Spectral triples from bimodule connections and Chern connections

TL;DR

This paper presents a geometric method to construct Connes spectral triples from bimodule connections, applying it to quantum spaces like the $q$-sphere and $q$-disk, and introduces a noncommutative Chern construction.

Contribution

It introduces a new geometric approach to building spectral triples from bimodule connections and extends the theory to quantum spaces with classical limits.

Findings

Successfully constructs spectral triples for quantum spaces

Shows the classical limit properties hold with a twisted isometry

Provides a noncommutative Chern construction for holomorphic bundles

Abstract

We give a geometrical construction of Connes spectral triples or noncommutative Dirac operators $D$ starting with a bimodule connection on the proposed spinor bundle. The theory is applied to the example of $M_2(\Bbb C)$, 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.