Hamilton Lie algebroids over Dirac structures and sigma models

TL;DR

This paper introduces a generalized Hamiltonian Lie algebroid framework over Dirac structures, extending classical concepts and applying it to construct new sigma models like the gauged Poisson and Dirac sigma models.

Contribution

It proposes a novel Hamiltonian Lie algebroid and momentum section over Dirac structures, extending existing geometric frameworks and developing related sigma models.

Findings

Defined a Hamiltonian Lie algebroid over Dirac structures.

Constructed gauged Poisson and Dirac sigma models using this framework.

Explored properties and applications of the new Hamiltonian structure.

Abstract

We propose a Hamiltonian Lie algebroid and a momentum section over a Dirac structure as a generalization of a Hamiltonian Lie algebroid over a pre-symplectic manifold and one over a Poisson manifold. A Hamiltonian Lie algebroid and a momentum section are generalizations of a Hamiltonian G-space and a momentum map over a symplectic manifold. We show some properties of a new Hamiltonian Lie algebroid, and construct the mechanics with this structure as an application, which are sigma models called the gauged Poisson sigma model and the gauged Dirac sigma model.