## Modules over the Noncommutative Torus and Elliptic Curves    [PDF]

Francesco D'Andrea, Gaetano Fiore, Davide Franco
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. We then conclude with some comments about formal deformations of vector bundles on the torus and twists based on the Lie algebra of the 3-dimensional Heisenberg group.
View original: http://arxiv.org/abs/1307.6802