Poisson geometry with a 3-form background

TL;DR

This paper explores twisted Poisson structures influenced by a closed 3-form, describing their geometric properties, group actions, and implications for deformation quantization and symplectic groupoids.

Contribution

It introduces a framework for understanding twisted Poisson structures via Dirac structures and Courant algebroids, extending classical Poisson geometry.

Findings

Twisted Poisson structures are described as Dirac structures in Courant algebroids.

The additive group of 2-forms acts on twisted Poisson structures.

Connections to deformation quantization and twisted symplectic groupoids are discussed.

Abstract

We study a modification of Poisson geometry by a closed 3-form. Just as for ordinary Poisson structures, these "twisted" Poisson structures are conveniently described as Dirac structures in suitable Courant algebroids. The additive group of 2-forms acts on twisted Poisson structures and permits them to be seen as glued from ordinary Poisson structures defined on local patches. We conclude with remarks on deformation quantization and twisted symplectic groupoids.