Courant morphisms and moment maps

TL;DR

This paper develops a unified framework for Hamiltonian spaces associated with Courant algebroids and Dirac structures, generalizing moment map theories and exploring their geometric and algebraic properties.

Contribution

It introduces a new approach using Courant morphisms to study moment maps, encompassing various existing theories and providing insights into their interrelations.

Findings

Unified description of moment maps via Courant morphisms

Reduction procedures for Hamiltonian spaces

Connections between quasi-Poisson and presymplectic groupoids

Abstract

We study Hamiltonian spaces associated with pairs (E,A), where E is a Courant algebroid and A\subset E is a Dirac structure. These spaces are defined in terms of morphisms of Courant algebroids with suitable compatibility conditions. Several of their properties are discussed, including a reduction procedure. This set-up encompasses familiar moment map theories, such as group-valued moment maps, and it provides an intrinsic approach from which different geometrical descriptions of moment maps can be naturally derived. As an application, we discuss the relationship between quasi-Poisson and presymplectic groupoids.