Symmetries of quaternionic Kähler manifolds with S1‐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. Contents

Four-dimensional Einstein manifolds with Heisenberg symmetry

“…Any complete Riemannian metric g on ℝ 4 invariant under a principal action of the Heisenberg group H can be brought to the form where ℝ 4 is identified with ℝ × H by an H-equivariant diffeomorphism and g t is a family of left-invariant metrics on H. This form is obtained by identifying the H-orbits by means of the normal geodesic flow, where t corresponds to the arc length parameter along a normal geodesic.…”

“…It has been recently shown [4] that all the known homogeneous quaternionic Kähler manifolds of negative Ricci curvature with the exception of the simplest examples, the quaternionic hyperbolic spaces, admit a canonical deformation to a complete quaternionic Kähler manifold with an isometric action of cohomogeneity one. The deformation is a special case of what is known as the one-loop deformation [8].…”

“…(The completeness requires the deformation parameter to be non-negative [2,Proposition 4]. ) Its isometry group is precisely O(2) ⋉ H , where H is the three-dimensional Heisenberg group [4].…”

Complete quaternionic Kähler manifolds with finite volume ends

“…The matter of lifting PSK automorphisms to the image ( N, g c ) under the c-map was taken up in [CST21b]. By definition, every one-parameter group of PSK automorphisms lifts to a one-parameter group of CASK automorphisms of the corresponding CASK manifold M .…”

“…In this article we show that complete quaternionic Kähler manifolds of negative scalar curvature can have ends of finite volume without being locally homogeneous. Our constructions are based on the recently proven fact that the known homogeneous quaternionic Kähler manifolds of negative scalar curvature and higher rank can be deformed into complete quaternionic Kähler manifolds with an isometric cohomogeneity one action [CST21b,Cor. 3.19].…”