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Connection (algebraic framework)

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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 written as a Koszul connection on the C ∞ ( X ) -module of sections of E → X .

Contents

Commutative algebra

Let A be a commutative ring and P an A-module. There are different equivalent definitions of a connection on P . Let D ( A ) be the module of derivations of a ring A . A connection on an A-module P is defined as an A-module morphism

∇ : D ( A ) ∋ u → ∇ u ∈ D i f f 1 ( P , P )

such that the first order differential operators ∇ u on P obey the Leibniz rule

∇ u ( a p ) = u ( a ) p + a ∇ u ( p ) , a ∈ A , p ∈ P .

Connections on a module over a commutative ring always exist.

The curvature of the connection ∇ is defined as the zero-order differential operator

R ( u , u ′ ) = [ ∇ u , ∇ u ′ ] − ∇ [ u , u ′ ]

on the module P for all u , u ′ ∈ D ( A ) .

If E → X is a vector bundle, there is one-to-one correspondence between linear connections Γ on E → X and the connections ∇ on the C ∞ ( X ) -module of sections of E → X . Strictly speaking, ∇ corresponds to the covariant differential of a connection on E → 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

∇ : D ( A ) ∋ u → ∇ u ∈ D i f f 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 .

References

Connection (algebraic framework) Wikipedia
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