On the proposed AdS dual of the critical O(N) sigma model for any dimension 2<d<4

TL;DR

The work tests the Klebanov–Polyakov AdS/CFT conjecture for the critical $O(N)$ vector model in $2<d<4$ by computing the auxiliary field ($\alpha$) four-point function at $O(1/N)$ and showing it is governed by the exchange of symmetric traceless tensor currents of even rank $l>0$, which correspond to massless higher-spin fields in AdS with $\Delta_{J^{(l)}}=d+l-2$. By matching the conformal partial-wave expansion and employing the shadow symmetry, the authors determine the couplings $\gamma_l$ of two $\alpha$ fields to $J^{(l)}$, finding a closed-form expression and a nontrivial algebraic check. They argue that shadow contributions are encoded in the tower of tensor exchanges and fix the cubic couplings of higher-spin fields to the scalar $\sigma$, consistent with gauge invariance and bulk graviton normalization, thereby providing evidence for the duality and a framework for extending it with explicit bulk interactions. The results illuminate higher-spin holography in a controlled CFT limit and lay groundwork for computing higher-point functions and bulk couplings in AdS.

Abstract

We evaluate the 4-point function of the auxiliary field in the critical O(N) sigma model at O(1/N) and show that it describes the exchange of tensor currents of arbitrary even rank l>0. These are dual to tensor gauge fields of the same rank in the AdS theory, which supports the recent hypothesis of Klebanov and Polyakov. Their couplings to two auxiliary fields are also derived.