Black Hole Microstate Counting and its Macroscopic Counterpart

TL;DR

The paper surveys exact microscopic counting of dyons in ${\cal N}=4$ string theories and its macroscopic interpretation via $AdS_2/CFT_1$, demonstrating that the protected helicity-index degeneracies $B_6(Q,P)$ (and its twisted CHL/type II generalizations) reproduce the exponential of the black hole entropy with controlled corrections. The macroscopic side leverages higher-derivative Wald entropy and the quantum entropy function to include both stringy and quantum effects, revealing horizon vs hair contributions and showing how $AdS_2$ saddles, including orbifolds, encode detailed microstate information such as the distribution of discrete quantum numbers. The results connect modular forms, notably the Igusa cusp form $\Phi_{10}$ and its twists, to exact microscopic degeneracies and validate the dual gravity description through walls of marginal stability and multi-centered configurations. The work also outlines practical uses of the quantum entropy function, including computations of twisted indices and logarithmic corrections, and discusses extensions to CHL models, twisted CHL, and type II compactifications, with implications for quantum gravity and future directions such as ${\cal N}=2$ generalizations and localization techniques.

Abstract

We survey recent results on the exact dyon spectrum in a class of N=4 supersymmetric string theories, and discuss how the results can be understood from the macroscopic viewpoint using AdS_2/CFT_1 correspondence. The comparison between the microscopic and the macroscopic results includes power suppressed corrections to the entropy, the sign of the index, logarithmic corrections and also the twisted index measuring the distribution of discrete quantum numbers among the microstates.