Classical and variational Poisson cohomology

Bojko Bakalov, Alberto De Sole, Reimundo Heluani, Victor G. Kac, Veronica Vignoli

Abstract: We prove that, for a Poisson vertex algebra $\mathcal{V}$, the canonical injective homomorphism of the variational cohomology of $\mathcal{V}$ to its classical cohomology is an isomorphism, provided that $\mathcal{V}$, 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.