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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 $E\to X$ 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 $M$ an A-module. There are different equivalent definitions of a connection on $M$ .^[2]

First definition

If $k\to A$ is a ring homomorphism, a $k$ -linear connection is a $k$ -linear morphism

\nabla :M\to \Omega _{A/k}^{1}\otimes _{A}M

which satisfies the identity

\nabla (am)=da\otimes m+a\nabla m

A connection extends, for all $p\geq 0$ to a unique map

\nabla :\Omega _{A/k}^{p}\otimes _{A}M\to \Omega _{A/k}^{p+1}\otimes _{A}M

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

Second definition

Let $D(A)$ be the module of derivations of a ring $A$ . A connection on an A-module $M$ is defined as an A-module morphism

\nabla :D(A)\to \mathrm {Diff} _{1}(M,M);u\mapsto \nabla _{u}

such that the first order differential operators $\nabla _{u}$ on $M$ obey the Leibniz rule

\nabla _{u}(ap)=u(a)p+a\nabla _{u}(p),\quad a\in A,\quad p\in M.

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 $M$ 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.

Notes

^ (Koszul 1950)
^ (Koszul 1950),(Mangiarotti & Sardanashvily 2000)
^ (Bartocci, Bruzzo & Hernández-Ruipérez 1991), (Mangiarotti & Sardanashvily 2000)
^ (Landi 1997)
^ (Dubois-Violette & Michor 1996),(Landi 1997)

References

Koszul, Jean-Louis (1950). "Homologie et cohomologie des algèbres de Lie" (PDF). Bulletin de la Société Mathématique de France. 78: 65–127. doi:10.24033/bsmf.1410.
Koszul, J. L. (1986). Lectures on Fibre Bundles and Differential Geometry (Tata University, Bombay, 1960). doi:10.1007/978-3-662-02503-1 (inactive 2024-04-26). ISBN 978-3-540-12876-2. S2CID 51020097. Zbl 0244.53026.{{cite book}}: CS1 maint: DOI inactive as of April 2024 (link)
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. arXiv:q-alg/9503020. doi:10.1016/0393-0440(95)00057-7. S2CID 15994413.
Landi, Giovanni (1997). An Introduction to Noncommutative Spaces and their Geometries. Lecture Notes in Physics. Vol. 51. arXiv:hep-th/9701078. doi:10.1007/3-540-14949-X. ISBN 978-3-540-63509-3. S2CID 14986502.
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". arXiv:0910.1515 [math-ph].

[1] (Koszul 1950)

[2] (Koszul 1950),(Mangiarotti & Sardanashvily 2000)

[3] (Bartocci, Bruzzo & Hernández-Ruipérez 1991), (Mangiarotti & Sardanashvily 2000)

[4] (Landi 1997)

[5] (Dubois-Violette & Michor 1996),(Landi 1997)

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