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 * -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 θ -deformation; in particular, they cover the θ -deformed quaternionic Hopf fibration C ∞ (S 7 θ ) ← C ∞ (S 4 θ ) as a noncommutative principal SU(2)-bundle. Contents

Geometric foundations for classical $\mathrm{U}(1)$-gauge theory on noncommutative manifolds

“…At last, we propose a refined notion of locally bounded commutator representation for κ-differentiable quantum principal U(1)-bundles with connection over B. When κ = 1, it reduces to a multigraded variation on a Dąbrowski-Sitarz's definition of principal U(1)-spectral triples [41] in the spirit of Ćaćić-Mesland [25]. supercommutes with π D (ϑ) and the remainder…”

“…The prototypical such construction is Connes-Rieffel's Yang-Mills gauge theory on irrational NC 2-tori [34], the first of many NC field theories built from a range of seemingly disparate variations on Connes's NC differential geometry [30,32]. Indeed, one can approach various aspects or special cases of NC U(1)-gauge theory in terms of quantum principal bundles [22,38], principal U(1)-spectral triples [41,19,25], or even the spectral action principle [42].…”

“…Fortunately, in our setting, we may obviate any resulting algebraic difficulties through the use of coherent 2-groups [7] and bar categories [12,43]. Moreover, following relevant applications of unbounded KK-theory [19,47,25], we obviate a wide range of analytic and algebraic difficulties through the systematic use of finite tight Parseval frames on (pre-)Hilbert modules [48].…”

An Atiyah Sequence for Noncommutative Principal Bundles

“…Additionally, noncommutative principal bundles are becoming increasingly prevalent in various applications of geometry (cf. [22,23,28,37,38]) and mathematical physics (see, e.g., [7,10,13,14,18,21,25,41] and references therein).…”

Curvature of differentiable Hilbert modules and Kasparov modules

“…Examples where curvature appears in the context of unbounded Kasparov theory is in the factorisation of Dirac operators on Riemannian submersions and G-spectral triples [7,9,29]. We will review and illustrate our notion of curvature for Riemannian submersions in Section 5.…”