3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients

Abstract

Using 3-Sasakian reduction techniques we obtain infinite families of new 3-Sasakian manifolds $\mathcal{M}(p_1,p_2,p_3)$ and $\mathcal{M}(p_1,p_2,p_3,p_4)$ in dimension 11 and 15 respectively. The metric cone on $\mathcal{M}(p_1,p_2,p_3)$ is a generalization of the Kronheimer hyperkähler metric on the regular maximal nilpotent orbit of $\scriptstyle{{\Got s}{\Got l}(3,\bbc)}$ whereas the cone on $\mathcal{M}(p_1,p_2,p_3,p_4)$ generalizes the hyperkähler metric on the 16-dimensional orbit of $\scriptstyle{{\Got s}{\Got o}(6,\bbc)}$. These are first examples of 3-Sasakian metrics which are neither homogeneous nor toric. In addition we consider some further $U(1)$-reductions of $\mathcal{M}(p_1,p_2,p_3)$. These yield examples of non-toric 3-Sasakian orbifold metrics in dimensions 7. As a result we obtain explicit families $\mathcal{O}(\Theta )$ of compact self-dual positive scalar curvature Einstein metrics with orbifold singularities and with only one Killing vector field.

22 pages