Coisotropic reduction in Multisymplectic Geometry

TL;DR

This paper explores coisotropic reduction in multisymplectic geometry, linking Hamiltonian multivector fields to Lagrangian submanifolds and extending classical symplectic results to multisymplectic bundles.

Contribution

It provides a new interpretation of Hamiltonian multivector fields and extends coisotropic reduction results from symplectic to multisymplectic geometry.

Findings

Hamiltonian multivector fields interpreted as Lagrangian submanifolds

k-coisotropic submanifolds induce a Lie subalgebra of Hamiltonian forms

Extension of symplectic projection results to multisymplectic bundles

Abstract

In this paper we study coisotropic reduction in multisymplectic geometry. On the one hand, we give an interpretation of Hamiltonian multivector fields as Lagrangian submanifolds and prove that $k$-coisotropic submanifolds induce a Lie subalgebra in the algebra of Hamiltonian $(k-1)$-forms, similar to how coisotropic submanifolds in symplectic geometry induce a Lie subalgebra under the Poisson bracket. On the other hand, we extend the classical result of symplectic geometry of projection of Lagrangian submanifolds in coisotropic reduction to bundles of forms, which naturally carry a multisymplectic structure.