Twistors and Black Holes

Motivated by black hole physics in N = 2, D = 4 supergravity, we study the geometry of quaternionic-Kähler manifolds M obtained by the c-map construction from projective special Kähler manifolds Ms. Improving on earlier treatments, we compute the Kähler 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 Σ of the special Kähler 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× CP (resp. M×R/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. Email: [email protected], [email protected], [email protected] Unité mixte de recherche du CNRS UMR 7589 Unité mixte de recherche du CNRS UMR 8549