On a noncommutative deformation of holomorphic line bundles on complex tori and the SYZ transform

TL;DR

This paper explores noncommutative deformations of holomorphic line bundles on complex tori and their mirror duals within the SYZ framework, extending previous constructions and analyzing their dual objects.

Contribution

It extends the construction of noncommutative holomorphic line bundles on complex tori and investigates their mirror dual objects in the SYZ mirror symmetry context.

Findings

Construction of noncommutative line bundles $L_{\theta}$ on deformed tori.

Extension of these constructions to more general settings.

Analysis of mirror dual objects corresponding to the noncommutative deformations.

Abstract

By regarding a given $n$-dimensional complex torus $X^n$ as the trivial torus fibration $X^n \to \mathbb{R}^n/\mathbb{Z}^n$, we can obtain a mirror dual complexified symplectic torus $\check{X}^n$ based on the SYZ construction. In the middle 2000s, as a part of the study on noncommutative deformations of $X^n$, Kajiura examined the noncommutative complex torus $X_{\theta}^n$ obtained via the (real) nonformal deformation quantization of $X^n \to \mathbb{R}^n/\mathbb{Z}^n$ by a Poisson bivector $\theta$ defined along the fibers. In particular, he constructed the noncommutative deformations $L_{\theta} \to X_{\theta}^n$ of holomorphic line bundles on $X^n$ and a curved dg-category consisting of them. On the other hand, associated to this noncommutative deformation, we can construct a non-trivial deformation of the trivial holomorphic line bundle on $X^n$ by twisting it with a suitable isomorphism. In this paper, from this point of view, we extend the construction of $L_{\theta}$ to the more general setting. Moreover, we also consider objects defined on a mirror partner of $X_{\theta}^n$ which are mirror dual to such extended noncommutative objects.