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
written as a Koszul connection on the
be a commutative ring and
There are different equivalent definitions of a connection on
is a ring homomorphism, a
-linear morphism which satisfies the identity A connection extends, for all
to a unique map satisfying
A connection is said to be integrable if
, or equivalently, if the curvature
be the module of derivations of a ring
connection on an A-module
is defined as an A-module morphism such that the first order differential operators
obey the Leibniz rule Connections on a module over a commutative ring always exist.
is defined as the zero-order differential operator on the module
is a vector bundle, there is one-to-one correspondence between linear connections
Strictly speaking,
corresponds to the covariant differential of a connection on
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.
is a noncommutative ring, connections on left and right A-modules are defined similarly to those on modules over commutative rings.
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.
is defined as a bimodule morphism which obeys the Leibniz rule