Asymptotic growth of the 4d $\mathcal N=4$ index and partially deconfined phases

TL;DR

The paper analyzes the Cardy-like and large-$N$ asymptotics of the 4d $\mathcal{N}=4$ superconformal index, uncovering partially deconfined infinite-temperature phases on the W wings and establishing a double-scaling leading behavior that sums over many holonomy configurations. It develops both integral and Bethe Ansatz (BA) formulations, revealing standard and previously unknown non-standard eBAE solutions; continua of BA roots for $N\ge3$ point to the need for a generalized BA measure and illuminate a deep link to vacua of compactified $\mathcal{N}=1^*$ theory. In finite $N$ cases (notably $N=3,4$) the leading Cardy-like asymptotics can be computed exactly, showing partial deconfinement, while in the large-$N$ limit a conjectured leading form emerges as a sum over $C$-pack configurations, with $S_C(J_1,J_2,Q_a)=S_{\mathrm{BH}}(J_1,J_2,Q_a)/C$ reflecting microstate entropies of potential multi-center black objects. Overall, the work bridges integral and BA approaches, clarifies phase structure via center symmetry, and advances the holographic understanding of black hole microstates in AdS$_5$/CFT$_4$.

Abstract

We study the Cardy-like asymptotics of the 4d $\mathcal N=4$ index and demonstrate the existence of partially deconfined phases where the asymptotic growth of the index is not as rapid as in the fully deconfined case. We then take the large-$N$ limit after the Cardy-like limit and make a conjecture for the leading asymptotics of the index. While the Cardy-like behavior is derived using the integral representation of the index, we demonstrate how the same results can be obtained using the Bethe ansatz type approach as well. In doing so, we discover new non-standard solutions to the elliptic Bethe ansatz equations including continuous families of solutions for $SU(N)$ theory with $N\ge3$. We argue that the existence of both standard and continuous non-standard solutions has a natural interpretation in terms of vacua of $\mathcal N=1^*$ theory on $\mathbb R^3\times S^1$.

Figures (8)

• Figure 1: The qualitative behavior of the pairwise potential for the holonomies, as a function of the pair's separation, for fixed $\Delta_{1,2}$ and fixed $\arg\beta\in(0,\pi/2)$, in the two complementary regions $-1<\Delta_{1},\Delta_{2},-1-\Delta_1-\Delta_2<0$ (lower-left) and $0<\Delta_{1},\Delta_{2},1-\Delta_1-\Delta_2<1$ (upper-right) of the space of the control-parameters $\Delta_{1,2}$ (taken to be inside $\mathbb{R}$). The M and W wings switch places if $\arg\beta$ is taken to be inside $(-\pi/2,0)$ instead---c.f.ArabiArdehali:2019tdm.
• Figure 2: The functions $C^{-3}\sum_{a=1}^3\kappa(C\Delta_a)$ for $C=1$ (brown), $C=2$ (green), $C=3$ (yellow), and $C=6$ (blue).
• Figure 3: The functions $C^{-3}\sum_{a=1}^3\kappa(C\Delta_a)$ for $C=1$ (brown), $C=2$ (green), and $C=4$ (blue). The take-over of the green curve signifies the partially deconfined phase in that region when $N=4$.
• Figure 4: The difference (scaled by a factor of 12) between the numerically maximized $Q_h$, and the $Q_h$ maximized over the divisor configurations $\boldsymbol{x}_d$, on the $\arg\beta>0$ W wing, for $N=5$ on the left, and for $N=6$ on the right. When the result is zero, it means the divisor configurations are maximizing $Q_h$ (hence minimizing $\mathcal{V}_{\mathrm{eff}}$). Note the big hole in the middle for $N=5$, and the small holes for $N=6$, signalling the failure of the divisor configurations to maximize $Q_h$ (hence to minimize $\mathcal{V}_{\mathrm{eff}}$).
• Figure 5: Numerical solutions to the eBAEs (\ref{['BAE']}) with $N=11$ and $\tau={\frac{1+23i}{230}}$. Note that (a) corresponds to Case 1 ($\Delta_1+\Delta_2\leq{\frac{1}{2}}$) and (b) corresponds to Case 2 ($\Delta_1+\Delta_2>{\frac{1}{2}}$).

Theorems & Definitions (4)

• Lemma 1
• Conjecture 1
• Lemma 2
• Conjecture 2