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 $TM+{\wedge }^{k}T{M}^{\ast }$ 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.

arXiv commentTo appear in IMRN

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Cited by in corpus (10)

• Reduction of multisymplectic manifolds
• The geometry of graded cotangent bundles
• Quantization of Polysymplectic Manifolds
• Quotients of multiplicative forms and Poisson reduction
• Higher omni-Lie algebroids
• Graded Poisson and Graded Dirac structures
• Coisotropic reduction in Multisymplectic Geometry
• Brackets in multicontact geometry and multisymplectization
• Multisymplectic actions of compact Lie groups on spheres
• Higher Courant-Dorfman algebras and associated higher Poisson vertex algebras