Connection (algebraic framework)
Geometry of quantum systems (e.g., noncommutative geometry and supergeometry) is mainly phrased in algebraic terms of modules and algebras. Connections on modules are generalization of a linear connection on a smooth vector bundle written as a Koszul connection on the -module of sections of .
Let be a commutative ring and a -module. There are different equivalent definitions of a connection on . Let be the module of derivations of a ring . A connection on an -module is defined as an -module morphism
such that the first order differential operators on obey the Leibniz rule
Connections on a module over a commutative ring always exist.
The curvature of the connection is defined as the zero-order differential operator
on the module for all .
If is a vector bundle, there is one-to-one correspondence between linear connections on and the connections on the -module of sections of . Strictly speaking, corresponds to the covariant differential of a connection on .
Graded commutative algebra
The notion of a connection on modules over commutative rings is straightforwardly extended to modules over a graded commutative algebra. This is the case of superconnections in supergeometry of graded manifolds and supervector bundles. Superconnections always exist.
If is a noncommutative ring, connections on left and right -modules are defined similarly to those on modules over commutative rings. However these connections need not exist.
In contrast with connections on left and right modules, there is a problem how to define a connection on an -bimodule over noncommutative rings and . There are different definitions of such a connection. Let us mention one of them. A connection on an -bimodule is defined as a bimodule morphism
which obeys the Leibniz rule
- Connection (vector bundle)
- Connection (mathematics)
- Noncommutative geometry
- Differential calculus over commutative algebras
- Koszul (1950)
- Koszul (1950), Mangiarotti (2000)
- Bartocci (1991), Mangiarotti (2000)
- Landi (1997)
- Dubois-Violette (1996), Landi (1997)
- Koszul, J., Homologie et cohomologie des algebres de Lie,Bulletin de la Societe Mathematique 78 (1950) 65
- Koszul, J., Lectures on Fibre Bundles and Differential Geometry (Tata University, Bombay, 1960)
- Bartocci, C., Bruzzo, U., Hernandez Ruiperez, D., The Geometry of Supermanifolds (Kluwer Academic Publ., 1991) ISBN 0-7923-1440-9
- Dubois-Violette, M., Michor, P., Connections on central bimodules in noncommutative differential geometry, J. Geom. Phys. 20 (1996) 218. arXiv:q-alg/9503020v2
- Landi, G., An Introduction to Noncommutative Spaces and their Geometries, Lect. Notes Physics, New series m: Monographs, 51 (Springer, 1997) ArXiv eprint, iv+181 pages.
- Mangiarotti, L., Sardanashvily, G., Connections in Classical and Quantum Field Theory (World Scientific, 2000) ISBN 981-02-2013-8