The three dimensional Fueter equation and divergence free frames

TL;DR

This paper explores the properties of divergence free frames on three manifolds through the Fueter equation, establishing analogies with classical conjectures and linking to gauge theories like Seiberg-Witten and Donaldson-Thomas equations.

Contribution

It introduces the concepts of regular and singular divergence free frames, and draws analogies with the Maslov cycle and Arnold conjecture, connecting these to gauge theory equations.

Findings

Regular frames satisfy an analogue of the Arnold conjecture.

Singular frames form an analogue of the Maslov cycle.

Seiberg-Witten and Donaldson-Thomas equations relate to the Fueter equation.

Abstract

A divergence free frame on a closed three manifold is called regular if every solution of the linear Fueter equation is constant and is called singular otherwise. The set of singular divergence free frames is an analogue of the Maslov cycle. Regular divergence free frames satisfy an analogue of the Arnold conjecture for flat hyperkaehler target manifolds. The Seiberg-Witten equations can be viewed as gauged versions of the Fueter equation, and so can the Donaldson-Thomas equations on certain seven dimensional product manifolds.