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

which satisfies the identity
 * $$ \nabla(am) = da \otimes m + a \nabla m $$

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


 * $$\nabla: \Omega^p_{A/k} \otimes_A M \to \Omega^{p+1}_{A/k} \otimes_A M$$

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. 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


 * $$ \nabla:D(A)\ni u\to \nabla_u\in \mathrm{Diff}_1(P,P)$$

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.$$