# Rough Outline of Heat Kernel Proof of Atiyah-Singer Index Theorem

## 1. Proof Strategy

Theorem 1 (Mckean-Singer Formula):
$$\text{ind}(D^+) = \text{Str}(e^{-tD^2}) = \int_{x \in M} \text{Str}(K(x,y)).$$

From this, we know Fredholm operator deformation invariance. At the same time, we need Chern-Weil theory.

Main Challenge:

1. In the expansion on heat kernel, we need to prove when $t \rightarrow 0$, the limit exists and find a way to calculate it.
2. Identify the limit as $t \rightarrow 0$.

Proof:
Our proof road: Mckean-Singer formula $\rightarrow$ local-index thm $\rightarrow$ A-S index thm $\rightarrow$ Riemann-Roch-Hirzebunch theorem. $\Box$

## 1.2. Preliminary Work

Superbundle: $E = E^+ \oplus E^-$.
On compact manifold $M$, $D: \Gamma(M,E) \rightarrow \Gamma(M,E)$ is a self-adjoint operator. $D = D^+, D^- = D|_{E^-}$.

Observe that:
$D$ is symmetric $\Longrightarrow$ eigenvalue space of $D^2$ is finite dimensional $\Longrightarrow$ in particular $\text{Ker} D^2$ is finite dimensional $\Longrightarrow$ $\text{Ker} D$ is finite dimensional.

Dimension of superspace $E = E^+ \oplus E^-$:
$$\text{dim} E = \text{dim} E^+ – \text{dim} E^-.$$
$$\text{Ker} D = \text{Ker} D^+ \oplus \text{Ker} D^-.$$

Definition:
$\text{ind} D^+ = \text{dim} \text{Ker} D^+ – \text{dim} \text{Ker} D^-$.

Lemma:

1. Let $D$ be a self-adjoint Dirac operator on a Clifford module $E$ over a compact manifold $M$, then
$$\Gamma(M,E^{\pm}) = \text{Ker} D^{\pm} \oplus \text{im} D^{\mp}.$$
In particular,
$$\text{ind} D^+ = \text{dim} \text{Ker} D^+ – \text{dim} \text{coker} D^+,$$
where coker $D^+ := \Gamma(M,E^-)/\text{im} D^+$.
2. Let $D$ be a differential operator acting on a $Z_2$-graded vector bundle $E$, then $\text{Str}[D,K] = 0$.

## 1.3. Mckean-Singer Formula

The formula is:
$$\text{ind}(D^+) = \text{Str}(e^{-tD^2}) = \int_{x \in M} \text{Str}(K(x,y)).$$

The expression of heat operator by spectral measure is:
$$e^{-tD^2} = \int_{0}^\infty d^{\lambda t}dE_{\lambda}. Proof 1: We have first eigenvalue estimate on compact manifold:$$
|Str(e^{-tD^2}-P_0)| \leq C\text{vol}(M)e^{-t\lambda}.

\Longrightarrow \lim_{t \rightarrow \infty} Str(e^{-tD^2}) = Str p_0 = \text{dim Ker} D^+ – \text{dim Ker} D^- = \text{ind} D^+.
$$On the other hand, we need to show Str e^{-tD^2} is independent with t, in fact:$$
\frac{d}{dt} Str (e^{-tD^2}) = -Str(D^2 e^{-tD^2}).
$$D odd parity \Longrightarrow D^2e^{-tD^2}=[D,D e^{-tD^2}]. [\ ,\ ] supercommutator \Longrightarrow \frac{d}{dt}Str (e^{-tD^2})= -Str[D,D e^{-tD^2}]=0. Q.E.D. Proof 2: By spectral decomposition of e^{-tD^2}:$$
\text{Str} (e^{-tD^2}) = \sum_{\lambda \geq 0}(n_{\lambda}^+ – n_{\lambda}^-)e^{-t\lambda}.
$$Observe that: n_{\lambda}^+=n_{\lambda}^- for \lambda \neq 0. \Longrightarrow \text{ind} D=n_0^+ – n_0^-. (detail in [BGV]) Q.E.D. Corollary: The index of a smooth on-parameter family of Dirac operator is constant. What we have proved is:$$
\text{ind} D = \text{Str} (e^{-tD^2}) = \int \text{Str}(k(x,y)).

