Modules over the Noncommutative Torus and Elliptic Curves

TL;DR

This paper explores the relationship between modules over the noncommutative torus and elliptic curves, using the Weil-Brezin-Zak transform to connect solid state physics, algebraic geometry, and noncommutative geometry.

Contribution

It introduces a novel interpretation of finitely-generated projective modules over the noncommutative torus as Moyal deformations of vector bundles over elliptic curves, linking deformation parameters.

Findings

Modules over the noncommutative torus relate to vector bundles on elliptic curves.

A condition links the deformation parameter and the modular parameter.

The approach bridges solid state physics and noncommutative geometry.

Abstract

Using the Weil-Brezin-Zak transform of solid state physics, we describe line bundles over elliptic curves in terms of Weyl operators. We then discuss the connection with finitely-generated projective modules over the algebra $A_\theta$ of the noncommutative torus. We show that such $A_\theta$-modules have a natural interpretation as Moyal deformations of vector bundles over an elliptic curve $E_\tau$, under the condition that the deformation parameter $\theta$ and the modular parameter $\tau$ satisfy a non-trivial relation.