Topologically Massive Gravity and the AdS/CFT Correspondence (0906.4926v2)

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.

• The paper establishes a holographic dictionary for Topologically Massive Gravity by deriving asymptotic solutions at the chiral point.
• It demonstrates that one- and two-point functions at μ=1 align with logarithmic CFT behavior through extensive holographic renormalization.
• It identifies negative energy modes for μ≠1, highlighting potential bulk instabilities and implications for TMG's viability as a fundamental theory.

Topologically Massive Gravity and the AdS/CFT Correspondence

This paper investigates the application of the AdS/CFT correspondence to Topologically Massive Gravity (TMG), a three-dimensional gravity theory that adds gravitational Chern-Simons terms to the Einstein-Hilbert action. The authors aim to establish a rigorous holographic dictionary for TMG, accounting for its unique properties, especially at the so-called "chiral point" where the dimensionless parameter $\mu$ equals 1, suggesting potentially chiral and stable behavior contrary to the typical instability introduced by higher-derivative terms.

Asymptotic Structure and Solution

The core of the paper focuses on adapting the holographic framework specific to TMG's properties by deriving asymptotic solutions for the field equations. This required exploring boundary conditions uniquely relevant to TMG. A significant finding is the presence of leading and subleading logarithms in the asymptotic structure, which indicate deviations from typical asymptotically Anti-de Sitter (AdS) behaviors. Particularly at the chiral point ($\mu = 1$), these solutions demonstrate characteristics of logarithmic conformal field theories (LCFTs).

Renormalization and Two-Point Functions

The paper elaborates on constructing the holographic dictionary for TMG, detailing how boundary fields correspond to sources for dual operators in the conformal field theory (CFT). This necessitates comprehensive holographic renormalization, including calculating one-point and two-point functions not only at the chiral point but also for $\mu \neq 1$. At $\mu = 1$, the two-point functions align with LCFT expectations, confirming the presence of a logarithmic sector in the boundary CFT. Notably, the paper identifies negative norm states and negative energy modes in the bulk at $\mu \neq 1$, revealing potential instabilities in the theory.

Implications and Theoretical Insights

The paper's findings have both theoretical implications and practical relevance. The identification of the LCFT structure at the chiral point amplifies the intriguing characteristics of TMG, suggesting potential applications in condensed matter systems where such dualities might be beneficial. The analysis posits that full TMG could face challenges as a fundamental theory due to issues with positivity and unitarity. Nevertheless, the paper opens avenues for considering purely right-moving sectors or exploring relaxations of boundary conditions to achieve theoretical consistency.

Future Directions

Given the intricate nature of TMG and its peculiar boundary behaviors, future work might extend toward warped AdS solutions or explore similarities with other gravitational theories like new massive gravity. Furthermore, refining the understanding of the holographic dual of TMG's right-moving sector could prove pivotal for unlocking applications beyond theoretical confines.

This research thus bridges crucial aspects of gravitational physics with advanced theoretical frameworks, laying groundwork for further investigations into the richly complex tapestry of holographic dualities in lower-dimensional gravity theories.