Mukai duality for gerbes with connection

TL;DR

This paper develops a framework for understanding gerbes with connection on stacks using noncommutative algebra, establishing dualities similar to Fourier-Mukai and T-duality, with applications to torus gerbes.

Contribution

It introduces a dg-category model for gerbes with connection, enabling the proof of dualities between gerbes and noncommutative spaces.

Findings

Gerbes with flat connection on a torus are dual to noncommutative holomorphic tori.

The dg-category provides a dg-enhancement of the derived category of coherent sheaves on gerbes.

Fourier-Mukai type dualities are established for gerbes using noncommutative algebraic methods.

Abstract

We study gerbes with connection over an etale stack via noncommutative algebras of differential forms on a groupoid presenting the stack. We then describe a dg-category of modules over any such algebra, which we claim represents a dg-enhancement of the derived category of coherent analytic sheaves on the gerbe in question. This category can be used to phrase and prove Fourier-Mukai type dualities between gerbes and other noncommutative spaces. As an application of the theory, we show that a gerbe with flat connection on a torus is dual (in a sense analogous to Fourier-Mukai duality or T-duality) to a noncommutative holomorphic dual torus.