Hyperkähler cones and orthogonal Wolf spaces

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