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 $\text{Heisenberg} \ltimes U(1)$. We finally discuss the realization of the latter by gauging appropriate $Sp(2,4)$ generators in $\mathcal{N}=2$ conformal supergravity.