Black Hole Entropy Function, Attractors and Precision Counting of Microstates

TL;DR

Ashoke Sen’s notes unify the macroscopic and microscopic perspectives on extremal black hole entropy through the entropy function formalism and exact dyon counting in N=4 theories. The approach reduces entropy computation to algebraic extremization of an entropy function, demonstrates attractor behavior independent of asymptotic moduli, and shows that higher-derivative corrections (e.g., Gauss–Bonnet) and AdS$_3$-based arguments yield controlled corrections consistent with dualities. The microscopic side computes precise dyon degeneracies via modular forms, holographic Cardy-type growth, and a full partition function that factors into KK-monopole, COM, and relative D1–D5 motion, including walls of marginal stability and duality transformations. The results show striking agreement between S_BH and S_stat across large charge regimes and clarify how multi-centered black holes account for jumps in degeneracy across walls. The framework provides a robust, duality-invariant bridge between gravitational entropy and stringy microstates, with implications for precision tests of holography and nonperturbative quantum gravity.

Abstract

In these lecture notes we describe recent progress in our understanding of attractor mechanism and entropy of extremal black holes based on the entropy function formalism. We also describe precise computation of the microscopic degeneracy of a class of quarter BPS dyons in N=4 supersymmetric string theories, and compare the statistical entropy of these dyons, expanded in inverse powers of electric and magnetic charges, with a similar expansion of the corresponding black hole entropy. This comparison is extended to include the contribution to the entropy from multi-centered black holes as well.