Submitted by Biblioteca do Instituto de Física (bif@ndc.uff.br) on 2017-03-07T18:35:27Z No. of bitstreams: 1 TESE.pdf: 797081 bytes, checksum: 36b77c687969ac7b12aeef2589d1d766 (MD5) / Made available in DSpace on 2017-03-07T18:35:27Z (GMT). No. of bitstreams: 1 TESE.pdf: 797081 bytes, checksum: 36b77c687969ac7b12aeef2589d1d766 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro / O confinamento de quarks e glúons continua sendo um dos maiores problemas da Física atual, mesmo depois de passados 50 anos da criação da cromodinâmica quântica. Existem diversas abordagens que procuram uma explicação para este comportamento. Um destes cenários consiste na supercondutividade dual, proposta por G. t’Hooft em 1978. Aqui, ele discute como a condensação de objetos cromomagnéticos poderia originar um potencial linear entre cargas cromoelétricas. Este mecanismo é um dos mais aceitos atualmente e nos dirige à algumas perguntas cruciais: como estes objetos poderiam se tornar relevantes em teorias de Yang-Mills puras? quais os tipos de objetos que devemos levar em consideração para gerar as propriedades do potencial confinante? Embora a primeira pergunta seja difícil de responder, a segunda pode ser atacada por técnicas diferentes, suportadas pelas descrições na rede e por descrições efetivas de ensembles 1. Nesta tese, me dedico a estudar uma classe de objetos que s˜ao bons candidatos a resolverem a segunda questão: monopólos e vórtices de centro. Quando estamos lidando com as teorias de Yang-Mills puras SU(N), o problema consiste que, em nível clássico, estes defeitos são singulares. Porém, recebendo suporte da rede (nosso laboratório em teoria quântica de campos), podemos imaginar que, devido a flutuações quânticas do vácuo, estes objetos poderiam adquirir algumas propriedades dimensionais, como tensão,rigidez e interações que ajudariam a caracterizar o ensemble magnético nos levando a descrições de campos efetivas, que podem ser utilizadas para extrair a corda elétrica confinante. Utilizando técnicas oriundas da física de polímeros obtivemos equações de difusão que representam objetos unidimensionais, como vórtices de centro em 3D ou monopólos em 4D. O surgimento de uma derivada covariante abeliana, no caso do ensemble de vórtices de centro e instantons correlacionados em 3D, e de uma derivada covariante não abeliana, no caso do ensemble de monopólos coloridos em 4D, foi fundamental paragerar os modelos efetivos correspondentes. Acreditamos que estas equações de difusão poderão ser úteis, no futuro, para relacionar as propriedades do potencial entre quarks e aquelas de seus possíveis ensembles correspondentes. / Nowadays, quark and gluon confinement continues to be one of the most important problems in Physics. It remains unsolved, although 50 years have passed since the foundations of quantum chromodynamics. There are various approaches aimed at explaining this behaviour. One of them is the dual superconductor scenario proposed by G. t’Hooft in 1978. The general idea is that the condensation of chromomagnetics objects could originate a linear potential between chromoelectric charges. This is a promising mechanism that posses some crucial questions: how could these objects be relevant in pure YangMills? what type of object would be needed in order to generate the properties of the confining potential? While the first question is very difficult, the second one can be approached by different techniques, guided by the lattice and effective ensemble descriptions. In this thesis, I’ve been working on some good candidates to solve the second question: monopoles and center vortices. When dealing with pure SU(N) Yang-Mills theory, the problem is that at the classical level these magnetic defects are singular. Nevertheless, supported by the lattice (our laboratory in quantum field theory), we can imagine that, due to quantum vacuum fluctuations, they could acquire dimensionful properties. The tension, stiffness, as well as possible interactions that characterize the magnetic ensemble lead to effective field descriptions, that could be used to extract the corresponding confining electric string. Based on techniques borrowed from the physics of polymers, we obtained diffusion equations that describe magnetic one-dimensional objects, such as center vortices in 3D and monopoles in 4D. The appearance of an Abelian covariant derivative, for an ensemble of chains in 3D, and a non Abelian one, in the case of coloured loops in 4D, was essential to generate the corresponding effective descriptions. We believe that these diffusion equations could be helpful in the future, to relate the properties of the interquark potential and those of the possible underlying ensembles.

Instantons on various spaces can be constructed via a generalization of the Fourier transform called the ADHM-Nahm transform. An explicit use of this construction, however, involves rather tedious calculations. Here we derive a simple formula for instantons on a space with one periodic direction. It simplifies the ADHM-Nahm machinery and can be generalized to other spaces.

