Abstract
We consider the question of loop corrections (i.e. $1/N$) in the vector model/higher spin duality following the recent work of Giombi and Klebanov [J. High Energy Phys. 12, 068 (2013)]. The purpose of this paper is to gain further a more precise comparison between the two sides of the duality. For CFTs given by 3d $O(N)$ or $U(N)$ vector models we evaluate the leading and one-loop partition functions in a variety of geometries. Our calculations are performed in the scheme of collective field theory which was seen in earlier studies to represent a bulk description of Vasiliev higher spin theory. The calculations presented provide data for the comparison of small fluctuation determinants giving further evidence for the one-to-one bulk identification between the bilocal and the AdS pictures. They also offer insight into the identification of coupling constants $G$ and $1/N$ of the two descriptions for models based on $O(N)$ symmetry.
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