In mathematics and mathematical physics, 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,[1] and also studied in a more general setting by Costello and Gwilliam to study quantum field theory.
[2] A factorization algebra is a prefactorization algebra satisfying some properties, similar to sheafs being a presheaf with extra conditions.
is a topological space, a prefactorization algebra
of vector spaces on
is an assignment of vector spaces
to open sets
, along with the following conditions on the assignment:
{\displaystyle {\begin{array}{lcl}&\bigotimes _{i}\bigotimes _{j}{\mathcal {F}}(U_{i,j})&\rightarrow &\bigotimes _{i}{\mathcal {F}}(V_{i})&\\&\downarrow &\swarrow &\\&{\mathcal {F}}(W)&&&\\\end{array}}}
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.
To define factorization algebras, it is necessary to define a Weiss cover.
an open set, a collection of opens
is a Weiss cover of
if for any finite collection of points
, there is an open set
Then a factorization algebra of vector spaces on
is a prefactorization algebra of vector spaces on
and every Weiss cover
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
While this formulation is related to the one given above, the relation is not immediate.
be a smooth complex curve.
A factorization algebra on
Any associative algebra
can be realized as a prefactorization algebra
To each open interval
An arbitrary open is a disjoint union of countably many open intervals,
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.