Heisenberg-invariant self-dual Einstein manifolds

TL;DR

This paper classifies all self-dual Einstein four-manifolds invariant under Heisenberg group actions, providing explicit metrics with arbitrary Einstein constants, and explores their completeness and physical interpretations.

Contribution

It offers a complete classification of Heisenberg-invariant self-dual Einstein 4-manifolds with explicit metrics and analyzes their geometric and physical properties.

Findings

Explicit metrics with arbitrary Einstein constants

Conditions for geodesic completeness

Connections to hypermultiplet solutions in physics

Abstract

We classify all self-dual Einstein four-manifolds invariant under a principal action of the three-dimensional Heisenberg group with non-degenerate orbits. The metrics are explicit and we find, in particular, that the Einstein constant can take any value. Then we study 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.