Gauge theory on noncommutative Riemannian principal bundles

We present a new, general approach to gauge theory on principal G-spectral triples, where G is a compact connected Lie group. We introduce a notion of vertical Riemannian geometry for G-C*-algebras and prove that the resulting noncommutative orbitwise family of Kostant's cubic Dirac operators defines a natural unbounded KKG-cycle in the case of a principal G-action. Then, we introduce a notion of principal G-spectral triple and prove, in particular, that any such spectral triple admits a canonical factorisation in unbounded KKG-theory with respect to such a cycle: up to a remainder, the total geometry is the twisting of the basic geometry by a noncommutative superconnection encoding the vertical geometry and underlying principal connection. Using these notions, we formulate an approach to gauge theory that explicitly generalises the classical case up to a groupoid cocycle and is compatible in general with this factorisation; in the unital case, it correctly yields a real affine space of noncommutative principal connections with affine gauge action. Our definitions cover all locally compact classical principal G-bundles and are compatible with theta-deformation; in particular, they cover the theta-deformed quaternionic Hopf fibration C-infinity(S-theta(7))hooked left arrow C-infinity(S-theta(4)) as a noncommutative principal SU(2)-bundle.Analysis and Stochastic

A reconstruction theorem for almost-commutative spectral triples

We propose an expansion of the definition of almost-commutative spectral triple that accommodates non-trivial fibrations and is stable under inner fluctuation of the metric, and then prove a reconstruction theorem for almost-commutative spectral triples under this definition as a simple consequence of Connes's reconstruction theorem for commutative spectral triples. Along the way, we weaken the orientability hypothesis in the reconstruction theorem for commutative spectral triples, and following Chakraborty and Mathai, prove a number of results concerning the stability of properties of spectral triples under suitable perturbation of the Dirac operator.Comment: AMS-LaTeX, 19 pp. V4: Updated version incorporating the erratum of June 2012, correcting the weak orientability axiom in the definition of commutative spectral triple, stengthening Lemma A.10 to cover the odd-dimensional case and the proof of Corollary 2.19 to accommodate the corrected weak orientability axio