## 1.4. Analytic Formula of ind$D^+$

From the discussion of heat kernel in section 2, we know the following result (section 2 only discusses the case of function but using the similar way we can get similar result on bundle):
$$K_t(x,y) \sim (4 \pi t)^{\frac{n}{2}} \sum_{i=0}^{+\infty} t^i K_i(x), K_i \in \Gamma(M,\text{End}(E)).$$
On the other hand:
$$D=\begin{bmatrix} 0 & D^- \ D^+ & 0 \end{bmatrix} \Longrightarrow D^2=\begin{bmatrix} D^-D^+ & 0 \ 0 & D^+D^- \end{bmatrix}.$$
And use the Mckean-Singer formula, we get:
$$\text{ind}(D^+) = \text{Str}(e^{-tD^2}) = \text{Tr}(e^{-tD^+D^-}) – \text{Tr}(e^{-tD^=D^-}) = \int_M \text{Tr}(K_t(x,y,D^-D^+)) – \int_M \text{Tr}(k_t(x,y,D^+D^-)) = \sum_{i=0}^{\infty} t^{i-\frac{n}{2}}a_i(D^+D^-) – \sum_{i=0}^{\infty} t^{i – \frac{n}{2}}a_i(D^+D^-).$$
Where $a_i$ is the heat trace invariants.
Take $t \rightarrow 0$, the only thing that makes sense is the series of order $\frac{n}{2}$, and we want to prove:
$$\text{ind}(D^+) = a_{\frac{n}{2}}(D^-D^+) – a_{\frac{n}{2}}(D^+D^-).$$
But the difficult thing is that the high order series is very hard to calculate.
Our strategy is as follows:

Step 1:
Proof $Str(K_t(x,y))$ has a limit as $t \rightarrow 0$, i.e. $Str(K_t(x,y))\stackrel{t \rightarrow 0}{\longrightarrow}$ index density.
Step 2:
Use a rescaling of space, time, Clifford bundles, to find a way that makes us only need to calculate the leader coefficient.

## 1.5. From the McKean-Singer Formula to the Index Theorem

Let $M$ be a compact oriented Riemannian manifold of even dimension $n$. We will write $k_t(x, y)$ for the heat kernel associated to $D^2$. The diagonal $k_t(x, x)$ is a section of $End(E)$ which is isomorphic to $Cl(M) \otimes End_{Cl(M)}(E)$. Using this isomorphism, we define a filtration on $End(E)$, induced by the filtration on $Cl(M)$. Elements of $End_{Cl(M)}(E)$ are given 0-degree. Denote by $Cl_i(M)$ the subbundle of $Cl(M)$ consisting of all elements of degree less or equal to $i$.

Theorem 1:
The following statements hold:

1. The coefficients $k_i$ have degree less or equal to $2i$. In other words, $k_i \in \Gamma (M, Cl_{2i}(M) \otimes End_{Cl(M)}(E))$.
2. If $\sigma(k):= \sum_{i=0}^{n/2}\sigma_{2i}(k_i) \in A(M,End_{Cl(M)}(E))$, where $\sigma_j :Cl_j(M) \hookrightarrow \wedge_j(M)$ denotes the $i=0$ restriction of the symbol map, then:
$$\sigma (k) = A\hat{A}(M) \exp(-\hat{E}/S).$$
$$\text{ind} (D^+) = \sum_{i=0}^{\infty} t^{i-\frac{n}{2}}a_i(D^+D^-) – \sum_{i=0}^{\infty} t^{i – \frac{n}{2}}a_i(D^+D^-).$$

The important observation is that the $Str$ of order less than $n$ all vanish, in other words:

Lemma 2:
For any quadratic space $v$ of dimension $n$, $CL_{n-1}(V) =[CL(V),CL(V)]$.

