A vertical exterior derivative in multisymplectic geometry and a graded Poisson bracket for nontrivial geometries

TL;DR

This paper introduces a vertical exterior derivative for multisymplectic manifolds over complex bundles, establishing a graded Poisson structure with proven properties and initial examples.

Contribution

It develops a vertical exterior derivative and a graded Poisson bracket framework for nontrivial multisymplectic geometries, expanding the mathematical tools available.

Findings

Defined a vertical exterior derivative for complex bundles

Proved properties of the graded Poisson bracket

Presented initial examples demonstrating the framework

Abstract

A vertical exterior derivative is constructed that is needed for a graded Poisson structure on multisymplectic manifolds over nontrivial vector bundles. In addition, the properties of the Poisson bracket are proved and first examples are discussed.