Black-hole thermodynamics and Riemann surfaces

Abstract

We use the analytic continuation procedure proposed in our earlier works to study the thermodynamics of black holes in 2 + 1 dimensions. A general black hole in 2 + 1 dimensions has g handles hidden behind h horizons. The result of the analytic continuation of a black-hole spacetime is a hyperbolic 3-manifold having the topology of a handlebody. The boundary of this handlebody is a compact Riemann surface of genus G = 2g + h − 1. Conformal moduli of this surface encode in a simple way the physical characteristics of the black hole. The moduli space of black holes of a given type (g, h) is then the Schottky space at genus G. The (logarithm of the) thermodynamic partition function of the hole is the Kähler potential for the Weil–Peterson metric on the Schottky space. The Bekenstein bound on the black-hole entropy leads us to conjecture a new strong bound on this Kähler potential.