# Anomalous Gravitation and its Positivity from Entanglement

## TL;DR

This paper investigates how gravitational phenomena, including anomalies and positivity, can emerge from quantum entanglement in holographic theories with gravitational anomalies, providing new insights into the duality between bulk gravity and boundary CFTs.

## Contribution

It derives the holographic dictionary for the energy-momentum tensor in topologically massive gravity and links entanglement properties to gravitational energy positivity.

## Key findings

- Holographic dictionary for TMG energy-momentum tensor derived.
- Linearized TMG equations obtained from entanglement considerations.
- Positivity of gravitational energy linked to relative entropy in anomalous CFTs.

## Abstract

We explore the emergence of gravitation from entanglement in holographic CFTs with gravitational anomalies. More specifically, the holographic correspondence between topologically massive gravity (TMG) with gravitational Chern-Simons term in the 3D bulk and its dual CFT with unbalanced left and right moving central charges on the 2D boundary, is studied from the quantum entanglement perspective. Using the first law of entanglement, we derive the holographic dictionary of the energy-momentum tensor in TMG, including the chiral case with logarithmic mode. Furthermore, we show that the linearized equation of motion of TMG can also be obtained from entanglement using the Wald-Tachikawa covariant phase space formalism. Finally, we identify a quasi-local gravitational energy in the entanglement wedge as the holographic dual of relative entropy in gravitationally anomalous CFTs. The positivity and monotonicity of relative entropy imply that such a gravitational energy should be positive definite and become larger when increasing the size of the entanglement wedge. These constraints from quantum information may be potentially used to discuss the UV inconsistent issues of TMG.