Unravelling Holographic Entanglement Entropy in Higher Spin Theories

TL;DR

The work addresses how to define and compute holographic entanglement entropy in AdS$_3$ higher spin gravity and shows that two seemingly different Wilson-line proposals are equivalent for SL$(N,\mathbb{R})$ CS theory. It develops two systematic methods to evaluate the Wilson line for arbitrary $N$, including a path-integral saddle-point framework and a small-interval expansion, and demonstrates exact agreement with the universal short-interval EE correction in a CFT$_2$ for finite-charge backgrounds. The results also extend to thermal entropy via Wilson loops and to finite-charge backgrounds under Drinfeld-Sokolov boundary conditions, providing a comprehensive check against CFT$_2$ expectations and known spin-3 cases. The paper thereby strengthens the interpretation of Wilson lines as a gauge-covariant generalization of geometric notions of entanglement and geometry in higher-spin holography, with implications for emergent geometry and universal EE behavior. Overall, it delivers practical tools and consistency checks for EE in higher-spin AdS$_3$/CFT$_2$ across embeddings and representations, paving the way for further explorations of entanglement in theories with infinitely many higher-spin fields.

Abstract

There are two proposals that compute holographic entanglement entropy in AdS$_3$ higher spin theories based on $SL(N,\mathbb{R})$ Chern-Simons theory. We show explicitly that these two proposals are equivalent. We also designed two methods that solve systematically the equations for arbitrary $N$. For finite charge backgrounds in AdS$_3$, we find exact agreement between our expressions and the short interval correction of the entanglement entropy for an excited state in a CFT$_2$.