Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields

TL;DR

The paper analyzes how higher-spin fields coupled to three-dimensional AdS gravity exhibit extended asymptotic symmetries beyond the Brown–Henneaux conformal symmetry. Using a Chern–Simons formulation with gauge algebras like SL$(3)\times$SL$(3)$ (spin-3) and more generally SL$(n)\times$SL$(n)$, it derives boundary conditions and performs a Drinfeld–Sokolov-type reduction to show that two copies of ${\cal W}$-algebras appear at the boundary, with central charge $c=\frac{3l}{2G}$ (the Brown–Henneaux value). The spin-3 case yields ${\cal W}_3\otimes{\cal W}_3$, with boundary currents ${\cal L}(\theta)$ and ${\cal W}(\theta)$ obeying a classical ${\cal W}_3$ algebra; the metric-like fields relate to frame-like CS variables and exhibit Brown–Henneaux fall-off. The results generalize to finite towers of higher spins, linking the asymptotic symmetries to DS reduction to ${\cal W}_n$ algebras for SL$(n)$, and establishing a robust boundary CFT/Toda-type structure as a 3D toy model for higher-spin holography.

Abstract

We discuss the emergence of W-algebras as asymptotic symmetries of higher-spin gauge theories coupled to three-dimensional Einstein gravity with a negative cosmological constant. We focus on models involving a finite number of bosonic higher-spin fields, and especially on the example provided by the coupling of a spin-3 field to gravity. It is described by a SL(3) \times SL(3) Chern-Simons theory and its asymptotic symmetry algebra is given by two copies of the classical W_3-algebra with central charge the one computed by Brown and Henneaux in pure gravity with negative cosmological constant.