Bilocal approach to the infra-red fixed point of O (N) invariant theories in 3d and its relation to higher spins

Abstract

The Klebanov-Polyakov Higher-Spin Anti-de Sitter/Conformal Field Theory conjecture posits that the free O(N) vector model is dual to the type A Vasiliev Higher Spin Gravity with the bulk scalar eld having conformal scaling dimension = 1. Similarly, the critical O(N) vector model in 3d is dual to type A Vasilev Higher Spin Gravity with bulk scalar having = 2 . This is a weak-weak duality and accordingly allows a setting where a reconstruction of bulk physics from the boundary CFT is possible. The Jevicki-Sakita collective eld theory provides an explicit realization of such a bulk reconstruction. In this thesis, we use the collective eld theory description of the large-N limit of vector models to study the O(N) infra-red interacting xed point. In particular, we compute the two-point functions for the non-linear sigma model (which is equivalent, in the infra-red, to the critical O(N) vector model) and the two-time bilocal propagator. The spectrum for the O(N) vector model is then obtained by looking at the poles of the connected Green's function. We then show that this same pole condition can be obtained from the homogeneous equation for the bilocal uctuations. We then discuss the single-time Hamiltonian formalism for the critical O(N) vector model. We derive a coupled integral equation for the single-time uctuations. This coupled integral equations allows us to write down the single-time pole condition. We show that the two-time pole condition is equivalent to the single-time pole condition. In addition, we also show that the two-time free bilocal propagator is equivalent to the single-time free bilocal propagator. A Lagrangian formulation of the single-time descripiv tion is given and we write down the single-time propagator. We then explain a puzzle which is that from our study of the non-linear sigma model and the the pole structure of both the two-time and the single-time propagators it would seem that both the = 1 and = 2 scalars are present. By studying the quadratic Hamiltonian determining the spectrum, we demonstrate how in the infra-red limit the state = 1 disappears from the spectum.