Canonical Formulation of $O(N)$ Vector/Higher Spin Correspondence

TL;DR

The work develops a time-like, canonical (collective-field) formulation of the Vector Model/Higher Spin duality in AdS$_4$, constructing bulk Higher Spin fields from bi-local bi- local fields via a canonical $1/N$-expanded Hamiltonian. It shows that a kernel implements an exact, invertible map between bi-local momenta and AdS$_4\times S^1$ bulk variables, preserving $SO(2,3)$ generators and canonical commutation relations to all orders in $1/N$, and produces a consistent bulk/boundary dictionary including the boundary currents $\mathcal{O}_s$. Linearization around the large-$N$ background yields a quadratic Hamiltonian whose bulk fields are built from bi-local oscillators, with explicit projection to physical higher-spin currents and a reduction to the physical subspace via polarization data. The construction provides a tractable, covariantizable canonical framework for Higher Spin holography, with potential applications to horizon physics and minimal-model dualities. The formalism ensures locality and unitarity in the bulk description, linking AdS bulk fields and CFT$_3$ currents in a mathematically controlled $1/N$ expansion.

Abstract

We discuss the canonical structure of the collective formulation of Vector Model/Higher Spin Duality in AdS$_4$. This involves a construction of bulk AdS Higher Spin fields through a time-like bi-local Map, with a Hamiltonian and canonical structure which are established to all orders in $1/N$.