Holonomies, anomalies and the Fefferman-Graham ambiguity in AdS3 gravity

TL;DR

This work shows that the Einstein-Hilbert action for AdS$_3$ gravity is equivalent to a Liouville action defined on the spatial boundary, once one passes through a Chern-Simons formulation and carefully accounts for boundary terms and holonomies. The Fefferman-Graham expansion is extended to curved boundary metrics, revealing a Liouville boundary mode whose dynamics are governed by a Liouville action $S_L$ with additional boundary contributions; the subleading FG data fix the boundary metric via $ ext{g}^{(2)}_{ab}$, and holonomies couple the boundary components. A central result is the precise relationship between the variation of the 3D diffeomorphism (gravity) action and the Weyl anomaly of the boundary Liouville theory, tying bulk diffeomorphism invariance to boundary conformal structure. The analysis suggests a generalized holographic perspective with multiple boundary components and holonomies encoding black hole microphysics, though quantum corrections and determinants require further study. Overall, the paper provides a concrete bridge between bulk AdS$_3$ gravity, boundary Liouville dynamics on curved backgrounds, and holonomy-induced couplings, with implications for holography and black hole entropy.

Abstract

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. Holonomies are described through multi-valued gauge and Liouville fields and are found to algebraically couple the fields defined on the disconnected components of spatial infinity. In the case of flat boundary metrics, explicit expressions are obtained for the fields and holonomies. 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.