U(1) x U(1) Quaternionic Metrics from Harmonic Superspace

TL;DR

This paper constructs a new class of quaternionic-K"ahler metrics with U(1) x U(1) symmetry using harmonic superspace, extending known hyper-K"ahler metrics and exploring their relation to Einstein-Maxwell solutions.

Contribution

It introduces a quaternionic-K"ahler extension of two-centre hyper-K"ahler metrics with U(1) x U(1) symmetry via harmonic superspace and quaternionic quotient methods.

Findings

Constructed explicit quaternionic-K"ahler metrics with U(1) x U(1) isometry.

Connected these metrics to Einstein-Maxwell solutions and self-dual Weyl solutions.

Analyzed relation to Calderbank-Pedersen ansatz for such metrics.

Abstract

We construct, using harmonic superspace and the quaternionic quotient approach, a quaternionic-K\"ahler extension of the most general two centres hyper-K\"ahler metric. It possesses $U(1)\times U(1)$ isometry, contains as special cases the quaternionic-K\"ahler extensions of the Taub-NUT and Eguchi-Hanson metrics and exhibits an extra one-parameter freedom which disappears in the hyper-K\"ahler limit. Some emphasis is put on the relation between this class of quaternionic-K\"ahler metrics and self-dual Weyl solutions of the coupled Einstein-Maxwell equations. The relation between our explicit results and the recent general ansatz of Calderbank and Pedersen for quaternionic-K\"ahler metrics with $U(1)\times U(1)$ isometries is traced in detail.