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

TL;DR

This paper demonstrates that the partition function of 4D superconformal gauge theories, computed via supersymmetric localization, is perturbatively protected and explains the growth of BPS microstates using large-charge saddle points.

Contribution

It introduces a matrix-integral representation for the protected partition function and develops a refined Cardy-like method for large-N and large-charge analysis.

Findings

Partition function is perturbatively protected and gauge coupling independent.

Microcanonical indices reproduce BPS state growth up to oscillations.

Large-charge saddle points accurately describe finite N microstates.

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 perturbatively protected and piecewise independent of the gauge coupling. We derive a matrix-integral representation of this observable at $g_{\text{YM}}=0$ for generic four-dimensional superconformal gauge theories. For $U(N)$ maximally supersymmetric Yang-Mills theory we study such a matrix integral and show that, even at finite $N$, it localizes to ensembles of superconformal indices near its essential singularities. The latter asymptotic localization projects out any potential discontinuity of the perturbatively protected partition function from zero to strong coupling and explains why single microcanonical indices reproduce the growth of the total number of BPS states in co-dimension one regions of large charges, up to large oscillations due to the insertion of $(-1)^F$. To compute quantum corrections to entropy at finite $N$ and small charges, the correct observable is the perturbatively protected partition function, which by definition is a positive quantity. 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-$N$ on-shell action of eigenvalue configurations that we call orbifold, dressed orbifold, and eigenvalue-instanton saddles. These are also saddle points of large-charge expansions at finite $N$. We test the conclusions obtained from such large-charge saddle-point analysis at $N=2$ using explicit Cauchy-residue evaluation.