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,^[1] and also studied in a more general setting by Costello 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 ${\displaystyle {ℱ}}$ of vector spaces on ${\displaystyle M}$ is an assignment of vector spaces ${\displaystyle {ℱ}(U)}$ 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 ${\displaystyle m_V^U:{ℱ}(U)\to {ℱ}(V)}$
• There is a linear map ${\displaystyle m_V^{U_1,\cdots ,U_n}:{ℱ}(U_1)\otimes \cdots \otimes {ℱ}(U_n)\to {ℱ}(V)}$ for each finite collection of open sets with each ${\displaystyle U_i\subset V}$ 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 ${\displaystyle U_{i,1}\bigsqcup \cdots \bigsqcup U_{i,n_i}\subset V_i}$ and ${\displaystyle V_1\bigsqcup \cdots V_n\subset W}$, the following diagram commutes.

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

So ${\displaystyle {ℱ}}$ 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 $${\displaystyle \underset{i,j}{⨁}{ℱ}(U_i\cap U_j)\to \underset{k}{⨁}{ℱ}(U_k)\to {ℱ}(U)\to 0}$$ is exact. That is, ${\displaystyle {ℱ}}$ 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 $${\displaystyle m_{U\bigsqcup V}^{U,V}:{ℱ}(U)\otimes {ℱ}(V)\to {ℱ}(U\bigsqcup V)}$$ 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 ${\displaystyle {{𝒱}}_{X,I}}$ 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 ${\displaystyle {\mathrm{\Delta }}_{J/I}^{*}{{𝒱}}_{X,J}\to {{𝒱}}_{X,I}}$ over ${\displaystyle X^I}$ for surjections ${\displaystyle J\to I}$.
• (Factorization) Functorial isomorphisms of quasicoherent sheaves

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

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

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, ${\displaystyle U=\underset{i}{⨆}{I}_i}$, and then set ${\displaystyle A^f(U)=\underset{i}{⨂}A}$. 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

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Factorization algebra. Read more