Homotopy Comomentum Maps in Multisymplectic Geometry

TL;DR

This paper develops new explicit constructions and classifications of homotopy comomentum maps in multisymplectic geometry, extending Hamiltonian actions to higher-degree forms with applications to hydrodynamics and knot theory.

Contribution

It provides a complete classification of group actions on multisymplectic spheres and constructs higher analogues of Poisson algebra embeddings, advancing the understanding of multisymplectic group actions.

Findings

Classified compact group actions on multisymplectic spheres.

Constructed explicit higher embeddings of Poisson algebras.

Related homotopy comomentum maps to hydrodynamics and knot theory.

Abstract

Homotopy comomentum maps are a higher generalization of the notion of moment map introduced to extend the concept of Hamiltonian actions to the framework of multisymplectic geometry. Loosely speaking, higher means passing from considering symplectic 2-form to consider differential forms in higher degrees. The goal of this thesis is to provide new explicit constructions and concrete examples related to group actions on multisymplectic manifolds admitting homotopy comomentum maps. The first result is a complete classification of compact group actions on multisymplectic spheres. The existence of a homotopy comomentum map about the latter depends on the dimension of the sphere and the transitivity of the group action. Several concrete examples of such actions are also provided. The second novel result is the explicit construction of the higher analogue of the embedding of the Poisson algebra of a given symplectic manifold into the corresponding twisted Lie algebroid. It is also proved a compatibility condition for such embedding for gauge-related multisymplectic manifolds in presence of a compatible Hamiltonian group action. The latter construction could play a role in determining the multisymplectic analogue of the geometric quantization procedure. Finally, a concrete construction of a homotopy comomentum map for the action of the group of volume-preserving diffeomorphisms on the multisymplectic 3-dimensional Euclidean space is proposed. This map can be naturally related to hydrodynamics. For instance, it transgresses to the standard hydrodynamical co-momentum map of Arnold, Marsden and Weinstein and others. A slight generalization of this construction to a special class of Riemannian manifolds is also provided. The explicitly constructed homotopy comomentum map can be also related to knot theory by virtue of the aforementioned hydrodynamical interpretation.