Modules over the Noncommutative Torus and Elliptic Curves

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.

16 pages, no figures; v2: minor corrections

References in corpus (2)

• Noncommutative tori and the Riemann-Hilbert correspondence
• On quantum mechanics with a magnetic field on R^n and on a torus T^n, and their relation

Cited by in corpus (3)

• Non-Associative Geometry of Quantum Tori
• On a symmetry of complex and real multiplication
• Topics in Noncommutative Geometry