On higher Dirac structures

TL;DR

This paper introduces higher Dirac structures as a generalization of Dirac and multisymplectic structures, exploring their geometric properties, relation to higher Poisson structures, and integration theory.

Contribution

It defines higher Dirac structures, analyzes their leafwise geometry, and connects them to higher Poisson structures and multisymplectic groupoids.

Findings

Higher Dirac structures generalize classical Dirac structures to higher degrees.

They are characterized by involutive subbundles satisfying a weak lagrangian condition.

The paper describes the leafwise geometry via a 1-cocycle and discusses their integration.

Abstract

We study higher-order analogues of Dirac structures, extending the multisymplectic structures that arise in field theory. We define higher Dirac structures as involutive subbundles of $TM+\wedge^k TM^*$ satisfying a weak version of the usual lagrangian condition (which agrees with it only when $k=1$). Higher Dirac structures transversal to $TM$ recover the higher Poisson structures introduced in [8] as the infinitesimal counterparts of multisymplectic groupoids. We describe the leafwise geometry underlying an involutive isotropic subbundle in terms of a distinguished 1-cocycle in a natural differential complex, generalizing the presymplectic foliation of a Dirac structure. We also identify the global objects integrating higher Dirac structures.