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Projective modules over non-commutative tori: classification of modules with constant curvature connection. (English) Zbl 0996.46031

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.

MSC:

46L87 Noncommutative differential geometry
81T75 Noncommutative geometry methods in quantum field theory
58B34 Noncommutative geometry (à la Connes)
46L08 \(C^*\)-modules