Emergence of spacetime from the algebra of total modular Hamiltonians (1812.02915v3)

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.