Factorization Algebra

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 and Gwilliam 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 $M$ is a topological space, a prefactorization algebra $\mathcal$ of vector spaces on $M$ is an assignment of vector spaces $\mathcal(U)$ 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 $m_V^U: \mathcal(U) \rightarrow \mathcal(V)$
* There is a linear map $m_V^: \mathcal(U_1)\otimes \cdots \otimes \mathcal(U_n) \rightarrow \mathcal(V)$ 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 $U_$, $V_i$ and an open $W$ satisfying $U_\sqcup \cdots \sqcup U_ \subset V_i$ and $V_1 \sqcup \cdots V_n \subset W$, the following diagram commutes.
$\begin & \bigotimes_i \bigotimes_j \mathcal(U_) & \rightarrow & \bigotimes_i \mathcal(V_i) & \\ & \downarrow & \swarrow & \\ & \mathcal(W) & & & \\ \end$

So $\mathcal$ 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 $\mathfrak = \$ is a Weiss cover of $U$ if for any finite collection of points $\$ in $U$, there is an open set $U_i \in \mathfrak$ such that $\ \subset U_i$.

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 $\$ of $U$, the sequence
$\bigoplus_ \mathcal(U_i \cap U_j) \rightarrow \bigoplus_k \mathcal(U_k) \rightarrow \mathcal(U) \rightarrow 0$
is exact. That is, $\mathcal$ 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
$m^_ : \mathcal(U)\otimes \mathcal(V) \rightarrow \mathcal(U \sqcup V)$
is an isomorphism.

Algebro-geometric formulation

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

Let $X$ be a smooth complex curve. A factorization algebra on $X$ consists of
* A quasicoherent sheaf $\mathcal_$ over $X^$ for any finite set $I$, with no non-zero local section supported at the union of all partial diagonals
* Functorial isomorphisms of quasicoherent sheaves $\Delta^*_\mathcal_ \rightarrow \mathcal_$ over $X^I$ for surjections $J \rightarrow I$.
* (''Factorization'') Functorial isomorphisms of quasicoherent sheaves
$j^*_\mathcal_ \rightarrow j^*_(\boxtimes_ \mathcal_)$ over $U^$.
* (''Unit'') Let $\mathcal = \mathcal_$ and $\mathcal_2 = \mathcal_$. A global section (the ''unit'') $1 \in \mathcal(X)$ with the property that for every local section $f \in \mathcal V(U)$ ($U \subset X$), the section $1 \boxtimes f$ of $\mathcal_2, _$ extends across the diagonal, and restricts to $f \in \mathcal \cong \mathcal_2, _\Delta$.

Example

Associative algebra

Any associative algebra $A$ can be realized as a prefactorization algebra $A^$ on $\mathbb$. To each open interval $(a,b)$, assign $A^f((a,b)) = A$. An arbitrary open is a disjoint union of countably many open intervals, $U = \bigsqcup_i I_i$, and then set $A^f(U) = \bigotimes_i A$. 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

References

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Abstract algebra