Symmetries of quaternionic Kähler manifolds with $S^1$-symmetry

Abstract

We study symmetry properties of quaternionic Kähler manifolds obtained by the HK/QK correspondence. To any Lie algebra g of infinitesimal automorphisms of the initial hyper-Kähler data we associate a central extension of g, acting by infinitesimal automorphisms of the resulting quaternionic Kähler manifold. More specifically, we study the metrics obtained by the one-loop deformation of the c-map construction, proving that the Lie algebra of infinitesimal automorphisms of the initial projective special Kähler manifold gives rise to a Lie algebra of Killing fields of the corresponding one-loop deformed c-map space. As an application, we show that this construction increases the cohomogeneity of the automorphism groups by at most one. In particular, if the initial manifold is homogeneous then the one-loop deformed metric is of cohomogeneity at most one. As an example, we consider the one-loop deformation of the symmetric quaternionic Kähler metric on SU (n, 2)/S(U (n) × U (2)), which we prove is of cohomogeneity exactly one. This family generalizes the so-called universal hypermultiplet (n = 1), for which we determine the full isometry group.