Factorization algebra

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 and applied in a more general setting by Costello and Gwilliam to formalize quantum field theory.

Definition

Prefactorization algebras

A factorization algebra is a prefactorization algebra satisfying some properties, similar to sheafs being a presheaf with extra conditions.
If is a topological space, a prefactorization algebra of vector spaces on is an assignment of vector spaces to open sets 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.

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 over for any finite set, with no non-zero local section supported at the union of all partial diagonals
Functorial isomorphisms of quasicoherent sheaves over for surjections.
Functorial isomorphisms of quasicoherent sheaves

over.

Let and. A global section 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, 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.