# Emergence of spacetime from the algebra of total modular Hamiltonians

## TL;DR

This paper explores how the algebra of total modular Hamiltonians in a CFT encodes the emergence of bulk spacetime, reconstructing bulk fields with spin and relating Lie algebra structures to the bulk metric.

## Contribution

It introduces the concept of weakly-maximal Lie subalgebras generated by modular Hamiltonians and links their structure to the bulk spacetime geometry in holography.

## Key findings

- Bulk spacetime parametrizes weakly-maximal Lie subalgebras.
- Bulk metric is invariant under transformations induced by these subalgebras.
- Holographic invariance of the CFT state under modular flow is established.

## Abstract

We study the action of the CFT total modular Hamiltonian on the CFT representation of bulk fields with spin. In the vacuum of the CFT the total modular Hamiltonian acts as a bulk Lie derivative, reducing on the RT surface to a boost perpendicular to the RT surface. This enables us to reconstruct bulk fields with spin from the CFT. On fields with gauge redundancies the total modular Hamiltonian acts as a bulk Lie derivative together with a compensating bulk gauge (or diffeomorphism) transformation to restore the original gauge. We consider the Lie algebra generated by the total modular Hamiltonians of all spherical CFT subregions and define weakly-maximal Lie subalgebras as proper subalgebras containing a maximal set of total modular Hamiltonians. In a CFT state with a bulk dual, we show that the bulk spacetime parametrizes the space of these weakly-maximal Lie subalgebras. Each such weakly-maximal Lie subalgebra induces Lorentz transformations at a particular point in the bulk manifold. The bulk metric dual to a pure CFT state is invariant at each point under this transformation. This condition fixes the metric up to a conformal factor that can be computed from knowledge of the equation parametrizing extremal surfaces. This gives a holographic notion of the invariance of a pure CFT state under CFT modular flow.