U(1)× U(1) quaternionic metrics from harmonic superspace

Abstract

We construct, using harmonic superspace and the quaternionic quotient approach, a quaternionic-Kähler extension of the most general two centres hyper-Kähler metric. It possesses U(1)× U(1) isometry, contains as special cases the quaternionic-Kähler extensions of the Taub-NUT and Eguchi–Hanson metrics and exhibits an extra one-parameter freedom which disappears in the hyper-Kähler limit. Some emphasis is put on the relation between this class of quaternionic-Kähler 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-Kähler metrics with U(1)× U(1) isometries is traced in detail.