A dissertation submitted to the Faculty of Science, University of the Witwatersrand,
Johannesburg, in fulfillment of the requirements for the degree of Doctor of Science.
Johannesburg, 2018The 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.MT 201
