In the book "Heat kernels and Dirac operators", I found the slogan by Quillen in the Introduction: "Dirac operators are a quantization of the theory of connections, and the supertrace of the heat kernel of the square of a Dirac operator is the quantization of the Chern character of the corresponding connection." In fact, Let $M$ be a manifold, $C(M)$ be the Clifford bundle over $M$, $\mathcal{E}$ a vector bundle with a $C(M)$ action, $D$ be the Dirac operator.

The strategy is first we proof that $e^{-tD^2}, t>0$ has a integral kernel $k_t(x,y)\in \Gamma(M\times M, \text{End}(\mathcal{E}))$. It is not difficult to prove that the integration of the super trace $\int_M\text{str}(k_t(x,x))|dx|$ is independent of $t$ and, when $t\rightarrow \infty$ the integration gives the analytic index of $D$ (Hence it gives the analytic index of $D$ for all $t>0$.)

The hard part is when $t\rightarrow 0$. For this we first get the asymptotic expansion $$ k_t(x,x)\sim(4\pi t)^{-n/2}\sum_{i=0}^{\infty}t^ik_i(x) $$ and then observe that $\text{End}(\mathcal{E})\cong C(M)\otimes \text{End}_{C(M)}(\mathcal{E})$. Moreover, the Clifford algebra $C(M)$ is a quantization of the exterior algebra $\mathcal{A}(M)$. Then we use the symbol map $\sigma$, which is the inverse of the quantization map, to get $\sigma(k)=\sum_{i=0}^{n/2}\sigma_{2i}(k_i)\in \mathcal{A}(M, \text{End}_{C(M)}(\mathcal{E}))$. After a careful rescaling we can proof that $$ \sigma(k)=\text{det}^{1/2}(\frac{R/2}{\sinh(R/2)})\exp(-F^{\mathcal{E}/S}) $$ which gives the local index theorem. For details of the proof see the first four chapters of "Heat kernels and Dirac operators", especially Chapter 4.

We see that involves the philosophy of quantization (Dirac operator, Clifford algebra, symbol map) as well as deformation (rescaling, $t\rightarrow 0$). On the other hand, the algebra of pseudo differential operators $\mathcal{D}(M)$ is a deformation quantization of the smooth functions $C^{\infty}(T^*M)$ (in fact, a subalgebra consists of functions satisfies some growing conditions)of the symplectic manifold $T^*M$. The $\textit{algebraic index theorem}$, developed by Fedosov, Nest-Tsygan, Dolgushev, ect., studies the general deformation quantization for any symplectic manifold. But their work is closer to Connes's approach to index theory, for example, method from cyclic homology are intensively used. (I'm not very familiar with algebraic index theorem, so please point out if I'm wrong.)

$\textbf{My question}$ is: do we have a more direct, clear relation between the heat kernel proof of the index theorem and deformation quantization? Do we also have a algebraic index theorem in this approach? Any references are very welcome!

  • $\begingroup$ This lecture notes of Antony Wassermann on the Atiyah-Singer index theorem, could help you. $\endgroup$ – Sebastien Palcoux Oct 7 '13 at 7:09

I think you should have a look at the various papers of Louis Boutet de Monvel. But there is actually a construction of star-products on a symplectic manifold which makes use of the index theorem, due to Richard Melrose.

Last but not least, you might also want to have a look at Appendix B of this paper by Engeli and Felder, where they use heat kernal methods while proving a HRR formula for traces of holomorphic differential operators.

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