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

Abstract: We systematically extend the elementary differential and Riemannian geometry of classical $\mathrm{U}(1)$-gauge theory to the noncommutative setting by combining recent advances in noncommutative Riemannian geometry with the theory of coherent $2$-groups. We show that Hermitian line bimodules with Hermitian bimodule connection over a unital pre-$\mathrm{C}^\ast$-algebra with $\ast$-exterior algebra form a coherent $2$-group, and we prove that weak monoidal functors between coherent $2$-groups canonically define bar or involutive monoidal functors in the sense of Beggs--Majid and Egger, respectively. Hence, we prove that a suitable Hermitian line bimodule with Hermitian bimodule connection yields an essentially unique differentiable quantum principal $\mathrm{U}(1)$-bundle with principal connection and vice versa; here, $\mathrm{U}(1)$ is $q$-deformed for $q$ a numerical invariant of the bimodule connection. From there, we formulate and solve the interrelated lifting problems for noncommutative Riemannian structure in terms of abstract Hodge star operators and formal spectral triples, respectively; all the while, we account precisely for emergent modular phenomena of geometric nature. In particular, it follows that the spin Dirac spectral triple on quantum $\mathbf{C}\mathrm{P}^1$ does not lift to a twisted spectral triple on $3$-dimensional quantum $\mathrm{SU}(2)$ with the $3$-dimensional calculus but does recover Kaad--Kyed's compact quantum metric space on quantum $\mathrm{SU}(2)$ for a canonical choice of parameters.