# Unitarity Problems in 3$D$ Gravity Theories

## TL;DR

This paper investigates the unitarity issues in 2+1 dimensional gravity theories, showing that extensions like f(R) do not resolve negative energy modes, and discusses potential solutions and the linearization instability of AdS spacetime.

## Contribution

It demonstrates that f(R) extensions of TMG do not fix unitarity problems and explores the behavior of perturbative modes in minimal massive gravity.

## Key findings

- f(R) extensions do not improve unitarity issues
- Massive modes have negative energy except in the chiral limit
- Possible linearization instability of AdS spacetime in the presence of log-modes

## Abstract

We revisit the problem of the bulk-boundary unitarity clash in 2 + 1 dimensional gravity theories, which has been an obstacle in providing a viable dual two-dimensional conformal field theory for bulk gravity in anti-de Sitter (AdS) spacetime. Chiral gravity, which is a particular limit of cosmological topologically massive gravity (TMG), suffers from pertur- bative log-modes with negative energies inducing a non-unitary logarithmic boundary field theory. We show here that any f(R) extension of TMG does not improve the situation. We also study the perturbative modes in the metric formulation of minimal massive gravity- originally constructed in a first-order formulation-and find that the massive mode has again negative energy except in the chiral limit. We comment on this issue and also discuss a possible solution to the problem of negative energy modes. In any of these theories, the infinitesimal dangerous deformations might not be integrable to full solutions; this suggests a linearization instability of AdS spacetime in the direction of the perturbative log-modes.