Motivated by black hole physics in N = 2,D = 4 supergravity, we study
the geometry of quaternionic-K¨ahler manifolds Mobtained by the c-map construction
from projective special Kähler manifolds Ms. Improving on earlier treatments, we
compute the Käahler potentials on the twistor space Z and Swann space S in the
complex coordinates adapted to the Heisenberg symmetries. The results bear a simple
relation to the Hesse potential E of the special Käahler manifold Ms, and hence to
the Bekenstein-Hawking entropy for BPS black holes. We explicitly construct the
“covariant c-map” and the “twistor map”, which relate real coordinates on M× CP1
(resp. M×R4/Z2) to complex coordinates on Z (resp. S). As applications, we solve for
the general BPS geodesic motion on M, and provide explicit integral formulae for the
quaternionic Penrose transform relating elements of H1(Z,O(−k)) to massless fields
on M annihilated by first or second order differential operators. Finally, we compute
the exact radial wave function (in the supergravity approximation) for BPS black holes
with fixed electric and magnetic charges
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