On toric Hermitian ALF gravitational instantons

TL;DR

This paper classifies toric Hermitian Ricci-flat ALF gravitational instantons in four dimensions, identifying smooth examples like hyperKähler metrics and others with conical singularities, providing explicit formulas and topological diversity.

Contribution

It offers the first comprehensive classification of toric Hermitian ALF gravitational instantons, including explicit formulas and topological classifications, extending previous work on smooth and singular cases.

Findings

Smooth examples include hyperKähler ALF metrics, Kerr, Taub-NUT, and Chen-Teo metrics.

Existence of examples with conical singularities and diverse topologies.

Explicit formulas for the classified metrics.

Abstract

We give a classification of toric, Hermitian, Ricci flat, ALF Riemannian metrics in dimension 4, including metrics with conical singularities. The only smooth examples are on one hand the hyperKaehler ALF metrics, on the other hand, the Kerr, Taub-NUT and Chen-Teo metrics. There are examples with conical singularities with infinitely many distinct topologies. We provide explicit formulas.