Path integrals, SUSY QM and the Atiyah-Singer index theorem for twisted Dirac

TL;DR

This paper rigorously constructs path integrals for supersymmetric quantum mechanics on Riemannian manifolds, linking them to the Atiyah-Singer index theorem through convergence analysis of short-time propagator approximations.

Contribution

It establishes conditions for convergence of path integrals in supersymmetric quantum mechanics and connects these to the Atiyah-Singer index theorem for twisted Dirac operators.

Findings

Path integrals can be rigorously defined for supersymmetric quantum mechanics.

The convergence of approximate propagators leads to a proof of the local Atiyah-Singer index theorem.

Includes analysis of twisted Dirac operators within the path integral framework.

Abstract

Feynman's time-slicing construction approximates the path integral by a product, determined by a partition of a finite time interval, of approximate propagators. This paper formulates general conditions to impose on a short-time approximation to the propagator in a general class of imaginary-time quantum mechanics on a Riemannian manifold which ensure these products converge. The limit defines a path integral which agrees pointwise with the heat kernel for a generalized Laplacian. The result is a rigorous construction of the propagator for supersymmetric quantum mechanics, with potential, as a path integral. Further, the class of Laplacians includes the square of the twisted Dirac operator, which corresponds to an extension of N=1/2 supersymmetric quantum mechanics. General results on the rate of convergence of the approximate path integrals suffice in this case to derive the local version of the Atiyah-Singer index theorem.