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

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 applied in a more general setting by Costello and Gwilliam to formalize quantum field theory. To define factorization algebras, it is necessary to define a Weiss cover. For U an open set, a collection of opens \mathfrak{U} = \{U_i | i \in I\} is a Weiss cover of U if for any finite collection of points \{x_1, \cdots, x_k\} in U , there is an open set U_i \in \mathfrak{U} such that \{x_1, \cdots, 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 \bigoplus_{i,j} \mathcal{F}(U_i \cap U_j) \rightarrow \bigoplus_k \mathcal{F}(U_k) \rightarrow \mathcal{F}(U) \rightarrow 0 is exact. That is, \mathcal{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^{U, V}_{U\sqcup V} : \mathcal{F}(U)\otimes \mathcal{F}(V) \rightarrow \mathcal{F}(U \sqcup V) is an isomorphism. j^*_{J/I}\mathcal{V}_{X, J} \rightarrow j^*_{J/I}(\boxtimes_{i \in I} \mathcal{V}_{X, p^{-1}(i)}) over U^{J/I} . (Unit) Let \mathcal{V} = \mathcal{V}_{X, \{1\}} and \mathcal{V}_2 = \mathcal{V}_{X, \{1, 2\}} . A global section (the unit) 1 \in \mathcal{V}(X) with the property that for every local section f \in \mathcal V(U) ( U \subset X ), the section 1 \boxtimes f of \mathcal{V}_2|_{U^2\Delta} extends across the diagonal, and restricts to f \in \mathcal{V} \cong \mathcal{V}_2|_\Delta . Any associative algebra A can be realized as a prefactorization algebra A^{f} on \mathbb{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 = \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.

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