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 $E\to X$ written as a Koszul connection on the $C^\infty(X)$-module of sections of $E\to X$.^[1]

Commutative algebra

Let $A$ be a commutative ring and $P$ an 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

$\nabla:D(A)\ni u\to \nabla_u\in \mathrm{Diff}_1(P,P)$

such that the first order differential operators $\nabla_u$ on $P$ obey the Leibniz rule

$\nabla_u(ap)=u(a)p+a\nabla_u(p), \quad a\in A, \quad p\in P.$

Connections on a module over a commutative ring always exist.

The curvature of the connection $\nabla$ is defined as the zero-order differential operator

$R(u,u')=[\nabla_u,\nabla_{u'}]-\nabla_{[u,u']} \,$

on the module $P$ for all $u,u'\in D(A)$.

If $E\to X$ is a vector bundle, there is one-to-one correspondence between linear connections $\Gamma$ on $E\to X$ and the connections $\nabla$ on the $C^\infty(X)$-module of sections of $E\to X$. Strictly speaking, $\nabla$ corresponds to the covariant differential of a connection on $E\to X$.

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

$\nabla:D(A)\ni u\to \nabla_u\in \mathrm{Diff}_1(P,P)$

which obeys the Leibniz rule

$\nabla_u(apb)=u(a)pb+a\nabla_u(p)b +apu(b), \quad a\in R, \quad b\in S, \quad p\in P.$

See also

• Connection (vector bundle)
• Connection (mathematics)
• Noncommutative geometry
• Supergeometry
• Differential calculus over commutative algebras

Notes

1. Koszul (1950)
2. Koszul (1950), Mangiarotti (2000)
3. Bartocci (1991), Mangiarotti (2000)
4. Landi (1997)
5. Dubois-Violette (1996), Landi (1997)

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 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

External links

• Sardanashvily, G., Lectures on Differential Geometry of Modules and Rings (Lambert Academic Publishing, Saarbrücken, 2012); arXiv: 0910.1515