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