Constructive Holography

TL;DR

This paper develops a detailed map between collective fields in a free matrix model and dual gravitational fields in AdS space, revealing how local collective modes correspond to bulk fields with spin in the holographic dual.

Contribution

It introduces a novel correspondence between k-local collective fields and fields in AdS spacetime, including their geometric interpretation and mode expansion leading to spinning bulk fields.

Findings

Mapped collective fields to dual gravitational fields in AdS.

Interpreted collective field coordinates as an extended spacetime.

Derived bulk spinning fields from harmonic expansion modes.

Abstract

We consider the collective field theory description of the singlet sector of a free and massless matrix field in $d$ dimensions. The $k$-local collective fields are functions of $(d-1)k+1$ coordinates. We provide a map between the collective fields and fields in the dual gravitational theory defined on AdS$_{d+1}$ spacetime. The coordinates of the collective field have a natural interpretation: the $k$-local collective field is a field defined on an AdS$_{d+1}\times$S$^{k-1}\times$S$^{(d-2)(k-2)}\times$S$^{d-3}$ spacetime. The modes of a harmonic expansion on the S$^{k-1}\times$S$^{(d-2)(k-2)}\times$S$^{d-3}$ portion of the spacetime leads to the spinning bulk fields of the dual gravity theory.