Hypermultiplets, Hyperkahler Cones and Quaternion-Kahler Geometry

TL;DR

This work establishes a one-to-one correspondence between hyperkähler cones and quaternion-Kähler spaces via the $oldsymbol{N=2}$ superconformal quotient, and provides a practical framework to classify $4(n-1)$-dimensional quaternion-Kähler spaces with $n$ abelian quaternionic isometries by dualizing $n$ tensor multiplets. It develops a concrete, contour-integral representation for the Lagrangians and derives explicit quaternion-Kähler metrics through reduction to the twistor space $oldsymbol{\mathcal Z}$ and subsequent gauge fixing, with the universal hypermultiplet serving as a central example. The results yield three inequivalent tensor descriptions of the universal hypermultiplet, connected to distinct commuting triholomorphic isometry sets, and demonstrate how these descriptions coherently reproduce the corresponding $X(n-1)$ Wolf spaces upon appropriate quotients. Overall, the paper provides a robust geometric toolkit for analyzing perturbative moduli spaces in type-II Calabi–Yau compactifications and offers a path to incorporating nonperturbative corrections via contour data and projective superspace formalisms.

Abstract

We study hyperkahler cones and their corresponding quaternion-Kahler spaces. We present a classification of 4(n-1)-dimensional quaternion-Kahler spaces with n abelian quaternionic isometries, based on dualizing superconformal tensor multiplets. These manifolds characterize the geometry of the hypermultiplet sector of perturbative moduli spaces of type-II strings compactified on a Calabi-Yau manifold. As an example of our construction, we study the universal hypermultiplet in detail, and give three inequivalent tensor multiplet descriptions. We also comment on the construction of quaternion-Kahler manifolds that may describe instanton corrections to the moduli space.