NONCOMMUTATIVE RIEMANNIAN PRINCIPAL BUNDLES

In this talk, I will present a notion of principal G-spectral triple, with G a compact Lie group, put forward in my joint work with B. Ćaćić (New Brunswick). Our notion connects the algebraic approach to noncommutative principal bundles via principal comodule algebras and strong connections to the analytic framework of morphisms of spectral triples via unbounded KK-theory. In particular, any such a spectral triple admits a canonical factorisation in unbounded equivariant KK^G-theory. Up to a remainder, the total geometry is the twisting of the basic geometry by a noncommutative super-connection encoding the vertical geometry and underlying principal connection. Such factorisations contain refined geometric information, such as curvature, and are compatible with index theory. This approach leads to a noncommutative gauge theory that  explicitly generalises the classical gauge theory and respects the aforementioned KK-factorisation. Our definitions cover all classical locally compact principal G-bundles, and are compatible with θ-deformations.