Emergence of spacetime from the algebra of total modular Hamiltonians

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Springer

Received: February 3, 2019 Accepted: April 23, 2019 Published: May 3, 2019

Daniel Kabata and Gilad Lifschytzb a

Department of Physics and Astronomy, Lehman College, City University of New York, Bronx NY 10468, U.S.A. b Department of Mathematics and Haifa Research Center for Theoretical Physics and Astrophysics, University of Haifa, Haifa 31905, Israel

E-mail: [email protected], [email protected] 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. Keywords: AdS-CFT Correspondence, Conformal Field Theory ArXiv ePrint: 1812.02915

c The Authors. Open Access, Article funded by SCOAP3 .

https://doi.org/10.1007/JHEP05(2019)017

JHEP05(2019)017

Emergence of spacetime from the algebra of total modular Hamiltonians

Contents 1

2 Modular Hamiltonian as a bulk Lie derivative 2.1 Scalars 2.2 Massive vectors 2.3 Gauge fields 2.4 Gravity

3 3 4 5 6

3 Emergence of the bulk spacetime 3.1 Special case: CFT vacuum

7 11

4 Conclusions

14

A Scalar fields A.1 ∆ = d case A.2 ∆ = d − 1 case

14 15 16

B Massive vector fields

16

C Reconstructing massive vectors in AdS3 C.1 Practicalities C.2 Example

17 19 20

D Gauge fields

20

E Gravity

21

1

Introduction

The AdS/CFT correspondence [1] re-packages boundary CFT properties into an effective higher-dimensional gravity theory. Understanding this equivalence from the CFT point of view has been the focus of many studies. A relationship [2] which underlies the present work is the identification between the bulk total modular Hamiltonian and the CFT total modular Hamiltonian.1 One aspect of this identification is that the CFT total modular Hamiltonian should act on CFT representations of bulk fields in the same way that the bulk total modular Hamiltonian acts on bulk fields in an e