HyperKähler potentials in cohomogeneity two. (Q2703579)

Hyper-Kähler manifolds are Riemannian manifolds which are Kähler with respect to three complex structures \(I, J, K\) satisfying the quaternion relations. They occur naturally in many geometric and physical contexts, e.g. as moduli spaces of solutions for some types of variational problems or as target spaces for certain supersymmetric field theories. An important example is the hyper-Kähler structure on adjoint orbits of complex semisimple Lie groups \(G\), which was discovered by Kronheimer. The nilpotent orbits (i.e. orbits of a nilpotent element in the Lie algebra of \(G\)) admit a hyper-Kähler metric which is defined by a so-called hyper-Kähler potential, i.e. a simultaneous Kähler potential for the three complex structures. It follows that these orbits fiber over quaternionic Kähler manifolds of positive scalar curvature. The maximal compact subgroup \(K\) of \(G\) acts on the nilpotent orbit preserving the hyper-Kähler structure. For the minimal orbits this action is of cohomogeneity one and is well understood [see \textit{A. Dancer} and \textit{A. Swann}, J. Geom. Phys. 21, 218--230 (1997; Zbl 0909.53032)].NEWLINENEWLINE In the paper under review the case of cohomogeneity two is investigated. The main result is an explicit formula for the hyper-Kähler potential. It is also shown that there is only one \(K\)-invariant hyper-Kähler metric on the nilpotent \(G\)-orbit admitting a hyper-Kähler potential. The paper contains valuable new algebraic information concerning the cohomogeneity two nilpotent orbits. Using it, the calculation of the potential and the proof of the above uniqueness statement are reduced to explicit calculations in the cases \(G = \text{SL}(2)\) and \(G = \text{SO}(4)\).