On Higher Dirac Structures

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.