On the Geometry of Multi-Dirac Structures and Gerstenhaber Algebras

TL;DR

This paper explores the geometric properties of multi-Dirac structures, establishing their algebraic framework, integrability conditions, and connections to graded Poisson brackets and multisymplectic geometry.

Contribution

It introduces a graded multiplication and multi-Courant bracket, demonstrating that multi-Dirac structures form a Gerstenhaber algebra and linking them to closed forms and multisymplectic brackets.

Findings

Multi-Dirac structures form a Gerstenhaber algebra.

Integrability of multi-Dirac structures corresponds to closed forms.

A graded Poisson bracket is defined on a subset of differential forms.

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.