A 4d N=1 Cardy Formula

TL;DR

This work extends the Cardy-type high-temperature analysis of supersymmetric indices to 4d ${\cal N}=1$ SCFTs by introducing a modified index with a phase $e^{\pi i R}$ and allowing complex chemical potentials. The leading asymptotics are encoded by conformal anomalies and expressed in closed form in terms of the central charges $a$ and $c$, yielding a universal Cardy formula and, via a Legendre transform, a microcanonical entropy that scales as $\mathrm{Re}(S) \sim (3c-2a)^{1/3} J^{2/3}$ for large angular momenta $J$. The approach is corroborated by a background-field derivation on $S^3$ and a comprehensive saddle-point analysis across a broad family of Lagrangian and non-Lagrangian theories, with the dominant saddle at the origin in holonomy space. For holographic theories with $a\approx c$, the results reproduce the Bekenstein-Hawking entropy of large AdS$_5$ black holes, and the formalism naturally incorporates flavor symmetries via anomaly coefficients. Overall, the paper provides a robust anomaly-based Cardy regime for ${\cal N}=1$ SCFTs and connects microscopic index data to macroscopic black hole entropy in AdS$_5$/CFT$_4$.

Abstract

We study the asymptotic behavior of the (modified) superconformal index for 4d $\mathcal{N} = 1$ gauge theory. By considering complexified chemical potential, we find that the `high-temperature limit' of the index can be written in terms of the conformal anomalies $3c-2a$. We also find macroscopic entropy from our asymptotic free energy when the Hofman-Maldacena bound $1/2 < a/c < 3/2$ for the interacting SCFT is satisfied. We study $\mathcal{N} = 1$ theories that are dual to AdS$_5 \times Y^{p, p}$ and find that the Cardy limit of our index accounts for the Bekenstein-Hawking entropy of large black holes.

Figures (7)

• Figure 1: $(\text{Re}\mathcal{F},\,\text{Im}\mathcal{F})|_{\Delta = -i \pi}$ of rank-1 SQCDs. The blue/orange line is $\text{Re}\mathcal{F}$/$\text{Im}\mathcal{F}$.
• Figure 2: The contour plots of $\text{Im}\mathcal{F}|_{\Delta = -i \pi}$ for rank-2 SQCDs. The brighter/darker region has bigger/smaller value. The $Sp(2)$ plots are omitted since ${\cal F}_{Sp(2)}^{N_f} - {\cal F}_{SO(5)}^{N_f} = \text{(const)}$. The white lines are located at the cusps at which the $\text{Li}_3$ function jumps between the branches of \ref{['eq:Li3id']}. The function is still smooth, and no additional saddle point exists on those lines.
• Figure 3: $(\text{Re}\mathcal{F},\,\text{Im}\mathcal{F})|_{\Delta = -i \pi}$ of rank-1 SYM with $N_a=2,3$ adjoint chiral multiplets.
• Figure 4: Contour plot of $\text{Im}\mathcal{F}|_{\Delta = -i \pi}$ for rank-2 SYM with $N_a=2,3$ adjoint chiral multiplets.
• Figure 5: $\text{Im}\mathcal{F}|_{\Delta = -i \pi}$ for $SU(2)$ ISS model, $SO(5)$ BCI model, and $SU(3)$ Pouliot model.