An operadic approach to vertex algebra and Poisson vertex algebra cohomology

TL;DR

This paper develops an operadic framework for vertex algebra and Poisson vertex algebra cohomology, translating geometric constructions into algebraic language and establishing a cohomology complex for these structures.

Contribution

It introduces an operadic approach to vertex algebra cohomology, connecting chiral operads with algebraic vertex algebras and classical operads with Poisson vertex algebras.

Findings

Constructs a vertex algebra cohomology complex from operads.

Relates the graded chiral operad to classical Poisson structures.

Links the new cohomology to variational Poisson cohomology.

Abstract

We translate the construction of the chiral operad by Beilinson and Drinfeld to the purely algebraic language of vertex algebras. Consequently, the general construction of a cohomology complex associated to a linear operad produces a vertex algebra cohomology complex. Likewise, the associated graded of the chiral operad leads to a classical operad, which produces a Poisson vertex algebra cohomology complex. The latter is closely related to the variational Poisson cohomology studied by two of the authors.