Evaluating the AdS dual of the critical O(N) vector model

TL;DR

The paper argues that the AdS$_4$ dual of the three-dimensional critical $O(N)$ vector model can be reconstructed using a Legendre-transform consistency condition between the UV free fixed point and the IR interacting fixed point. By adopting a minimal bulk scalar action and matching bulk correlators to boundary data, it fixes the bulk self-interactions up to quartic order and determines the normalization from the boundary central charge $C_T$. In $D=3$ the cubic coupling vanishes ($g_3=0$), while the quartic coupling is fixed at $g_4=-3$, as inferred from leading-log analyses of four-point functions and higher-spin exchange contributions. These results have important implications for higher-spin holography and subleading-$N$ dualities, providing concrete tests for hs(4) and guiding how UV/IR fixed points are encoded in a bulk AdS theory.

Abstract

We argue that the AdS dual of the three dimensional critical O(N) vector model can be evaluated using the Legendre transform that relates the generating functionals of the free UV and the interacting IR fixed points of the boundary theory. As an example, we use our proposal to evaluate the minimal bulk action of the scalar field that it is dual to the spin-zero ``current'' of the O(N) vector model. We find that the cubic bulk self interaction coupling vanishes. We briefly discuss the implications of our results for higher spin theories and comment on the bulk-boundary duality for subleading N.

Figures (1)

• Figure 1: Graphical representation of $\Pi_4(x_1,x_2,x_3,x_4)$. The solid lines correspond to the scalar and the dotted lines to the graviton.