Formal symplectic realizations

TL;DR

This paper presents a general formula for formal symplectic realizations of Poisson manifolds on ^n, connecting rooted trees, Bernoulli numbers, and Kontsevich's star product, unifying several constructions.

Contribution

It introduces a unified formula for formal symplectic realizations using rooted trees and elementary differentials, extending previous methods and linking them to Kontsevich's quantization.

Findings

The formula generalizes Bernoulli numbers for arbitrary Poisson structures.

It coincides with Weinstein's original construction in Darboux form.

Coefficients match Kontsevich weights via iterated integrals.

Abstract

We study the relationship between several constructions of symplectic realizations of a given Poisson manifold. Our main result is a general formula for a formal symplectic realization in the case of an arbitrary Poisson structure on $\R^n$. This formula is expressed in terms of rooted trees and elementary differentials, building on the work of Butcher, and the coefficients are shown to be a generalization of Bernoulli numbers appearing in the linear Poisson case. We also show that this realization coincides with a formal version of the original construction of Weinstein, when suitably put in global Darboux form, and with the realization coming from tree-level part of Kontsevich's star product. We provide a simple iterated integral expression for the relevant coefficients and show that they coincide with underlying Kontsevich weights.