Journal of Noncommutative Geometry
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Gauge theory for spectral triples and the unbounded Kasparov productSimon Brain, Bram Mesland and Walter D. van Suijlekom (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