The Yang-Mills duals of small AdS black holes

TL;DR

This work analyzes the large-$N$ matrix model for the 4d $\uN=4$ SYM index and its truncations to illuminate the holographic AdS$_5$ BH spectrum. By combining truncated-model numerics with exact analytic saddle points at small charges, the authors connect confinement/deconfinement physics to known BH phases and derive precise small-BPS BH entropies, including BMPV-like cases with spin. They show that, in the full YM limit, the small AdS BH entropy $S(q)= sy \, ext{pi}\,ig(q^3/(27N^2)ig)^{1/2}$ (and its spinning generalization $S(q,j)= ext{pi}\,ig(q^3/(27N^2)-j^2ig)^{1/2}$) emerges from a universal Cardy-like scaling, thereby counting microstates of asymptotically flat 5d BHs regulated by AdS. They also uncover thermodynamic instabilities of BMPV-like BHs embedded in AdS due to graviton hair and extend the small-BH analysis to three charges, obtaining $S(q_1,q_2,q_3;j)= ext{pi}\,ig(q_1 q_2 q_3/N^2 - j^2ig)^{1/2}$. The results provide a principled QFT framework for understanding small AdS BH microstates and their instabilities, linking gauge theory saddle points to gravitational thermodynamics and offering a path to richer BH phenomenology through multi-cut and higher-$p$ analyses.

Abstract

We study the large $N$ matrix model for the index of 4d $\mathcal{N}=4$ Yang-Mills theory and its truncations to understand the dual AdS$_5$ black holes. Numerical studies of the truncated models provide insights on the black hole physics, some of which we investigate analytically with the full Yang-Mills matrix model. In particular, we find many branches of saddle points which describe the known black hole solutions. We analytically construct the saddle points dual to the small black holes whose sizes are much smaller than the AdS radius. They include the asymptotically flat BMPV black holes embedded in large AdS with novel thermodynamic instabilities.

Figures (9)

• Figure 1: Illustrating how the critical point $\theta_\ast$ (yellow dots) satisfying $\rho(\theta_\ast)=0$ can destroy the cut as $x$ crosses a wall. The blue lines are ${\rm Im}[s(\theta)]=0$ lines, red dots the branch points $\theta=\pm\theta_0$, and the black dots the origin $\theta=0$. The background contour plots are those for ${\rm Im}[s(\theta)]$. (Green lines are the branch cuts of $s(\theta)$ chosen by mathematica's convention.)
• Figure 2: Illustrating why (\ref{['cut-wall']}) is only a necessary condition for the wall, by the critical point meeting the ${\rm Im}[s(\theta)]=0$ line not through the cut. Color conventions are all same as Fig. \ref{['cut-change']}.
• Figure 3: The contour plots of: (a) $\frac{1}{N^2}{\rm Re}(\log Z_+)$, (b) $\frac{1}{N^2}{\rm Re}(\log Z_{\rm BH})$. The red curves denote ${\rm Re}(\log Z_+)=0$ lines. The black dotted curve is the tachyon threshold ${\rm Re}(a_1-1)=0$, right of which is the tachyonic region. The dashed blue curves denote ${\rm Re}(\log Z_{\rm BH})=0$ lines.
• Figure 4: The contour plots of $\frac{1}{N^2}{\rm Re}(\log Z_-)$ for the saddle point $g_-$. The solid red curves denote ${\rm Re}(\log Z_-)=0$ lines. The dashed red curve is the deconfinement line ${\rm Re}(\log Z_+)=0$.
• Figure 5: Saddles of Legendre transformation at real charges $0<q<\infty$ for the $g_+$ saddle (solid red) and the black holes (solid blue). We also show the red/blue dashed lines for the deconfinement and the Hawking-Page transitions, respectively. Plots shown only near the tachyonic region of $\rho_1$ ($\frac{2\pi}{3}<\phi<\pi$), whose boundary is the black dotted line.