Projective modules over non-commutative tori: classification of modules with constant curvature connection

Abstract

We study finitely generated projective modules over noncommutative tori. We prove that for every module $E$ with constant curvature connection the corresponding element $[E]$ of the K-group is a generalized quadratic exponent and, conversely, for every positive generalized quadratic exponent $\mu$ in the K-group one can find such a module $E$ with constant curvature connection that $[E] = \mu $. In physical words we give necessary and sufficient conditions for existence of 1/2 BPS states in terms of topological numbers.