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 ${\displaystyle E\to X}$ written as a Koszul connection on the ${\displaystyle C^{\infty }(X)}$-module of sections of ${\displaystyle E\to X}$.¹

Let ${\displaystyle A}$ be a commutative ring and ${\displaystyle M}$ an A-module. There are different equivalent definitions of a connection on ${\displaystyle M}$.²

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

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

which satisfies the identity

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

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

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

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

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

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

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

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

Connections on a module over a commutative ring always exist.

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

${\displaystyle R(u,u')=[\nabla _{u},\nabla _{u'}]-\nabla _{[u,u']}\,}$

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

If ${\displaystyle E\to X}$ is a vector bundle, there is one-to-one correspondence between linear connections ${\displaystyle \Gamma }$ on ${\displaystyle E\to X}$ and the connections ${\displaystyle \nabla }$ on the ${\displaystyle C^{\infty }(X)}$-module of sections of ${\displaystyle E\to X}$. Strictly speaking, ${\displaystyle \nabla }$ corresponds to the covariant differential of a connection on ${\displaystyle E\to X}$.

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.

If ${\displaystyle A}$ 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 ${\displaystyle P}$ is defined as a bimodule morphism

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

which obeys the Leibniz rule

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