A Derivation of AdS/CFT for Vector Models

TL;DR

This paper explicitly derives a duality between free or critical vector models and a high-spin gravity theory on anti-de Sitter space, providing a concrete path integral mapping and potential for loop calculations.

Contribution

It presents an explicit non-local action for high-spin fields on AdS space derived from vector models, extending the AdS/CFT correspondence to include detailed bulk field mappings.

Findings

Explicit path integral mapping from vector models to AdS high-spin fields

Derivation of a non-local high-spin action on AdS space

Potential extension of the duality to finite N theories

Abstract

We explicitly rewrite the path integral for the free or critical $O(N)$ (or $U(N)$) bosonic vector models in $d$ space-time dimensions as a path integral over fields (including massless high-spin fields) living on ($d+1$)-dimensional anti-de Sitter space. Inspired by de Mello Koch, Jevicki, Suzuki and Yoon and earlier work, we first rewrite the vector models in terms of bi-local fields, then expand these fields in eigenmodes of the conformal group, and finally map these eigenmodes to those of fields on anti-de Sitter space. Our results provide an explicit (non-local) action for a high-spin theory on anti-de Sitter space, which is presumably equivalent in the large $N$ limit to Vasiliev's classical high-spin gravity theory (with some specific gauge-fixing to a fixed background), but which can be used also for loop computations. Our mapping is explicit within the $1/N$ expansion, but in principle can be extended also to finite $N$ theories, where extra constraints on products of bulk fields need to be taken into account.