Mathematics

The non-commutative torus is commonly described as a cocycle quantization of the group (C^*-)algebra of the abelian group Z^2. In the first part of the talk I will explain how, using the WBZ transform of solid state physics, finitely generated projective modules over the NC-torus can be interpreted as deformations of vector bundles on elliptic curves by the action of a 2-cocycle, provided that the deformation parameter of the NC-torus and the modular parameter of the elliptic curve satisfy a non-trivial relation. I will then discuss the relation between (formal) deformations of vector bundles on the torus and cochain twists based on the Lie algebra of the 3-dimensional Heisenberg group. Based on a joint work with G. Fiore and D. Franco.