On the geometry of multi-Dirac structures and Gerstenhaber algebras

Abstract

We dedicate this paper to the memory of Jerrold E. Marsden, our friend and mentor. AbstractIn 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.

The Hamilton-Pontryagin principle and multi-Dirac structures for classical field theories

“…As d M = 0, this multi-Dirac structure turns out be integrable; see Ref. 38. From Theorem III.4, we have that the implicit field equations can be written in terms of a partial multivector field X and the generalized energy E as…”

“…These geometric structures were introduced in Ref. 38 as a graded version of the standard concept of Dirac structures, and their relevance for field theory lies in the fact that they generalize the concept of the "graph of a multisymplectic structure. "…”

“…In the companion paper Ref. 38, we develop the algebraic and geometric properties of multi-Dirac structures.…”

“…In order to keep the technical side of the paper to a minimum, we focus on the applications of multi-Dirac structures to field theories, and we leave proofs and a more thorough discussion to Ref. 38. We also point out that there are alternative approaches to the construction of a "field-theoretic analogue" of Dirac structures; see for instance Refs.…”

“…38, we define a graded version of the Courant bracket, and we refer to integrable multi-Dirac structures as almost multi-Dirac structures which are closed under the graded Courant bracket.…”