Asymptotic symmetries of three-dimensional higher-spin gravity: the metric approach

TL;DR

This paper investigates the asymptotic symmetries of three-dimensional higher-spin gravity using the metric approach, revealing a non-linear W-algebra structure consistent with previous Chern-Simons results.

Contribution

It provides a detailed analysis of the asymptotic symmetry algebra in the metric formulation, including boundary conditions and the role of higher-spin gauge transformations.

Findings

Asymptotic symmetries form a non-linear W-algebra.

Boundary conditions are precisely defined for higher-spin fields.

Emergence of W-extended conformal structure at the boundary.

Abstract

The asymptotic structure of three-dimensional higher-spin anti-de Sitter gravity is analyzed in the metric approach, in which the fields are described by completely symmetric tensors and the dynamics is determined by the standard Einstein-Fronsdal action improved by higher order terms that secure gauge invariance. Precise boundary conditions are given on the fields. The asymptotic symmetries are computed and shown to form a non-linear W-algebra, in complete agreement with what was found in the Chern-Simons formulation. The W-symmetry generators are two-dimensional traceless and divergenceless rank-s symmetric tensor densities of weight s (s = 2, 3, ...), while asymptotic symmetries emerge at infinity through the conformal Killing vector and conformal Killing tensor equations on the two-dimensional boundary, the solution space of which is infinite-dimensional. For definiteness, only the spin 3 and spin 4 cases are considered, but these illustrate the features of the general case: emergence of the W-extended conformal structure, importance of the improvement terms in the action that maintain gauge invariance, necessity of the higher spin gauge transformations of the metric, role of field redefinitions.