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

TL;DR

This paper examines the dual descriptions of a two charge system in string theory, highlighting that small black holes and horizonless solutions are not dual to each other but contribute separately to the system's entropy, impacting the understanding of BPS state degeneracies.

Contribution

It demonstrates that small black holes and horizonless solutions are not dual descriptions in the same frame, affecting how their contributions are combined for entropy calculations.

Findings

Small black holes have zero entropy in supergravity but gain finite entropy from higher derivative corrections.

Horizonless solutions represent microstates but are not dual to small black holes in any frame.

Contributions of black holes and horizonless solutions must be summed for duality-invariant degeneracy.

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.