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

TL;DR

This work develops the metric-like formulation of three-dimensional higher-spin AdS gravity, deriving explicit boundary conditions and showing that the asymptotic symmetry algebra is a nonlinear $W$-algebra (e.g., $W_3$, $W_4$) that matches the Chern-Simons results. By analyzing spin-3 and spin-4 couplings to gravity, it reveals how conformal Killing tensors on the boundary parameterize symmetry generators and how higher-spin interactions induce nonlinearities in the charge algebra, controlled by cubic vertices and field redefinitions. The central charge remains consistent with pure gravity, and the wedge subalgebras reproduce the expected infinite-dimensional structure, supporting a robust AdS$_3$/CFT$_2$ interpretation and providing a blueprint for extending to higher spins via hs[$$]-type algebras. The results underscore the essential role of boundary conditions, improvement terms, and compensating gauge transformations in obtaining a coherent metric-based realization of higher-spin asymptotic symmetries. Overall, the paper clarifies how metric-like approaches corroborate CS findings and lay the groundwork for broader holographic applications in 3D higher-spin gravity.

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.