Classical gauge theory on quantum principal bundles

Abstract

We propose a conceptually economical and computationally tractable completion of the foundations of gauge theory on quantum principal bundles à la Brzeziński--Majid to the case of general differential calculi and strong bimodule connections. In particular, we use explicit groupoid equivalences 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. We therefore construct gauge-equivariant moduli spaces of all suitable first- and second-order total differential calculi, respectively, compatible with these choices. As a first illustration, we relate the gauge theory of a crossed product algebra qua trivial quantum principal bundle to lazy Sweedler and Hochschild cohomology with coefficients. As a second illustration, we show that a noncommutative $2$-torus with real multiplication is the base space of a non-trivial $\mathrm{U}_{q^2}(1)$-gauge theory admitting Connes's constant curvature connection as a $q$-monopole connection, where -- in the spirit of Manin's Alterstraum -- one must take $q$ to be the norm-positive fundamental unit of the corresponding real quadratic field.