A Rigorous Path Integral for N=1 Supersymmetic Quantum Mechanics on a Riemannian Manifold

TL;DR

This paper constructs a rigorous path integral formulation for N=1 supersymmetric quantum mechanics on Riemannian manifolds, demonstrating convergence to the heat kernel and analyzing short-time behavior.

Contribution

It provides a rigorous short-time approximation of the propagator as a path integral on Riemannian manifolds, connecting it to the heat kernel and supersymmetric quantum mechanics.

Findings

Path integral converges uniformly to the heat kernel.

Steepest descent approximation reproduces expected short-time behavior.

Constructs a rigorous framework for supersymmetric quantum mechanics on manifolds.

Abstract

Following Feynman's prescription for constructing a path integral representation of the propagator of a quantum theory, a short-time approximation to the propagator for imaginary time, N=1 supersymmetric quantum mechanics on a compact, even-dimensional Riemannian manifold is constructed. The path integral is interpreted as the limit of products, determined by a partition of a finite time interval, of this approximate propagator. The limit under refinements of the partition is shown to converge uniformly to the heat kernel for the Laplace-Beltrami operator on forms. A version of the steepest descent approximation to the path integral is obtained, and shown to give the expected short-time behavior of the supertrace of the heat kernel.