Quaternionic metrics from harmonic superspace : lagrangian approach and quotient construction

TL;DR

This paper develops a harmonic superspace approach to quaternionic-Kahler sigma models, deriving their actions, exploring flat limits, and constructing specific metrics like Taub-NUT and Eguchi-Hanson with enhanced symmetries.

Contribution

It introduces a new harmonic superspace framework for quaternionic-Kahler metrics and provides novel quotient constructions for important examples.

Findings

Derived general bosonic sigma model action for QK manifolds.

Identified flat limits leading to hyper-Kahler models.

Constructed QK extensions of Taub-NUT and Eguchi-Hanson metrics.

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 4n dimensional quaternionic-Kahler (QK) manifold as the target space. The action is determined by the analytic harmonic QK potential. We find out this action to have two flat limits. One gives the hyper-Kahler sigma model with a 4n dimensional target manifold, while another yields a conformally invariant sigma model with 4(n+1) dimensional hyper-Kahler 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 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 isometry group to SU(3) or SU(2,1) at the special relations between its parameters : the Einstein constant and the "mass".