ROOM THREE: 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 these induce 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. 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 θ-deformation; in particular, they cover the θ-deformed quaternionic Hopf fibration as a noncommutative principal SU(2)-bundle.
This is joint work with Bram Mesland. This work was supported by NSERC Grant RGPIN-2017-04249.