the heat-kernel proof of the index theorem

The Atiyah-Singer index theorem equates a count from analysis (how many solutions an elliptic equation has, more precisely dim kernel minus dim cokernel) with a number from topology (an integral of characteristic curvature). Why on earth should those two things be equal? The heat-kernel proof gives a strikingly physical answer: run the heat equation associated to the operator and watch how heat dissipates as time runs to zero. A particular combination of heat traces is independent of time, so you can compute it two ways — at large time it sees only the analytic index, at small time it sees only local curvature — and equating the two yields the theorem.

Here is the method in plain steps. Let D be a Dirac-type operator with adjoint D*, splitting spinors into + and - parts, so D: S^+ -> S^- and the index is dim ker D - dim ker D*. (1) Form the two heat operators e^{-t D* D} and e^{-t D D*}. The McKean-Singer formula says index(D) = Tr(e^{-t D* D}) - Tr(e^{-t D D*}) for EVERY t > 0 — the supertrace is exactly time-independent because nonzero eigenvalues of D*D and DD* pair up and cancel, leaving only the kernels. (2) Take t -> infinity: only the zero eigenvalues survive, and you literally read off the index. (3) Take t -> 0: the heat kernel has a short-time asymptotic expansion whose coefficients are local curvature polynomials, so the supertrace becomes the integral of a specific curvature density over M. (4) Because the answer cannot depend on t, the small-time local integral must equal the index. Identifying that local integral as exactly the A-hat genus (or the relevant characteristic class) is the hard analytic step; the cleanest version is Getzler's rescaling, which uses a clever scaling of the Clifford variables so the leading short-time term assembles into the A-hat form with almost no computation.

Why this proof is loved: it is local and explicit — it shows the topological side literally emerges from a curvature computation, and it generalizes to families, equivariant settings, and gives the local index theorem (a pointwise, not just integrated, statement). The honest cautions. First, time-independence of the supertrace is exact, but extracting the right characteristic class from the small-time expansion is genuinely delicate; the 'fantastic cancellations' that Atiyah hoped for are real but were only tamed by Getzler's rescaling and Patodi's earlier work. Second, the heat-kernel method computes the cohomological index formula, and it is not a curvature identity in disguise — the equality is between an integer (analytic index) and an integral that happens to be an integer; do not say Atiyah-Singer 'is' a Gauss-Bonnet-type curvature law, even though Gauss-Bonnet is one special case it reproduces.