A gerby deformation of complex tori and the homological mirror symmetry

TL;DR

This paper constructs a gerby deformation of complex tori and explores its implications for homological mirror symmetry, focusing on deformations of objects and their mirror counterparts within this framework.

Contribution

It introduces a new gerby deformation of complex tori and analyzes the resulting deformations of objects and mirror pairs in homological mirror symmetry.

Findings

Constructed a flat gerbe deformation of complex tori.

Analyzed deformations of line bundles and Lagrangian sections under gerby deformation.

Discussed the impact of gerby deformation on mirror symmetry correspondence.

Abstract

Let $(X,\check{X})$ be a mirror pair of a complex torus $X$ and its mirror partner $\check{X}$. This mirror pair is described as the trivial special Lagrangian torus fibrations $X\rightarrow B$ and $\check{X}\rightarrow B$ on the same base space $B$ by SYZ construction. Then, we can associate a holomorphic line bundle $E(s,\mathcal{L})\rightarrow X$ to a pair $(s,\mathcal{L})$ of a Lagrangian section $s$ of $\check{X}\rightarrow B$ and a unitary local system $\mathcal{L}$ along it. In this paper, we first construct the deformation $X_{\mathcal{G}}$ of $X$ by a certain flat gerbe $\mathcal{G}$ and its mirror partner $\check{X}_{\mathcal{G}}$ from the mirror pair $(X,\check{X})$, and discuss deformations of objects $E(s,\mathcal{L})$ and $(s,\mathcal{L})$ over the deformed mirror pair $(X_{\mathcal{G}},\check{X}_{\mathcal{G}})$.