light fields of arbitrarily high spin.

There is one example of the $AdS$/CFT correspondence in which both sides are weakly coupled in the large $N$ limit; this is the conjectured duality [@Sundborg:2000wp; @Sezgin:2002rt; @Klebanov:2002ja] between the singlet sector of the $O(N)$ vector model (namely, $N$ free real scalar fields) in three space-time dimensions, and Vasiliev’s higher-spin gravity theory on $AdS_4$ [@Fradkin:1987ks] (see [@Vasiliev:1999ba] for a review). While the gravitational side of this duality is only understood at the classical level, and it is not yet known how to give it a quantum completion, in the classical gravity limit (governed by tree-level diagrams in the bulk) this provides an example of the $AdS$/CFT correspondence in which both sides are weakly coupled. This allows many detailed tests of the correspondence to be performed in this case [@Giombi:2009wh; @Giombi:2010vg; @Giombi:2011ya], and it also suggests that this could be an ideal toy model for which a derivation of the $AdS$/CFT correspondence could be found (and perhaps then generalized to more complicated cases). Indeed, there are several suggestions in the literature [@Das:2003vw; @Koch:2010cy; @Douglas:2010rc; @Jevicki:2011ss] for how to derive the $AdS$/CFT correspondence explicitly in this example.

In this paper we study a small deformation of 