Equivariant Gerbes on Complex Tori

TL;DR

This paper develops a new framework for understanding symmetries of holomorphic gerbes on complex tori, extending classical concepts like the theta group to a higher categorical setting and analyzing their representations.

Contribution

It introduces the concept of a theta group for gerbes, calculates its structure as a Picard groupoid, and explores obstructions to equivariance, advancing the representation theory of gerbes on complex tori.

Findings

Explicit calculation of the theta group of a gerbe as a central extension

Identification of obstructions to gerbe equivariance

Survey of representation types of the symmetry group on twisted sheaves

Abstract

We explore a new direction in representation theory which comes from holomorphic gerbes on complex tori. The analogue of the theta group of a holomorphic line bundle on a (compact) complex torus is developed for gerbes in place of line bundles. The theta group of symmetries of the gerbe has the structure of a Picard groupoid. We calculate it explicitly as a central extension of the group of symmetries of the gerbe by the Picard groupoid of the underlying complex torus. We discuss obstruction to equivariance and give an example of a group of symmetries of a gerbe with respect to which the gerbe cannot be equivariant. We survey various types of representations of the group of symmetries of a gerbe on the stack of sheaves of modules on the gerbe and the associated abelian category of sheaves on the gerbe (twisted sheaves).