Discrete Information from CHL Black Holes

TL;DR

This work tests the AdS_2/CFT_1 prediction for CHL black holes by computing a protected $Z_N$ twisted helicity index, showing that $|d(\vec{q})| \sim \exp[S_{BH}(\vec{q})/N]$ with $S_{BH}(\vec{q}) \simeq \pi\sqrt{Q^2 P^2 - (Q\cdot P)^2}$. The author constructs the microscopic index via a D1-D5-KK monopole system, expressing the twisted partition function as $Z(\rho,\sigma,v) = -1/\widetilde{\Phi}(\rho,\sigma,v)$, where $\widetilde{\Phi}$ is a modular form encoding the orbifold data and RR-sector twists. In the large-charge limit, the growth is dominated by poles of $1/\widetilde{\Phi}$, leading to $|d(\vec{q})| \sim \exp[\pi\sqrt{Q^2 P^2 - (Q\cdot P)^2}/N]$, in exact agreement with $S_{BH}(\vec{q})/N$ and confirming that black holes retain information about microstate distributions beyond degeneracy. The results reinforce the view that string theory around black hole backgrounds captures refined microstate structure and illustrate the utility of protected indices in linking microscopic counts to macroscopic entropy in CHL models.

Abstract

AdS_2/CFT_1 correspondence predicts that the logarithm of a Z_N twisted index over states carrying a fixed set of charges grows as 1/N times the entropy of the black hole carrying the same set of charges. In this paper we verify this explicitly by calculating the microscopic Z_N twisted index for a class of states in the CHL models. This demonstrates that black holes carry more information about the microstates than just the total degeneracy.