Hidden symmetries of generalised gravitational instantons

TL;DR

This paper develops a framework for solving field equations of conformally Kähler four-manifolds with symmetries, leading to new solutions in gravitational instanton theory through Toda equations.

Contribution

It introduces a novel approach to derive and solve Toda equations for conformally Kähler manifolds, extending known gravitational instanton classes with Einstein-Maxwell generalizations.

Findings

Derived generic curvature identities without field equations.

Formulated $SU( abla)$ Toda equations for key instanton classes.

Constructed new conformally self-dual Einstein-Maxwell geometries.

Abstract

For conformally K\"ahler Riemannian four-manifolds with a Killing field, we present a framework to solve the field equations for generalised gravitational instantons corresponding to conformal self-duality and to cosmological Einstein-Maxwell. After deriving generic identities for the curvature of such manifolds without assuming field equations, we obtain $SU(\infty)$ Toda formulations for the Page-Pope, Plebanski-Demianski, and Chen-Teo classes, we show how to solve the (modified) Toda equation, and we use this to find conformally self-dual and Einstein-Maxwell generalisations of these geometries.