Fueter equations and the search for a higher dimensional Hamiltonian Floer theory I: analytical foundations and compactness

TL;DR

This paper develops analytical foundations for a Floer-theoretic approach to harmonic maps from the two-torus into Kähler manifolds, focusing on compactness of moduli spaces of Fueter maps using complex-regularized polysymplectic formalism.

Contribution

It introduces a new analytical framework for Fueter equations in higher dimensions and proves compactness results for moduli spaces in both flat and non-flat cases.

Findings

Proved relative compactness of moduli spaces for compact quotients of complex hyperbolic space.

Established $L^ abla$-estimates in the flat case.

Outlined perturbative strategies for non-flat Kähler manifolds.

Abstract

We study a Floer-theoretic approach to harmonic maps from the two-torus into non-flat K\"ahler manifolds. Building on the complex-regularized polysymplectic (CRPS) formalism of [BF24], which provides a Hamiltonian description of harmonic maps for which the associated equations are elliptic, we analyze the compactness of the associated moduli spaces of Fueter maps. For compact quotients $Q$ of complex hyperbolic space, we exploit the structure of the Biquard-Gauduchon hyperk\"ahler metric to prove relative compactness under suitable smallness assumptions on the Hamiltonian. In the flat case, we establish the necessary quantitative $L^\infty$-estimates and outline a perturbative strategy for the non-flat setting.