o t h p e r i o -

d i c vector fields. If X,Y6TxM ~ , we define the scalar product

1

(3.21) < X,Y> = f < Xs,Ys > ds o

~ has a Riemannian structure.

In the sequel, we will do as if the Brownian measure was carried by

smooth paths. Note that although Brownian motion is not smooth, all

the standard operations like parallel transportation are well defined

41

and are limits of the corresponding operations on approximating smooth

paths [16, Chapters 1- 3].

S 1 acts isometrically on M ~ by the mapping k s defined by

ksX = XS+ "

k s acts isometrically on M ~ . The generating Killing vector field

X(x) is given by

dx

(3.22) X(x) = s ds

The associated one form X' is given by

(3.23)

]

X' (Y) : f <Y,dx > O

dX' is the 2-form given by

(3.24) 1

DY dX' (Y,Z) : 2 ~ < ~,Z> ds .

o

DY In (3.24), ~ is the covariant differential of Y along x .

In particular

(d + ix)X' [ i dx dX' (3 .251 2 =~ I ~ 12ds + 2

0

Now Atiyah considers in {]4] the eigenvalue problem on Y6TxM

DY (3.26) - ~<Y .

Ds

Recall that Yo =YI " (3.26) can be put in the equivalent form

dZ (3.27) d-~ = kZ

o TIZ 1 = Z O

If 81,..- 8£ are the angles of

takes the values

o <i , it is trivial to verify that

(3.28) + 2ivm + iO,

42

The Pfaffian of dX ' 2

is formally given by

(3.29)

+~ f dX' Pf~--~-] = ~ 9 ~ [4~ 2 m 2 -9~]

j=i ] M=l ]

By dividing formally by the infinite normalizing constant

@- oo

H (4~ 2 m 2) £ , we get m=l

f dx' Pf ~---i-- ]

(3.30) ('~ 4-rr2 m 2) £

1

~, +~ e2 It e . H [i- J

j=l 3 m=l 4~2 m 2

Z 0 = ff 2 sin --/o

j=l

The idea of Atiyah and Witten is to use the equality of (3.20) and(3.30)

to rewrite (2.28) in a different way. Namely if Q~O is the law of the

scaled Brownian bridge starting and ending at x ° , if Pt(x,y) is

the heat kernel on M , they write formally

i I~12 (3.31) Pt(Xo,Xo)dXo doXo(x) 1 exo{-fo -~-ds}dD(x)

t (~2~t)d -

In (3.31), d is the dimension of M ~ (d=+ ~ ) and dD(x) is the

"volume element" of M ~

Using (2.23) , (3.20), (3.30) , (3.31) and neglecting

exp{-8f~K(Xs) dS}, we can write the Index of D+ as

@-oo

(~ m 2 ) £ i £ 1 dx

(3.32) IndD+ £ fexp { ~°l~12ds 2t }Pf[- ]dD (x) (2~)

NOW if d' <+ ~ , if A is a(d',d') antisymmetric matrix d' ^ff

(3.33) A d'= A BfA dxlA ... idx d e (d,/2) ! =

We now use (3.33) formally, and so we obtain

(~ m2) ~ i ~ i (3.34) Ind m fexp

+ ~ 2t (2~)

(d+~ X'

43

Since L X' =0 , we have X

(d+ x!X : ( d + i ) e x p -

X 2 t 0 .

Now Atiyah and Witten [4] apply formula (3.8) in a formal way. They

note that

M = (X =0 )

and so we should get +~

(Km2) Z i Z i 1

(3.35) Ind D+= i2~) Z M JX + RN

P f [ ~ ]

Now at x6M,N x is the set of f takina values in T M such that d - x

]~ _ fds = 0 . Also JX =~s " The eigenvalues of Jx are + 2i~m I

Proceeding as inAtiyah [4], we find that if A is a (n,n) antisymme-

tric matrix with diagonal entries [-4 o xi] , then

J + A +~ x 2 (3.36) pf X = I] [m 2 ...... l ]

2~ m=l 4~ 2

so that

(3.37)

+~ z x i (~ m2 ) ~ n --f

1 1 1

+ ~ x2~ +~ x 2 x N [m 2 ....... ! ] N [i - l ] sin _~l

m= 1 4z 2 m= 1 472 m2 2

and so using (3.35), we should get

(3.38) Ind D+ = f ~(_RR) M 2~

(3.38) is of course the correct answer.

