Abstract:
In the 1980’s, Alvarez-Gaume, following Witten, used heuristic path integrals in supersymmetric quantum mechanics to give arguments for index theorems including the Gauss-Bonnet-Chern, Hirzebruch, and Atiyah-Singer theorems. In this talk, I summarize the arguments, and describe a mathematical construction of the relevant path integrals. This provides a new construction of the heat kernel for a class of Laplacians broad enough to include the square of the twisted Dirac operator, which corresponds to an extension of N=1/2 supersymmetric quantum mechanics. In this case, general results on the rate of convergence of the approximate path integrals suffice to derive the local version of the Atiyah-Singer index theorem.