Proof:
Let $e_1,…,e_n$ be a basis of $V$. For any multi-index $I \subset {1,…,n}$, denote by $c_I$ the Clifford product $\prod_{i \in I} c$. Then the set ${c_I}$ is a basis for $CL(V)$. If $|I|<n$, there is at least one $j$ such that $j \notin I$, and we have
$$c(e_I)=-\frac{1}{2}[c_j,c_j c_I].$$
Q.E.D.

So take $t \rightarrow 0$, we get:
$$\text{ind} (D) =(4 \pi)^{\frac{n}{2}} \sum_{i = \frac{n}{2}}^{\infty} t^{i- \frac{n}{2}}Str(k_i(x)).$$

Now we want to identify the term $Str(k_{\frac{n}{2}}(x))$ as a characteristic form on $M$:

Lemma 3:
Let $V$ be a Euclidean space. There is, up to a constant factor, a unique supertrace on $Cl(V)$, equal to $T \circ \sigma$ where $\sigma$ denotes the symbol map and $T$ is the projection of $lpha \in \wedge V$ onto the coefficient of $e_1 \wedge, … ,\wedge e_n$ if $e_1,…,e_n$ form an oriented orthonormal basis of $V$. Furthermore, the supertrace defined above equals:
$$Str(a) = (-2i)^{\frac{n}{2}} (T \circ \sigma(a)).$$

Remark:
The map $T$ is also called the canonical Berezin integral.

Proof:
The dimension of $Str$ space is one because of $CL_{n-1}(V)=[CL(V),CL(V)]$ and it never be empty because there is a naturally defined supertrace on $CL(V)$. So the only thing we need to do is to determine the constant, in fact we only need to calculate the supertrace on any non-zero element. For instance, the chirality operator $\Gamma$. We have that $Str(\Gamma) = \text{dim} (\wedge P) = 2^{n/2}$ and therefore $Str(a) = (-2i)^{n/2} T \circ \sigma (a)$ for all $a \in Cl(M)$.
Q.E.D.

So we know:
For all section $a \otimes b \in \Gamma(M \otimes CL(M) \otimes End_{CL(M)}(E)$:
$$Str_E( a \otimes b)(x) = (-2i)^{\frac{n}{2}} \sigma_n(a(x))Str {E/S}(b(x)).$$ $$Str_E(k{\frac{n}{2}})(x) = (-2i)^{\frac{n}{2}} Str {E/S}(\sigma_n(k{\frac{n}{2}})). Theorem 2 (Local Index Theorem): Let M be a compact, oriented even-dimensional manifold and let E be a Clifford module with Clifford connection \nabla E. Let D be the associated Dirac operator. Then \lim_{t \rightarrow 0} Str(k_t(x, x))|dx| exists and is obtained by taking the n-th form piece of$$
(2\pi i)^{-\frac{n}{2}} \hat{A}(M)ch(E /S).

Theorem 3 (Atiyah-Singer Index Theorem):
Let $M$ be a compact, oriented, even-dimensional manifold and let $D$ be a Dirac operator on a Clifford module $E$. Then the index of $D$ is given by:
$$\text{ind} D = (2 \pi i)^{-\frac{n}{2}}\int_{M} \hat{A}(M) ch(E/S).$$
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 the appendix, where 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 $x$ around a point $x_0 \in M$. Near the diagonal, we use parallel transport to pull back the heat kernel $k_t(x, x_0)$, which is a section of the vector bundle $E_x \otimes E_x^$, and define a new kernel $k(t, x) := \iota(x_0, x)k_t(x, x_0)$, a section of $End(E_{x_0} ) \cong Cl(V^) \otimes End(W)$ for some twisting space $W$. Using the symbol isomorphism $arphi$, we can look at $k(t, x)$ as a section of $\wedge(V^*) \otimes E(W)$.
2. We use Lichnerowicz’s formula to get the explicit form of the operator $L$ such that the kernel $k(t, x)$ satisfies the heat equation $(\partial t + L)k(t, x) = 0$.
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 $\sigma(k)$ 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.