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Connection (algebraic framework)

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Source updated September 4, 2025

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 $E\to X$ .

01Commutative 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

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

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 $E\to X$ .

02Graded 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.

03Noncommutative 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

$\nabla :D(A)\ni u\to \nabla _{u}\in \mathrm {Diff} _{1}(P,P)$

which obeys the Leibniz rule

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

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Connection (algebraic framework)

01Commutative algebra

First definition

Second definition

02Graded commutative algebra

03Noncommutative algebra

Sources and credits

Related topics

Connection (vector bundle)

Connection (mathematics)

Noncommutative geometry