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 $E \to X$ .

Commutative algebra

Let $A$ be a commutative ring and $M$ an A-module. There are different equivalent definitions of a connection on $M$ .

First definition

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

\nabla : M \to Ω_{A / k}^{1} \otimes_{A} M

which satisfies the identity

\nabla (a m) = d a \otimes m + a \nabla m

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

\nabla : Ω_{A / k}^{p} \otimes_{A} M \to Ω_{A / k}^{p + 1} \otimes_{A} M

satisfying $\nabla (ω \otimes f) = d ω \otimes f + (- 1)^{p} ω \land \nabla f$ . A connection is said to be integrable if $\nabla \circ \nabla = 0$ , or equivalently, if the curvature $\nabla^{2} : M \to Ω_{A / k}^{2} \otimes M$ 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 {D i f f}_{1} (M, M); u \mapsto \nabla_{u}

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

\nabla_{u} (a p) = u (a) p + a \nabla_{u} (p), a \in A, 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 $Γ$ 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. 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. 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. Let us mention one of them. A connection on an R-S-bimodule $P$ is defined as a bimodule morphism

\nabla : D (A) ∋ u \to \nabla_{u} \in {D i f f}_{1} (P, P)

which obeys the Leibniz rule

\nabla_{u} (a p b) = u (a) p b + a \nabla_{u} (p) b + a p u (b), a \in R, b \in S, p \in P .

Notes

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 11 July 2025). ISBN 978-3-540-12876-2. S2CID 51020097. Zbl 0244.53026.{{cite book}}: CS1 maint: DOI inactive as of July 2025 (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. Bibcode:1996JGP....20..218D. 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)