A dissertation submitted in fulfillment of the requirements for the degree of Master of Science. August 2016. / The six scalar fields in N = 4 super Yang-Mills theory enjoy a global SO(6) symmetry, and large N but non-planar limits of this theory are well-described by adopting a group representation approach. Studies have shown that the one-loop dilatation operator is highly determined by the action of the su(2)=su(3) subalgebras on restricted Schur polynomials. These actions involve the traces of products of projection operators. In this dissertation, exact analytical formulae for these traces are found which in turn are used to find the exact action of these algebras on restricted Schur polynomials. The potential of the su(2) algebra to determine the one-loop dilatation operator is also explored. This is done by exploiting necessary symmetry conditions and moving to a continuum limit in order to derive a number of partial differential equations which determine the dilatation operator. The ultimate goal of this work is to provide tools to find the exact one-loop dilatation operator in the non-planar limit. / LG2017

A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. 14 May 2013 / We study the action of the dilatation operator on restricted Schur polynomials labeled by Young diagrams with p long columns or p long rows. A new version of Schur-Weyl duality provides a powerful approach to the computation and manipulation of the symmetric group operators appearing in the restricted Schur polynomials. Using this new technology, we are able to evaluate the action of the one loop dilatation operator. The result has a direct and natural connection to the Gauss Law constraint for branes with a compact world volume. Generalzing previous results, we find considerable evidence that the dilatation operator reduces to a decoupled set of harmonic oscillators. This strongly suggests that integrability in N = 4 super Yang-Mills theory is not just a feature of the planar limit, but extends to other large N but non-planar limits.

In this thesis we explore certain limits of the AdS/CFT correspondence for integrability. This is done by calculating the action of the dilatation operator on operators known as restricted Schur polynomials, which are AdS/CFT dual to D3-branes known as giant gravitons. We focus on operators in N = 4 super-Yang-Mills theory, which is dual to type IIB string theory on an AdS5×S5 background. We find that, in various cases, this theory is integrable in a large N non-planar limit.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. February 2015. / We explore the non-perturbative physics of giant gravitons in type IIB string theory on the AdS5 ⇥ S5 background in this thesis. The gauge theory dual is N = 4 super Yang-Mills theory with a U(N) gauge group. We diagonalise the one and two-loop dilatation operators acting on the restricted Schur polynomial basis. These operators are dual to a system of giant gravitons with strings attached. Hence, we present evidence for integrability in certain non-planar sectors of the gauge theory. In the second half of the thesis, we turn our focus to N = 4 super Yang-Mills theory with an SO(N) gauge group. In this case, the geometry of the dual gravity theory is AdS5 ⇥RP5. The non-planar physics of the SO(N) theory is distinct from that of the U(N) theory. To pursue the goal of searching for non-planar integrability in the SO(N) gauge theory, one might try to generalise the restricted Schur basis to the SO(N) case. We propose such a basis and evaluate their two-point functions exactly in the free theory. Further, we develop techniques to compute correlation functions of multi-trace operators involving two scalar fields exactly. Lastly, we extend these results to the theory with an Sp(N) gauge group.

Planar maximally supersymmetric Yang-Mills theory (N = 4 SYM) is a special quantum fi eld theory. A few of its remarkable features are conformal symmetry at the quantum level, evidence of integrability and, moreover, it is a prime example of the AdS/CFT duality. Triggered by Witten's twistor string theory [1], the past 15 years have witnessed enormous progress in reformulating this theory to make as many of these special features manifest, from the choice of convenient variables to recursion relations that allowed new mathematical structures to appear, like the Grassmannian [2]. These methods are collectively referred to as on-shell methods. The ultimate hope is that, by understanding N = 4 SYM in depth, one can learn about other, more realistic quantum fi eld theories. The overarching theme of this thesis is the investigation of how on-shell methods can aid the computation of quantities other than scattering amplitudes. In this spirit we study form factors and correlation functions, said to be partially and completely off-shell quantities, respectively. More explicitly, we compute form factors of half-BPS operators up to two loops, and study the dilatation operator in the SO(6) and SU(2j3) sectors using techniques originally designed for amplitudes. A second part of the work is dedicated to the study of scattering amplitudes beyond the planar limit, an area of research which is still in its infancy, and not much is known about which special features of the planar theory survive in the non-planar regime. In this context, we generalise some aspects of the on-shell diagram formulation of Arkani-Hamed et al. [3] to take into account non-planar corrections.

Orientador: Abraham Hirsz Zimerman / Banca: Waldyr Alves Rodrigues Junior / Banca: Cesar Augusto Linhares / Banca: Juan Alberto Mignaco / Banca: Ruben Aldrovandi / Doutor

Thesis (Ph. D.)--University of Chicago, Dept. of Physics, June 1999. / Includes bibliographical references. Also available on the Internet.

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