Two Charge System Revisited: Small Black Holes or Horizonless Solutions?

TL;DR

The paper analyzes the two-charge BPS system in string theory, showing that in any fixed duality frame the microstates are captured either by small black holes or by smooth horizonless solutions, but not both. Consequently, the duality-invariant macroscopic degeneracy is obtained by summing contributions from both descriptions, with the dominant description depending on the frame. It develops a framework relating degeneracy and index, clarifies how smooth solutions and horizon-based counting contribute in different frames, and discusses puzzles such as the absence of small black holes in certain Type II compactifications. Additionally, it proposes a mechanism by which information about unprotected quantities can be lost through mixing with non-BPS states, shedding light on how protected indices relate to broader microscopic data.

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.