nLab connection in noncommutative geometry

In noncommutative geometry there are several versions of noncommutative bundle theory, e.g. considering vector bundles as finitely generated projective modules and the theory of noncommutative principal bundles as Hopf-Galois extensions and their coalgebra and global analogues. Each of these formalism can be a setup in whcih one can try to develop a noncommutative analogue of the theory of connections on a bundle. The connection theory on noncommutative spaces is of course, the basis of gauge theories on noncommutative spaces. A remarkable distinction between the commutative and noncommutative case of connections on Hopf-Galois extensions is the difference between generic connection on a generic and so-called strong connections, as discovered by P. Hajac. There is an approach close to Koszul’s for dga-s, by defining the connections by action on noncommutative differential forms of Karoubi or by an appropriate analogue in cyclic homology.

There is a rather vast literature on the subject and we should list the more important ones.

• Max Karoubi, Connexions, courbures et classes caractéristiques en K-théorie algébrique, Current trends in algebraic topology, Part I, vol. 2, 19-27, London, Ontario 1981, pdf
• Alain Connes, Noncommutative geometry, Academic Press 1994, 661 p. PDF
• Tomasz Brzeziński, Flat connections and (co)modules, math/0608170
• Tomasz Brzeziński, A note on flat noncommutative connections, arxiv/1109.0858
• Tomasz Brzeziński, Shahn Majid, Quantum group gauge theory on quantum spaces, Commun. Math. Phys. 157:591-638, 1993, hep-th/9208007, doi; Erratum 167:235, 1995,
• Piotr M. Hajac, Strong connections on quantum principal bundles, Comm. Math. Phys. 182 (1996), no. 3, 579–617, MR98e:58022, euclid
• S. Majid, Quantum and braided group Riemannian geometry, J. Geom. Phys. 30 (1999) 113-146, q-alg/9709025
• M. Dubois-Violette, P. Michor, Connections on central bimodules in noncommutative differential geometry, J. Geom. Phys. 20 (1996) 218
• M. Dubois-Violette, J. Madore, T. Masson, J. Mourad, On curvature in noncommutative geometry, q-alg/9512004

In framework of spectral triples, see

• Branimir Ćaćić, Bram Mesland, Gauge theory on noncommutative Riemannian principal bundles, arxiv/1912.04179