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 and Gwilliam to study quantum field theory.[2]

Definition

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Prefactorization algebras

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

If   is a topological space, a prefactorization algebra   of vector spaces on   is an assignment of vector spaces   to open sets   of  , along with the following conditions on the assignment:

  • For each inclusion  , there's a linear map  
  • There is a linear map   for each finite collection of open sets with each   and the   pairwise disjoint.
  • The maps compose in the obvious way: for collections of opens  ,   and an open   satisfying   and  , the following diagram commutes.

 

So   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

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To define factorization algebras, it is necessary to define a Weiss cover. For   an open set, a collection of opens   is a Weiss cover of   if for any finite collection of points   in  , there is an open set   such that  .

Then a factorization algebra of vector spaces on   is a prefactorization algebra of vector spaces on   so that for every open   and every Weiss cover   of  , the sequence   is exact. That is,   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  , the structure map   is an isomorphism.

Algebro-geometric formulation

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

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

  • A quasicoherent sheaf   over   for any finite set  , with no non-zero local section supported at the union of all partial diagonals
  • Functorial isomorphisms of quasicoherent sheaves   over   for surjections  .
  • (Factorization) Functorial isomorphisms of quasicoherent sheaves

  over  .

  • (Unit) Let   and  . A global section (the unit)   with the property that for every local section   ( ), the section   of   extends across the diagonal, and restricts to  .

Example

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Associative algebra

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Any associative algebra   can be realized as a prefactorization algebra   on  . To each open interval  , assign  . An arbitrary open is a disjoint union of countably many open intervals,  , and then set  . The structure maps simply come from the multiplication map on  . Some care is needed for infinite tensor products, but for finitely many open intervals the picture is straightforward.

See also

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References

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  1. ^ Beilinson, Alexander; Drinfeld, Vladimir (2004). Chiral algebras. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-3528-9. Retrieved 21 February 2023.
  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)