Two charge system revisited: small black holes or horizonless solutions?

Abstract

A two charge system in string theory preserving eight supercharges can be described as a small black hole that has zero entropy in the supergravity approximation, but classical higher derivative corrections produce a finite entropy in accordance with the prediction of microstate counting. On the other hand for the same system one can construct smooth horizonless classical solutions whose geometric quantization describes the individual microstates which contribute to the entropy. In this note we point out that there is no duality frame in which the system admits both these classical descriptions. Thus in a given duality frame horizonless classical solutions and small black holes are not alternate descriptions of the same system; their contributions must be added to get a duality invariant result for the macroscopic degeneracy. We discuss the significance of this observation for the macroscopic computation of the degeneracy of BPS states for a general system. We also discuss the relationship between the degeneracy and the index computation and address the puzzle regarding absence of small black holes in toroidally compactified type II string theory.