Factorization algebra

Short description: Algebraic structure in mathematical physics

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 ${ℱ}$ of vector spaces on $M$ is an assignment of vector spaces ${ℱ}(U)$ 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 $m_V^U:{ℱ}(U)\to {ℱ}(V)$
• There is a linear map $m_V^{U_1,\cdots ,U_n}:{ℱ}(U_1)\otimes \cdots \otimes {ℱ}(U_n)\to {ℱ}(V)$ 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 $U_{i,1}\bigsqcup \cdots \bigsqcup U_{i,n_i}\subset V_i$ and $V_1\bigsqcup \cdots V_n\subset W$, the following diagram commutes.

$\begin{array}{lclll}& \underset{i}{⨂}\underset{j}{⨂}{ℱ}(U_{i,j})& \to & \underset{i}{⨂}{ℱ}(V_i)& \\ & \downarrow & \swarrow & \\ & {ℱ}(W)& & & \\ \end{array}$

So ${ℱ}$ 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 $$\underset{i,j}{⨁}{ℱ}(U_i\cap U_j)\to \underset{k}{⨁}{ℱ}(U_k)\to {ℱ}(U)\to 0$$ is exact. That is, ${ℱ}$ 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 $$m_{U\bigsqcup V}^{U,V}:{ℱ}(U)\otimes {ℱ}(V)\to {ℱ}(U\bigsqcup V)$$ is an isomorphism.

Algebro-geometric formulation

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

Let $X$ be a smooth complex curve. A factorization algebra on $X$ consists of

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

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

• (Unit) Let ${𝒱}={{𝒱}}_{X,\{1\}}$ and ${{𝒱}}_2={{𝒱}}_{X,\{1,2\}}$. A global section (the unit) $1\in {𝒱}(X)$ with the property that for every local section $f\in {𝒱}(U)$ ($U\subset X$), the section $1⊠f$ of ${{𝒱}}_2{|}_{U^2\mathrm{\Delta }}$ extends across the diagonal, and restricts to $f\in {𝒱}\cong {{𝒱}}_2{|}_{\mathrm{\Delta }}$.

Example

Associative algebra

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

See also

• Vertex algebra