Reconstructing bulk equation of motion using CFT modular Hamiltonians

TL;DR

This paper explores how CFT modular Hamiltonians influence the bulk equations of motion in holography, revealing constraints that ensure gauge and diffeomorphism invariance for scalar fields interacting with gravity or gauge fields.

Contribution

It demonstrates how the action of CFT modular Hamiltonians constrains and helps reconstruct the bulk equations of motion, including gauge invariance considerations, at first order in 1/N.

Findings

Modular Hamiltonians impose constraints on bulk equations of motion.

Bulk gauge and diffeomorphism invariance are consistent with modular Hamiltonian actions.

First-order 1/N corrections are crucial for fixing bulk dynamics.

Abstract

In the framework of bulk reconstruction, we elucidate the relationship between the action of CFT modular Hamiltonians on bulk operators, the possible equation of motion for the bulk operators, and the charge distribution at infinity corresponding to such bulk fields. In particular for scalar fields interacting with gravity or with gauge fields, we show how CFT considerations of the action of the modular Hamiltonian constrain the possible bulk equation of motion to be consistent with bulk gauge invariance and diffeomorphism invariance. In fact we show that requiring that the action of the modular Hamiltonian on a dressed bulk scalar operator be compatible with some unknown simple equation of motion, fixes, under reasonable assumptions, both the equation of motion and the action of the modular Hamiltonian once the first order $\frac{1}{N}$ terms are known.