Misplaced Pages

This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (February 2023) (Learn how and when to remove this message)

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 and Gwilliam 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 ${\displaystyle M}$ 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.

Factorization algebras

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.

Algebro-geometric formulation

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 ${\displaystyle f\in {\mathcal {V}}\cong {\mathcal {V}}_{2}|_{\Delta }}$.

Example

Associative algebra

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.

See also

• Vertex algebra

References

1. Beilinson, Alexander; Drinfeld, Vladimir (2004). Chiral algebras. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-3528-9. Retrieved 21 February 2023.
2. Costello, Kevin; Gwilliam, Owen (2017). Factorization algebras in quantum field theory, Volume 1. Cambridge. ISBN 9781316678626.{{cite book}}: CS1 maint: location missing publisher (link)

• Abstract algebra