AdS/CFT correspondence for the $O(N)$ invariant critical $\varphi^4$ model in 3-dimensions by the conformal smearing

TL;DR

This paper constructs a 4D bulk space from a 3D $O(N)$ invariant critical $^4$ model using conformal smearing, revealing an asymptotic AdS geometry and encoding the boundary operator dimensions through bulk propagators.

Contribution

It demonstrates how to derive an asymptotic AdS bulk geometry and scalar propagator from a 3D critical $^4$ model, including interaction effects, using conformal smearing and large N expansion.

Findings

Bulk metric describes asymptotic AdS space at UV and IR fixed points.

Bulk-to-boundary propagator encodes $^2$ conformal dimensions at fixed points.

Consistency with GKP-Witten relation including interaction effects.

Abstract

We investigate a structure of a 4-dimensional bulk space constructed from the $O(N)$ invariant critical $\varphi^4$ model in 3-dimension using the conformal smearing. We calculate a bulk metric corresponding to the information metric and the bulk-to-boundary propagator for a composite scalar field $\varphi^2$ in the large $N$ expansion. We show that the bulk metric describes an asymptotic AdS space at both UV (near boundary) and IR (deep in the bulk) limits, which correspond to the asymptotic free UV fixed point and the Wilson-Fisher IR fixed point of the 3-dimensional $\varphi^4$ model, respectively. The bulk-to-boundary scalar propagator, on the other hand, encodes $\Delta_{\varphi^2}$ (the conformal dimension of $\varphi^2$) into its $z$ (a coordinate in the extra direction of the AdS space) dependence. Namely it correctly reproduces not only $\Delta_{\varphi^2}=1$ at UV fixed point but also $\Delta_{\varphi^2}=2$ at the IR fixed point for the boundary theory. Moreover, we confirm consistency with the GKP-Witten relation in the interacting theory that the coefficient of the $z^{\Delta_{\varphi^2}}$ term in $z\to 0$ limit agrees exactly with the two-point function of $\varphi^2$ including an effect of the $\varphi^4$ interaction. }