From Euclidean sources to Lorentzian spacetimes in holographic conformal field theories

Mark Van Raamsdonk,Charles Rabideau,Ali Izadi Rad,Onkar Parrikar,Donald Marolf

doi:10.1007/jhep06(2018)077

Abstract

We consider states of holographic conformal field theories constructed by adding sources for local operators in the Euclidean path integral, with the aim of investigating the extent to which arbitrary bulk coherent states can be represented by such Euclidean path-integrals in the CFT. We construct the associated dual Lorentzian spacetimes perturbatively in the sources. Extending earlier work, we provide explicit formulae for the Lorentzian fields to first order in the sources for general scalar field and metric perturbations in arbitrary dimensions. We check the results by holographically computing the Lorentzian one-point functions for the sourced operators and comparing with a direct CFT calculation. We present evidence that at the linearized level, arbitrary bulk initial data profiles can be generated by an appropriate choice of Euclidean sources. However, in order to produce initial data that is very localized, the amplitude must be taken small at the same time otherwise the required sources diverge, invalidating the perturbative approach.

Highlights

In this paper, following various earlier works [4,5,6,7], we consider a large class of states in a holographic conformal field theory obtained by adding sources for local, primary operators to the Euclidean path integral defining the vacuum state of the CFT
To obtain a classical bulk, here and below we take λ to be parametrically of the same magnitude as the CFT action in the holographic large N limt; e.g. λ is of order the central charge c for d = 2 CFTs and of order N 2 for N = 4 super Yang-Mills in d = 4. In light of these results, it is clearly of interest to understand in more detail the map between Euclidean path-integral states (1.1) and Lorentzian spacetimes, provide additional checks that the states (1.1) are holographic, and investigate to what extent an arbitrary classical bulk solution can be described by a state of the form (1.1)
We could choose to restrict the form of γ in order to avoid these redundancies; we will leave the form of γ general so that our results are applicable to any chosen description of the boundary geometry

Summary

Introduction

In this paper, following various earlier works [4,5,6,7], we consider a large class of states in a holographic conformal field theory obtained by adding sources for local, primary operators (dual to classical bulk fields) to the Euclidean path integral defining the vacuum state of the CFT. To obtain a classical bulk, here and below we take λ to be parametrically of the same magnitude as the CFT action in the holographic large N limt; e.g. λ is of order the central charge c for d = 2 CFTs and of order N 2 for N = 4 super Yang-Mills in d = 4 In light of these results, it is clearly of interest to understand in more detail the map between Euclidean path-integral states (1.1) and Lorentzian spacetimes, provide additional checks that the states (1.1) are holographic, and investigate to what extent an arbitrary classical bulk solution can be described by a state of the form (1.1). Can we associate a path-integral state to an arbitrary Lorentzian geometry? Are there multiple path-integral states which correspond to the same geometry? Are there some CFT states with a good classical gravity description that cannot be described or well-approximated in this way?

Objectives

Results

Conclusion