# Vertex algebras and factorization algebras

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?

• As far as I know only vertex algebras that are "universal envelopes" or related have been constructed this way. In particular there's no known construction of the irreducible affine Kac-Moody, rational W-algebras, chiral Hecke algebras, lattice vertex algebras, etc. in the Costello/Gwilliam framework. However, in answering to the title in the question, In Beilinson and Drinfeld there's an algebraic definition of factorization algebra. When looking to translation equivariant FA (a la BD) on the line these are equivalent to Vertex algebras. There's no known construction BD <-> CG Commented Jan 21, 2019 at 10:59
• It is not known whether W algebras arise this way. Also, I think you need a vertex operator algebra structure to induce a BD chiral/factorisation algebra. Commented Mar 24, 2022 at 13:55

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.