Poisson Vertex Cohomology and Tate Lie Algebroids

TL;DR

This paper explores the cohomology of Poisson vertex algebras on holomorphic loops, linking it to Tate Lie algebroids and de Rham cohomology, especially in symplectic cases, providing new computational tools.

Contribution

It introduces a new description of the deformation complex of Poisson vertex algebras using Tate Lie algebroids and computes their cohomology in symplectic cases.

Findings

Cohomology of the complex is described via de Rham-Lie cohomology of Tate Lie algebroids.

Explicit computation of cohomology when the Poisson structure is symplectic.

Connection established between Poisson vertex algebra deformations and classical de Rham cohomology.

Abstract

We study sheaves on holomorphic spaces of loops and apply this to the study of the complex, defined in \cite{BdSHK}, governing deformations of the \emph{Poisson vertex algebra} structure on the space of holomorphic loops into a Poisson variety. We describe this complex in terms of the (continuous) de Rham-Lie cohomology of an associated Lie algebroid object in locally linearly compact topological (alias \emph{Tate}) sheaves of modules on $\mathcal{L}^{+}M$. In particular this allows us to easily compute the cohomology of the above in the case where $\pi$ is symplectic - we obtain de Rham cohomology of $M$.