On the Geometry of Multi-Dirac Structures and Gerstenhaber Algebras

Abstract

In a companion paper, we introduced a notion of multi-Dirac structures, a graded version of Dirac structures, and we discussed their relevance for classical field theories. In the current paper we focus on the geometry of multi-Dirac structures. After recalling the basic definitions, we introduce a graded multiplication and a multi-Courant bracket on the space of sections of a multi-Dirac structure, so that the space of sections has the structure of a Gerstenhaber algebra. We then show that the graph of a k-form on a manifold gives rise to a multi-Dirac structure and also that this multi-Dirac structure is integrable if and only if the corresponding form is closed. Finally, we show that the multi-Courant bracket endows a subset of the ring of differential forms with a graded Poisson bracket, and we relate this bracket to some of the multisymplectic brackets found in the literature.

arXiv comment19 pages; v2: different convention for degree

References in corpus (8)

• Generalized complex geometry
• L-infinity algebras from multisymplectic geometry
• Categorified Symplectic Geometry and the Classical String
• Covariant Poisson Brackets in Geometric Field Theory
• Hamiltonian Multivector Fields and Poisson Forms in Multisymplectic Field Theory
• The Hamilton-Pontryagin Principle and Multi-Dirac Structures for Classical Field Theories
• Graded geometry and Poisson reduction
• From the Poincaré-Cartan form to a Gerstenhaber algebra of Poisson brackets in field theory

Cited by in corpus (7)

• Brackets
• The Hamilton-Pontryagin Principle and Multi-Dirac Structures for Classical Field Theories
• Precanonical quantization and the Schrödinger wave functional revisited
• Remarks on Hamiltonian Structures in G_2-Geometry
• Graded Poisson and Graded Dirac structures
• Coisotropic reduction in Multisymplectic Geometry
• Poisson color algebras of arbitrary degree