On higher Dirac structures

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 + ∧^kTM^∗ 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 Bursztyn et al. [8] as the infinitesimal counterparts of multisymplectic groupoids. We describe the leaf-wise 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.