In [4], Atiyah suggested that these formal considerations should be

proved. The proofs of Berline-Vergne [13], [14], Duistermaat-Heckman

[28] were in fact impossible to extend in infinite dimensions.

44

d) The heat equation pr0babilistic proof as a ri@orous proof of

localization.

Our idea was that the heat equation proof of the Index Theorem should

be taken as a model proof of localization in infinite dimensions, which

could be extended in finite dimensions.

This is the reason why we gave the new proof of Theorem 3.2.

Remarkably enough, even the intermediary steps of the proof of Theorem

2.10 ~nd Theorem 3.2 are closely related. In particular the intermediary

formulas (2.39) and (3.17), the key steps (2.47) and (3.18) can be easi -~

ly compared.

Note that the normalizations which had to be explicitly done in (3.34)

are automatically performed by Brownian motion.

This point of view is fully developed in our papers [20] and [25].

e) Extension of the forma!ism of Atiyah and Witten to twi~te ~ spin

complexes.

We now briefly explain how in [20], we extended the formalism of

Atiyah -Witten [4] to twisted spin complexes.

Namely we used formula (2.41). We find from (2.41) that

n(n-l)t 2- ) (3.39) Tr trl't =EY[Trs[VII 't lo't]Tr[V~'tTl't ] s UI o T O ] exp( 16

Now doing as if y was differentiable and proceeding as in (3.30),

we find that if r Y is the (random) 2-form on T ~ ~ x

1 ry(y,z) =So< TSYs,e i >< s Zs,ej >dy ~

then

.£ dX' 1 mf[--~- - tr Y] (3.40)

1 4~2m2)~

2 Forgetting about exp[ n(n-l)t ]

16

=+- Trs[V~'t Tl't]o

dX' we obtain formally that if ~= 2

(3.41) Ind D +

4S

( U m 2 ) ~ i £

(2~) exp{

1 I%I 2

2t

Pf [- (~ +trY)]

} t d

Tr[V~ 't

Proceeding as in (3.33), we get

( 1 ~ m 2 ) ~ i B (3.42) Ind D+ Q~ #_~oexp

(2~) M

1 T 't]dP(x) dP' (y). o

-(d + ix)X' 2t [~exp(-rY)T~[V2"t Tl'ho

dP' (y) ]

Now the final term in (3.42) can be explicitly computed using the

stochastic calculus. We briefly summarize the results in [20].

Definition 3.3. H denotes the solution of the differential equation S

(3.43) dH = H T s L ds . S S O X

s

H = I • O

(3.43) should be interpreted as follows.

toS Lx is a 2-form on Tx ~ with values in End(< x ) defined by s o

--~ ÷ S L (Ys,Zs)r~ (Y'Z)6TxM o X S

H sould then be expanded as a series

v I v 2 H = I + fs Tv L o dv +S T L x AT L dv I dv 2 + ... s o o x v o o x

v o ~ v I < v 2 ~ s v I v 2

1 In [20], we find that if ~ =Tr[HI ~0 ] ' then

(~ m2) ~ i ~ -(d + ix)X' (3.44) Ind D+ = i Z fexp 2t

(2~)

Now we proved in [20] that 8 is X equivariantly closed i.e.

(3.45) (d + i X ) 8 = 0

46

appears as an infinite dimensional characteristic class associa-

ted with a bundle whose structure group is a Kac-Moody group.

f) Applications and extensions.

It should now be clear that there is indeed a theory of differen-

tial forms on the loop space L~°(M) , and that Brownian motion produces

as such all the required normalizations} so that this theory makes sense.

The very comparison of the proof of Theorems 2.10 and 3.2 shows that this

is not just a formal remark, but that computations reflect this in de-

tail.

The comparison of (2.23)-(2.28) with (3.43) is instructive. Brownian

motion selects by itself the form of maximal degree (which is here in-

finite) which is to be integrated. It does so by using spinors (which

are finite dimensional objects), instead of differential forms. However

by doing so, Brownian motion hides the deep algebraic structure which

reveals itself in the non rigorous formu]a (3.44).

