Extremal Black Holes as Bound States

TL;DR

The paper addresses whether certain extremal dilatonic black holes in string theory can be interpreted as bound states of fundamental $a=\sqrt{3}$ black holes with zero binding energy. It constructs a static multi-centered solution comprising four $a=\sqrt{3}$ constituents charged under different U(1) fields, and demonstrates a no-force condition that allows arbitrary separations and yields no net binding energy. The solution interpolates among the standard dilaton couplings $a=\sqrt{3}, 1, 1/\sqrt{3}, 0$ as holes sit at infinity, thereby proving the bound-state hypothesis for these cases. It then generalizes this interpretation to broader classes of dyonic black holes in toroidally compactified string theory, including cases not captured by a single scalar truncation, demonstrating a wide applicability of the bound-state picture and clarifying the role of scalar forces in extremal solutions.

Abstract

We consider a simple static extremal multi-black hole solution with constituents charged under different $U(1)$ fields. Each of the constituents by itself is an extremal dilatonic black hole of coupling $a=\srt$. For a special case with two electrically and two magnetically charged black holes the multi-black hole solution interpolates between the familiar $a=\sqrt{3},1,\frac{1}{\sqrt{3}}$ and $0$ solutions, depending on how many black holes are placed at infinity. This proves the hypothesis that black holes with the above dilaton couplings arise in string theory as bound states of fundamental $a=\sqrt{3}$ states with zero binding energy. We also generalize the result to states where the action does not admit a single scalar truncation and show that a wide class of dyonic black holes in toroidally compactified string theory can be viewed as bound states of fundamental $a=\srt$ black holes.