On higher analogues of Courant algebroids

TL;DR

This paper explores higher analogues of Courant algebroids on manifolds, establishing their relation to Nambu-Poisson and multisymplectic structures, and revealing new algebraic structures like Leibniz and Lie algebroids.

Contribution

It introduces higher Courant algebroid structures on $TM igoplus igwedge^n T^*M$ and connects them to Nambu-Poisson and multisymplectic geometries, including new algebraic frameworks.

Findings

Graph of an $(n+1)$-vector field is closed under the higher Dorfman bracket iff it is Nambu-Poisson.

Graph of an $(n+1)$-form is closed iff it is premultisymplectic.

Induces Leibniz and Lie algebroid structures related to multisymplectic geometry.

Abstract

In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle $TM\oplus\wedge^nT^*M$ for an $m$-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an $(n+1)$-vector field $\pi$ is closed under the higher-order Dorfman bracket iff $\pi$ is a Nambu-Poisson structure. Consequently, there is an induced Leibniz algebroid structure on $\wedge^nT^*M$. The graph of an $(n+1)$-form $\omega$ is closed under the higher-order Dorfman bracket iff $\omega$ is a premultisymplectic structure of order $n$, i.e. $\dM\omega=0$. Furthermore, there is a Lie algebroid structure on the admissible bundle $A\subset\wedge^{n}T^*M$. In particular, for a 2-plectic structure, it induces the Lie 2-algebra structure given in \cite{baez:classicalstring}.