Such observations can be pushed much beyond what is done here.

First of all, the Lefschetz fixed point formulas of Atiyah-Bott [5],

Atiyah-Singer [8] have been proved probabilistically in Bismut [19]

Another proof has been given in Berline-Vergne [14]. It was shown in

[20] that the equivariant cohomology formalism also applies in this

case. It G is the group of isometries of M , G acts obviously on

M , and the action of G commutes with the action of S 1 S 1 is

then replaced by SIXG . Also the infinitesimal Lefschetz formulas

were proved in [21] - [22] using heat equation. The proof was inspired

by a formal representation of the type (3.44) .

We want to mention the recent proof by us [24] of the Index Theorem

for families of Dirac operators. Let M be a compact connected mani,

fold, f a submersion from M onto the compact connected manifold B

Let Gy be the fiber Gy = f-l({y}) For each yEB,Dy is a Dirac

operator operating along the fiber Gy The Index Theorem of Atiyah-

Singer for families calculates Ker D+ -Ker D_ as an element of K(B) .

In [24], we gave two heat equation proofs of the computation of

(3.46) ch(Ker D+-Ker D_)

47

The proofs in [24] are based on two facts :

i). The extension to infinite dimensions of the formalism of Quillen

[49]. In [49], Quillen had considered the case of a Z 2 graded vec-

tor bundle E = Eo~ E 1 endowed with an "odd" linear mapping u exchan-

ging E ° and E 1 He had produced non trivial representatives of

ch(E o-E 1 ) using explicitly the linear mapping u. If v is the restric-

tion of u to E , clearly in the sense of K theory O

Kerv -Coker v = E -E 1 O

In particular in Quillen's formalism, ordinary connections ? on

E ° @E 1 (which respect the splitting) and odd linear mappings like u

are on the same footing. V +u is called a superconnection by Quil!en

[49]. If R (?+u) 2 [e -R/2i~] still represents = , Quillen shows that Tr s

ch(E ° -E 1 ) =ch(Ker v-Coker v)

H ~ Cco Now for y£B , let H+ , _ be the set of sections over the

fiber G of the considered fiber bundles. The whole purpose of Index Y

Theory for families is to make sense of the equality

co (3.47) Ker D+-Coker D+ = H+-H_ ~

However, contrary to what happens in finite dimensions, the r.h.s.

can be well defined only if the operator D+ is given. Two unequivalent

co- H ~ So in infinite elliptic operators of course produce different H+ - .

dimensions, (3.47) is tautological.

In [24], we could extend Quillen's formalism to the infinite dimen-

H ~ sional H+ , _ . Using D instead of u we proved that indeed

ch(Ker D+-Ker D_) was given by the infinite dimensional analogue of

Quillen's formula.

2). Using formal formulas like (3.44), we could find the right objects

to be considered so as to calculate ch(Ker D+ -Coker D+) explicitly.

This makes the formula (2.23) for Ind D + D 2

2 (3.48) Ind D+ = T r s e

as crying to be considered asaformula for the Chern character of

48

Ker D+-Coker D+ (this is irrelevant if B is a point ~). Also D 2 should

now be viewed as a curvature.

For the full proofs of these facts, we refer to [24]. The proof has

been explained in detail in [25].

4. The Witten complex and the Morse inequalities.

In this section, we briefly discuss the proof by Witten [56] of the

Morse inequalities for a Morse function having isolated critical points.

In [56], Witten has shown how to construct other complexes than the com-

plex of the Rham which have the same cohomology as the de Rham complex.

In [56] Witten introduced a small parameter t++0 . By studying the lo-

wer Dart of the spectrum of a modified Laplacian, he proved the

Morse inequalities.

In this section, we will show how to use instead the small time be-

havior of the heat kernel. Equality of the Index with a heat equation

trace is now replaced by an inequality. The proof then becomes

closely related to the heat equation proof of the Lefschetz fixed point

formulas when the fixed points are isolated (see [15], [19] for the gene-

ral case). This point of view has been developed by us in [23].

