Locality, bulk equations of motion and the conformal bootstrap

Abstract

We develop an approach to construct local bulk operators in a CFT to order $1/N^2$. Since 4-point functions are not fixed by conformal invariance we use the OPE to categorize possible forms for a bulk operator. Using previous results on 3-point functions we construct a local bulk operator in each OPE channel. We then impose the condition that the bulk operators constructed in different channels agree, and hence give rise to a well-defined bulk operator. We refer to this condition as the "bulk bootstrap." We argue and explicitly show in some examples that the bulk bootstrap leads to some of the same results as the regular conformal bootstrap. In fact the bulk bootstrap provides an easier way to determine some CFT data, since it does not require knowing the form of the conformal blocks. This analysis clarifies previous results on the relation between bulk locality and the bootstrap for theories with a $1/N$ expansion, and it identifies a simple and direct way in which OPE coefficients and anomalous dimensions determine the bulk equations of motion to order $1/N^2$.

37 pages, LaTeX. v2: added references and comments on OPE convergence, version to appear in JHEP

References in corpus (6)

• Holography from Conformal Field Theory
• Writing CFT correlation functions as AdS scattering amplitudes
• Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks
• A Stereoscopic Look into the Bulk
• cMERA as Surface/State Correspondence in AdS/CFT
• Bulk-Boundary Duality, Gauge Invariance, and Quantum Error Correction

Cited by in corpus (18)

• Flat space physics from AdS/CFT
• Exact Virasoro Blocks from Wilson Lines and Background-Independent Operators
• An Exact Operator That Knows Its Location
• Local bulk physics from intersecting modular Hamiltonians
• Bulk metric reconstruction from boundary entanglement
• AdS/CFT Correspondence in Operator Formalism
• Equivalent Equations of Motion for Gravity and Entropy
• Bulk Locality in AdS/CFT Correspondence
• Explicit reconstruction of the entanglement wedge
• The AdS${}_3$ Propagator and the Fate of Locality
• The Bulk-to-Boundary Propagator in Black Hole Microstate Backgrounds
• Renormalization group flows in AdS and the bootstrap program
• Classical Limit of Large N Gauge Theories with Conformal Symmetry
• Does boundary quantum mechanics imply quantum mechanics in the bulk?
• Dressing bulk fields in AdS${}_3$
• AdS${}_3$ Reconstruction with General Gravitational Dressings
• From conformal correlators to analytic S-matrices: CFT${}_1$/QFT${}_2$
• Reconstructing bulk equation of motion using CFT modular Hamiltonians