Holonomies , anomalies and the Fefferman - Graham ambiguity in AdS 3 gravity

Using the Chern-Simon formulation of (2+1) gravity, we derive, for the general asymptotic metrics given by the Fefferman-Graham-Lee theorems, the emergence of the Liouville mode associated to the boundary degrees of freedom of (2+1) dimensional anti de Sitter geometries. Special attention is paid to holonomies, which are shown to relate the non-connected components of spatial infinity and are explicitly expressed in terms of the Liouville field in case of flat boundary metric. We also show the link between the variation under diffeomorphisms of the Einstein theory of gravitation and the Weyl anomaly of the conformal theory at infinity.