Gravitational instantons of type $D_k$ and a generalization of the Gibbons-Hawking Ansatz

TL;DR

This paper introduces a quaternionic-based generalization of the Gibbons-Hawking Ansatz to construct explicit hyperk"ahler metrics with hidden symmetries, specifically applying it to derive gravitational instantons of type $D_k$.

Contribution

It presents a novel quaternionic Ansatz that extends the Gibbons-Hawking method for hyperk"ahler metrics, enabling explicit construction of $D_k$-type gravitational instantons.

Findings

Derived explicit $D_k$ gravitational instanton metrics

Generalized Gibbons-Hawking Ansatz using quaternionic structures

Identified hidden symmetries in the new class of metrics

Abstract

We describe a quaternionic-based Ansatz generalizing the Gibbons-Hawking Ansatz to a class of hyperk\"ahler metrics with hidden symmetries. We then apply it to obtain explicit expressions for gravitational instanton metrics of type $D_k$.