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.

No description available.

Esta tese estuda as contribuições proporcionais a In(T) da função de três pontos na teoria de Yang-Mills no calibre axial temporal na ordem de um loop no limite de altas temperaturas. Nós provamos que tais contribuições satisfazem uma identidade de Ward abeliana que as relaciona com o tensor de polarização do glúon, concluindo que são invariantes de Lorentz e têm a mesma estrutura dos pólos ultravioleta que ocorrem a temperatura zero. Usando uma simples prescrição para as constantes de renormalização e as equações do grupo de renormalização a temperatura finita, foi possível mostrar que a constante de acoplamento efetiva decai logaritmicamente em função da temperatura, de acordo com a liberdade assintótica, e é idêntica ao resultado obtido numa classe geral de calibres covariantes. / This thesis studies the In(T) contributions of the three-point function in the Yang-Mills theory in the temporal axial gauge at one-loop level in high-temperature limit. We proved that such contributions satisfy an abelian Ward identity which relates them with the gluon polarization tensor, concluding that they are Lorentz invariant and have the same structure of the ultraviolet poles which occur at zero temperature. Using a simple prescription for the renormalization constants and the finite-temperature renormalization group equations, it was possible to show that the effective coupling constant decreases logarithmically as a function of temperature, in accordance with asymptotic freedom, and is identical to the results obtained in a general class of covariant gauges.

Orientador: Ricardo Antonio Mosna / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin / Made available in DSpace on 2018-09-24T14:09:39Z (GMT). No. of bitstreams: 1 Tavares_GustavoMarques_M.pdf: 695474 bytes, checksum: c437bafa3afb0c0768437e1a139eea12 (MD5) Previous issue date: 2010 / Resumo: Neste trabalho estudamos os instantons da teoria de Yang-Mills nos espaços de Schwarzs-child e de Reissner-Nordstrom com grupo de gauge SU(2).Instantons são soluções clássicas da teoria de Yang-Mills definida em um espaço com métrica riemanniana (positiva-definida) e com ação finita. Primeiramente revisamos a formulação geométrica da teoria de Yang-Mills em uma variedade 4-dimensional,identificando os campos de gauge com conexões em um fibrado principal. Em seguida apresentamos os principais resultados clássicos relacionados aos instantons no espaço plano. Na segunda parte da dissertação realizamos um estudo sistemático das soluções da teoria de Yang-Mills nos espaços de Schwarzschild e de Reissner-Nordstrom euclidianos. Esta abordagem nos permitiu descobrir novas famílias de instantons neste contexto.Ainda,os resultados obtidos mostram que o número de famílias de instantons no espaço de Reissner- Nordstrom depende diretamente da carga elétrica que caracteriza esta geometria / Abstract: In this work we study instanton solutions of the Yang-Mills theory in Schwarzschild and Reissner-Nordstrom spaces with gauge group SU(2).Instantons are solutions to the classical field equations of Yang-Mills theory defined in a space with Riemannian (positive de finite)metric with finite action. We begin with a review of the geometric setting of Yang-Mills theory on a four dimensional manifold,which relates the gauge fields to connections on a fiber bundle.We proceed by presenting the main results related to instantons in flat space. In the second part of this thesis we perform a systematic study of the solutions of Yang-Mills theory in Euclidian Schwarzschild and Reissner-Nordstrom spaces.This approach led us to discover a new family of instantons de fined in those backgrounds. Moreover, our results show that the number of instanton families in the Reissner-Nordstrom space depends directly on the eletric charge which caracterizes this geometry / Mestrado / Física das Particulas Elementares e Campos / Mestre em Física

Η πιο κάτω εργασία έχει σκοπό να περιγράψει την κβαντική μηχανική. Θα γίνει μια προσπάθεια συνδυασμού με την σχετικότητα σαν μια ενιαία θεωρία. Στη συνέχεια θα συζητηθεί η κβαντικη θεωρία πεδίων. Τελος θα συζητηθεί το ηλεκτρομαγνητικό πεδίο οι θεωρίες βαθμίδος και το πεδίο Yang-Mills. / The following essay will discuss the quantum theorem. It will present the field theorem and later we will discuss the Yang-Millw field.

