1. rough outline of heat kernel proof of Atiyah-singer index theorem

1.1. proof strategy

Theorem 1 (Mckean-Singer formula.)

from this we know Fredholm operator deformation invariance,in the same time we need chern-weil theory.

Main challenge:

1. in the expansion on heat kernel ,we need to proof when , the limit exist and find a way to calculate it.

2. indentify the limit as .

Proof: Our proof road锛� mckean-singer formula local-index thm A-S index thm Riemann-roch-Hirzebunch theorem.

1.2. preliminary work

superbundle: .

on compact manifold , is a self-adjoint operator . . .

observe that:

is symmetric eigenvalue space of is finite dimention in particular is finite dimesion is finite dimension.

dimention of superspace :

Def: .

Lemma:

1.Let be a self-adjoint Dirac operator on a clifford module over a compact manifold ,then

in particular,

where coker .

2.Let be a differential operator acting on a -graded vector bundle ,then .

the proof of this two lemma is easy,leave sas exercise.

1.3. Mckean-Singer formula

the formula is:

the expression of heat operator by spectral measure is:

proof 1:

we have first eigenvalue estimate on compact manifold:

on the other hand ,we need to show is independent with ,in fact:

odd parity : .

supercommunater

q.e.d

proof2:

by spectral decompositon of :

observe that: for . . (detail in [BGV])

q.e.d

Corallary: the index of a smooth on-parameter family of Dirac operator is constant.

what we have proved is:

1.4. analytic formula of ind

from the discuss of heat kernel in section 2,we know following result(section 2 only discuss the case of function but use the similar way we can get similar result on bundle):

on the other hand: .

and use the Mckean-Singer formula,we get: ind (D^+)&=&Str(e^{-tD^2})

&=&Tr(e^{-tD^+D^-})-Tr(e^{-tD^=D^-})

&=&\int\limits_M Tr(K_t(x,y,D^-D^+)) – \int\limits_M Tr(k_t(x,y,D^+D^-))

&=&\sum\limits_{i=0}^{\infty} t^{i-\frac{n}{2}}a_i(D^+D^-) – \sum\limits_{i=0}^{\infty} t^{i – \frac{n}{2}}a_i(D^+D^-). where is the heat trace invariants.

take ,the only thing make sense is the series of order ,and we want to proof:

But the difficult thing is that the high order series is very hard ro calculate….

our strategy is following:

Step1: proof has a limit as i.e index density.

step2:use a rescaling of space,time,clifford bundles ,to find a way that make us only need to calculate the leader coefficient.

1.5. From the McKean鈥揝inger formula to the index theorem

Let be a compact oriented Riemannian manifold of even dimension . We will write for the heat kernel associated to . The diagonal is a section of which is iso- morphic to . Using this isomorphism, we define a filtration on , induced by the filtration on . Elements of are given 0-degree. Denote by the subbundle of consisting of all elements of degree less or equal to .

the following theorem hold:

Theorem 1. The following statements hold:

1. The coefficients have degree less or equal to . In other words, . 2. If ,where denotes the restriction of the symbol map, then:

the important observation is that the of order less than n all vanish,in the other word:

lemma2:for any quadratic space of dimension ,.

Proof: Let be a basis of . For any multi-index , denote by the Clifford product .Then the set is a basis for .If,there is at least one such that , and we have

q.e.d

so take , we get:

now we want to identify the term as a characteristic form on :

Lemma 3. Let be a Euclidean space. There is, up to a constant factor, a unique supertrace on , equal to where denotes the symbol map and is the projection of onto the coefficient of if form an oriented orthonormal basis of . Furthermore, the supertrace defined above equals:

rmk:(The map is also called the canonical Berezin integral.)

proof:

the dimension of space is one because of and it never be empty because there is a natural defined supertrace on .so the only thing we need to do is to determine the constant,in face we only need to calculate the supertarce on any non-zero element .for instance the chirality operator . We have that and therefore for all .

q.e.d

so we know:

section :

theorem1 implies then the following theorem for the index of a Dirac operator associated to a Clifford connection which is known as the local index theorem.

Theorem 2. (Local index theorem) Let be a compact, oriented even-dimensional manifold and let be a Clifford module with Clifford connection . Let be the associated Dirac operator. Then exists and is obtained by taking the -th form piece of

rmk:This theorem only holds for Dirac operators associated to Clifford connections, which are those which are compatible with the Clifford action. However, since the index of a Dirac operator is independent of the Clifford superconnection used to define it, we get Atiyah鈥揝inger index formula for any Dirac operator.

Theorem 3. (Atiyah鈥揝inger Index Theorem) Let be a compact, oriented, even-dimensional manifold and let be a Dirac operator on a Clifford module . Then the index of is given by :

hence theorem 3 is a consequence of theorem 1.

up to now,to prove index theorem ,we only need to prove theorem 1.

1.6. idea of the proof of theorem 1

we give the clear proof in appendix锛宼here we explain the idea:

To prove Theorem 4.11 we mainly follow Chapter 4 of [BGV] but rearrange the different steps in order to make the proof clearer, at least for us. Let us summarize the first part of the proof:

1. The idea of the proof is to work in normal coordinates around a point . Near the diagonal, we use parallel transport to pull back the heat kernel which is a section of the vector bundle and define a new kernel , a section of for some twisting space . Using the symbol isomorphism , we can look at as a section of .

2. We use Lichnerowicz鈥� formula to get the explicit form of the operator such that the kernel satisfies the heat equation .

3. In a third step, we define a rescaling of space, time and the Clifford algebra, introduced by Getzler. This rescaling has the effect that the leading coefficient of the asymptotic expansion of the rescaled kernel is exactly the differential form of theorem 1 which leads to a reformulation of Theorem 1.

\appendix

2. Chern-Weil theory

some text in Appendix A

3. Complete proof of theorem 1

some text in Appendix B