Quaternionic metrics from harmonic superspace: Lagrangian approach and quotient construction

Abstract

Starting from the most general harmonic superspace action of self-interacting Q + hypermultiplets in the background of N=2 conformal supergravity, we derive the general action for the bosonic sigma model with a generic 4 n-dimensional quaternionic-Kähler (QK) manifold as the target space. The action is determined by the analytic harmonic QK potential and supplies an efficient systematic procedure of the explicit construction of QK metrics by the given QK potential. We find out this action to have two flat limits. One gives the hyper-Kähler (HK) sigma model with a 4 n-dimensional target manifold, while another yields a conformally-invariant sigma model with 4( n+1)-dimensional HK target. We work out the harmonic superspace version of the QK quotient construction and use it to give a new derivation of QK extensions of four-dimensional Taub–NUT and Eguchi–Hanson metrics. We analyze in detail the geometrical and symmetry structure of the second metric. The QK sigma model approach allows us to reveal the enhancement of its SU(2)⊗ U(1) isometry to SU(3) or SU(1,2) at the special relations between its free parameters: the Sp(1) curvature (“Einstein constant”) and the “mass”.