- Connection (algebraic framework)
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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 .[1]
Contents
Commutative algebra
Let A be a commutative ring and P a A-module. There are different equivalent definitions of a connection on P.[2] Let D(A) be the module of derivations of a ring A. A connection on an A-module P is defined as an A-module morphism
such that the first order differential operators on P 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 P for all .
If is a vector bundle, there is one-to-one correspondence between linear connections Γ 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.[3] This is the case of superconnections in supergeometry of graded manifolds and supervector bundles. Superconnections always exist.
Noncommutative algebra
If A is a noncommutative ring, connections on left and right A-modules are defined similarly to those on modules over commutative rings.[4] 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 R − S-bimodule over noncommutative rings R and S. There are different definitions of such a connection.[5] Let us mention one of them. A connection on an R − S-bimodule P is defined as a bimodule morphism
which obeys the Leibniz rule
See also
- Connection (vector bundle)
- Noncommutative geometry
- Supergeometry
- Differential calculus over commutative algebras
Notes
References
- 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 0792314409
- Dubois-Violette, M., Michor, P., Connections on central bimodules in noncommutative differential geometry, J. Geom. Phys. 20 (1996) 218
- Landi, G., An Introduction to Noncommutative Spaces and their Geometries, Lect. Notes Physics, New series m: Monographs, 51 (Springer, 1997)
- Mangiarotti, L., Sardanashvily, G., Connections in Classical and Quantum Field Theory (World Scientific, 2000) ISBN 9810220138
External links
- Sardanashvily, G., Lectures on differential geometry of modules and rings, arXiv: 0910.1515
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