A Chern-Simons transgression formula for supersymmetric path integrals on spin manifolds

TL;DR

This paper establishes a Chern-Simons transgression formula for supersymmetric path integrals on spin manifolds, linking geometric analysis with topological invariants like the A-hat genus.

Contribution

It introduces a canonical Chern-Simons current associated with metric families, providing a new transgression formula for supersymmetric path integrals on spin manifolds.

Findings

The supersymmetric path integral induces a differential topological invariant.

A Chern-Simons current is constructed from metric families on spin manifolds.

The invariant relates to the A-hat genus of the manifold.

Abstract

Earlier results show that the N = 1/2 supersymmetric path integral on a closed even dimensional Riemannian spin manifold (X,g) can be constructed in a mathematically rigorous way via Chen differential forms and techniques from non-commutative geometry, if one considers it as a current on the smooth loop space of X. This construction admits a Duistermaat-Heckman localization formula. In this note, fixing a topological spin structure on X, we prove that any smooth family of Riemannian metrics on X canonically induces a Chern-Simons current which fits into a transgression formula for the supersymmetric path integral. In particular, this result entails that the supersymmetric path integral induces a differential topological invariant on X, which essentially stems from the A-hat-genus of X.