Factorization algebra explained

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]

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 l{F} of vector spaces on M is an assignment of vector spaces l{F}(U) to open sets U of M , along with the following conditions on the assignment:

• For each inclusion

U \subset V , there's a linear map m ^{U:} _{V} l{F}(U) → l{F}(V)

• There is a linear map

m ^{U _{1,} … , U _{n}} _{V} : l{F}(U _{1) ⊗} … ⊗ l{F}(U _{n)} → l{F}(V) for each finite collection of open sets with each U _{i} \subset V and the U _{i} pairwise disjoint.

• The maps compose in the obvious way: for collections of opens

U _{i,} , V _{i} and an open W satisfying U _{i,1} \sqcup … \sqcup U _{i, n _{i}} \subset V _{i} and V _{1} \sqcup … V _{n} \subset W , the following diagram commutes.\begin{array}{lcl} & o times _{i} o times _{j} l{F}(U _{i,j} ) & → & o times _{i} l{F}(V _{i)} & \\ & \downarrow & \swarrow & \\ & l{F}(W) & & & \\ \end{array}

So

l{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 ak{U} = \{U _{i} | i \in I\} is a Weiss cover of U if for any finite collection of points \{x _{1,} … , x _{k\}} in U , there is an open set U _{i} \in ak{U} such that \{x _{1,} … , x _{k\}} \subset U _{i} .

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 \{U _{i} | i \in I\} of U , the sequence$$\backslash bigoplus\_\; \backslash mathcal(U\_i\; \backslash cap\; U\_j)\; \backslash rightarrow\; \backslash bigoplus\_k\; \backslash mathcal(U\_k)\; \backslash rightarrow\; \backslash mathcal(U)\; \backslash rightarrow\; 0$$is exact. That is, l{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 \subset M , the structure map$$m^\_\; :\; \backslash mathcal(U)\backslash otimes\; \backslash mathcal(V)\; \backslash rightarrow\; \backslash mathcal(U\; \backslash sqcup\; V)$$is an isomorphism.

Algebro-geometric formulation

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

Let

X be a smooth complex curve. A factorization algebra on X consists ofl{V} _{X,} over X ^{I} for any finite set I , with no non-zero local section supported at the union of all partial diagonals

• Functorial isomorphisms of quasicoherent sheaves

\Delta ^{*} _{J/I} l{V} _{X,} → l{V} _{X,} over X ^{I} for surjections J → I .

• (Factorization) Functorial isomorphisms of quasicoherent sheaves

$$j^*\_\backslash mathcal\_\; \backslash rightarrow\; j^*\_(\backslash boxtimes\_\; \backslash mathcal\_)$$ over

U ^{J/I} .

• (Unit) Let

l{V} = l{V} _{X,} } and l{V} _{2} = l{V} _{X,} }. A global section (the unit) 1 \in l{V}(X) with the property that for every local section f \in l V(U) (U \subset X ), the section 1 \boxtimes f of l{V} _{2|} ^{U ^{2\Delta}} extends across the diagonal, and restricts to f \in l{V} \cong l{V} _{2|} _{\Delta} .

Example

Associative algebra

See also: associative algebra. Any associative algebra

A can be realized as a prefactorization algebra A ^{f} on R . To each open interval (a,b) , assign A ^{f((a,b))} = A . An arbitrary open is a disjoint union of countably many open intervals, U = s qcup _{i} I _{i} , and then set A ^{f(U)} = o times _{i} A . 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

• Vertex algebra

Notes and References

1. Book: Beilinson . Alexander . Drinfeld . Vladimir . Chiral algebras . 2004 . American Mathematical Society . Providence, R.I. . 978-0-8218-3528-9 . 21 February 2023.
2. Book: Costello . Kevin . Gwilliam . Owen . Factorization algebras in quantum field theory, Volume 1 . 2017 . Cambridge . 9781316678626.