Explicit holography for vector models at finite N, volume and temperature

In previous work we constructed an explicit mapping between large $N$ vector models (free or critical) in $d$ dimensions and a non-local high-spin gravity theory on $AdS_{d+1}$, such that the gravitational reproduces field correlation functions order by $1/N$. this paper discuss three aspects of mapping. First, our original was not valid non-perturbatively $1/N$, since it did include correlations fields which appear at finite $N$. We show using bi-local $G-\Sigma$ type formalism similar to one used SYK model, can construct exact bulk is also The contains additional auxiliary implement constraints. Second, generalization $S^d$, particular how sphere free energy matches exactly two sides, be consistently regularized. Finally, temperature, low-temperature phase mapped thermal AdS space.