Hyperkahler cones and orthogonal Wolf spaces

Abstract

We construct the hyperk¨ahler cones corresponding to the quaternion-K¨ahler orthogonal Wolf spaces SO(n+4)/ SO(n)×SO(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 hyperk¨ahler potential, which we compute from an SU(2) hyperk¨ahler quotient of flat space. We derive the Killing vectors and moment maps for the SO(n + 4) isometry group on the hyperk¨ahler 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 hyperk¨ahler quotient via quiver diagrams, we deduce the existence of a new eight dimensional ALE space.