We discuss large N rules of the Sachdev-Ye-Kitaev model and the bi-local representation of holography of this theory. This is done by establishing 1/N Feynman rules in terms of bi-local propagators and vertices, which can be evaluated following the recent procedure of Polchinski and Rosenhaus. These rules can be interpreted as Witten type diagrams of the dual AdS theory, which we are able to define at IR fixed point and off.

We pursue the construction of higher-spin theory in ${\mathrm{AdS}}_4$ from ${\mathrm{CFT}}_3$ of the O(N) vector model in terms of canonical collective fields. In null-plane quantization an exact map is established between the two spaces. The coordinates of the ${\mathrm{AdS}}_4$ space-time are generated from the collective coordinates of the bi-local field. This, in the light-cone gauge, provides an exact one-to-one reconstruction of bulk ${\mathrm{AdS}}_4$ space-time and higher-spin fields.

We propose that the Euclidean bilocal collective field theory of critical large-N vector models provides a complete definition of the proposed dual theory of higher spin fields in anti–de Sitter spaces. We show how this bilocal field can be decomposed into an infinite number of even spin fields in one more dimension. The collective field has a nontrivial classical solution which leads to an $O\left(N\right)\mathrm{}$ thermodynamic entropy characteristic of the lower dimensional theory, as required by general considerations of holography. A subtle cancellation of the entropy coming from the bulk fields in one higher dimension with $O\left(1\right)\mathrm{}$ contributions from the classical solution ensures that the subleading terms in thermodynamic quantities are of the expected form. While the spin components of the collective field transform properly under dilatational, translational, and rotational isometries of AdS, special conformal transformations mix fields of different spins indicating a need for a nonlocal map between the two sets of fields. We discuss the nature of the propagating degrees of freedom through a Hamiltonian form of collective field theory and argue that nonsinglet states which are present in an Euclidean version are related to nontrivial backgrounds.

We study in detail the extension of the generalized conformal symmetry proposed previously for D-particles to the case of supersymmetric Yang-Mills matrix models of $\mathrm{D}p$-branes for general p. It is demonstrated that such a symmetry indeed exists both in Yang-Mills theory and in the corresponding supergravity backgrounds produced by $\mathrm{D}p$-branes. On the Yang-Mills side, we derive field-dependent special conformal transformations for the collective coordinates of $\mathrm{D}p$-branes in the one-loop approximation, and show that they coincide with transformations on the supergravity side. These transformations are powerful in restricting the forms of the effective actions of probe D-branes in the fixed backgrounds of source D-branes. Furthermore, our formalism enables us to extend the concept of (generalized) conformal symmetry to arbitrary configurations of D-branes, which can still be used to restrict the dynamics of D-branes. For such general configurations, however, a simple classical space-time interpretation cannot be given at least in the static gauge adopted in the present formulation of D-branes.

We present in detail the basic ingredients contained in bi-local holography, representing a constructive scheme for reconstructing AdS bulk theories in Vectorial/AdS duality. Explicit Mapping to bulk AdS and higher spin fields is seen to be given by a double Fourier transform. All order interactions are explicitly specified through the collective action. This generates bulk Feynman (Witten) diagrams (at tree and loop level). We give details of the four-point case evaluation. It is noted that the bi-local construction goes beyond the assumptions in various discussions of non-locality.