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Since the summary below is already a bit long, I thought I'd add a few lines at the beginning as a guide. The proofs all proceed as follows:

• Identify the quantity of interest (like the Euler characteristic) as the index of an operator going from an 'even' bundle to an 'odd' bundle.

• Use Hodge theory to write the index in terms of the dimensions of harmonic sections, i.e., kernels of Laplacians.

• Use the heat evolution operator for the Laplacians and 'supersymmetry' to rewrite this as a 'supertrace.'

• Write the heat evolution operator in terms of the heat kernel to express the supertrace as the integral of a local density.

• Use the eigenfunction expansion of the heat kernel to identify the constant (in time) part of the local density.

• Most of this is general nonsense, and the difficult step is 5. By and large, the advances made after the seventies all had to do with finding interpretations of this last step that employed intuition arising from physics.

I suffered over this proof quite a bit in my pre-arithmetic youth and wrote upterms with negative $i$. (I think it was fashionable to refer to this computation cancellation as 'miraculous cancellations.')Thereforemiraculous,' which it is, compared to the easy cancellation above.At this point, Patodi could take a limit

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an integral of local (point-wise) traces, and similarly for $Tr(e^{-t\Delta^-})$. One needs therefore, techniques to evaluate the density

Up to here the discussion was completely general, but then the proof begins to involve special cases, or Subsequent interpretations of the proof (more precisely, the supertrace), in terms of supersymmetry, path integrals, loop spaces, etc., were tremendously influential to many areas of mathematics and physics, but the mathematical core of the index theorem itself appears to have remained largely unchanged for almost forty years. In particular, the terminology I've used myself above, the super- things, didn't occur at all in the original papers of Patodi, Atiyah-Bott-Patodi, or Gilkey.

literally approaches the identity operator on all even forms (except for the fact that it diverges). That is, the heat kernel interpolates between the identity and the projection to the harmonic forms, in some genuine sense expressing the diffusion of heat from a point distribution to a harmonic steady-state. A similar discussion holds for the odd forms as well. I can't justify this next point even vaguely at the moment, but one should therefore think of $$[K^+_t(x,y)-K^-_t(x,y)]dvol(y)$$ as regularizing the current on $M\times M$ given by the diagonal $M\subset M\times M$. Thus, the integral of $$[K^{+}_t(x,x)-K^-_t(x,x)]dvol(x)$$ $[TrK^{+}_t(x,x)-TrK^-_t(x,x)]dvol(x)$$ends up computing a deformed self-intersection number of the diagonal in M\times M. From this perspective, it shouldn't be too surprising that the Euler class, representing exactly this self-intersection, shows up. Added: I forgot to mention that the Riemann-Roch case is where$$P=\bar{\partial}+\bar{\partial}^*$$going from the even to the odd part of the Dolbeault resolution associated to a holomorphic vector bundle. The limit of the local density is a differential form representing the top degree portion of the Chern character of the bundle multiplied by the Todd class of the tangent bundle. Perhaps it's worth stressing that these special cases all go through the general argument outlined above. 3 added 1667 characters in body in terms of finite-dimensional eigenspaces for the Laplacians with non-negative eigenvalues. And then, the identitiesit's easy to see that the only contributions to the trace are from the kernels of the plus and minus Laplacians. This is the 'easy cancellation' that occurs in this proof.But thereon the zero eigenspaces, the heat evolution operators are clearly the identity, allowing us to identify the supertrace with the index.as a precise differential form representative for a characteristic class is referred to sometimes as a {\em local index theorem}theorem, a statementmore refined than the topological formula for the global index. There is even a beautiful version of a local {\em families index theorem} Subsequent {\em interpretations} of the proof (more precisely, the supertrace), supersymmetry, path integrals, loop spaces, etc., were tremendously influential to many areas of mathematics and physics, but the mathematical core of the index theorem itself appears to have remained largely unchanged for almost forty years. Added: Here is just a little bit of geometric-physical intuition regarding the heat kernel in the Gauss-Bonnet case, which I'm sure is completely banal for most people. The density$$\sum_ie^{-t\lambda_i}||\phi^+_i(x)||^2 dvol(x)-\sum_ie^{-t\mu_i}||\phi^-_i(x)||^2 dvol(x)$$expresses the heat kernel in terms of orthonormal bases for the even and odd forms. When t\rightarrow \infty all terms involving the positive eigenvalues decay to zero, leaving only contributions from orthonormal harmonic forms. This is one way to to see that the integral of this density, which is independent of t, must be the Euler characteristic. On the other hand, as t\rightarrow 0, the operator$$K^+_t(x,y)dvol(y)=[\sum_i e^{-t\lambda_i } \phi^+_i(x)\otimes \phi^+_i(y)]dvol(y)$$literally approaches the identity operator on all even forms (except for the fact that it diverges). That is, the heat kernel interpolates between the identity and the projection to the harmonic forms. A similar discussion holds for the odd forms as well. I can't justify this next point even vaguely at the moment, but one should therefore think of$$[K^+_t(x,y)-K^-_t(x,y)]dvol(y)$$as regularizing the current on M\times M given by the diagonal M\subset M\times M. Thus, the integral of$$[K^{+}_t(x,x)-K^-_t(x,x)]dvol(x)$$ends up computing a deformed self-intersection number of the diagonal in$M\times M\$. From this perspective, it shouldn't be too surprising that the Euler class, representing exactly this self-intersection, shows up.

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