Dirac structures, moment maps and quasi-Poisson manifolds

TL;DR

This paper extends the relationship between Poisson maps and symplectic groupoid actions to Dirac geometry, illustrating how quasi-Poisson manifolds relate to twisted Dirac structures through an inversion process.

Contribution

It introduces a framework connecting quasi-Poisson manifolds with Dirac structures, generalizing existing concepts in Poisson geometry.

Findings

Established a correspondence between quasi-Poisson bivectors and twisted Dirac structures.

Extended the Poisson map and symplectic groupoid action relationship to Dirac geometry.

Provided a construction method linking quasi-Poisson and Dirac structures.

Abstract

We extend the correspondence between Poisson maps and actions of symplectic groupoids, which generalizes the one between momentum maps and hamiltonian actions, to the realm of Dirac geometry. As an example, we show how hamiltonian quasi-Poisson manifolds fit into this framework by constructing an ``inversion'' procedure relating quasi-Poisson bivectors to twisted Dirac structures.