Heisenberg-invariant self-dual Einstein manifolds

We classify all self-dual Einstein four-manifolds invariant under a principal action of the three-dimensional Heisenberg group with non-degenerate orbits. This is of interest in the study of quantum-corrected gravitational physics, since they naturally arise as scalar manifolds of particular Lorentzian and Euclidean supergravities with one-loop corrections. The metrics are explicit and we find, in particular, that the Einstein constant can take any value. Then we examine when the corresponding (Riemannian or neutral-signature) metrics are (geodesically) complete. Finally, we exhibit the solutions of non-zero Ricci-curvature as different branches of one-loop deformed universal hypermultiplets in Riemannian and neutral signature.