AdS black holes and finite N indices

TL;DR

This paper numerically investigates the index of 4d $ ext{N}=4$ SYM with $U(N)$ gauge group, revealing its agreement with black hole entropy and supporting complex saddle point analyses for finite $N$ in the AdS/CFT correspondence.

Contribution

It provides numerical evidence for the finite $N$ index capturing black hole entropy and supports the use of complex saddle points and oscillating signs in the index analysis.

Findings

Index matches Bekenstein-Hawking entropy at moderate N.

Complex saddle points explain oscillating patterns in the index.

Universal behavior observed in a model related to the MacMahon function.

Abstract

We study the index of 4d $\mathcal{N}=4$ Yang-Mills theory with $U(N)$ gauge group, focussing on the physics of the dual BPS black holes in $AdS_5\times S^5$. Certain aspects of these black holes can be studied from finite $N$ indices with reasonably large $N^2$. We make numerical studies of the index for $N\leq 6$, by expanding it up to reasonably high orders in the fugacity. The entropy of the index agrees very well with the Bekenstein-Hawking entropy of the dual black holes, say at $N^2=25$ or $36$. Our data clarifies and supports the recent ideas which allowed analytic studies of these black holes from the index, such as the complex saddle points of the Legendre transformation and the oscillating signs in the index. In particular, the complex saddle points naturally explain the $\frac{1}{N}$-subleading oscillating patterns of the index. We also illustrate the universality of our ideas by studying a model given by the inverse of the MacMahon function.