On the positivity of black hole degeneracies in string theory

Abstract

Certain helicity trace indices of charged states in N = 4 and N = 8 superstring theory have been computed exactly using their explicit weakly coupled microscopic descrip- tion. These indices are expected to count the exact quantum degeneracies of black holes carrying the same charges. In order for this interpretation to be consistent, these indices should be positive integers. We prove this positivity property for a class of four/five dimen- sional black holes in type II string theory compactified on T 6 /T 5 and on K3 ×T 2 /S 1 . The proof relies on the mock modular properties of the corresponding generating functions. 1. Introduction and statement of results The study of supersymmetric black holes in string theory has been very effective in shedding light on the issue of black hole entropy. The strength of the string theoretic approach lies in the fact that there are two related descriptions of charged black holes. The first (strong string coupling, macroscopic) description is a low energy effective description as general relativity coupled to a set of matter fields. In the second (weak string coupling, microscopic) descrip- tion, a generic state of the theory with the same charges as the black hole is identified as a collection of fundamental objects of string theory namely fundamental strings and branes. The fluctuations of these objects make up the elementary excitations (microstates) of the theory, which can be described by conventional quantum field theoretic methods. The key idea is to identify these two descriptions valid at strong and weak coupling, respectively. At strong coupling, the excitations of the strings and branes exert a gravitational force on each other, and the black hole can be thought of as a quantum mechanical bound state of these microstates. In a class of supersymmetric string theories with sixteen or more unbroken supercharges we now have a practically complete understanding of the spectrum of BPS states (see (27) for a relatively recent review). One can therefore subject the above idea to high precision tests, by comparing the statistical entropy of the ensemble of states and an appropriately defined thermodynamic entropy of the corresponding BPS black hole, beyond a large charge approximation. Since we know the microscopic degeneracies exactly, one can even aim for an