Asymptotic symmetries of three-dimensional higher-spin gravity: the   metric approach
  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. 
Andrea Campoleoni, Marc Henneaux
hep-th gr-qc