Entanglement Entropy and Higher Spin Holography in AdS$_3$

TL;DR

The paper develops a Wilson-line based holographic entanglement entropy functional for AdS_3 higher spin theories, enabling EE calculations in CFTs with W_N symmetry. It recovers standard EE results in the pure gravity limit (N=2) and, via carefully chosen representations linked to the embedding (principal vs diagonal), reproduces the correct thermal entropy in high-temperature limits. The authors apply the framework to SL(3,R)xSL(3,R) backgrounds (RG flow, charged BTZ, and spin-3 black holes), deriving explicit EE expressions across BTZ, high-T, zero-T, and short-distance regimes, and they discuss strong subadditivity and potential breakdowns at short distances. The work offers a concrete bulk procedure for entanglement in higher spin holography, highlights the relationship between canonical and holomorphic entropy formulations, and outlines future directions including Rényi entropies and hs[λ] generalizations.

Abstract

A holographic correspondence has been recently developed between higher spin theories in three-dimensional anti-de Sitter space (AdS_3) and two-dimensional Conformal Field Theories (CFTs) with extended symmetries. A class of such dualities involves SL(N,R)\times SL(N,R) Chern-Simons gauge theories in the (2+1)-dimensional bulk spacetime, and CFTs with W_N symmetry algebras on the (1+1)-dimensional boundary. The topological character of the Chern-Simons theory forces one to reconsider standard geometric notions such as black hole horizons and entropy, as well as the usual holographic dictionary. Motivated by this challenge, in this note we present a proposal to compute entanglement entropy in the W_N CFTs via holographic methods. In particular, we introduce a functional constructed from Wilson lines in the bulk Chern-Simons theory that captures the entanglement entropy in the CFTs dual to standard AdS_3 gravity, corresponding to SL(2,R)\times SL(2,R) gauge group, and admits an immediate generalization to the higher spin case. We explicitly evaluate this functional for several known solutions of the Chern-Simons theory, including charged black holes dual to thermal CFT states carrying higher spin charge, and show that it reproduces expected features of entanglement entropy, study whether it obeys strong subadditivity, and moreover show that it reduces to the thermal entropy in the appropriate limit.

Figures (6)

• Figure 1: Minimal surface with non-trivial bulk topology in the R-T prescription. The interior of the black hole horizon is represented by the grey shaded area. Left figure: for a small boundary region (yellow), the minimal surface (red) is given by a connected geodesic. Right figure: for a large boundary region (yellow), the minimal surface (red) is disconnected and includes a component that wraps around the horizon, effectively computing its area and hence the black hole thermal entropy.
• Figure 2: Relevant configurations for two disjoint intervals on the boundary. The pairing $(a_{1},b_{1})\,$ and $(a_{2},b_{2})$ is excluded by a condition on the homology of the bulk configuration.
• Figure 3: Configuration for overlapping intervals.
• Figure 4: Left: $F\left(\frac{\Delta x}{\beta},C\right)$ as a function of $\pi\frac{\Delta x}{\beta}$ for fixed $\mu/\beta$ (fixed $C$). The red curve corresponds to the result in the absence of higher spin charges, $F\left(\frac{\Delta x}{\beta},\infty\right)=\sinh\left(\pi\frac{\Delta x}{\beta}\right)$. The blue curves correspond to the higher spin result for different values of $C \in \left[10,1000\right]$ ($\mu/\beta \in \left[0.0038,0.035\right]$). Right: Zoom into the short-distance regime suggesting a redefinition of the cutoff.
• Figure 5: Dots: Numerically-determined values of $\Delta x/\beta$ at which the entanglement expression breaks down, as a function of $1/C$. Solid line: $\mu /\beta$ as a function of $1/C$.