Factorization algebra

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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[1] and applied in a more general setting by Costello and Gwilliam to formalize quantum field theory.[2]

## Contents

* 1 Definition
* 1.1 Prefactorization algebras
* 1.2 Factorization algebras
* 1.3 Algebro-geometric formulation
* 2 Example
* 2.1 Associative algebra
* 3 See also
* 4 References

## 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⊂V, there's a linear map mVU:ℱ(U)->ℱ(V)
* There is a linear map mVU1,⋯,Un:ℱ(U1)⊗⋯⊗ℱ(Un)->ℱ(V) for each finite collection of open sets with each Ui⊂V and the Ui pairwise disjoint.
* The maps compose in the obvious way: for collections of opens Ui,j, Vi and an open W satisfying Ui,1⊔⋯⊔Ui,ni⊂Vi and V1⊔⋯Vn⊂W, the following diagram commutes.

⨂i⨂jℱ(Ui,j)->⨂iℱ(Vi)↓↙ℱ(W)

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 𝔘={Ui|i∈I} is a Weiss cover of U if for any finite collection of points {x1,⋯,xk} in U, there is an open set Ui∈𝔘 such that {x1,⋯,xk}⊂Ui.

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 {Ui|i∈I} of U, the sequence ⨁i,jℱ(Ui∩Uj)->⨁kℱ(Uk)->ℱ(U)->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⊂M, the structure map mU⊔VU,V:ℱ(U)⊗ℱ(V)->ℱ(U⊔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 𝒱X,I over XI for any finite set I, with no non-zero local section supported at the union of all partial diagonals
* Functorial isomorphisms of quasicoherent sheaves ΔJ/I*𝒱X,J->𝒱X,I over XI for surjections J->I.
* (Factorization) Functorial isomorphisms of quasicoherent sheaves

jJ/I*𝒱X,J->jJ/I*(⊠i∈I𝒱X,p−1(i)) over UJ/I.

* (Unit) Let 𝒱=𝒱X,{1} and 𝒱2=𝒱X,{1,2}. A global section (the unit) 1∈𝒱(X) with the property that for every local section f∈𝒱(U) (U⊂X), the section 1⊠f of 𝒱2|U2Δ extends across the diagonal, and restricts to f∈𝒱≅𝒱2|Δ.

## Example

### Associative algebra

Any associative algebra A can be realized as a prefactorization algebra Af on ℝ. To each open interval (a,b), assign Af((a,b))=A. An arbitrary open is a disjoint union of countably many open intervals, U=⨆iIi, and then set Af(U)=⨂iA. 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

* Algebraic quantum field theory
* Vertex algebra

## References

1. ↑ Beilinson, Alexander; Drinfeld, Vladimir (2004). Chiral algebras. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-3528-9. https://books.google.com/books?id=yHZh3p-kFqQC. Retrieved 21 February 2023.
2. ↑ Costello, Kevin; Gwilliam, Owen (2017). Factorization algebras in quantum field theory, Volume 1. Cambridge. ISBN 9781316678626.

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