Symmetries of one-loop deformed q-map spaces
Vicente Cortés, Alejandro Gil-García, Danu Thung
TL;DR
The paper determines a universal symmetry group for one-loop deformed q-map spaces by tracing the symmetry from rigid c-map spaces through the HK/QK twist. It shows that the isometry group contains a semidirect product structure combining affine automorphisms of a PSR hypersurface with a Heisenberg group, specifically $((\mathbb{R}_{>0}\times\mathrm{Aut}(\mathcal{H}))\ltimes\mathbb{R}^{n-1})\ltimes(\mathrm{Heis}_{2n+1}/\mathcal{F})$ for spaces of real dimension $4n$, after quotienting by a central subgroup $\mathcal{F}$ to ensure effectiveness. The construction proceeds by analyzing canonical lifts and fiber translations on the rigid c-map $N=T^*M$, passing to the HK/QK twist to obtain $\bar{N}$, and then describing both the infinitesimal and global actions on the twisted space. This provides a unified framework linking PSR geometry, conical affine special Kähler structure, and the one-loop deformation, with implications for homogeneous and cohomogeneity-one constructions in quaternionic Kähler geometry and string-theoretic moduli spaces.
Abstract
Q-map spaces form an important class of quaternionic Kähler manifolds of negative scalar curvature. Their one-loop deformations are always inhomogeneous and have been used to construct cohomogeneity one quaternionic Kähler manifolds as deformations of homogeneous spaces. Here we study the group of isometries in the deformed case. Our main result is the statement that it always contains a semidirect product of a group of affine transformations of $\mathbb{R}^{n-1}$ with a Heisenberg group of dimension $2n+1$ for a q-map space of dimension $4n$. The affine group and its action on the normal Heisenberg factor in the semidirect product depend on the cubic affine hypersurface which encodes the q-map space.