Classical and variational Poisson cohomology

TL;DR

This paper establishes an isomorphism between variational and classical Poisson cohomology for certain vertex algebras, using Hochschild and Harrison cohomology complexes, advancing the understanding of vertex algebra cohomology.

Contribution

It proves the isomorphism between variational and classical cohomology for Poisson vertex algebras under specific conditions, introducing new cohomology complexes and vanishing theorems.

Findings

Proves the isomorphism between variational and classical cohomology.

Introduces sesquilinear Hochschild and Harrison cohomology complexes.

Establishes a vanishing theorem for symmetric Harrison cohomology.

Abstract

We prove that, for a Poisson vertex algebra V, the canonical injective homomorphism of the variational cohomology of V to its classical cohomology is an isomorphism, provided that V, viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables.