2021/08/31 by Branimir Ćaćić, Ćaćić, Branimir
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2108.13789
openalex publication_date 2021/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Uq2(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.