BPS phases and fortuity in higher spin holography

TL;DR

This paper analyzes the ABJ vector model $U(N)_k\times U(1)_{-k}$ at weak coupling to extract a BPS spectrum that interfaces with AdS$_4$ higher-spin gravity. Using a cohomological framework, it shows that although higher-spin currents are anomalous for $\lambda\neq 0$, multi-particle BPS bound states (“fortuitous cohomologies”) arise from trace relations, and it constructs a new heavy BPS state at $N=2$. It then studies the large-$N$ index as a matrix-model problem, uncovering a high-temperature 1-cut saddle with a deconfinement threshold $j_c$ and a low-temperature 2-cut phase described by complex eigenvalue distributions and holomorphic anomaly, including an eigenvalue-instanton analysis that yields a lower bound on the transition temperature. The results illuminate how BPS microstates in a higher-spin/vector framework replicate black-hole-like thermodynamics, while revealing subtleties such as background independence and non-holomorphic corrections in the large-$N$ regime, and offering a potential bulk interpretation in terms of giant gravitons and the small/large black hole spectrum. Together, these findings advance our understanding of BPS spectra and phase structures in higher-spin holography and their connections to black-hole physics in AdS$_4$.

Abstract

We study the BPS states of $U(N)_k\times U(1)_{-k}$ vector Chern-Simons theory on a sphere at weak coupling $λ=\frac{N}{k}\ll 1$, dual to an AdS$_4$ higher spin gravity. Higher spin currents are well known to be anomalous at $λ\neq 0$. We show that these non-BPS higher spin particles form multi-particle `BPS bounds' at low energy, and interpret them as a primordial form of small black hole states. We also construct a new heavy BPS operator at $N=2$. We study the BPS phases of this system from the large $N$ index at Planckian `temperatures'. The deconfined saddles at high temperature exist only above a threshold, similar to the BTZ black holes. The low temperature saddles are given by novel 2-cut eigenvalue distributions. Their phase transition involves subtle issues like the holomorphic anomaly and the background independence, whose studies we initiate. In particular, we obtain a lower bound on the critical temperature by studying the eigenvalue instantons.

Figures (13)

• Figure 1: The cut $C$ for the single cut saddles at various values of $\gamma(j)$: $j=100$ (black), $j=0.13$ (blue), $j=0.05$ (purple). The red dots are the branch points $\theta=\pm\frac{\pi}{2}$. ($\gamma(j)$ is determined by Legendre transformation at fixed charge $j$: see below for explanations.)
• Figure 2: Plots of $\theta_0(j)$ (left) and $\gamma(j)$ (right). The red curve is for $j_c<j<500$; solid blue for $j_0<j<j_c$; dashed light blue for $0<j<j_0$ ($j_c=0.017674$, $j_0=0.013924$). For $j<j_c$, the cut $C$ does not exist but we have shown the formal results using the function (\ref{['free-1-cut']}). The black dot on the left figure is the branch point $\theta_0=\frac{\pi}{2}$.
• Figure 3: Plot of $\frac{1}{N^2}{\rm Re}[S(j)]$: colors/dash of the curve denote the same ranges as in Fig. \ref{['theta0-legendre']}.
• Figure 4: (Left) Plot of the 'real temperature' ${\rm Re}(\gamma)^{-1}$ vs. the free energy $\frac{1}{N^2}{\rm Re}(\log Z)$. (Right) Zoom-in to the cusp region $0.007<j<0.02$. (Colors/dash mean the same as in Fig. \ref{['theta0-legendre']}.)
• Figure 5: Examples of the double cut $C=C_1\cup C_2$. At fixed $\theta_2$, the dashed curve shows those $\theta_1$ which admit the saddle point solutions. The tables show the filling fractions $\nu_1\equiv \nu$, $\nu_2\equiv 1-\nu$ for the cuts and $\gamma=N\beta$. (On the left figure, $C_2$ for the three chosen $\theta_1$ are almost degenerate. On the right figure, only parts of $C_1$ and $C_2$ are shown.)