Conserved quantities on multisymplectic manifolds

TL;DR

This paper explores conserved quantities on multisymplectic manifolds, extending classical symplectic results to a broader geometric context and establishing new links with symmetry groups and homotopy co-momentum maps.

Contribution

It generalizes the concept of conserved quantities to multisymplectic manifolds and demonstrates their behavior under symmetries with homotopy co-momentum maps.

Findings

Conserved quantities are differential forms with exact Lie derivatives.

Integrals of conserved quantities remain constant under evolution.

Presence of symmetry groups yields families of conserved quantities.

Abstract

Given a vector field on a manifold M, we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the Kelvin circulation theorem being a well-known special case. More generally, conserved quantities are well-behaved under transgression to spaces of maps into M. We focus on the case of multisymplectic manifolds and Hamiltonian vector fields. We show that in the presence of a Lie group of symmetries admitting a homotopy co-momentum map, one obtains a whole family of globally conserved quantities. This extends a classical result in symplectic geometry. We carry this out in a general setting, considering several variants of the notion of globally conserved quantity.