Aspects of three-dimensional spin-4 gravity

Abstract

We discuss some interesting holographical aspects of three dimensional higher-spin gravity with a negative cosmological constant in the framework of SL(4, $\mathbb{R}$) × SL(4, $\mathbb{R}$) 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 ${\mathcal{W}_4}$ 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, $\mathbb{R}$) × SL(N, $\mathbb{R}$) Chern-Simons theories.