Scalar Multiplet Recombination at Large N and Holography

TL;DR

The paper analyzes how coupling a free scalar to a large-$N$ CFT via a double-trace deformation drives an RG flow that, at leading order in $1/N$, exhibits multiplet recombination when one operator approaches the unitarity bound $\Delta_1=\frac{d}{2}-1$. Using both field-theory Hubbard–Stratonovich techniques and a holographic AdS description, it shows that in the IR the would-be short multiplets merge into a single long AdS multiplet: in field theory, $\tilde{O}_1=fO_2$ with $\tilde{O}_1=f^{-1}\square\tilde{O}_2$; in holography, a boundary singleton mixes with a bulk scalar so that only one boundary degree of freedom remains. The analysis is extended to a general double-trace deformation, and a holographic singleton limit clarifies how the IR spectrum simplifies to one primary of dimension $\Delta^{IR}=d-\Delta_2$. They also verify a generalized F-theorem: the flow decreases the appropriate free-energy functional $\tilde{F}$ between UV and IR fixed points. The work highlights a boundary-condition–driven Higgs-like mechanism in AdS without bulk higher-spin fields and suggests future extensions to fermionic operators and other settings.

Abstract

We consider the coupling of a free scalar to a single-trace operator of a large N CFT in d dimensions. This is equivalent to a double-trace deformation coupling two primary operators of the CFT, in the limit when one of the two saturates the unitarity bound. At leading order, the RG-flow has a non-trivial fixed point where multiplets recombine. We show this phenomenon in field theory, and provide the holographic dual description. Free scalars correspond to singleton representations of the AdS algebra. The double-trace interaction is mapped to a boundary condition mixing the singleton with the bulk field dual to the single-trace operator. In the IR, the singleton and the bulk scalar merge, providing just one long representation of the AdS algebra.