A class of locally inhomogeneous complete quaternionic Kähler manifolds
Vicente Cortés, Alejandro Gil-García, Arpan Saha
TL;DR
The work addresses whether the one-loop deformation of c-map spaces remains locally homogeneous. By deriving curvature formulas for rigid c-map spaces and applying the HK/QK correspondence and twist construction to pass to the deformed quaternionic Kähler setting, the authors prove that the deformation is locally inhomogeneous for any c-map space when the deformation parameter $c>0$. This leads to the corollary that the full isometry group of the deformed space has cohomogeneity exactly one for homogeneous starting data (excluding quaternionic hyperbolic spaces). The results hinge on showing that a curvature-related invariant on the hyper-Kähler side is non-constant under the deformation, which transfers to non-constancy of the curvature on the quaternionic Kähler side. Overall, the paper clarifies the geometry of deformed c-map manifolds and the structure of their isometry groups, with implications for the global and local symmetry properties of these moduli spaces.
Abstract
We prove that the one-loop deformation of any quaternionic Kähler manifold in the class of c-map spaces is locally inhomogeneous. As a corollary, we obtain that the full isometry group of the one-loop deformation of any homogeneous c-map space has precisely cohomogeneity one.