Moduli Spaces of Standard Holomorphic Bundles on a Noncommutative Complex Torus

TL;DR

This paper explores the moduli space of standard holomorphic bundles on a noncommutative complex torus, revealing its connection to stable bundles on elliptic curves and proposing a mirror symmetry interpretation involving foliations.

Contribution

It establishes a natural identification between the moduli space of holomorphic bundles on a noncommutative torus and stable bundles on an elliptic curve, and links this to mirror symmetry concepts.

Findings

Moduli space of holomorphic bundles on noncommutative torus is identified with that on elliptic curves.

Mirror reflection of the noncommutative torus corresponds to an elliptic curve with a foliation.

Moduli space of super cycles on the mirror matches the holomorphic bundle moduli space.

Abstract

In this paper we study the moduli space of standard holomorphic structures on a noncommutative complex two torus. It will be shown that the moduli space is naturally identified with the moduli space of stable bundles on an elliptic curve. We also propose that the mirror reflection of the noncommutative complex torus is the mirror reflection of the elliptic curve together with a linear foliation. From this we identify the moduli space of super cycles on the mirror reflection with the moduli space of standard holomorphic bundles on a noncommutative complex torus.