Geometric foundations for classical U(1)-gauge theory on noncommutative manifolds
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Date
2024-08-22
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Springer
Abstract
We systematically extend the elementary differential and Riemannian geometry of classical 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-C∗-algebra with ∗-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. Using this, we prove that a suitable Hermitian line bimodule with Hermitian bimodule connection yields an essentially unique differentiable quantum principal U(1)-bundle with principal connection and vice versa; here, U(1) is q-deformed for q a numerical invariant of the bimodule connection. Finally, 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. Thus, the spin Dirac spectral triple on quantum CP1 does not lift to a non-pathological twisted spectral triple on 3-dimensional quantum SU(2), but its formal lift nonetheless induces Kaad–Kyed’s compact quantum metric space on quantum SU(2) for a canonical choice of parameters