On the AdS Higher Spin / O(N) Vector Model Correspondence: degeneracy of the holographic image

TL;DR

This work tests the Klebanov-Polyakov holographic duality by reorganizing the free four-point function of the scalar singlet in the $O(N)$ vector model into a crossing-symmetric conformal partial wave expansion that involves only the minimal-twist higher-spin currents, thereby aligning with a massless HS bulk in AdS$_4$. It analytically derives fusion coefficients and two- and three-point data for HS currents, demonstrates closure of the singlet bilinear sector at leading order in $1/N$, and extends the CPWE analysis to the IR fixed point via an amputation/Legendre-transformation framework. In 3d ($d=3$), the IR theory exhibits holographic degeneracy, with explicit IR fusion coefficients and an IR CPWE that matches known amplitudes, including the vanishing of certain three-point functions. The paper further clarifies the relation between CPWs and AdS exchange graphs at $d=3$, showing that a single bulk HS theory with a consistent truncation can reproduce the full connected four-point function, supporting the view that the HS bulk provides a faithful holographic image of the free UV and interacting IR CFTs for $2<d<4$. Overall, it supplies analytic control over HS/CFT data, clarifies the role of the $oldsymbol{ riangle_-}$ branch, and strengthens the case for a degenerate hologram relating UV and IR CFTs through HS bulk dynamics.

Abstract

We explore the conjectured duality between the critical O(N) vector model and minimal bosonic massless higher spin (HS) theory in AdS. In the boundary free theory, the conformal partial wave expansion (CPWE) of the four-point function of the scalar singlet bilinear is reorganized to make it explicitly crossing-symmetric and closed in the singlet sector, dual to the bulk HS gauge fields. We are able to analytically establish the factorized form of the fusion coefficients as well as the two-point function coefficient of the HS currents. We insist in directly computing the free correlators from bulk graphs with the unconventional branch. The three-point function of the scalar bilinear turns out to be an "extremal" one at d=3. The four-leg bulk exchange graph can be precisely related to the CPWs of the boundary dual scalar and its shadow. The flow in the IR by Legendre transforming at leading 1/N, following the pattern of double-trace deformations, and the assumption of degeneracy of the hologram lead to the CPWE of the scalar four-point function at IR. Here we confirm some previous results, obtained from more involved computations of skeleton graphs, as well as extend some of them from d=3 to generic dimension 2<d<4.

Figures (5)

• Figure 1: Connected part of the free four-point function of the spin-zero current $J$.
• Figure 2: Free four-point function as a sum partial waves of minimal twist operators.
• Figure 3: IR four-point function as a sum of partial waves of minimal twist operators at $d=3$.
• Figure 4: Free three-point function and star Witten graph at $d\rightarrow 3$.
• Figure 5: Scalar exchange Witten graph VS. scalar CPWs in the limit $d\rightarrow 3$