Gauging Isometries on Hyperkahler Cones and Quaternion-Kahler Manifolds

TL;DR

This work extends the link between quaternion-Kähler manifolds and hyperkähler cones by detailing how isometries, moment maps, and scalar potentials descend from the HKC to the QK space via the $N=2$ superconformal quotient. It provides a systematic gauging framework for triholomorphic and Sp(1) isometries, yielding explicit potential formulas and clarifying the role of the gauge fields and projective invariance in four dimensions. The unitary Wolf spaces, including the universal hypermultiplet, serve as concrete examples with complete HKC-to-QK reduction and explicit moment-map/Killing-vector data. The results enhance the construction of gauged $N=2$ supergravity and illuminate moduli-space structures arising in string compactifications.

Abstract

We extend our previous results on the relation between quaternion-Kahler manifolds and hyperkahler cones and we describe how isometries, moment maps and scalar potentials descend from the cone to the quaternion-Kahler space. As an example of the general construction, we discuss the gauging and the corresponding scalar potential of hypermultiplets with the unitary Wolf spaces as target spaces. This class includes the universal hypermultiplet.