In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and
mathematical physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
, a factorization algebra is an algebraic structure first introduced by
Beilinson and
Drinfel'd in an
algebro-geometric setting as a reformulation of
chiral algebras,
and also studied in a more general setting by
Costello and Gwilliam to study
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
.
Definition
Prefactorization algebras
A factorization algebra is a prefactorization algebra satisfying some properties, similar to
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics)
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open s ...
s being a
presheaf
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the d ...
with extra conditions.
If
is a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, a prefactorization algebra
of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s on
is an assignment of vector spaces
to
open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s
of
, along with the following conditions on the assignment:
* For each inclusion
, there's a
linear map
* There is a linear map
for each finite collection of open sets with each
and the
pairwise disjoint.
* The maps compose in the obvious way: for collections of opens
,
and an open
satisfying
and
, the following diagram commutes.
So
resembles a
precosheaf, except the vector spaces are
tensored rather than
(direct-)summed.
The category of vector spaces can be replaced with any
symmetric monoidal category In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict sen ...
.
Factorization algebras
To define factorization algebras, it is necessary to define a Weiss
cover. For
an open set, a collection of opens
is a Weiss cover of
if for any finite collection of points
in
, there is an open set
such that
.
Then a factorization algebra of vector spaces on
is a prefactorization algebra of vector spaces on
so that for every open
and every Weiss cover
of
, the sequence
is
exact. That is,
is a factorization algebra if it is a cosheaf with respect to the Weiss topology.
A factorization algebra is ''multiplicative'' if, in addition, for each pair of disjoint opens
, the structure map
is an isomorphism.
Algebro-geometric formulation
While this formulation is related to the one given above, the relation is not immediate.
Let
be a
smooth complex curve. A factorization algebra on
consists of
* A
quasicoherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
over
for any finite set
, with no non-zero local
section
Section, Sectioning, or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
supported at the union of all partial diagonals
*
Functorial isomorphisms of quasicoherent sheaves
over
for surjections
.
* (''Factorization'') Functorial isomorphisms of quasicoherent sheaves
over
.
* (''Unit'') Let
and
. A global section (the ''unit'')
with the property that for every local section
(
), the section
of
extends across the diagonal, and restricts to
.
Example
Associative algebra
Any associative algebra
can be realized as a prefactorization algebra
on
. To each
open interval
In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
, assign
. An arbitrary open is a disjoint union of countably many open intervals,
, and then set
. The structure maps simply come from the multiplication map on
. Some care is needed for infinite tensor products, but for finitely many open intervals the picture is straightforward.
See also
*
Vertex algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven usef ...
References
{{reflist
Abstract algebra