Gauge Theory for Spectral Triples and the Unbounded Kasparov Product

TL;DR

This paper develops a bundle-theoretic framework for gauge theories within noncommutative geometry using unbounded KK-theory, providing new insights into the structure of spectral triples and their gauge groups.

Contribution

It introduces a natural bundle formulation of gauge theories from spectral triples and extends the gauge group concept to include unitary endomorphisms, with detailed examples.

Findings

Unitary group of spectral triple as endomorphisms of a Hilbert bundle

Inner fluctuations split into connections and endomorphisms

Extended gauge group of unitary endomorphisms introduced

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.