Hyperinstantons, the beltrami equation, and triholomorphic maps

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 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 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.