  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. The
purpose of this paper is to gain further 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
comparison of small fluctuation determinants giving further evidence for the
one-to-one bulk identification between the bi-local and the AdS picture. 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.
