1/N and Loop Corrections in Higher Spin AdS_4/CFT_3 Duality

TL;DR

This work analyzes 1/N loop corrections in the vector model–higher-spin duality by computing leading and one-loop partition functions in various backgrounds using a bi-local collective field framework that mirrors the AdS bulk. The authors show that the quadratic fluctuations in the bi-local theory reproduce the same Laplacian determinants as the higher-spin bulk, with a crucial role played by the functional measure, which can induce a finite redefinition of the bulk coupling (e.g., $G'=1/(N-1)$ for $O(N)$). Across geometries such as $S^1 imes ext{R}^2$, $S^3$, and thermal AdS$_4$, the one-loop contributions from determinants cancel against measure terms in the $O(N)$ case, yielding consistent $1/N$ expansions with the expected single-scalar-like results when regularization is applied. The results strengthen the bulk–boundary dictionary, clarify the coupling-constant identification, and point to the importance of the measure in Vasiliev-type theories for perturbative corrections and potential nonperturbative aspects.

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. 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.