Hyperinstantons, the Beltrami Equation, and Triholomorphic Maps

TL;DR

This paper links solutions of the Beltrami equation in hydrodynamics to triholomorphic maps in a 4D hyperKähler sigma model, proposing a topological approach for their classification and enumeration.

Contribution

It introduces a novel interpretation of Beltrami solutions as instantons in a 4D hyperKähler sigma model and develops a topological framework for their classification.

Findings

Solutions correspond to triholomorphic maps between hyperKähler manifolds.

Counting solutions can be simplified using discrete symmetries.

Reformulation as a topological sigma model aids enumeration of solutions.

Abstract

We consider the Beltrami equation for hydrodynamics and we show that its solutions can be viewed as instanton solutions of a more general system of equations. The latter are the equations of motion for an ${\cal N}=2$ sigma model on 4-dimensional worldvolume (which is taken locally HyperK\"ahler) with a 4-dimensional HyperK\"ahler target space. By means of the 4D twisting procedure originally introduced by Witten for gauge theories and later generalized to 4D sigma-models by Anselmi and Fr\'e, we show that the equations of motion describe triholomophic maps between the worldvolume and the target space. Therefore, the classification of the solutions to the 3-dimensional Beltrami equation can be performed by counting the triholomorphic maps. The counting is easily obtained by using several discrete symmetries. Finally, the similarity with holomorphic maps for ${\cal N}=2$ sigma on Calabi-Yau space prompts us to reformulate the problem of the enumeration of triholomorphic maps in terms of a topological sigma model.