Holography from Conformal Field Theory

TL;DR

This work investigates how bulk locality at sub-AdS scales emerges from boundary CFT data. By framing locality in terms of the four-point function and enforcing conformal bootstrap constraints, the authors show that, to leading nontrivial order in $1/N^2$, a large-$N$ planar CFT with a sufficient gap in single-trace operator dimensions yields a local bulk dual; the number and structure of CFT solutions precisely match those of local bulk quartic interactions. They develop a counting argument, solve crossing in $d=2$ and $d=4$, and connect bulk Witten diagrams with the conformal partial-wave expansion, including Regge-limit behavior and locality-related singularities. The results provide a nonperturbative derivation of sub-AdS bulk locality from CFT data and offer a framework for applying AdS/CFT to systems where a concrete bulk string construction is not available.

Abstract

The locality of bulk physics at distances below the AdS length is one of the remarkable aspects of AdS/CFT duality, and one of the least tested. It requires that the AdS radius be large compared to the Planck length and the string length. In the CFT this implies a large-N expansion and a gap in the spectum of anomalous dimensions. We conjecture that the implication also runs in the other direction, so that any CFT with a planar expansion and a large gap has a local bulk dual. For an abstract CFT we formulate the consistency conditions, most notably crossing symmetry, and show that the conjecture is true in a broad range of CFT's, to first nontrivial order in 1/N^2: any CFT with a gap and a planar expansion is generated via the AdS/CFT dictionary from a local bulk interaction. We establish this result by a counting argument on each side, and also investigate various properties of some explicit solutions.

Figures (6)

• Figure 1: Four-point correlator with wavepackets aligned to intersect in the bulk.
• Figure 2: Quartic interactions of spin $l=2a$ and with $2k$ derivatives. There are $1+a$ interactions of even spin $l=2a$, with the number of derivatives given by $k = 2a, 2a+1, \ldots , 3a$. The total number of interactions with spin at most $L$ is $(L+2)(L+4)/8$.
• Figure 3: Open circles are $\gamma(n,l)$ not determined by the equations while filled circles correspond to $\gamma(n,l)$ determined by the equations (\ref{['pq']}) with $p=0,1,\dots,n$ and $q=0,1, \dots,{\rm min}(p-1,L/2)$. We see that up to spin $L$ we have $(L/2+1)(L/2+2)/2$ undetermined $\gamma(n,l)$.
• Figure 4: Witten diagram for the CFT four-point function associated to a quartic contact interaction in AdS. The boundary points $P_i$ are connected to the bulk interaction point $X$ by bulk to boundary propagators. In general, the quartic vertex at $X$ includes derivatives acting on the bulk to boundary propagators.
• Figure 5: External points $P_i$ in the boundary of conformally compactified AdS. The Regge limit corresponds to $P_3 \to -P_1$ or $P_2 \to -P_4$. In this limit the dominant contribution to the four-point function comes from the AdS region around the future lightcone of $P_1$ and past lightcone of $P_4$, in particular, from their intersection at the $(d-1)$-dimensional hyperboloid shown in blue.