# Journal of Operator Theory

Volume 46, Issue 3, Supplementary 2001 pp. 619-634.

Projective modules over non-commutative tori: classification of modules with constant curvature connection**Authors**: Alexander Astashkevich (1), and Albert Schwarz (2)

**Author institution:**(1) Renaissance Technologies

(2) Department of Mathematics, University of California, Davis, USA

**Summary:**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 a module $E$ with constant curvature connection such that $[E] = \mu $. In physical words we give necessary and sufficient conditions for existence of 1/2 BPS states in terms of topological numbers.

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