On Higher Spin Gauge Theory and the Critical O(N) Model

TL;DR

This work establishes a perturbative bridge between the free and critical $O(N)$ vector models in three dimensions by exploiting a simple factorization identity for the bulk scalar propagator in AdS4. By comparing $\Delta=1$ (free) and $\Delta=2$ (critical) boundary conditions in Vasiliev’s higher-spin theory, the authors show that all discrepancies between the two sets of correlators arise from the scalar intermediate channel and can be accounted for by the boundary-to-bulk propagator relations, enabling an all-orders $1/N$ match. They develop explicit results for three- and four-point functions and extend the argument to general $n$-point functions via a cutting procedure, clarifying how higher-spin symmetry is broken in the bulk and how anomalous dimensions emerge at finite $N$. The findings support the conjectured duality between higher-spin gauge theory in $AdS_4$ and the critical $O(N)$ vector model and illuminate the role of the Lagrange multiplier field $\alpha$ in mediating this duality, with implications for ultraviolet completions of higher-spin gravity.

Abstract

We show that the differences between correlators of the critical O(N) vector model in three dimensions and those of the free theory are precisely accounted for by the change of boundary condition on the bulk scalar of the dual higher spin gauge theory in AdS4. Thus, the conjectured duality between Vasiliev's theory and the critical O(N) model follows, order by order in 1/N, from the duality with free field theory on the boundary.