Higher Multi-Courant Algebroids

TL;DR

This paper extends the concept of Courant algebroids to higher multi-Courant algebroids using graded symplectic geometry, establishing a geometric framework for n-ary brackets and their algebraic properties.

Contribution

It constructs a higher geometric version of Keller-Waldmann Poisson algebra and defines higher multi-Courant algebroids within graded symplectic geometry.

Findings

Defined higher multi-Courant algebroids as functions of degree n on graded symplectic manifolds.

Extended the algebraic Keller-Waldmann Poisson algebra to a geometric setting.

Provided a new geometric perspective on n-ary brackets in Courant algebroids.

Abstract

The binary bracket of a Courant algebroid structure on $(E,\langle \cdot,\cdot \rangle)$ can be extended to a $n$-ary bracket on $\Gamma(E)$, yielding a multi-Courant algebroid. These $n$-ary brackets form a Poisson algebra and were defined, in an algebraic setting, by Keller and Waldmann. We construct a higher geometric version of Keller-Waldmann Poisson algebra and define higher multi-Courant algebroids. As Courant algebroid structures can be seen as degree $3$ functions on a graded symplectic manifold of degree $2$, higher multi-Courant structures can be seen as functions of degree $n\geq 3$ on that graded symplectic manifold.