Topologically Massive Gravity and the AdS/CFT Correspondence

TL;DR

This work constructs the AdS/CFT dictionary for three-dimensional topologically massive gravity, treating the third-order bulk dynamics on equal footing with the Einstein term. By performing a Fefferman-Graham analysis and holographic renormalization, it identifies two boundary sources: the boundary metric and an extrinsic-curvature datum, which sources a new operator that becomes the logarithmic partner of the stress tensor at the chiral point μ=1. At μ=1, the boundary theory is a logarithmic CFT with c_L=0 and c_R=3/G_N, with a logarithmic partner characterized by b = -3/G_N; away from μ=1, the dual theory exhibits negative-norm states reflecting bulk instabilities. The paper also computes BTZ-related charges and a full set of two-point functions, showing exact agreement with LCFT expectations in the μ→1 limit and providing a detailed map between bulk massive gravitons and boundary operators, including energy and anomaly structures. These results illuminate the viability and limitations of TMG as a holographic model and suggest avenues for chiral truncations and applications to condensed-matter systems.

Abstract

We set up the AdS/CFT correspondence for topologically massive gravity (TMG) in three dimensions. The first step in this procedure is to determine the appropriate fall off conditions at infinity. These cannot be fixed a priori as they depend on the bulk theory under consideration and are derived by solving asymptotically the non-linear field equations. We discuss in detail the asymptotic structure of the field equations for TMG, showing that it contains leading and subleading logarithms, determine the map between bulk fields and CFT operators, obtain the appropriate counterterms needed for holographic renormalization and compute holographically one- and two-point functions at and away from the 'chiral point' (mu = 1). The 2-point functions at the chiral point are those of a logarithmic CFT (LCFT) with c_L = 0, c_R = 3l/G_N and b = -3l/G_N, where b is a parameter characterizing different c = 0 LCFTs. The bulk correlators away from the chiral point (mu \neq 1) smoothly limit to the LCFT ones as mu \to 1. Away from the chiral point, the CFT contains a state of negative norm and the expectation value of the energy momentum tensor in that state is also negative, reflecting a corresponding bulk instability due to negative energy modes.