Twistors and Black Holes

TL;DR

This paper explores the geometry of quaternionic-Kahler manifolds derived from special Kahler manifolds in supergravity, linking them to black hole entropy, and provides explicit formulas for BPS black hole wave functions.

Contribution

It introduces the covariant c-map and twistor map, relating complex and real coordinates, and computes exact wave functions for BPS black holes in supergravity.

Findings

Computed Kahler potentials on twistor and Swann spaces.

Established relations between Hesse potential and black hole entropy.

Derived explicit formulas for BPS black hole wave functions.

Abstract

Motivated by black hole physics in N=2, D=4 supergravity, we study the geometry of quaternionic-Kahler manifolds M obtained by the c-map construction from projective special Kahler manifolds M_s. Improving on earlier treatments, we compute the Kahler 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 \Sigma of the special Kahler manifold M_s, 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 x CP^1 (resp. M x R^4/Z_2) 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 H^1(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.