In [56], Witten also suggested that instanton considerations could

prove analytically that a geometrically constructed finite dimensional

complex associated with h gives the real (and even the integer) coho-

mology of M , as shown by Smale [64]. This has been shown analytical-

ly by Helffer and Sj~strand [38] using remarkably efficient techniques

to study tunnelling problems (which are problems where energy wells

interact). The analysis of Helffer-Sj~strand goes much beyond what is

done here. However note that probabilistic work of Ventcell-Freidlin

[54] - [55], was also applied in Venteell [65] to eigenvalue problems

and may have a direct bearing on this problem. Ventcell-Freidlin tech-

niques were used in tunnelling problems by Jona Lasinio-Martinelli-

Seoppola in [41].

a) Assumptions and notations.

M is a connected compact Riemannian manifold of dimension n .

49

? denotes the covariant differentiation operator for the Levi-Civita

connection.

AP(M) denotes the set of p-forms on TM.A(M) is the exterior alge-

bra n

A(M) = ~A p(M). o

C ~ The operator d acts on sections of A(M) Let ~ be its

adjoint for the standard Riemannian structure on differential forms.

The Laplacian [] is defined by

[] = (d+6) 2 : d6 +6d .

Hodge's theory shows that if D is the restriction of [] to sec ~ P th

tions of AP(M) , Ker [] is isomorphic to the p real cohomology P

group HP(M;R)

B (o <p<n) are the Betti numbers of M , i.e. B =dim HP(M;R) P P

oo

h is a C function on M with values in R.

If A is a (n,n) tensor, A acts as a derivation on A(M) , so

that if ~6AI (M) , X6TM

(4.1) As (X) = -a(AX)

Following Witten [56], we now define the operators

(4.2) d h = e -h de h

~h = e h ~ e-h

Dh = d h ~h+6h d h

As pointed out by Witten [56], the cohomology of the operator d h

is the same as the cohomology of d . Hodge's theory still holds, so

that if D h is the restriction of ~h to the section AP(M) P

= dim[Ker D h ]. (4.3) Bp P

Clearly

5O

(4.4) d h = d+dhA

6h = ~ + id h .

In (4.4), dh6A£(M) is identified with an element of TM by the

metric, and so idh is well-defined.

An elementary computation shows that

(4.5) O h = [3+ idhl 2 - Ah - 2v.dh

b) The basic inequality.

For a >o , t >o , let Pt(x,y)

~t~ h/t

2 w i t h t h e o p e r a t o r e

eo

be the C kernel associated

~(x,x) actinq on AP(M) Jp(~,t,x) denotes the trace of Pt - x

K (e,t) is defined by P

K (a,t) = f Jp(~,t,x)dx P M

Definition 4.1. If (ap)o < p < n

real numbers, we write

(bp)o<p<n are two sequences of

(4.6) (ap) > (bp)

if for any q(o<q~<n)

(4.7) a -a + ... >i b -b q q-i q q-I

with equality in (4.7) for q =n .

+ ..°

We now have

Theorem 4.2.

(4.8)

Proof : For

For any ~ > 0 , t >0

(Kp(~,t)) ~ (Bp)

>0 , let FI be the eigenspace of ~tD h/t . F~

51

splits into

n (4.9) F 1 = • F~

o

where F~ a r e t h e c o r r e s p o n d i n g f o r m s o f d e g r e e p .

Also the sequence

(4.10) 0 + F O ÷ F 1 .. ÷ F n ÷ 0

dh/t Idh/t I + . dh/t

is exact. Indeed if ~6F p , ~t• h/t ~ = lw , if

~td h/t 6h/t

1

d h/t ~ = 0 then

and so d h/t is exact on F 1

Trivially, this implies that

(4.11) (dim F~) > 0

Now

(4.12) K (~,t)-K p p-i

~tGh/t ~t[] h/t _ p - p-i

2 2 (~,t)+... =Tr e - Tr e +...

= B B p- p-i

-At +... + Z e

l>0

D-I F-D- dim F: ...]. [dim l

Using (4.11), (4.8) is obvious. []

Remark i. The inequality (4.8) is the analogue of the equality (2.23)

in Index Theory.

c) The Morse inequa!.ities : the non degenerate case.

h is now assumed to be a Morse function, i.e. h has a finite number

of critical point Xl,... x£ at which d2h is non degenerate. Recall

that we use the convention (4.1) .

