A bilocal description of the conformal algebra at the critical point in 3 dimensions

Abstract

Klebanov and Polyakov conjectured in 2002 a duality between O(N) vector models and Vasiliev higher spin theory, providing a useful probe of the AdS/CFT Correspondence[1]. When considering this duality in d = 2 + 1 dimensions, the free 3d O(N) vector model is dual to type A minimal Vasiliev higher spin gauge theory with a = 1 scalar at the boundary, and the critical 3d O(N) vector model is dual to type A minimal Vasiliev higher spin theory with scalar having = 2. In particular, in the presence of a relevant interaction, the theory ows from an unstable UV xed point with a scalar of dimension = 1 to an IR xed point where = 2. Strong evidence for this conjectured duality was provided by Giombi and Yin around 2010 where they performed explicit tree level and one-loop calculations. For example they showed that at tree level, the three-point functions on both sides of the conjecture match for both = 1 and = 2, and that the one-loop 3-sphere free energy on both sides of the conjecture agree[2, 3, 4]. Further evidence has been the reconstruction of the bulk by de Mello Koch, Jevicki, Jin and Rodrigues[5] (for another construction see Ref. [6]). In 2018 Mulokwe and Rodrigues showed that, making use of a bilocal eld approach (taking a 1=N expansion) to investigate a three-dimensional O(N) invariant bosonic model with N (aa)2 interaction at the critical point / infrared xed point, there was indeed a state identi ed to correspond to a = 2 scalar state; the = 1 state was found to vanish from the spectrum in agreement with Polyakov and Klebanov[7].

In this thesis, we make use of Collective eld theory, where bilocals are used to encode the invariance of the theory explicitly so that it is the invariant variables which are described by the theory[8]. We build on the constructive approach developed by de Mello Koch, Jevicki, Jin, Rodrigues and Yoon between 2010 and 2015 in both the light-cone and temporal gauge for the free theory, wherein an explicit map between the conformal eld theory in d = 2 + 1 dimensions and the higher spin theory in AdS4 S1 was established[5, 9, 10]. In the Hamiltonian approach, the 1 + 2 + 2 = 5 coordinates of the equal time bilocals map (in phase space) to the coordinates of AdS4 S1, and in this thesis we investigate the applicability of this map to the interacting theory. Using the Hamiltonian approach in time-like gauge, it is found that the quartic interaction contributes linearly in the bilocal elductuation equations so that the spectrum problem is that of a potential scattering problem. The scattering state solution takes a universal form at the critical point. Using a change of variables from bilocal momenta to bulk momenta (as dictated by the map), and instituting a eld rede nition to de ne bulk higher spin elds, we are able to obtain a bulk description of these boundary scattering states. It is rather remarkably shown directly in the bulk that for the interacting theory, the s = 0 state (corresponding to the = 0 + 1 = 1 state) is precisely removed in agreement with Ref. [7]. It is then shown that this approach is equivalent to that taken in Ref. [10], by showing that the bulk algebra agrees with that found in [10] both at the free and interacting critical point.