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

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

Large N bilocals at the infrared fixed point of the three dimensional O(N) invariant vector theory with a quartic interaction

We study the three dimensional O(N) invariant bosonic vector model with a $\frac{\lambda}{N}(\phi^{a}\phi^{a})^{2}$ interaction at its infrared fixed point, using a bilocal field approach and in an $1/N$ expansion. We identify a (negative energy squared) bound state in its spectrum about the large $N$ conformal background. At the critical point this is identified with the $\Delta=2$ state. We further demonstrate that at the critical point the $\Delta=1$ state disappears from the spectrum.Comment: 30 page

Large N Matrix Hyperspheres and the Gauge-Gravity Correspondence

The large N dynamics of a subsector of d=0 interacting complex multi matrix systems, which is naturally parametrized by a matrix valued radial coordinate, and which embodies the canonical AdS/CFT relationship between 't Hooft's coupling constant and radius, is obtained. Unlike the case of the single complex matrix, for two or more complex matrices a new repulsive logarithmic potential is present, as a result of which the density of radial eigenvalues has support on an hyper annulus. For the single complex matrix, the integral over the angular degrees of freedom of the Yang-Mills interaction can be carried out exactly, and in the presence of an harmonic potential, the density of radial eigenvalues is shown to be of the Wigner type.Comment: 15 pages - v2: minor text changes in abstract and introductio

Constructing the bulk at the critical point of three-dimensional large N vector theories

Please read abstract in the article.The National and Mandelstam Institutes for Theoretical Physics; a Postdoctoral Fellowship from the National Research Foundation (NRF) and the National Institute for Theoretical and Computational Sciences.http://www.elsevier.com/locate/physletbhj2023Physic

Six-membered ring systems: with O and/or S atoms

A large variety of publications involving O- and S-6-membered ring systems have appeared in 2017. The importance of these heterocyclic compounds is highlighted by the huge number of publications on the total synthesis of natural oxygen derivatives and of other communications dedicated to synthetic derivatives. Reviews on stereoselective organocatalytic synthesis of tetrahydropyrans (17EJO4666), of tetrahydropyrans and their application in total synthesis of natural products (17CSR1661), on the synthesis of the less thermodynamically stable 2,6-trans-tetrahydropyrans (17S4899), on enantioselective synthesis of polyfunctionalized pyran and chromene derivatives (17TA1462), and on enantioselective and racemic total synthesis of camptothecins, including the formation of their pyran-2-one ring (17SL1134), have appeared. Advances in the transition metal-catalyzed synthesis of pyran-2/4-ones (17TL263), N-heterocyclic carbene (NHC)-catalyzed achiral synthesis of pyran-2-one, coumarin and (thio)chromone derivatives (17OBC4731), on the synthesis and transformation of 2H-pyran-2-ones (17T2529) and 2-styrylchromones (17EJO3115) into other heterocyclic compounds, have been surveyed. The strategies to build up the tetrahydropyranyl core of brevisamide (17H(95)81) and the reactions of ketyl radicals, generated from carbonyl derivatives under transition-metal photoredox-catalyzed conditions, leading to isochromen- and chroman-type compounds (17CC13093) were disclosed. Developments in the synthesis of pentafluorosulfanyl(chromene and coumarin) derivatives (17TL4803), photoswitchable D9-tetrahydrocannabinol derivatives (17JA18206), and aminobenzopyranoxanthenes with nitrogen-containing rings (17JOC13626) have been studied.info:eu-repo/semantics/publishedVersio

The large-N limit of matrix models and AdS/CFT

Random matrix models have found numerous applications in both Theoretical Physics and Mathematics. In the gauge-gravity duality, for example, the dynamics of the half- BPS sector can be fully described by the holomorphic sector of a single complex matrix model. In this thesis, we study the large-N limit of multi-matrix models at strong-coupling. In particular, we explore the significance of rescaling the matrix fields. In order to investigate this, we consider the matrix quantum mechanics of a single Hermitian system with a quartic interaction. We “compactify” this system on a circle and compute the first-order perturbation theory correction to the ground-state energy. The exact ground-state energy is obtained using the Das-Jevicki-Sakita Collective Field Theory approach. We then discuss the multi-matrix model that results from the compactification of the Higgs sector of N = 4 SYM on S4 (or T S3). For the radial subsector, the saddle-point equations are solved exactly and hence the radial density of eigenvalues for an arbitrary number of even Hermitian matrices is obtained. The single complex matrix model is parametrized in terms of the matrix valued polar coordinates and the first-order perturbation theory density of eigenstates is obtained. We make use of the Harish-Chandra- Itzykson-Zuber (HCIZ) formula to write down the exact saddle-point equations. We then give a complementary approach - based on the Dyson-Schwinger (loop) equations formalism - to the saddle-point method. We reproduce the results obtained for the radial (single matrix) subsector. The two-matrix integral does not close on the original set of variables and thus we map the system onto an auxiliary Penner-type two matrix model. In the absence of a logarithmic potential we derive a radial hemispherical density of eigenvalues. The system is regulated with a logarithm potential, and the Dobroliubov-Makeenko-Semenoff (DMS) loop equations yield an equation of third degree that is satisfied by the generating function. This equation is solved at strong coupling and, accordingly, we obtain the radial density of eigenvalues

Constructing the bulk at the critical point of three-dimensional large NN vector theories

In the context of the $AdS_{4}/CFT_{3}$ correspondence between higher spin fields and vector theories, we use the constructive bilocal fields based approach to this correspondence, to demonstrate, at the $IR$ critical point of the interacting vector theory and directly in the bulk, the removal of the $\Delta=1$ ($s=0$) state from the higher spins field spectrum, and to exhibit simple Klein-Gordon higher spin Hamiltonians. The bulk variables and higher spin fields are obtained in a simple manner from boundary bilocals, by the change of variables previously derived for the $UV$ critical point (in momentum space), together with a field redefinition.Comment: Derivation of the \Delta=2 Hamiltonian added to Section 3.3; 19 pages including reference

NaHSO₄–SiO₂-catalyzed aza-Diels–Alder reaction of <i>o</i>-hydroxybenzaldimines with 2,3-dihydrofuran: Diastereoselective synthesis of furanobenzopyrans

<p>NaHSO₄ supported on silica gel catalyzes the aza-Diels–Alder reaction of <i>o</i>-hydroxybenzaldimines with 2,3-dihydrofuran to provide furanobenzopyrans in reasonable yields and diastereoselectivity. The catalyst is recyclable and reusable up to two times without a significant loss of activity.</p