# On higher Dirac structures

Research paper by Henrique Bursztyn, Nicolas Martinez Alba, Roberto Rubio

Indexed on: 07 Nov '16Published on: 07 Nov '16Published in: arXiv - Mathematics - Symplectic Geometry

#### 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.