Reconstructing the spacetime dual to a free matrix

TL;DR

This paper explores how the collective field theory of a free matrix in 2+1 dimensions can be interpreted as a dual gravitational spacetime, involving higher-dimensional AdS and spherical geometries.

Contribution

It introduces a novel interpretation of $k$-local collective fields as fields on an AdS$_4$ with additional spherical dimensions, linking collective modes to bulk gravity fields.

Findings

$k$-local collective fields correspond to fields on AdS$_4\times$S$^{k-2}\times$S$^{k-1}$.

Harmonic expansion modes relate to spinning bulk gravity fields.

Provides a geometric interpretation of collective field modes in holography.

Abstract

In this paper we consider the collective field theory description of the singlet sector of a free matrix field in 2+1 dimensions. This necessarily involves the study of $k$-local collective fields, which are functions of $2k+1$ coordinates. We argue that these coordinates have a natural interpretation: the $k$-local collective field is a field defined on an AdS$_4\times$S$^{k-2}\times$S$^{k-1}$ spacetime. The modes of a harmonic expansion on the S$^{k-2}\times$S$^{k-1}$ portion of the spacetime leads to the spinning bulk fields of the dual gravity theory.