Exact Microscopic Entropy of Non-Supersymmetric Extremal Black Rings

TL;DR

Emparan demonstrates that the horizon entropy of non-supersymmetric extremal black rings is exactly reproduced by the MSW CFT of the four-charge MSW string, with central charge $c=6|n_1 n_2 n_3|$ and oscillator level chosen so that $n_p=-J_1$, yielding the Cardy entropy $S_{CFT}=2\pi\sqrt{\frac{c|q_0|}{6}}$ that matches $S_{BH}$ (e.g., $S_{BH}=2\pi\sqrt{|J_1 n_1 n_2 n_3|}$ in the dipole case). The mass is radius-dependent and can be interpreted as a Newtonian self-energy correction, consistent with an attractor-like renormalization. For extremal non-BPS charged rings the entropy formula involves the charges $N_i$ and dipoles $n_i$, and the MSW Cardy count reproduces the full expression with a specific oscillator level $q_0$; this supports the MSW dual as the microscopic description but also reveals puzzles related to the full eight-parameter freedom and the role of both CFT sectors. Overall the work provides a stringent test of holographic microscopic entropy in non-supersymmetric settings and highlights the need to understand ring excitations and spontaneous superradiance within the MSW framework.

Abstract

In this brief note we show that the horizon entropy of the largest known class of non-supersymmetric extremal black rings, with up to six parameters, is exactly reproduced for all values of the ring radius using the same conformal field theory of the four-charge four-dimensional black hole. A particularly simple case is a dipole black ring without any conserved charges. The mass gets renormalized, but the first corrections it receives can be easily understood as an interaction potential energy. Finally, we stress that even if the entropy is correctly reproduced, this only implies that one sector of chiral excitations has been identified, but an understanding of excitations in the other sector is still required in order to capture the black ring dynamics.