Heisenberg symmetry and hypermultiplet manifolds

We study the emergence of Heisenberg (Bianchi II) algebra in hyper-K"ahler and quaternionic spaces. This is motivated by the r^ole these spaces with this symmetry play in $\mathcal{N}=2$ hypermultiplet scalar manifolds. We show how to construct related pairs of hyper-K"ahler and quaternionic spaces under general symmetry assumptions, the former being a zooming-in limit of the latter at vanishing cosmological constant. We further apply this method for the two hyper-K"ahler spaces with Heisenberg algebra, which is reduced to $U(1)\times U(1)$ at the quaternionic level. We also show that no quaternionic spaces exist with a strict Heisenberg symmetry – as opposed to $$\text{Heisenberg}$ \ltimes U(1)$\mathrm{.}Wefinallydiscusstherealizationofthelatterbygaugingappropriate$Sp(2,4)$generatorsin$$\mathcal{N}$=2$ conformal supergravity.