Modular Classes and Supersymmetric Berezin Volumes

TL;DR

This paper explores how modular classes of Q-manifolds can be used to determine the existence of supersymmetric Berezin volumes in supergeometric representation theory, providing a cohomological criterion for invariance under supercharges.

Contribution

It introduces a cohomological coherence criterion based on modular classes for the existence of invariant supersymmetric Berezin volumes in supergeometry.

Findings

Modular classes serve as an effective tool in supergeometric analysis.

A cohomological criterion for Berezin volume invariance is established.

The approach simplifies the verification of supersymmetric volume existence.

Abstract

We argue that modular classes of Q-manifolds provide an efficient method for addressing the existence of supersymmetric Berezin volumes in the supergeometric representation theory of the $\mathcal{N}=2$ $d=1$ supertranslation algebra. We establish a cohomological coherence criterion for the existence of a Berezin volume that is invariant under both of the supercharges.