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 also studied in a more general setting by Costello to study 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 $M$ is a topological space, a prefactorization algebra ${\mathcal {F}}$ of vector spaces on $M$ is an assignment of vector spaces ${\mathcal {F}}(U)$ to open sets $U$ of $M$, along with the following conditions on the assignment:

• For each inclusion $U\subset V$, there's a linear map $m_{V}^{U}:{\mathcal {F}}(U)\rightarrow {\mathcal {F}}(V)$
• There is a linear map $m_{V}^{U_{1},\cdots ,U_{n}}:{\mathcal {F}}(U_{1})\otimes \cdots \otimes {\mathcal {F}}(U_{n})\rightarrow {\mathcal {F}}(V)$ for each finite collection of open sets with each $U_{i}\subset V$ and the $U_{i}$ pairwise disjoint.
• The maps compose in the obvious way: for collections of opens $U_{i,j}$, $V_{i}$ and an open $W$ satisfying $U_{i,1}\sqcup \cdots \sqcup U_{i,n_{i}}\subset V_{i}$ and $V_{1}\sqcup \cdots V_{n}\subset W$, the following diagram commutes.

${\begin{array}{lcl}&\bigotimes _{i}\bigotimes _{j}{\mathcal {F}}(U_{i,j})&\rightarrow &\bigotimes _{i}{\mathcal {F}}(V_{i})&\\&\downarrow &\swarrow &\\&{\mathcal {F}}(W)&&&\\\end{array}}$

So ${\mathcal {F}}$ 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 $U$ an open set, a collection of opens ${\mathfrak {U}}=\{U_{i}|i\in I\}$ is a Weiss cover of $U$ if for any finite collection of points $\{x_{1},\cdots ,x_{k}\}$ in $U$, there is an open set $U_{i}\in {\mathfrak {U}}$ such that $\{x_{1},\cdots ,x_{k}\}\subset U_{i}$.

Then a factorization algebra of vector spaces on $M$ is a prefactorization algebra of vector spaces on $M$ so that for every open $U$ and every Weiss cover $\{U_{i}|i\in I\}$ of $U$, the sequence $$\bigoplus _{i,j}{\mathcal {F}}(U_{i}\cap U_{j})\rightarrow \bigoplus _{k}{\mathcal {F}}(U_{k})\rightarrow {\mathcal {F}}(U)\rightarrow 0$$ is exact. That is, ${\mathcal {F}}$ 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 $U,V\subset M$, the structure map $$m_{U\sqcup V}^{U,V}:{\mathcal {F}}(U)\otimes {\mathcal {F}}(V)\rightarrow {\mathcal {F}}(U\sqcap V)$$ is an isomorphism.

See also

Vertex algebra