A Poisson bracket on the space of Poisson structures

TL;DR

This paper introduces a Poisson bracket on the space of Poisson structures on a manifold, explores its Hamiltonian flow, and defines invariants that detect properties like unimodularity, connecting to symplectic cohomology.

Contribution

It constructs a new Poisson bracket on the space of Poisson structures depending on a volume form and introduces invariants related to flow fixed points and unimodularity.

Findings

Invariant detects unimodularity in regular Poisson 3-manifolds.

Flow acts by volume-preserving diffeomorphisms.

Invariant relates to symplectic cohomology groups.

Abstract

Let $M$ be a smooth closed orientable manifold and $\mathcal{P}(M)$ the space of Poisson structures on $M$. We construct a Poisson bracket on $\mathcal{P}(M)$ depending on a choice of volume form. The Hamiltonian flow of the bracket acts on $\mathcal{P}(M)$ by volume-preserving diffeomorphism of $M$. We then define an invariant of a Poisson structure that describes fixed points of the flow equation and compute it for regular Poisson 3-manifolds, where it detects unimodularity. For unimodular Poisson structures we define a further, related Poisson bracket and show that for symplectic structures the associated invariant counting fixed points of the flow equation is given in terms of the $d d^\Lambda$ and $d+ d^\Lambda$ symplectic cohomology groups defined by Tseng and Yau.