Symmetries of One-loop Deformed q-map 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.