Journal of Noncommutative Geometry

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Volume 11, Issue 2, 2017, pp. 669–701
DOI: 10.4171/JNCG/11-2-7

Published online: 2017-06-29

Spectral triples from bimodule connections and Chern connections

Edwin Beggs[1] and Shahn Majid[2]

(1) Swansea University, UK
(2) Queen Mary University of London, UK

We give a geometrical construction of Connes spectral triples or noncommutative Dirac operators $\slashed D$ starting with a bimodule connection on the proposed spinor bundle. The theory is applied to the example of $M_2(\mathbb C)$, and also applies to the standard $q$-sphere and the $q$-disk with the right classical limit and all properties holding except for $\mathcal J$ now being a twisted isometry. We also describe a noncommutative Chern construction from holomorphic bundles which in the $q$-sphere case provides the relevant bimodule connection.

Keywords: Dirac operator, noncommutative differential calculus, bimodule connections, Chern connections, spectral triple, quantum sphere

Beggs Edwin, Majid Shahn: Spectral triples from bimodule connections and Chern connections. J. Noncommut. Geom. 11 (2017), 669-701. doi: 10.4171/JNCG/11-2-7