Esta tese estuda as contribuições proporcionais a In(T) da função de três pontos na teoria de Yang-Mills no calibre axial temporal na ordem de um loop no limite de altas temperaturas. Nós provamos que tais contribuições satisfazem uma identidade de Ward abeliana que as relaciona com o tensor de polarização do glúon, concluindo que são invariantes de Lorentz e têm a mesma estrutura dos pólos ultravioleta que ocorrem a temperatura zero. Usando uma simples prescrição para as constantes de renormalização e as equações do grupo de renormalização a temperatura finita, foi possível mostrar que a constante de acoplamento efetiva decai logaritmicamente em função da temperatura, de acordo com a liberdade assintótica, e é idêntica ao resultado obtido numa classe geral de calibres covariantes. / This thesis studies the In(T) contributions of the three-point function in the Yang-Mills theory in the temporal axial gauge at one-loop level in high-temperature limit. We proved that such contributions satisfy an abelian Ward identity which relates them with the gluon polarization tensor, concluding that they are Lorentz invariant and have the same structure of the ultraviolet poles which occur at zero temperature. Using a simple prescription for the renormalization constants and the finite-temperature renormalization group equations, it was possible to show that the effective coupling constant decreases logarithmically as a function of temperature, in accordance with asymptotic freedom, and is identical to the results obtained in a general class of covariant gauges.

Neste trabalho nós estudamos alguns sistemas de spins e vértices exatamente solúveis em duas dimensões. A solubilidade exata está ligada ao fato de existirem soluções não triviais das equações de fatorização, o que nos permite obter a energia livre no limite termodinâmico. Introduzimos e resolvemos pelo método de espalhamento inverso, um modelo de dez vértices assimétrico com dois e três estados nas ligações. Obtemos o diagrama de fases e mostramos que o sistema exibe uma transição de fase de primeira ordem. Analisamos um modelo de oito vértices de férmions livres e propomos uma nova relação funcional que nos permite calcular a energia livre por vértice. Mostramos que este sistema de vértices corresponde ao modelo de Ising na rede Union Jack. Apresentamos um método de solução de modelos de spin em redes triangulares a partir da solução do mesmo modelo na rede quadrada. O método se aplica sempre que o modelo de spins envolver interação de primeiros vizinhos e satisfizer a relação triângulo-estrela. Estendemos para a rede triangular, as soluções autoduais de Fateev e Zamolodchikov para a rede quadrada, de modelos de spin com simetria Z(N). Analisamos as conjecturas existentes sobre a criticalidade do modelo de Potts definido na rede de Kagomé. Baseados na simetria e nas degenerescências dessa rede conjecturamos uma expressão para a sua linha crítica. / We study some spin and vertex systems which are exactly solvable in two dimensions. The exact solubility is connected to the existence of non trivial solutions of the factorization equations which allow us to determine the free energy in the thermodynamic limit. We introduce and solve by the inverse scattering method, a ten vertex model with two and three states on the links. We get the phase diagram of the system and show that it exhibits a first order phase transition. Analysing a free fermion eight vertex model, we propose a new functional relation which permit us to get the free energy per vertex. We also show that this system is equivalent to the Ising model in a Union Jack lattice. We present a method to solve spin models on triangular lattices from the known solution of the same model on square lattices. The method applies whenever the model involves first neighbours interactions and satisfies the star triangle relation. We extend to the triangular lattice the self dual solutions of Fateev and Zamolodchikov for Z(N) invariant spin systems. We also analyse the conjectures made before for the critical Potts model on a Kagomé lattice. Based on symmetry and on the collapses of this lattice we conjecture an expression for their critical line.

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.

張之洞「書目答問」有關清代古文家，列有不立宗派、桐城派、陽湖派之專門研究， 則尚付闕如。本論文之研究目的，即在介紹陽湖文派之主要人物，與探討其形成北景 、構成形態及其文學理論之內容分析。 對於本論文之撰述，曾盡力蒐集相關人物之文集、其他著作及傳記資料。雖公私珍藏 ，亦必設法影印，以求文獻之充實完整。 本論文之研究方法，除一般之整理歸納外，特重比較分析，以顯示陽湖派文學理論之 特色，此亦本論文之用心所在也。至於論述陽湖派文人之傳記及其師友關係、學術造 詣與文學主張，靡不一本實事求是之態度，亦即博徵於文獻而可據，衷之於己而弗安 ，然後敢筆之於書，雖未必皆當，要無敢不慎也。 本論文共分六章，約九萬餘言，茲概述各章內容如下：第一章：「緒論」。說明本論 文之研究動機及目的，以及研究內容及範圍。並對研究之方法及流程加以確定，以為 本論文研究之依據。第二章：「歷代古文的流變」。概述先秦至清代陽湖派以前之散 文發展情況，並申述歷代文論之主流。第三章：「陽湖派形成的北景」。論述陽湖派 之時代環境、與桐城派之關係、影響陽湖派之人物、陽湖派之得名及前人對陽湖派能 否獨立成派之看法。第四章：「陽湖派二祖及其他文人述略」。洛述陽湖派創始者惲 敬、張惠言及二人之同調者、弟子、私淑者，並作其師承關係表。第五章：「陽湖派 的文學理論」。將陽湖派主要人物論文之原理、體裁、創作理論等歸納出重點，並由 後人之評論，探討其文學理論之特色。第六章：「結論」。對陽湖文派構成形態及其 文學理論之價值等研究重心，作一總結。

