Twistors and Black Holes

Abstract

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.