Constructive Holography

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.