Hyperkahler Cones and Orthogonal Wolf Spaces

TL;DR

This paper constructs hyperkahler cones for orthogonal Wolf spaces related to N=2 supergravity, deriving their geometry, symmetries, and dual descriptions, and introduces a new eight-dimensional ALE space using quiver diagrams.

Contribution

It provides explicit constructions of hyperkahler cones for orthogonal Wolf spaces and their non-compact versions, including their isometries and dual formulations.

Findings

Explicit hyperkahler potential computed from hyperkahler quotient.

Derived Killing vectors and moment maps for isometry groups.

Identified a new eight-dimensional ALE space via quiver diagram methods.

Abstract

We construct the hyperkahler cones corresponding to the Quaternion-Kahler orthogonal Wolf spaces SO(n+4)/(SO(n)xSO(4)) and their non-compact versions, which appear in hypermultiplet couplings to N=2 supergravity. The geometry is completely encoded by a single function, the hyperkahler potential, which we compute from an SU(2) hyperkahler quotient of flat space. We derive the Killing vectors and moment maps for the SO(n+4) isometry group on the hyperkahler cone. For the non-compact case, the isometry group SO(n,4) contains n+2 abelian isometries which can be used to find a dual description in terms of n tensor multiplets and one double-tensor multiplet. Finally, using a representation of the hyperkahler quotient via quiver diagrams, we deduce the existence of a new eight dimensional ALE space.