AdS_4/CFT_3 Construction from Collective Fields

TL;DR

The paper constructs an explicit Hamiltonian, light-cone formulation of the $O(N)$ vector model’s bi-local collective field and demonstrates an exact canonical map to the AdS$_4$ higher-spin theory. It provides a detailed 1/$N$ expansion, explicit conformal generators in the collective picture, and a canonical transformation that expresses AdS$_4$ coordinates, including the radial direction $z$, in terms of bi-local variables: $z^2 = \frac{(x_1-x_2)^2 p_1^+ p_2^+}{(p_1^++p_2^+)^2}$ and $\theta = 2 \arctan\sqrt{\frac{p_2^+}{p_1^+}}$. The results establish a concrete bulk–boundary dictionary at the Hamiltonian level for higher-spin holography and illuminate the origin of the extra dimension as a collective degree of freedom, with potential routes to compare bulk interactions with Vasiliev’s theory. This framework clarifies how bulk AdS$_4$ and its infinite tower of higher-spin fields emerge from boundary collective dynamics in a controlled $1/N$ expansion.

Abstract

We pursue the construction of higher-spin theory in AdS_4 from CFT_3 of the O(N) vector model in terms of canonical collective fields. In null plane quantization an exact map is established between the two spaces. The coordinates of the AdS_4 space-time are generated from the collective coordinates of the bi-local field. This, in the light cone gauge, provides an exact one to one reconstruction of bulk AdS_4 space-time and higher-spin fields.