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

Commutative algebra

Let [math]\displaystyle{ A }[/math] be a commutative ring and [math]\displaystyle{ M }[/math] an A-module. There are different equivalent definitions of a connection on [math]\displaystyle{ M }[/math].^[2]

First definition

If [math]\displaystyle{ k \to A }[/math] is a ring homomorphism, a [math]\displaystyle{ k }[/math]-linear connection is a [math]\displaystyle{ k }[/math]-linear morphism

[math]\displaystyle{ \nabla: M \to \Omega^1_{A/k} \otimes_A M }[/math]

which satisfies the identity

[math]\displaystyle{ \nabla(am) = da \otimes m + a \nabla m }[/math]

A connection extends, for all [math]\displaystyle{ p \geq 0 }[/math] to a unique map

[math]\displaystyle{ \nabla: \Omega^p_{A/k} \otimes_A M \to \Omega^{p+1}_{A/k} \otimes_A M }[/math]

satisfying [math]\displaystyle{ \nabla(\omega \otimes f) = d\omega \otimes f + (-1)^p \omega \wedge \nabla f }[/math]. A connection is said to be integrable if [math]\displaystyle{ \nabla \circ \nabla = 0 }[/math], or equivalently, if the curvature [math]\displaystyle{ \nabla^2: M \to \Omega_{A/k}^2 \otimes M }[/math] vanishes.

Second definition

Let [math]\displaystyle{ D(A) }[/math] be the module of derivations of a ring [math]\displaystyle{ A }[/math]. A connection on an A-module [math]\displaystyle{ M }[/math] is defined as an A-module morphism

[math]\displaystyle{ \nabla:D(A) \to \mathrm{Diff}_1(M,M); u \mapsto \nabla_u }[/math]

such that the first order differential operators [math]\displaystyle{ \nabla_u }[/math] on [math]\displaystyle{ M }[/math] obey the Leibniz rule

[math]\displaystyle{ \nabla_u(ap)=u(a)p+a\nabla_u(p), \quad a\in A, \quad p\in M. }[/math]

Connections on a module over a commutative ring always exist.

The curvature of the connection [math]\displaystyle{ \nabla }[/math] is defined as the zero-order differential operator

[math]\displaystyle{ R(u,u')=[\nabla_u,\nabla_{u'}]-\nabla_{[u,u']} \, }[/math]

on the module [math]\displaystyle{ M }[/math] for all [math]\displaystyle{ u,u'\in D(A) }[/math].

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

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 [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ P }[/math] is defined as a bimodule morphism

[math]\displaystyle{ \nabla:D(A)\ni u\to \nabla_u\in \mathrm{Diff}_1(P,P) }[/math]

which obeys the Leibniz rule

[math]\displaystyle{ \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. }[/math]

Notes

↑ (Koszul 1950)
↑ (Koszul 1950),(Mangiarotti Sardanashvily)
↑ (Bartocci Bruzzo), (Mangiarotti Sardanashvily)
↑ (Landi 1997)
↑ (Dubois-Violette Michor),(Landi 1997)

References

Koszul, Jean-Louis (1950). "Homologie et cohomologie des algèbres de Lie". Bulletin de la Société Mathématique de France 78: 65–127. doi:10.24033/bsmf.1410. http://www.numdam.org/item/10.24033/bsmf.1410.pdf.
Koszul, J. L. (1986). Lectures on Fibre Bundles and Differential Geometry (Tata University, Bombay, 1960). doi:10.1007/978-3-662-02503-1. ISBN 978-3-540-12876-2.
Bartocci, Claudio; Bruzzo, Ugo; Hernández-Ruipérez, Daniel (1991). The Geometry of Supermanifolds. doi:10.1007/978-94-011-3504-7. ISBN 978-94-010-5550-5.
Dubois-Violette, Michel; Michor, Peter W. (1996). "Connections on central bimodules in noncommutative differential geometry". Journal of Geometry and Physics 20 (2–3): 218–232. doi:10.1016/0393-0440(95)00057-7.
Landi, Giovanni (1997). An Introduction to Noncommutative Spaces and their Geometries. Lecture Notes in Physics. 51. doi:10.1007/3-540-14949-X. ISBN 978-3-540-63509-3.
Mangiarotti, L.; Sardanashvily, G. (2000). Connections in Classical and Quantum Field Theory. doi:10.1142/2524. ISBN 978-981-02-2013-6.

External links

Sardanashvily, G. (2009). Lectures on Differential Geometry of Modules and Rings.

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[1] (Koszul 1950)

[2] (Koszul 1950),(Mangiarotti Sardanashvily)

[3] (Bartocci Bruzzo), (Mangiarotti Sardanashvily)

[4] (Landi 1997)

[5] (Dubois-Violette Michor),(Landi 1997)