Chiral Gravity, Log Gravity and Extremal CFT

TL;DR

This work analyzes chiral gravity, a special case of three‑dimensional topologically massive gravity at the chiral point, around the $AdS_3$ vacuum. It identifies a linearization instability: nonchiral linearized modes produce infrared divergences at second order and cannot be extended to exact solutions, leaving a positive‑energy, right‑moving spectrum and a Birkhoff‑like result that stationary, axially symmetric solutions are BTZ black holes. The authors then introduce log gravity with relaxed boundary conditions, show it is nonchiral yet finite in charges, and demonstrate a decoupled chiral gravity sector within it; they compute the Euclidean torus partition function and show it matches the holomorphic extremal CFT partition function, suggesting a consistent quantum theory of chiral gravity linked to extremal CFTs. The paper also outlines how log gravity could correspond to a logarithmic extremal CFT and discusses the holographic implications, including the modular invariance and integrality of the expansion coefficients, which support a quantum dual description. Together, these results connect classical perturbative stability, nonlinear constraints, and quantum modular structure to propose a coherent picture in which chiral gravity (or its logarithmic extension) may realize a novel 2D conformal field theory dual, with BTZ microstates counting captured by an extremal (or log extremal) CFT.

Abstract

We show that the linearization of all exact solutions of classical chiral gravity around the AdS3 vacuum have positive energy. Non-chiral and negative-energy solutions of the linearized equations are infrared divergent at second order, and so are removed from the spectrum. In other words, chirality is confined and the equations of motion have linearization instabilities. We prove that the only stationary, axially symmetric solutions of chiral gravity are BTZ black holes, which have positive energy. It is further shown that classical log gravity-- the theory with logarithmically relaxed boundary conditions --has finite asymptotic symmetry generators but is not chiral and hence may be dual at the quantum level to a logarithmic CFT. Moreover we show that log gravity contains chiral gravity within it as a decoupled charge superselection sector. We normally evaluate the Euclidean sum over geometries of chiral gravity and show that it gives precisely the holomorphic extremal CFT partition function. The modular invariance and integrality of the expansion coefficients of this partition function are consistent with the existence of an exact quantum theory of chiral gravity. We argue that the problem of quantizing chiral gravity is the holographic dual of the problem of constructing an extremal CFT, while quantizing log gravity is dual to the problem of constructing a logarithmic extremal CFT.