[Paper Review] Bulk Locality and Boundary Creating Operators

Yu Nakayama, Hirosi Ooguri|arXiv (Cornell University)|Jul 15, 2015

Black Holes and Theoretical Physics4 citations

TL;DR

This paper formulates a minimal compatibility condition between conformal symmetry in CFT and bulk isometry in AdS, showing that bulk local operators are precisely linear superpositions of twisted Ishibashi states—boundary operators with imaginary time-dilatation—generalizing Miyaji et al.'s 3D construction to arbitrary dimensions and $1/N$ corrections. The key result is that such superpositions satisfy free field equations in AdS and can be systematically corrected to include interactions via perturbative modifications of the bulk-boundary map.

ABSTRACT

We formulate a minimum requirement for CFT operators to be localized in the dual AdS. In any spacetime dimensions, we show that a general solution to the requirement is a linear superposition of operators creating spherical boundaries in CFT, with the dilatation by the imaginary unit from their centers. This generalizes the recent proposal by Miyaji et al. for bulk local operators in the three dimensional AdS. We show that Ishibashi states for the global conformal symmetry in any dimensions and with the imaginary dilatation obey free field equations in AdS and that incorporating bulk interactions require their superpositions. We also comment on the recent proposals by Kabat et al., and by H. Verlinde.

Motivation & Objective

• To identify a minimal, physically consistent condition for CFT operators to represent bulk local operators in AdS.
• To generalize Miyaji et al.'s construction of bulk local operators in 3D AdS to arbitrary spacetime dimensions and $1/N$ corrections.
• To clarify the role of Ishibashi states and imaginary time-dilatation in defining bulk locality in AdS/CFT.
• To reconcile the bulk-boundary map with microscopic causality in interacting AdS gravity.
• To explore the non-perturbative consistency of the compatibility condition with causality.

Proposed method

• Formulate a compatibility condition (2) between CFT conformal generators and bulk isometry generators via Lie derivatives.
• Solve the compatibility condition at the origin of AdS, showing that bulk local operators must be eigenstates of $P_a + K_a$ and $M_{ab}$ with specific quantum numbers.
• Construct solutions as twisted Ishibashi states $|\phi(t,\rho,\vec{x})\rangle\rangle$ via eigenstates of $H$ and $P^2$, expressed using Bessel functions.
• Relate the twisted Ishibashi states to the KLL bulk-boundary map, showing equivalence at leading order in $1/N$.
• Perturbatively modify the bulk-boundary map to restore microscopic causality in interacting theories, consistent with OPE coefficients.
• Demonstrate that all such corrections remain within the superposition framework of twisted Ishibashi states.

Experimental results

Research questions

• RQ1 What is the minimal condition that ensures a CFT operator corresponds to a bulk local operator in AdS?
• RQ2 How can the 3D construction of Miyaji et al. be generalized to arbitrary spacetime dimensions?
• RQ3 Why is imaginary time-dilatation ($e^{-i\pi H/2}$) necessary for bulk locality in the CFT?
• RQ4 Can perturbative corrections to the bulk-boundary map be consistently expressed as superpositions of twisted Ishibashi states?
• RQ5 Is the compatibility condition (2) compatible with microscopic causality at the non-perturbative level?

Key findings

• A general solution to the compatibility condition (2) is a linear superposition of Ishibashi states with imaginary time-dilatation, generalizing Miyaji et al.'s proposal to any dimension.
• Twisted Ishibashi states for global conformal symmetry with imaginary dilatation satisfy free field equations in AdS.
• The leading-order KLL bulk-boundary map corresponds exactly to the twisted Ishibashi state construction, establishing equivalence.
• Perturbative corrections to restore microscopic causality in interacting AdS gravity are consistent with the superposition framework and can be computed order-by-order via OPE coefficients.
• The imaginary dilatation is essential: Verlinde's proposal without it corresponds to a restricted choice of superposition coefficients, missing contributions from other Virasoro representations.
• The framework remains consistent with the BTZ black hole geometry in $d=2$, suggesting broader applicability beyond pure AdS.