From Lorentz-Chern-Simons to Massive Gravity in 2+1 Dimensions

TL;DR

This paper generalizes Chiral Gravity in 2+1 dimensions using a Chern-Simons approach, introducing a scalar mode related to torsion, and explores its dual conformal field theory and exact solutions beyond standard models.

Contribution

It introduces a novel generalization of Chiral Gravity incorporating torsion and scalar modes, expanding the solution space and dual CFT interpretations.

Findings

Includes Chiral Gravity as a special case.

Finds new exact solutions not present in TMG or GR.

Proposes a dual CFT with asymmetric central charges.

Abstract

We propose a generalization of Chiral Gravity, which follows from considering a Chern-Simons action for the spin connection with anti-symmetric contorsion. The theory corresponds to Topologically Massive Gravity at the chiral point non-minimally coupled to an additional scalar mode that gathers the torsion degree of freedom. In this setup, the effective cosmological constant (the inverse of the curvature radius of maximally symmetric solutions) is either negative or zero, and it enters as an integration constant associated to the value of the contorsion at infinity. We explain how this is not in conflict with the Zamolodchikov's $c$-theorem holding in the dual boundary theory. In fact, we conjecture that the theory formulated about three-dimensional Anti-de Sitter space is dual to a two-dimensional conformal field theory whose right- and left-moving central charges are given by $c_{R}=24k$ and $c_{L}=0$, respectively, being $k$ the level of the Chern-Simons action. We study the classical theory both at the linear and non-linear level. In particular, we show how Chiral Gravity is included as a special sector. In addition, the theory has other sectors, which we explore; we exhibit analytic exact solutions that are not solutions of Topologically Massive Gravity (and, consequently, neither of General Relativity) and still satisfy Brown-Henneaux asymptotically AdS$_{3}$ boundary conditions.