Hypermultiplets, hyper-Kähler cones and quaternion-Kähler geometry

We study hyper-Kähler cones and their corresponding quaternion-Kähler spaces. We present a classification of 4(n - 1)-dimensional quaternion-Kähler spaces with n abelian quaternionic isometrics, 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-Kähler manifolds that may describe instanton corrections to the moduli space.