This is a study of Yang Changji (1871-1920), whose thought exerted a profound influence on the shaping of intellectual trends in the early twentieth-century China, notably the ideology of Mao Zedong, who was taught by Yang for five years. Yang, well-versed in the Confucian and Neo-Confucian traditions, spent ten years studying Western moral philosophy and education in Japan (1903-1909), Scotland (1909-1912) and Germany (1912-1913). After returning to China he devoted the rest of his life to teaching ethics and education firstly at the First Normal School in Changsha (1913-17) and latterly at Beijing University (1918-1920), and to introducing Western philosophical, ethical and educational thought through translation and writings. How Yang Changji adopted and incorporated various Western elements, such as Kantian and Neo-Kantian ethics, the British idealism of T.H. Green, the humanistic and liberal tradition instigated by J. Rousseau, and Spencerian utilitarianism, into his socio-political and ethical thoughts, while retaining the framework of Confucian humanism, is one of the principal aims of this study. This study is divided into three parts, each of which consists of three chapters. A narrative account of the Hunanese intellectual tradition and the main trends of thought prevalent in nineteenth-century China, with particular reference to Hunan, will be outlined in chapter I. The starting point of Yang's intellectual-spiritual quest was the achievement of sagehood and self-cultivation, a goal was based on a threefold humanistic concern: man's ultimate potential as an individual, the individual's relationship to society and the realisation of man's ultimate potential. Mind-cultivation and altering natural character were particularly emphasised, and his methodology was characterised by quietness, reverence and the floating mind. During this painstaking process of self-cultivation Yang's metaphysical views of man, mind and human nature were formed, influenced mainly by Zhou Dunyi, Zhang Zai and Wang Fuzhi (see chapter 2). Between 1897 and 1902 Yang reached his intellectual maturity. His reformist thought can be seen as a syncretism of the Confucian humanistic principle of "Perpetual Renewal of Life" and Western liberal democratic ideas, such as "popular sovereignty," "people's rights" and "individual rights." These new democratic ideals, together with nineteenth-century Western evolutionary theory, provided a new authority for Yang to challenge the Chinese monarchical system, and to call for political reform. However, on a practical level, Yang committed himself to an intellectualistic-educational approach mainly influenced by his idea of seeking for radical solution and gradualism (see chapter 3). Yang Changji's life and study abroad are studied in the contexts of the movement of Chinese students abroad at the turn of the twentieth century and of cultural communication between China and Scotland. The experience in Japan was crucial for providing Yang with his first contact with Western philosophy, ethics and education. The intellectual influence of Aberdeen University can be seen in Yang's systematic exposure to the history of Western ethics and modem currents of British and German ethics, such as utilitarian and evolutionary ethics and T.H. Green's concept of self-realisation. In chapter 7 of Part Ill Yang's reappraisal of Confucianism, from the perspectives of Confucianism as religion and his attitude toward traditional culture, are discussed. In chapter 8 Yang's intellectual-education approach to China's modernisation is characterised in six aspects. His social criticism is distinctive for its application of Western humanistic values, particularly the concepts of person and personality in Kantian ethical thought. Furthermore, Yang was probably the first Chinese to introduce and advocate the idea of "sound and wealthy middle class." The influence of Western thought can also be seen in shaping the core of Yang's thought, that is, two distinctive but inseparable ideas; valuing the self and comprehending the present reality, which are the subject of the final chapter. Yang incorporated pivotal ideas and values of Western liberal individualism, particularly Kantian notions of autonomy, respect for the self and person, and subjectivity, etc., into his notion of valuing the self. While freedom was the most fundamental concern in Kantian ethics and humanism, the independence of the self or an individual was at the centre of Yang's idea of valuing the self. However, his metaphysical view of the self and person remains largely a Confucian one. His notion of comprehending the present reality shows his profound concern with reality and an overwhelming emphasis on "strenuous action." The Individual's self-realisation should be applied here and now. Underlying Yang's two ideas was Confucian threefold concern with humanity. The Confucian ideal of the sage-king or junzi still loomed large in both the form and content, of each of Yang's two ideas.