Tamarkin-Tsygan Calculus and Chiral Poisson Cohomology

TL;DR

This paper develops a vertex algebra-based framework to extend Poisson homology and cohomology into a chiral setting, connecting non-commutative geometry with vertex algebra theory.

Contribution

It introduces a chiral calculus for Poisson varieties using vertex algebras, establishing foundational theorems in this new context.

Findings

Constructed vertex theoretic invariants for Poisson varieties.

Extended Poisson (co)homology theorems to the chiral setting.

Linked non-commutative geometry with vertex algebra structures.

Abstract

We construct and study some vertex theoretic invariants associated to Poisson varieties, specialising in the conformal weight $0$ case to the familiar package of Poisson homology and cohomology. In order to do this conceptually we sketch a version of the \emph{calculus}, in the sense of \cite{TT}, adapted to the context of vertex algebras. We obtain the standard theorems of Poisson (co)homology in this \emph{chiral} context. This is part of a larger project related to promoting non-commutative geometric structures to chiral versions of such.