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

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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 μ in the K-group one can find such a module E with constant curvature connection that [E] = μ. In physical words we give necessary and sufficient conditions for existence of 1/2 BPS states in terms of topological numbers.