# An analogue of the Gibbons-Hawking Ansatz for quaternionic K\"ahler spaces

## TL;DR

This paper extends the Gibbons-Hawking Ansatz to higher-dimensional quaternionic K"ahler spaces with symmetries, providing a new explicit construction method and applications to twistor space completions in mathematical physics.

## Contribution

It introduces a Gibbons-Hawking-like description for quaternionic K"ahler spaces with symmetries, generalizing four-dimensional constructions to higher dimensions.

## Key findings

- Provides a new geometric framework for quaternionic K"ahler spaces.
- Explicitly constructs equivariant twistor space completions.
- Connects to the local c-map in string theory.

## Abstract

We show that the geometry of $4n$-dimensional quaternionic K\"ahler spaces with a locally free $\mathbb{R}^{n+1}$-action admits a Gibbons-Hawking-like description based on the Galicki-Lawson notion of quaternionic K\"ahler moment map. This generalizes to higher dimensions a four-dimensional construction, due to Calderbank and Pedersen, of self-dual Einstein manifolds with two linearly independent commuting Killing vector fields. As an application, we use this new Ansatz to give an explicit equivariant completion of the twistor space construction of the local c-map proposed by Ro\v{c}ek, Vafa and Vandoren.