Toric hyperkahler manifolds with quaternionic Kahler bases and supergravity solutions

TL;DR

This paper constructs explicit examples of eight-dimensional toric hyperkahler manifolds, explores their geometric structures, and lifts them to supergravity solutions, revealing new links between hyperkahler geometry and string theory backgrounds.

Contribution

It provides explicit hyperkahler metrics with toric symmetry, connects quaternionic Kahler and hyperkahler geometries, and constructs supergravity solutions from these geometries using eigenfunctions.

Findings

Explicit eight-dimensional hyperkahler metrics with $U(1) imes U(1)$ symmetry.

Connection between Calderbank-Pedersen metrics and quaternionic Kahler structures.

Supergravity solutions derived from hyperkahler examples using dualities.

Abstract

In the present work some examples of toric hyperkahler metrics in eight dimensions are constructed. First it is described how the Calderbank-Pedersen metrics arise as a consequence of the Joyce description of selfdual structures in four dimensions, the Jones-Tod correspondence and a result due to Tod and Przanowski. It is also shown that any quaternionic Kahler metric with $T^2$ isometry is locally isometric to a Calderbank-Pedersen one. The Swann construction of hyperkahler metrics in eight dimensions is applied to them to find hyperkahler examples with $U(1)\times U(1)$ isometry. The connection with the Pedersen-Poon toric hyperkahler metrics is explained and it is shown that there is a class of solutions of the generalized monopole equation in $\mathbb{R}^2 \otimes Im\mathbb{H}$ related to eigenfunctions of certain linear equation. This hyperkahler examples are lifted to solutions of the D=11 supergravity and type IIA and IIB backgrounds are found by use of dualities. As before, all the description is achieved in terms of a single eigenfunction F. Some explicit F are found, together with the Toda structure corresponding to the trajectories of the Killing vectors of the Calderbank-Pedersen bases.