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arXiv · 1306.1951

Gauge Theory for Spectral Triples and the Unbounded Kasparov Product

Simon Brain·Bram Mesland·Walter D. van Suijlekom
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Abstract

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.

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BibTeXRIS

Simon Brain, Bram Mesland, Walter D. van Suijlekom. 2013-06-08. Gauge Theory for Spectral Triples and the Unbounded Kasparov Product. https://doi.org/10.4171/jncg/230

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