TITLE: What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?
QUESTION [104 upvotes]: 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.

REPLY [5 votes]: This is a wonderful question, but I think people are having a hard time answering it because even in the simplest settings there is a lot going on and it is not clear what not a lot of background means.  The simplest case would have to be proving GBC for a surface (nothing I say will be any easier for a surface, unfortunately!).  First you have to know that heat kernel here means heat kernel for the Hodge Laplace operator, which is the Laplace operator on forms $(d+d^*)^2.$  Second it involves the supertrace of the heat kernel, which means the trace, except you add a minus sign for the parts taking odd forms to odd forms.  A previous post mentions McKean and Singer, whose argument I will not repeat because it cannot be improved upon, but it argues that the eigenforms of this Laplacian with nonnegative eigenvalue cancel out, so only the $0$ eigenfunctions contribute, so the quantity is $t$-independent and in fact the Euler number.  But if it is $t$-independent, we may as well look at this quantity as $t \to 0.$  Since the heat kernel is approaching a delta function on the diagonal, the small $t$ behavior is a local question, and can be addressed by comparing it to the flat space heat kernel.  This reasonably straightforward calculation gets you the curvature, or in higher dimensions the Pfaffian of the of the curvature (the supertrace itself contained an integral which I have not mentioned explicitly). If you are looking for a principle or technique to bring with you to other problems, I would say it is this:  Show some quantity you can compute from the time evolution kernel is somehow topological and therefore time independent, then equate the small time calculation of it to the large time calculation and see a connection between two things that look unrelated.