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Quantum Hall effect on the hyperbolic plane. (English) Zbl 0916.46057

Summary: We study both the continuous model and the discrete model of the Quantum Hall Effect (QHE) on the hyperbolic plane. Hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables. We define a twisted analogue of the Kasparov map, which enables us to use the pairing between \(K\)-theory and cyclic cohomology theory, to identify this geometric invariant with a topological index, thereby proving the integrality of the hall conductivity in this case.

MSC:

46N50 Applications of functional analysis in quantum physics
46L80 \(K\)-theory and operator algebras (including cyclic theory)
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
81T70 Quantization in field theory; cohomological methods
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