Gauge theory on noncommutative Riemannian principal bundles

Abstract

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 $G$-$C^{\ast }$-algebras and prove that the resulting noncommutative orbitwise family of Kostant's cubic Dirac operators defines a natural unbounded $K{K}^G$-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 $K{K}^G$-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 $C^{\infty }(S_{\theta }^7)\hookleftarrow C^{\infty }(S_{\theta }^4)$ as a noncommutative principal $\mathrm{SU}(2)$-bundle.

Final version to appear in Commun. Math. Phys. encompassing various clarifications and corrections including thorough revisions of Prop. 2.35, Prop. 2.36, and Lemma 2.45 and a correction to Def. B.2. The authors thank the anonymous reviewers for their extraordinarily thoughtful, thorough, and useful feedback

References in corpus (3)

• Connes-Landi Deformation of Spectral Triples
• On globally non-trivial almost-commutative manifolds
• Homotopy equivalence in unbounded KK-theory

Cited by in corpus (6)

• Gysin sequences and SU(2)-symmetries of C*-algebras
• Geometric foundations for classical $\mathrm{U}(1)$-gauge theory on noncommutative manifolds
• A Note on Twisted Crossed Products and Spectral Triples
• Cubic Dirac operator for $U_q({\mathfrak{sl}}_2)$
• An Atiyah Sequence for Noncommutative Principal Bundles
• Realizations of free actions via their fixed point algebras