Localization and Stationary Phase Approximation on Supermanifolds

TL;DR

This paper extends localization techniques and stationary phase approximation to supermanifolds, providing formulas for integrals involving odd vector fields and exploring non-degenerate conditions on the vanishing locus.

Contribution

It introduces a localization formula for integrals on supermanifolds with odd vector fields and extends stationary phase and Morse-Bott lemmas to the super setting.

Findings

Derived a formula for integrals on supermanifolds with non-degenerate vanishing loci.

Extended stationary phase approximation to supermanifolds.

Generalized Morse-Bott Lemma to the supergeometric context.

Abstract

Given an odd vector field $Q$ on a supermanifold $M$ and a $Q$-invariant density $\mu$ on $M$, under certain compactness conditions on $Q$, the value of the integral $\int_{M}\mu$ is determined by the value of $\mu$ on any neighborhood of the vanishing locus $N$ of $Q$. We present a formula for the integral in the case where $N$ is a subsupermanifold which is appropriately non-degenerate with respect to $Q$. In the process, we discuss the linear algebra necessary to express our result in a coordinate independent way. We also extend stationary phase approximation and the Morse-Bott Lemma to supermanifolds.