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Definition

Prefactorization algebras

A factorization algebra is a prefactorization algebra satisfying some properties, similar to sheafs being a presheaf with extra conditions.

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

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

⨂ i ⨂ j F ( U i , j ) → ⨂ i F ( V i ) ↓ ↙ F ( W ) {\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 F {\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 U {\displaystyle U} an open set, a collection of opens U = { U i | i ∈ I } {\displaystyle {\mathfrak {U}}=\{U_{i}|i\in I\}} is a Weiss cover of U {\displaystyle U} if for any finite collection of points { x 1 , ⋯ , x k } {\displaystyle \{x_{1},\cdots ,x_{k}\}} in U {\displaystyle U} , there is an open set U i ∈ U {\displaystyle U_{i}\in {\mathfrak {U}}} such that { x 1 , ⋯ , x k } ⊂ U i {\displaystyle \{x_{1},\cdots ,x_{k}\}\subset U_{i}} .

Then a factorization algebra of vector spaces on M {\displaystyle M} is a prefactorization algebra of vector spaces on M {\displaystyle M} so that for every open U {\displaystyle U} and every Weiss cover { U i | i ∈ I } {\displaystyle \{U_{i}|i\in I\}} of U {\displaystyle U} , the sequence ⨁ i , j F ( U i ∩ U j ) → ⨁ k F ( U k ) → F ( U ) → 0 {\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, F {\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 U , V ⊂ M {\displaystyle U,V\subset M} , the structure map m U ⊔ V U , V : F ( U ) ⊗ F ( V ) → F ( U ⊔ V ) {\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 X {\displaystyle X} be a smooth complex curve. A factorization algebra on X {\displaystyle X} consists of

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

j J / I ∗ V X , J → j J / I ∗ ( ⊠ i ∈ I V X , p − 1 ( i ) ) {\displaystyle j_{J/I}^{*}{\mathcal {V}}_{X,J}\rightarrow j_{J/I}^{*}(\boxtimes _{i\in I}{\mathcal {V}}_{X,p^{-1}(i)})} over U J / I {\displaystyle U^{J/I}} .

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

Example

Associative algebra

See also: associative algebra

Any associative algebra A {\displaystyle A} can be realized as a prefactorization algebra A f {\displaystyle A^{f}} on R {\displaystyle \mathbb {R} } . To each open interval ( a , b ) {\displaystyle (a,b)} , assign A f ( ( a , b ) ) = A {\displaystyle A^{f}((a,b))=A} . An arbitrary open is a disjoint union of countably many open intervals, U = ⨆ i I i {\displaystyle U=\bigsqcup _{i}I_{i}} , and then set A f ( U ) = ⨂ i A {\displaystyle A^{f}(U)=\bigotimes _{i}A} . The structure maps simply come from the multiplication map on A {\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. 978-0-8218-3528-9 ↩

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) 9781316678626 ↩