Question

I know the following facts. (Don't assume I know much more than the following facts.)

• The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem.
• The Atiyah-Singer index theorem can be proven using heat kernels.

This implies that both Riemann-Roch and Gauss-Bonnet can be proven using heat kernels. Now, I don't think I have the background necessary to understand the details of the proofs, but I would really appreciate it if someone briefly outlined for me an extremely high-level summary of how the above two proofs might go. Mostly what I'm looking for is physical intuition: when does one know that heat kernel methods are relevant to a mathematical problem? Is the mathematical problem recast as a physical problem to do so, and how?

(Also, does one get Riemann-Roch for Riemann surfaces only or can we also prove the version for more general algebraic curves?)

Edit: Sorry, the original question was a little unclear. While I appreciate the answers so far concerning how one gets from heat kernels to the index theorem to the two theorems I mentioned, I'm wondering what one can say about going from heat kernels directly to the two theorems I mentioned. As Deane mentions in this comments, my hope is that this reduces the amount of formalism necessary to the point where the physical ideas are clear to someone without a lot of background.

Answer 1

Here is how the heat kernel proof of Atiyah-Singer goes at a high level. Let $(\partial_t - \Delta)u = 0$ and define the heat kernel (HK) or Green function via $\exp(-t\Delta):u(0,\cdot) \rightarrow u(t,\cdot)$. The HK derives from the solution of the heat equation on the circle:

$u(t,\theta) = \sum_n a_n(t) \exp(in\theta) \implies a_n(t) = a_n(0)\cdot \exp(-tn^2)$

For a sufficiently nice case the solution of the heat equation is $u(t,\cdot) = \exp(-t\Delta) * u(0,\cdot)$.

The hard part is building the HK: we have to compute the eigenstuff of $\Delta$ (this is the Hodge theorem). But once we do that, a miracle occurs and we get the

Atiyah-Singer Theorem: The supertrace of the HK on forms is constant: viz.

$Tr_s \exp(-t\Delta) = \sum_k (-1)^k Tr \exp(-t\Delta^k) = const$

For $t$ large, this can be evaluated topologically; for small $t$, it can be evaluated analytically as an integral of a characteristic class.

Edit per Qiaochu's clarification

This article of Kotake (really in here as the books seem to be mixed up) proves Riemann-Roch directly using the heat kernel.

Answer 2

In my opinion, the most "physical" recasting of the index theorem is the Witten index for supersymmetric quantum theories. It is superficially similar to the heat kernel, but much more general. The Witten index of a supersymmetric quantum mechanical system is the regularised supertrace $$\mathrm{Tr} (-1)^F \exp(-\beta H)$$ where $(-1)^F$ is the grading operator which is $-1$ on fermionic states and $+1$ on bosonic states and $H$ is the hamiltonian. The trace is taken over the Hilbert space of states.

One can show that this does not depend on $\beta$ and hence can be evaluated both at small $\beta$ ("large temperature expansion") or large $\beta$ ("small temperature expansion"). In one expansion one sees that it computes the trace of $(-1)^F$ on zero modes of the hamiltonian, since for a supersymmetric system the dimensions of the bosonic and fermionic positive-energy eigenstates are equal. In the other expansion one writes the Witten index in terms of a functional integral, which (when formally manipulated) becomes a geometric integral. The formal manipulations can be justified as in Getzler's proof of the local Atiyah-Singer index theorem.

The relation with the heat kernel comes from taking a particular supersymmetric model in which the hamiltonian is the laplacian. The relation with the Gauss-Bonnet theorem comes from considering a supersymmetric sigma model in which the hamiltonian is the Hodge laplacian acting on ($L^2$) differential forms. The Witten index then is computing the index of the elliptic operator $d + \delta$ from the odd to the even rank differential forms, which is the Euler characteristic of the manifold.

There are supersymmetric models for which the Witten index computes the index of a generalised Dirac operator as in the original work of Atiyah and Singer.

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Witten introduced "his" index in order to study supersymmetry breaking. A nonzero value of the index shows that there is an imbalance of fermionic and bosonic vacua (=states of minimal energy), hence supersymmetry is broken.

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There is yet another relation between the heat kernel and the index theorem and it comes from a certain regularisation of the functional integral measure as in Fujikawa's celebrated derivation of the chiral anomaly.

Answer 3

By the way, the first heat kernel proof was given - if I am not mistaken - by Peter Gilkey!
The following MSRI workshop also had some introductory remarks about index theory
http://www.msri.org/calendar/workshops/WorkshopInfo/443/show_workshop

I personally would recommend Gerd Grubbs first lecture; what you are looking for is on slide 5 and onwards (up to slide 7) http://jessica2.msri.org/attachments/12917/pages/129170005.htm

Paul Loya's lectures are also interesting, by the way!
edit:
Gilkey writes in
http://mmf.ruc.dk/~Booss/recoll.pdf (an interesting article, you should read it!)
"During the course, he said that there was this wonderful invariant that Bob Seeley had constructed analytically and, ‘here is Bott’s proof of the index theorem, and somebody should actually show that this gives a heat equation proof of the index theorem.’ That struck me as a fun problem, so I went home that night and gave the heat equation proof to the Gauss-Bonnet-Theorem."

Answer 4

For a completely elementary description of the relation between the Euler characteristic of a domain in $\mathbb{R}^2$ and the heat kernel on the domain, you could start with Mark Kac's "Can you hear the shape of a drum". At the very end of the paper he observes that the constant term in the expansion of the trace of the heat kernel contains topological information, namely the Euler characteristic of the domain. He doesn't prove this, but indicates how you could get this from what he explained before in the paper. He left his formula as a conjecture (in 1966).

Kac, Mark (1966), "Can one hear the shape of a drum?", American Mathematical Monthly 73 (4, part 2): 1–23

Answer 5

For surfaces everything is quite simple: The starting point is of course the McKean Singer formula for the index. Next step is to compute an assymptotic expansion of the heat kernel, but this can be done for surfaces by hand (for the terms which are important for the index, i.e. up to order 1): it is more or less the curvature of the corresponing bundle-> integrating this up gives you an heat equation proof of RR or Gauss Bonnet. (Details can be found in Roe s Elliptic operators,... book)