Conformal geometry and half-integrable spacetimes

Using a combination of techniques from conformal and complex geometry, we show the potentialization of 4-dimensional closed Einstein-Weyl structures which are half-algebraically special and admit a "half-integrable" almost-complex structure. That is, we reduce the Einstein-Weyl equations to a single, conformally invariant, non-linear scalar equation, that we call the "conformal HH equation", and we reconstruct the conformal structure (curvature and metric) from a solution to this equation. We show that the conformal metric is composed of: a conformally flat part, a conformally half-flat part related to certain "constants" of integration, and a potential part that encodes the full non-linear curvature, and that coincides in form with the Hertz potential from perturbation theory. We also study the potentialization of the Dirac-Weyl, Maxwell (with and without sources), and Yang-Mills systems. We show how to deal with the ordinary Einstein equations by using a simple trick. Our results give a conformally invariant, coordinate-free, generalization of the hyper-heavenly construction of Plebanski and collaborators.