Branimir Cacic
(University of New Brunswick)
October 20, 2021
Organized by Europe GNCG Seminar.

Classical gauge theory on quantum principal bundles

In his very first note on NC differential geometry, Connes introduced constant curvature connections on Heisenberg bimodules over an irrational NC $2$-torus $\mathcal{A}_\theta$. When $\theta$ is a quadratic irrationality, these Heisenberg bimodules include non-trivial NC line bundles—what, then, is the underlying $U(1)$-gauge theory? We use this case study to demonstrate how approaches to non-universal quantum principal bundles introduced by Brzeziński–Majid and Đurđević, respectively, can be fruitfully synthesized to reframe the basic concepts of gauge theory—gauge transformation, gauge potential, and field strength — in terms of reconstruction of calculi on the total space (to second order) from given calculi on the structure quantum group and base, respectively. In particular, we obtain gauge-equivariant moduli spaces of all suitable first- and second-order total differential calculi, respectively, compatible with these choices.

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