Abstract
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.
Highlights
We begin to explore the potential pathologies associated with higher spin fields in AdS by generalizing Weinberg’s soft theorems to AdS/CFT
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
Recent work has shown that Mellin space provides an extremely natural language for discussing CFT correlation functions dual to AdS quantum field theories [5, 11, 12]
Summary
Let us begin by reviewing Mellin amplitudes for the boundary correlators of AdS effective field theories, in order to see why they automatically satisfy the three criteria discussed in the introduction. We recall that the n-point Mellin amplitude from a simple scalar contact interaction LAdS = gφn is just the coupling times a normalization constant, independent of δij: Mn(δij) = gλn The simplificity of this result is extremely non-trivial and powerful, since as we will see it means that the Gamma functions in definition of the Mellin amplitude, equation (2.1), completely account for the shift in the OPE coefficients and anomalous dimensions of double-trace operators. By direct calculation, one sees that AdS effective field theories produce tree-level Mellin amplitudes that are bounded by polynomials with a maximum degree determined by the highest dimension interaction in the effective Lagrangian. One may prefer to think of effective theories as having an infinite number of operators of increasingly large dimensions, suppressed by increasing powers of the cutoff In this case, it remains obvious that the Mellin amplitude will satisfy our weaker criteria 3’ — it will have an EFT expansion. We will be able to show that our three conditions are necessary, and sufficient, so that every CFT obeying them must be described by an AdS EFT Lagrangian
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
