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Modular Classes and Supersymmetric Berezin Volumes

Andrew James Bruce

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.

Modular Classes and Supersymmetric Berezin Volumes

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 supertranslation algebra. We establish a cohomological coherence criterion for the existence of a Berezin volume that is invariant under both of the supercharges.

Paper Structure

This paper contains 1 section, 2 theorems, 25 equations.

Table of Contents

  1. Acknowledgements

Key Result

Theorem 1

Let $(M, Q_i, P, \hbox{\boldmath$\rho$})$ be a $\mathcal{N} =2$ supermanifold with a volume. Assume that both modular classes $\mathsf{Mod}(M, Q_i)$ vanish, i.e., there exists Berezin volumes $\hbox{\boldmath$\rho$}_i$, such that $\mathop{\mathrm{Div}}\nolimits_{\hbox{\boldmath$\rho$}_i}Q_i =0$. The

Theorems & Definitions (8)

  • Definition 1
  • Theorem 1: Cohomological Coherence Criterion
  • proof
  • Corollary 1
  • Example 1
  • Example 2
  • Example 3
  • Example 4