Existence and unicity of co-moments in multisymplectic geometry

TL;DR

This paper investigates the conditions for the existence and uniqueness of homotopy co-moment maps in multisymplectic geometry, extending classical symplectic concepts to higher structures and providing new existence results.

Contribution

It provides a cohomological framework for understanding homotopy co-moment maps and applies it to specific cases like exact manifolds and Lie groups, deriving new existence results.

Findings

Cohomological criteria for existence and uniqueness of homotopy co-moment maps

Application to exact multisymplectic manifolds and simple Lie groups

Derivation of conditions for partial co-moment maps

Abstract

Given a multisymplectic manifold $(M,\omega)$ and a Lie algebra $\frak{g}$ acting on it by infinitesimal symmetries, Fregier-Rogers-Zambon define a homotopy (co-)moment as an $L_{\infty}$-algebra-homomorphism from $\frak{g}$ to the observable algebra $L(M,\omega)$ associated to $(M,\omega)$, in analogy with and generalizing the notion of a co-moment map in symplectic geometry. We give a cohomological characterization of existence and unicity for homotopy co-moment maps and show its utility in multisymplectic geometry by applying it to special cases as exact multisymplectic manifolds and simple Lie groups and by deriving from it existence results concerning partial co-moment maps, as e.g. covariant multimomentum maps and multi-moment maps.