It is often said that vertex algebras are a special case of factorization algebras. In particular, in their book "FAs in QFT" Costello/Gwilliam construct a functor from a certain class of 2d "holomorphic" factorization algebras to vertex algebras, by taking the cohomology. Does every vertex algebra arise in this way?

## 1 Answer

Yes, every $\mathbf{Z}$-graded vertex algebra arises from a factorization algebra on $\mathbf{C}$ with meromorphic operator product expansion. See Vertex Algebras and Costello-Gwilliam Factorization Algebras.

This factorization algebra takes values in the symmetric monoidal category of *complete* bornological vector spaces.
I don't know if the underlying precosheaf of vector spaces satisfies codescent or homotopy codescent, but the precosheaf of complete bornological vector spaces does satisfy codescent.
The factorization algebra depends functorially on the $\mathbf{Z}$-graded vertex algebra, so there is a one-sided inverse to the functor from (certain) factorization algebras to $\mathbf{Z}$-graded vertex algebras.

operatoralgebra structure to induce a BD chiral/factorisation algebra. $\endgroup$