Aspects of Three-dimensional Spin-4 Gravity

TL;DR

This work analyzes three-dimensional higher-spin gravity with negative cosmological constant in the $SL(4,\mathbb{R})\times SL(4,\mathbb{R})$ Chern-Simons framework, focusing on spin-4 dynamics and ${\cal W}_4$ symmetry. It constructs a spin-4 generalization of the BTZ solution and derives the corresponding Ward identities and OPEs from bulk equations with spin-3 and spin-4 sources, establishing a holographic dictionary for higher-spin operators. By implementing a Wilson-loop holonomy prescription, it shows that the thermodynamics of the spin-4 configuration is consistent with the BTZ limit and exhibits integrability relations that tie bulk charges to boundary data. The analysis also covers non-principal $SL(2,\mathbb{R})$ embeddings, their AdS vacua, and the smoothness of higher-spin fields under appropriate gauge choices, providing concrete evidence for the broader program of viewing spacetime geometry in higher-spin gravity as encoded in CS holonomies. These results bolster the holographic interpretation of higher-spin spacetimes and offer a concrete testbed for the Gaberdiel-Gopakumar duality in a finite-N setting with $N=4$ and beyond.

Abstract

We discuss some interesting holographical aspects of three-dimensional higher-spin gravity with a negative cosmological constant in the framework of SL(4, R) \times SL(4, R) Chern-Simons theory. Using a recently found technique, we construct explicitly a solution that can be interpreted as spin-4 generalization of the BTZ solution, and demonstrate how W_4 symmetry and the higher-spin Ward identities arise from the bulk equations of motion coupled to spin-3 and spin-4 currents. We match the eigenvalues of a Wilson loop along the time-like direction of the BTZ to that of the spin-4 solution, and show that this yields remarkably consistent gravitational thermodynamics for the latter. This furnishes an important, concrete supporting example for a recent proposal to understand spacetime geometries in three-dimensional higher-spin gravity formulated via SL(N, R) \times SL(N, R) Chern-Simons theories.