Factorization algebra

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, and also studied in a more general setting by Costello to study quantum field theory.

Definition

Prefactorization algebras

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

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

• For each inclusion [math]\displaystyle{ U \subset V }[/math], there's a linear map [math]\displaystyle{ m_V^U: \mathcal{F}(U) \rightarrow \mathcal{F}(V) }[/math]
• There is a linear map [math]\displaystyle{ m_V^{U_1, \cdots, U_n}: \mathcal{F}(U_1)\otimes \cdots \otimes \mathcal{F}(U_n) \rightarrow \mathcal{F}(V) }[/math] for each finite collection of open sets with each [math]\displaystyle{ U_i \subset V }[/math] and the [math]\displaystyle{ U_i }[/math] pairwise disjoint.
• The maps compose in the obvious way: for collections of opens [math]\displaystyle{ U_{i, j} }[/math], [math]\displaystyle{ V_i }[/math] and an open [math]\displaystyle{ W }[/math] satisfying [math]\displaystyle{ U_{i,1}\sqcup \cdots \sqcup U_{i, n_i} \subset V_i }[/math] and [math]\displaystyle{ V_1 \sqcup \cdots V_n \subset W }[/math], the following diagram commutes.

[math]\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} }[/math]

So [math]\displaystyle{ \mathcal{F} }[/math] 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 [math]\displaystyle{ U }[/math] an open set, a collection of opens [math]\displaystyle{ \mathfrak{U} = \{U_i | i \in I\} }[/math] is a Weiss cover of [math]\displaystyle{ U }[/math] if for any finite collection of points [math]\displaystyle{ \{x_1, \cdots, x_k\} }[/math] in [math]\displaystyle{ U }[/math], there is an open set [math]\displaystyle{ U_i \in \mathfrak{U} }[/math] such that [math]\displaystyle{ \{x_1, \cdots, x_k\} \subset U_i }[/math].

Then a factorization algebra of vector spaces on [math]\displaystyle{ M }[/math] is a prefactorization algebra of vector spaces on [math]\displaystyle{ M }[/math] so that for every open [math]\displaystyle{ U }[/math] and every Weiss cover [math]\displaystyle{ \{U_i | i \in I\} }[/math] of [math]\displaystyle{ U }[/math], the sequence [math]\displaystyle{ \bigoplus_{i,j} \mathcal{F}(U_i \cap U_j) \rightarrow \bigoplus_k \mathcal{F}(U_k) \rightarrow \mathcal{F}(U) \rightarrow 0 }[/math] is exact. That is, [math]\displaystyle{ \mathcal{F} }[/math] 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 [math]\displaystyle{ U, V \subset M }[/math], the structure map [math]\displaystyle{ m^{U, V}_{U\sqcup V} : \mathcal{F}(U)\otimes \mathcal{F}(V) \rightarrow \mathcal{F}(U \sqcup V) }[/math] is an isomorphism.

Algebro-geometric formulation

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

Let [math]\displaystyle{ X }[/math] be a smooth complex curve. A factorization algebra on [math]\displaystyle{ X }[/math] consists of

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

[math]\displaystyle{ j^*_{J/I}\mathcal{V}_{X, J} \rightarrow j^*_{J/I}(\boxtimes_{i \in I} \mathcal{V}_{X, p^{-1}(i)}) }[/math] over [math]\displaystyle{ U^{J/I} }[/math].

• (Unit) Let [math]\displaystyle{ \mathcal{V} = \mathcal{V}_{X, \{1\}} }[/math] and [math]\displaystyle{ \mathcal{V}_2 = \mathcal{V}_{X, \{1, 2\}} }[/math]. A global section (the unit) [math]\displaystyle{ 1 \in \mathcal{V}(X) }[/math] with the property that for every local section [math]\displaystyle{ f \in \mathcal V(U) }[/math] ([math]\displaystyle{ U \subset X }[/math]), the section [math]\displaystyle{ 1 \boxtimes f }[/math] of [math]\displaystyle{ \mathcal{V}_2|_{U^2\Delta} }[/math] extends across the diagonal, and restricts to [math]\displaystyle{ f \in \mathcal{V} \cong \mathcal{V}_2|_\Delta }[/math].

Example

Associative algebra

Any associative algebra [math]\displaystyle{ A }[/math] can be realized as a prefactorization algebra [math]\displaystyle{ A^{f} }[/math] on [math]\displaystyle{ \mathbb{R} }[/math]. To each open interval [math]\displaystyle{ (a,b) }[/math], assign [math]\displaystyle{ A^f((a,b)) = A }[/math]. An arbitrary open is a disjoint union of countably many open intervals, [math]\displaystyle{ U = \bigsqcup_i I_i }[/math], and then set [math]\displaystyle{ A^f(U) = \bigotimes_i A }[/math]. The structure maps simply come from the multiplication map on [math]\displaystyle{ A }[/math]. Some care is needed for infinite tensor products, but for finitely many open intervals the picture is straightforward.

See also

• Vertex algebra