Supersymmetric quantum mechanics and the index theorem

Abstract

The basic features of supersymmetric quantum mechanics are reviewed and illustrated by examples from physics and geometry (the hydrogen atom, and massless fields in curved space). Using a discrete approximation to the path integral in the associated supersymmetric quantum mechanics, the Atiyah-Singer Index Theorem is derived for the twisted Diraf operator. Specializations of this in four dimensions include the Gauss-Bonnet theorem, and the Hirzebruch signature theorem. The relationship of the index theorem to anomalies, and their cancellation in the standard model and beyond, is briefly discussed.