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

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

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

∇ : M → Ω A / k 1 ⊗ A M {\displaystyle \nabla :M\to \Omega _{A/k}^{1}\otimes _{A}M}

which satisfies the identity

∇ ( a m ) = d a ⊗ m + a ∇ m {\displaystyle \nabla (am)=da\otimes m+a\nabla m}

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

∇ : Ω A / k p ⊗ A M → Ω A / k p + 1 ⊗ A M {\displaystyle \nabla :\Omega _{A/k}^{p}\otimes _{A}M\to \Omega _{A/k}^{p+1}\otimes _{A}M}

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

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

∇ : D ( A ) → D i f f 1 ( M , M ) ; u ↦ ∇ u {\displaystyle \nabla :D(A)\to \mathrm {Diff} _{1}(M,M);u\mapsto \nabla _{u}}

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

∇ u ( a p ) = u ( a ) p + a ∇ u ( p ) , a ∈ A , p ∈ M . {\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

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

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

If E → X {\displaystyle E\to X} is a vector bundle, there is one-to-one correspondence between linear connections Γ {\displaystyle \Gamma } on E → X {\displaystyle E\to X} and the connections ∇ {\displaystyle \nabla } on the C ∞ ( X ) {\displaystyle C^{\infty }(X)} -module of sections of E → X {\displaystyle E\to X} . Strictly speaking, ∇ {\displaystyle \nabla } corresponds to the covariant differential of a connection on E → X {\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 A {\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 P {\displaystyle P} is defined as a bimodule morphism

∇ : D ( A ) ∋ u → ∇ u ∈ D i f f 1 ( P , P ) {\displaystyle \nabla :D(A)\ni u\to \nabla _{u}\in \mathrm {Diff} _{1}(P,P)}

which obeys the Leibniz rule

∇ u ( a p b ) = u ( a ) p b + a ∇ u ( p ) b + a p u ( b ) , a ∈ R , b ∈ S , p ∈ P . {\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.}

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

• Sardanashvily, G. (2009). "Lectures on Differential Geometry of Modules and Rings". arXiv:0910.1515 [math-ph].