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 $C^{\mathrm{\infty }}(X)$-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

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

which satisfies the identity

$\mathrm{\nabla }(am)=da\otimes m+a\mathrm{\nabla }m$

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

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

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

$\mathrm{\nabla }:D(A)\to {\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_1(M,M);u\mapsto {\mathrm{\nabla }}_u$

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

${\mathrm{\nabla }}_u(ap)=u(a)p+a{\mathrm{\nabla }}_u(p),a\in A,p\in M.$

Connections on a module over a commutative ring always exist.

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

$R(u,u^{\prime })=[{\mathrm{\nabla }}_u,{\mathrm{\nabla }}_{u^{\prime }}]-{\mathrm{\nabla }}_{[u,u^{\prime }]}$

on the module $M$ for all $u,u^{\prime }\in D(A)$.

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 $\mathrm{\nabla }$ on the $C^{\mathrm{\infty }}(X)$-module of sections of $E\to X$. Strictly speaking, $\mathrm{\nabla }$ 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

$\mathrm{\nabla }:D(A)\ni u\to {\mathrm{\nabla }}_u\in {\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_1(P,P)$

which obeys the Leibniz rule

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

See also

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