Geometry of
quantum systems (e.g.,
noncommutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...
and
supergeometry
Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part and parcel of many classical and quantum field theories involving odd fields, e.g., SUSY field theory, ...
) is mainly
phrased in algebraic terms of
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
and
algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
. Connections on modules are
generalization of a linear
connection on a smooth
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
written as a
Koszul connection on the
-module of sections of
.
Commutative algebra
Let
be a commutative
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
and
an ''A''-
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
. There are different equivalent definitions
of a connection on
.
First definition
If
is a ring homomorphism, a
-linear connection is 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
vanishes.
Second definition
Let
be the module of
derivations
Derivation may refer to:
Language
* Morphological derivation, a word-formation process
* Parse tree or concrete syntax tree, representing a string's syntax in formal grammars
Law
* Derivative work, in copyright law
* Derivation proceeding, a proc ...
of a ring
. A
connection on an ''A''-module
is defined
as an ''A''-module morphism
:
such that the first order
differential operators
on
obey the Leibniz rule
:
Connections on a module over a commutative ring always exist.
The curvature of the connection
is defined as
the zero-order differential operator
:
on the module
for all
.
If
is a vector bundle, there is one-to-one
correspondence between
linear
connections on
and the
connections
on the
-module of sections of
. Strictly speaking,
corresponds to
the
covariant differential
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differ ...
of a
connection on
.
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
Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part and parcel of many classical and quantum field theories involving odd fields, e.g., SUSY field theory, ...
of
graded manifold
In algebraic geometry, graded manifolds are extensions of the concept of manifolds based on ideas coming from supersymmetry and supercommutative algebra. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commuta ...
s and
supervector bundles.
Superconnections always exist.
Noncommutative algebra
If
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 In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in t ...
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
is defined as a bimodule
morphism
:
which obeys the Leibniz rule
:
See also
*
Connection (vector bundle)
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The mo ...
*
Connection (mathematics) In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a ''parallel'' and consistent manner. There are var ...
*
Noncommutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...
*
Supergeometry
Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part and parcel of many classical and quantum field theories involving odd fields, e.g., SUSY field theory, ...
*
Differential calculus over commutative algebras In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this ...
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