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

Last revised on December 10, 2019 at 18:08:57. See the history of this page for a list of all contributions to it.