A note on vertex algebras and Costello-Gwilliam factorization algebras
Yusuke Nishinaka
TL;DR
The paper addresses the correspondence between vertex algebras and Costello-Gwilliam factorization algebras on the complex plane, showing that the discreteness constraint on weight spaces is unnecessary for constructing vertex algebras from holomorphic prefactorization algebras valued in locally convex spaces. It then builds locally constant factorization algebras from commutative vertex algebras and proves a precise adjunction and equivalence between commutative algebras and locally constant holomorphic factorization algebras on $\mathbb{C}$, including holomorphic refinements. A general construction is provided to obtain vertex algebras from holomorphic PFAs, along with a converse direction yielding locally constant PFAs from commutative vertex algebras, and a jet-algebra compatibility result connecting to $\mathcal{J}A$ in the geometric study of vertex algebras. The work relies on developing calculus for functions with values in locally convex spaces and establishes a coherent framework linking vertex algebras, factorization algebras, and jet schemes, with implications for chiral and geometric vertex algebra theory.
Abstract
We show that the construction of vertex algebras from Costello-Gwilliam factorization algebras on $\mathbb{C}$ can be achieved without the discreteness condition on the weight spaces. Furthermore, we construct locally constant factorization algebras from commutative vertex algebras, and discuss the relationship between this construction and the jet algebras of commutative algebras.