Multisymplectic actions of compact Lie groups on spheres
Antonio Michele Miti, Leonid Ryvkin
TL;DR
The paper investigates the existence of homotopy comoment maps for high-dimensional spheres under compact group actions preserving volume, establishing a sharp criterion: such a comoment exists if and only if the sphere dimension $n$ is even or the action is non-transitive. The authors develop an intrinsic obstruction theory via a bi-graded cochain complex and connect it to invariant and equivariant cohomology, providing explicit constructions where possible. They prove a non-transitive action on $S^n$ always admits a comoment and give explicit $f_1$ and $f_2$ components for the transitive case of $SO(n+1)$ on $S^n$ (with higher components posed as open). Concrete examples include the $SO(n)$-action on $S^n$ and a detailed $G_2$-on-$S^6$ comoment, showcasing the practical computation of comoments and illuminating the role of cohomological obstructions in multisymplectic symmetry. The work thus extends prior results, providing explicit formulas and guiding principles for constructing comoments in multisymplectic geometry with compact group symmetries.
Abstract
We investigate the existence of homotopy comoment maps (comoments) for high-dimensional spheres seen as multisymplectic manifolds. Especially, we solve the existence problem for compact effective group actions on spheres and provide explicit constructions for such comoments in interesting particular cases.