Why Indices Count the Total Number of Black Hole Microstates (at large N)

TL;DR

This work shows that the $Z_{BPS}$ partition function, counting BPS states without $(-1)^F$ grading, is a protected observable independent of the gauge coupling and admits a matrix-integral representation for generic 4D SCFTs. In $ ext{U}(N)$ $ ext{N}=4$ SYM, $Z_{BPS}$ localizes at large $N$ to ensembles of superconformal indices near essential singularities, providing a mechanism by which the ungraded growth of BPS states is captured without large sign oscillations. The authors identify new large-$N saddles—orbifold, dressed orbifold, and eigenvalue-instanton configurations—and develop a refined Cardy-like expansion to compute their perturbatively exact leading on-shell actions, revealing how these saddles organize the microcanonical counting. At finite $N$, the positive $Z_{BPS}$ is the natural observable for entropy corrections, while the microcanonical index can oscillate; the analysis also links the field-theory localization results to gravitational extended-BPS limits and near-horizon AdS$_2$ physics, strengthening the AdS$_5$/CFT$_4$ picture of black hole microstate counting.

Abstract

Using supersymmetric localization, we show that the partition function of four-dimensional superconformal gauge theories -- computed as a trace over BPS states without the insertion of $(-1)^F$ -- is protected and independent of the gauge coupling $g_{YM}$. We derive a matrix-integral representation of this observable for generic four-dimensional superconformal gauge theories. For $U(N)$ maximally supersymmetric Yang-Mills theory, we study such matrix integral and show that it localizes to ensembles of superconformal indices near its essential singularities. The latter asymptotic localization explains why a single microcanonical index reproduces the growth of the total number of BPS states in a large-$N$ expansion at charges of order $N^2$, despite exhibiting large sign-oscillations due to the insertion of $(-1)^F$. To compute quantum corrections to entropy, at finite $N$, the correct observable is the protected partition function which by definition is a positive quantity. To study this protected observable, we propose and test an improvement of the Cardy-like method that allows us to identify and compute perturbatively exact expressions for the leading large-$N$ onshell action of eigenvalue-configurations that we call orbifold, dressed orbifold, and eigenvalue-instanton saddles.