Modular Hamiltonian in flat holography in the framework of generalized minimal massive gravity

TL;DR

This paper extends the holographic understanding of modular Hamiltonians from Einstein gravity to generalized minimal massive gravity in flat spacetimes, deriving explicit forms and entanglement laws in BMS-invariant field theories.

Contribution

It derives the modular Hamiltonian for BMSFTs within the GMMG model and connects it to gravitational charges, generalizing previous Einstein gravity results.

Findings

Modular Hamiltonian for BMSFTs in GMMG derived

Holographic duals identified as gravitational charges

First law of entanglement entropy established

Abstract

Recently a general prescription for determining the vacuum modular flow generator and the corresponding modular Hamiltonian in the BMS-invariant field theories (BMSFTs) have provided by Apolo et. al \cite{Apolo}. According to this paper Einstein gravity in asymptotically flat three-dimensional spacetimes is dual to a BMSFT. In the present paper we extend this study to the generalized minimal massive gravity model (GMMG). So we study some aspects of $Flat_3/BMSFT$ holography in the framework of this model. We derive the modular Hamiltonian for single intervals in thermal and non-thermal BMSFTs and show its holographic dual is the gravitational charge associated with isometries that preserve the asymptotic form of the metric. Also using the modular Hamiltonian we derive the first law of entanglement entropy. Our results for GMMG model reduces to the result for Einstein gravity in $2+1$ dimension, when $\mu , m^2 \to \infty $, and $\sigma=-1$. These limiting cases, are a place in parameter space of model, where GMMG model reduces to the Einstein gravity.