Reconstructing the spacetime dual to a free matrix

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.