A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifold

TL;DR

This paper rigorously constructs the supersymmetric path integral on the loop space of a compact spin manifold, providing a solid mathematical foundation for supersymmetric proofs of the Atiyah-Singer index theorem.

Contribution

It introduces a rigorous integral map for differential forms on loop space, connecting supersymmetric path integrals with the non-commutative loop space Chern character.

Findings

Constructs a rigorous supersymmetric path integral for compact spin manifolds.

Establishes a comparison between the path integral and the non-commutative loop space Chern character.

Provides a mathematical foundation for formal proofs of the Atiyah-Singer index theorem using supersymmetry.

Abstract

We give a rigorous construction of the path integral in N=1/2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler-Jones-Petrack. Via the iterated integral map, we compare our path integral to the non-commutative loop space Chern character of G\"uneysu and the second author. Our theory provides a rigorous background to various formal proofs of the Atiyah-Singer index theorem using supersymmetric path integrals, as investigated by Alvarez-Gaum\'e, Atiyah, Bismut and Witten.