  We show that the geometry of $4n$-dimensional quaternionic K\"ahler spaces
with a locally free $\mathbb{R}^{n+1}$-action admits a Gibbons-Hawking-like
description based on the Galicki-Lawson notion of quaternionic K\"ahler moment
map. This generalizes to higher dimensions a four-dimensional construction, due
to Calderbank and Pedersen, of self-dual Einstein manifolds with two linearly
independent commuting Killing vector fields. As an application, we use this new
Ansatz to give an explicit equivariant completion of the twistor space
construction of the local c-map proposed by Ro\v{c}ek, Vafa and Vandoren.

Differential Geometry;High Energy Physics - Theory