We now claim [23, Theorem 1.4].

52

Theorem 4.3. As t++0 , Kp(e,t) has a limit K (~) given by P

Tr exp(~V.dh(x i) ) (4.13) K (e)= E P

P i=l I det (I-exp (-~V. dh (xi)) I

Proof : WeitzenbSck's formula shows that

[] = -A H + L

where L is a 0 order matrix valued operator.

We now prove (4.13) with ~=I. With a general ~ the proof is iden-

tical.

t t s be the process of u. , x. are taken as in (2.31), (2.32). Let U t

linear operators acting on A x (M) given by o

(4.14) dU t t s (XsS-[t o = Us[T ° Vdh 1 s L(x t) ]ds

U t = I o

Then for any Xo£M , for any k6N

(4.15) Jp(l,t,Xo) =f 1 (~2~t) n

Idhl2(<)ds 1 Ah(Xs )ds exp{- I ........ +

o 2t o 2

exp{ Iv 2 ~tdwl,Xo) 12}T r [U~T~,t]G(t,dwl)dPl (wl)

2t P det[ C' (,~dw I ,Xo)] + o(tk).

Here v2(~tdwl,xo ) is explicitly depending on x O

Also det[C' (~tdwl,xo)] is the determinant appearing in (1.35).

As h++0 , since Idhl~0 out of the x. , and since a Brownian 1

bridge with parameter t escapes with small probability far from the

starting point x ° , we find that if V is a small neighborhood of

the {x i} , as t++0

(4.16) f J (l,t,xe)dX ° ~ ~ Jp(l,t,xo)dX M P o

53

Taking geodesic coordinates around each x. and doing the change 1

of variables X =~tX' i n T x i M , we f i n a l l y f i n d t h a t i f u 1 . . . u ~

are orthogormal frames at x I ... x~

1 (4.17) liD ~Jp(l,t,xo)dXo = Z ~exp{~ Ah(x i) -

t~+0 i=l

1 dP 1 (w I ) dc i

ifoVci+uiwl s dh(xi) 12 ds}Trp[eXpV.dh(xi)] (w/2~)n

Now if b is a one dimensional Bridge starting and ending at x

(at time i), by [66, p.206]

(4.18) E exp{-~flIbI2ds} =~ ~ exD[-Bx 2 th~ ] sh~ - "

o

Putting ?.dh(x i) in diagonal form and using (4.18), we get (4.13)

Remark 2. (4.18) is directly related to the harmonic oscillator. In

[56] Witten instead studies directly the lower Dart of the spectrum of [] h/t

Also note that using the Morse Lemma, we could as well assume that

on a neighborhood of each x , h is quadratic, and the metric is i Dh

flat, so that on this neighborhood, is exactly the harmonic oscil-

lator.

Let us recall that x 1

eigenvalues of V dh(x i)

is of index p if the number of negative

is exactly p .

Let M be the number of x. of index p . p l

We now have [25, Theorem 1.5]

Theorem 4.4. The following relations holds.

(4.19) lim K (~) = M ~÷+ ~ P P

Proof : The argument is now directly related to Witten [56].

Let ~I'''" Un be the eigenvalues of V.dh(x i) Using the conven-

tion (4.1), we know that the eigenvalues of V.dh(x i) acting on

54

A p (M) are all the sums x, l

P - ~ ~j

k=l k

where jl <j2< ..~< jp

given by exp{- ~ ~ Zj } k=l k

The eigenvalues of exp(eV.dh(xi)) are then

If the index of x. is p , the correspon- 1

ding term in (4.13) tends to 1 as ~ ÷+~ . Otherwise it tends to 0

Using Theorems 4.2 and 4.4, we now find the Morse inequalities.

Theorem 4.5. (Mp) >> (Bp)

Remark 3. In [56], Witten suggested that his argument extended to the

case where the critical points of h form a submanifold of M , in

order to obtain the degenerate Morse inequalities of Bott. In [23], we

found out that in the case of degenerate critical points, the metric

of M has to be directly related in a non trivial way to h We were

able to prove the degenerate Morse inequalities using the existence of

the Thom class of the normal bundle of the critical point set.

The proof is not a trivial extension of what has been done before.

55

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