Floer sections in multisymplectic geometry

TL;DR

This paper extends Floer theory to multisymplectic geometry, introducing a new framework that generalizes elliptic PDE methods from symplectic geometry to higher-dimensional Hamiltonian field theories.

Contribution

It develops a novel multisymplectic framework with pseudo-Fueter curves, generalizing Floer theory to higher dimensions and proving a Darboux theorem in this context.

Findings

Introduces a multisymplectic framework compatible with hyperkähler structures.

Shows that gradient lines are pseudo-Fueter curves.

Proves a Darboux theorem in multisymplectic geometry.

Abstract

In symplectic geometry, Floer theory is the most important tool to prove the existence of time-periodic solutions in Hamiltonian mechanics. The core observation is that the $L^2$-gradient lines of the symplectic action functional are pseudo-holomorphic curves, enabling the use of elliptic PDE methods. Multisymplectic geometry is the geometric framework underlying Hamiltonian field theory, where the time line is replaced by higher-dimensional manifolds. In the case of two dimensions and using complex structures, we introduce a novel multisymplectic framework that is fit for the generalization of the elliptic methods from symplectic geometry. Besides proving a Darboux theorem, we show that the $L^2$-gradient lines of our multisymplectic action functional are now pseudo-Fueter curves defined using a compatible almost hyperk\"ahler structure.