AdS Field Theory from Conformal Field Theory

TL;DR

This work identifies three key criteria—a perturbative 1/N expansion, a finite-gap spectrum of single-trace primaries, and Mellin amplitudes polynomially bounded at large Mellin variables—that together guarantee a local AdS effective field theory dual for a conformal field theory. Using Mellin space, it connects CFT poles with AdS exchange diagrams, showing how conformal blocks can be repackaged into bulk processes without explicit Lagrangian construction, thereby realizing an S-matrix–like program for AdS/CFT. The authors demonstrate a constructive extension to all orders in 1/N via unitarity and discuss how higher-spin currents are constrained by a CFT version of Weinberg’s soft theorems, highlighting loopholes due to the AdS IR cutoff. They further relate the flat-space S-matrix analyticity to Mellin-space properties, discuss potential pathologies, and outline future directions involving higher-spin theories and integrability to deepen the holographic understanding of locality.

Abstract

We provide necessary and sufficient conditions for a Conformal Field Theory to have a description in terms of a perturbative Effective Field Theory in AdS. The first two conditions are well-known: the existence of a perturbative `1/N' expansion and an approximate Fock space of states generated by a finite number of low-dimension operators. We add a third condition, that the Mellin amplitudes of the CFT correlators must be well-approximated by functions that are bounded by a polynomial at infinity in Mellin space, or in other words, that the Mellin amplitudes have an effective theory-type expansion. We explain the relationship between our conditions and unitarity, and provide an analogy with scattering amplitudes that becomes exact in the flat space limit of AdS. The analysis also yields a simple connection between conformal blocks and AdS diagrams, providing a new calculational tool very much in the spirit of the S-Matrix program. We also begin to explore the potential pathologies associated with higher spin fields in AdS by generalizing Weinberg's soft theorems to AdS/CFT. The AdS analog of Weinberg's argument constrains the interactions of conserved currents in CFTs, but there are potential loopholes that are unavailable to theories of massless higher spin particles in flat spacetime.

Figures (5)

• Figure 1: This figure shows what happens when one drops the exponentially growing part of the Mellin amplitude for a spin $\ell$ conformal block. The block turns into an AdS exchange Feynman diagram for a spin $\ell$ particle plus AdS contact interactions.
• Figure 2: This figure gives a schematic depiction of the relationship between the AdS cutting rules (left) and the unitarity or bootstrap equations (right) in the CFT Unitarity. Both connect one order in $1/N$ perturbation theory to the next, so once the AdS and CFT theories agree to leading non-trivial order in $1/N$, their difference will be strongly constrained to all orders.
• Figure 3: Irrelevant interactions in AdS like $(\partial \phi)^4$ create both Mellin amplitudes that grow at large $\delta_{ij}$ and anomalous dimensions of double-trace operators ${\cal O}_{n,\ell}$ that grow at large dimension, eventually violating unitarity in perturbation theory. Alternatively, starting only with the Mellin amplitude, one can directly calculate the anomalous dimensions without a priori identifying the corresponding bulk interaction.
• Figure 4: This figure shows how the differential operator $(D_{si})^\ell$ maps a certain class of scalar CFT correlation functions into correlators with a current ${\cal J}_s$. These are also the AdS Feynman diagrams relevant for the AdS/CFT analog of the Weinberg soft limits. Imposing current conservation on the sum of these diagrams provides strong constraints on the couplings of conserved higher spin currents.
• Figure 5: This figure shows how imposing current conservation by acting with the differential operator $\mathcal{D}_s$ transforms AdS diagrams relevant in the soft limit. Their propagators collapse, creating a contact interaction with an effective operator of spin $\ell - 1$. The left hand diagram is represented by equation (\ref{['eq:SpinLDiagram']}) while the right diagram corresponds to equation (\ref{['eq:SpinLDiagramContact']}). This process is formally and conceptually analogous to the S-Matrix manipulations that use Lorentz invariance (or gauge invariance) to obtain equation (\ref{['eq:SoftThmConstraint']}) from equation (\ref{['eq:SoftLimitScattering']}) in the soft limit.