Floer sections in multisymplectic geometry
Ronen Brilleslijper, Oliver Fabert
TL;DR
The paper develops a complex-regularized multisymplectic framework to generalize Floer theory from symplectic manifolds to field-theoretic settings. It introduces CRMS forms on extended multimomentum bundles and establishes a Darboux-type theorem, enabling local standardization of the CRMS structure. Floer theory is formulated via the L^2-gradient of a CRMS action, yielding a perturbed Fueter equation for Floer curves and linking to elliptic PDE methods, with implications for elliptic techniques in multisymplectic geometry. The work lays foundational machinery for extending Floer-style existence results from 1D Hamiltonian dynamics to higher-dimensional field theories.
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 hyperkähler structure.