Factorization algebra

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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.

If ${\displaystyle M}$ is a topological space, a prefactorization algebra ${\displaystyle {\mathcal {F}}}$ of vector spaces on ${\displaystyle M}$ is an assignment of vector spaces ${\displaystyle {\mathcal {F}}(U)}$ to open sets ${\displaystyle U}$ of ${\displaystyle M}$, along with the following conditions on the assignment:

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

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

So ${\displaystyle {\mathcal {F}}}$ 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.

To define factorization algebras, it is necessary to define a Weiss cover. For ${\displaystyle U}$ an open set, a collection of opens ${\displaystyle {\mathfrak {U}}=\{U_{i}|i\in I\}}$ is a Weiss cover of ${\displaystyle U}$ if for any finite collection of points ${\displaystyle \{x_{1},\cdots ,x_{k}\}}$ in ${\displaystyle U}$, there is an open set ${\displaystyle U_{i}\in {\mathfrak {U}}}$ such that ${\displaystyle \{x_{1},\cdots ,x_{k}\}\subset U_{i}}$.

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

While this formulation is related to the one given above, the relation is not immediate.

Let ${\displaystyle X}$ be a smooth complex curve. A factorization algebra on ${\displaystyle X}$ consists of

• A quasicoherent sheaf ${\displaystyle {\mathcal {V}}_{X,I}}$ over ${\displaystyle X^{I}}$ for any finite set ${\displaystyle I}$, with no non-zero local section supported at the union of all partial diagonals
• Functorial isomorphisms of quasicoherent sheaves ${\displaystyle \Delta _{J/I}^{*}{\mathcal {V}}_{X,J}\rightarrow {\mathcal {V}}_{X,I}}$ over ${\displaystyle X^{I}}$ for surjections ${\displaystyle J\rightarrow I}$.
• (Factorization) Functorial isomorphisms of quasicoherent sheaves

$${\displaystyle j_{J/I}^{*}{\mathcal {V}}_{X,J}\rightarrow j_{J/I}^{*}(\boxtimes _{i\in I}{\mathcal {V}}_{X,p^{-1}(i)})}$$ over ${\displaystyle U^{J/I}}$.

• (Unit) Let ${\displaystyle {\mathcal {V}}={\mathcal {V}}_{X,\{1\}}}$ and ${\displaystyle {\mathcal {V}}_{2}={\mathcal {V}}_{X,\{1,2\}}}$. A global section (the unit) ${\displaystyle 1\in {\mathcal {V}}(X)}$ with the property that for every local section ${\displaystyle f\in {\mathcal {V}}(U)}$ (${\displaystyle U\subset X}$), the section ${\displaystyle 1\boxtimes f}$ of ${\displaystyle {\mathcal {V}}_{2}|_{U^{2}\Delta }}$ extends across the diagonal, and restricts to ${\displaystyle f\in {\mathcal {V}}\cong {\mathcal {V}}_{2}|_{\Delta }}$.

Any associative algebra ${\displaystyle A}$ can be realized as a prefactorization algebra ${\displaystyle A^{f}}$ on ${\displaystyle \mathbb {R} }$. To each open interval ${\displaystyle (a,b)}$, assign ${\displaystyle A^{f}((a,b))=A}$. An arbitrary open is a disjoint union of countably many open intervals, ${\displaystyle U=\bigsqcup _{i}I_{i}}$, and then set ${\displaystyle A^{f}(U)=\bigotimes _{i}A}$. The structure maps simply come from the multiplication map on ${\displaystyle A}$. Some care is needed for infinite tensor products, but for finitely many open intervals the picture is straightforward.

• Vertex algebra