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Inmathematics andmathematical physics, afactorization algebra is an algebraic structure first introduced byBeilinson andDrinfel'd in analgebro-geometric setting as a reformulation ofchiral algebras^[1] and applied in a more general setting byCostello and Gwilliam to formalizequantum field theory.^[2]

A factorization algebra is a prefactorization algebra satisfying some properties, similar tosheafs being apresheaf with extra conditions.

If${\displaystyle M}$ is atopological space, aprefactorization algebra${\displaystyle \mathcal{F}}$ ofvector spaces on${\displaystyle M}$ is an assignment of vector spaces${\displaystyle \mathcal{F}(U)}$ toopen sets${\displaystyle U}$ of${\displaystyle M}$, along with the following conditions on the assignment:

• For each inclusion${\displaystyle U\subset V}$, there's alinear map${\displaystyle m_V^U:\mathcal{F}(U)\to \mathcal{F}(V)}$
• There is a linear map${\displaystyle m_V^{U_1,\cdots ,U_n}:\mathcal{F}(U_1)\otimes \cdots \otimes \mathcal{F}(U_n)\to \mathcal{F}(V)}$ 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${\displaystyle U_{i,1}\bigsqcup \cdots \bigsqcup U_{i,n_i}\subset V_i}$ and${\displaystyle V_1\bigsqcup \cdots V_n\subset W}$, the following diagram commutes.

So${\displaystyle \mathcal{F}}$ resembles aprecosheaf, except the vector spaces aretensored rather than(direct-)summed.

The category of vector spaces can be replaced with anysymmetric monoidal category.

To define factorization algebras, it is necessary to define a Weisscover. For${\displaystyle U}$ an open set, a collection of opens${\displaystyle \mathfrak{U}=\{U_i|i\in I\}}$ is aWeiss 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 afactorization 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$${\displaystyle \underset{i,j}{⨁}\mathcal{F}(U_i\cap U_j)\to \underset{k}{⨁}\mathcal{F}(U_k)\to \mathcal{F}(U)\to 0}$$isexact. That is,${\displaystyle \mathcal{F}}$ is a factorization algebra if it is a cosheaf with respect to the Weiss topology.

A factorization algebra ismultiplicative if, in addition, for each pair of disjoint opens${\displaystyle U,V\subset M}$, the structure map$${\displaystyle m_{U\bigsqcup V}^{U,V}:\mathcal{F}(U)\otimes \mathcal{F}(V)\to \mathcal{F}(U\bigsqcup V)}$$is an isomorphism.

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

Let${\displaystyle X}$ be asmoothcomplex curve. Afactorization algebra on${\displaystyle X}$ consists of

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

$${\displaystyle j_{J/I}^{\ast }{\mathcal{V}}_{X,J}\to j_{J/I}^{\ast }({⊠}_{i\in I}{\mathcal{V}}_{X,p^{-1}(i)})}$$ 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 (theunit)${\displaystyle 1\in \mathcal{V}(X)}$ with the property that for every local section${\displaystyle f\in \mathcal{V}(U)}$ (${\displaystyle U\subset X}$), the section${\displaystyle 1⊠f}$ of${\displaystyle {\mathcal{V}}_2{|}_{U^2\mathrm{\Delta }}}$ extends across the diagonal, and restricts to${\displaystyle f\in \mathcal{V}\cong {\mathcal{V}}_2{|}_{\mathrm{\Delta }}}$.

Any associative algebra${\displaystyle A}$ can be realized as a prefactorization algebra${\displaystyle A^f}$ on${\displaystyle \mathbb{R}}$. To eachopen 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.

• Algebraic quantum field theory
• Vertex algebra

• Abstract algebra

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