Non-commutative tori and Fourier-Mukai duality

TL;DR

This paper extends Fourier-Mukai duality to non-commutative and gerbe deformations of complex tori, establishing an equivalence between these generalized dual objects.

Contribution

It introduces a novel duality between non-commutative deformations and gerbe deformations of complex tori, expanding the classical Fourier-Mukai framework.

Findings

Non-commutative deformation of a complex torus via a holomorphic Poisson structure.

Gerbe deformation of the dual complex torus via a B-field.

The two deformations are shown to be Fourier-Mukai equivalent.

Abstract

The classical Fourier-Mukai duality establishes an equivalence of categories between the derived categories of sheaves on dual complex tori. In this article we show that this equivalence extends to an equivalence between two dual objects. Both of these are generalized deformations of the complex tori. In one case, a complex torus is deformed formally in a non-commutative direction specified by a holomorphic Poisson structure. In the other, the dual complex torus is deformed in a B-field direction to a formal gerbe. We show these two deformations are Fourier-Mukai equivalent.