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

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

which satisfies the identity

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

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

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

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

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

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

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

Connections on a module over a commutative ring always exist.

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

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

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

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

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

which obeys the Leibniz rule

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

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

• Sardanashvily, G. (2009). Lectures on Differential Geometry of Modules and Rings. 

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