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, coordinatefree, generalization of the hyper-heavenly construction of Plebański and collaborators.