Journal of Noncommutative Geometry

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Volume 10, Issue 1, 2016, pp. 135–206
DOI: 10.4171/JNCG/230

Gauge theory for spectral triples and the unbounded Kasparov product

Simon Brain[1], Bram Mesland[2] and Walter D. van Suijlekom[3]

(1) Radboud Universiteit Nijmegen, Netherlands
(2) Leibniz Universität Hannover, Germany
(3) Radboud Universiteit Nijmegen, Netherlands

We explore factorizations of noncommutative Riemannian spin geometries over commutative base manifolds in unbounded KK-theory. After setting up the general formalism of unbounded KK-theory and improving upon the construction of internal products, we arrive at a natural bundle-theoretic formulation of gauge theories arising from spectral triples. We find that the unitary group of a given noncommutative spectral triple arises as the group of endomorphisms of a certain Hilbert bundle; the inner fluctuations split in terms of connections on, and endomorphisms of, this Hilbert bundle. Moreover, we introduce an extended gauge group of unitary endomorphisms and a corresponding notion of gauge fields. We work out several examples in full detail, to wit Yang–Mills theory, the noncommutative torus and the $\theta$-deformed Hopf fibration over the two-sphere.

Keywords: Gauge theories, unbounded KK-theory

Brain Simon, Mesland Bram, van Suijlekom Walter: Gauge theory for spectral triples and the unbounded Kasparov product. J. Noncommut. Geom. 10 (2016), 135-206. doi: 10.4171/JNCG/230