Symmetries of quaternionic Kähler manifolds with S1‐symmetry

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