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

Commutative algebra

Let $A$ be a commutative ring
and $M$ an ''A''-module. There are different equivalent definitions
of a connection on $M$.

First definition

If $k \to A$ is a ring homomorphism, a $k$-linear connection is a $k$-linear morphism
: $\nabla: M \to \Omega^1_ \otimes_A M$
which satisfies the identity
: $\nabla(am) = da \otimes m + a \nabla m$
A connection extends, for all $p \geq 0$ to a unique map

: $\nabla: \Omega^p_ \otimes_A M \to \Omega^_ \otimes_A M$

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

Second definition

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

: $\nabla:D(A) \to \mathrm_1(M,M); u \mapsto \nabla_u$

such that the first order differential operators $\nabla_u$ on
$M$ obey the Leibniz rule

: $\nabla_u(ap)=u(a)p+a\nabla_u(p), \quad a\in A, \quad p\in M.$

Connections on a module over a commutative ring always exist.

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

: R(u,u')= nabla_u,\nabla_\nabla_ \,

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

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

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.Dubois-Violette
(1996), Landi (1997) Let us mention one of them. A connection on an
''R''-''S''-bimodule $P$ is defined as a bimodule
morphism

: $\nabla:D(A)\ni u\to \nabla_u\in \mathrm_1(P,P)$

which obeys the Leibniz rule

: $\nabla_u(apb)=u(a)pb+a\nabla_u(p)b +apu(b), \quad a\in R, \quad b\in S, \quad p\in P.$

See also

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

Notes

References

* Koszul, J., Homologie et cohomologie des algebres de Lie,''Bulletin de la Société Mathématique'' 78 (1950) 65
* Koszul, J., ''Lectures on Fibre Bundles and Differential Geometry'' (Tata University, Bombay, 1960)
* Bartocci, C., Bruzzo, U., Hernandez Ruiperez, D., ''The Geometry of Supermanifolds'' (Kluwer Academic Publ., 1991)
* Dubois-Violette, M., Michor, P., Connections on central bimodules in noncommutative differential geometry, ''J. Geom. Phys.'' 20 (1996) 218.
* Landi, G., ''An Introduction to Noncommutative Spaces and their Geometries'', Lect. Notes Physics, New series m: Monographs, 51 (Springer, 1997) , iv+181 pages.
* Mangiarotti, L., Sardanashvily, G., ''Connections in Classical and Quantum Field Theory'' (World Scientific, 2000)

External links

* Sardanashvily, G., ''Lectures on Differential Geometry of Modules and Rings'' (Lambert Academic Publishing, Saarbrücken, 2012); {{arXiv, 0910.1515

Connection (mathematics)
Noncommutative geometry