A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, 2016. / The AdS/CFT correspondence is a conjectured exact duality between type IIB string theory on the AdS5 S5 background and N = 4 Super Yang-Mills theory, a conformal eld theory (CFT), on the boundary of the AdS space. A speci c observable of the CFT, which can be read from the two point correlation function, is the anomalous dimension. In this dissertation we will compute spectra of anomalous dimensions of excited giant gravitons up to two loops in a speci c limit. We are interested in the anomalous dimensions because the AdS/CFT correspondence associates them with energies of states in quantum gravity. We study operators constructed using n Z elds and m Y elds with n << m: In this case m n is a small parameter. At the leading order in m n and at large N, the problem of determining the anomalous dimensions can be mapped into the dynamics of m non-interacting magnons. The subleading terms at two loops, computed for the rst time in this dissertation, induce interactions between the magnons. Even after including this new correction, we nd the BPS operators remain BPS. / MT2016

Abstract The collective field theory technique provides a method of tackling problems with two N × N matrices in the large N limit. The collective field background from one matrix is first found, then the second matrix is introduced into this background as an impurity. Within the context of the AdS/CFT correspondence, this technique can be used to describe gauge theory states in the BMN limit. This dissertation starts by developing the collective field theory technique, firstly in general variables, then for one matrix, and subsequently for two matrices. It goes on to introduce a Yang-Mills interaction term, where two variable identifications are considered. The first is the more traditional angular momentum eigenstate model. The second is a model that directly uses two of the Higgs scalars. This model has been mentioned in the literature, but has not been considered in great depth. The exact two impurity spectrum is found, and the multi-impurity spectrum is found to first order. The resulting energy values match a spectrum that has been found for giant magnons.

L’intérêt majeur des travaux exposés dans cette thèse est d’explorer la chevelure des trous noirs dans des cadres plus généraux que celui de la Relativité Générale en tenant compte de la présence d’une constante cosmologique, de dimensions supplémentaires, de champs de matière exotiques ou de termes de courbure de rang plus élevé. Ces extensions de la Relativité Générale peuvent s’inscrire dans le cadre de la théorie des cordes. C’est en étudiant des extensions naturelles de la Relativité Générale que nous pouvons aussi mieux comprendre la théorie d’Einstein. Dans un premier temps, nous exposerons la théorie de la Relativité Générale avec notamment les principes sur lesquelles elle s’appuie et nous donnerons les éléments mathématiques dont nous avons besoin pour la suite. Puis, une première extension sera présentée avec l’introduction de dimensions supplémentaires et de champs de p-formes qui constituent la généralisation naturelle de l’interaction électromagnétique. Nous construirons dans ce cadre de nouvelles solutions statiques de trous noirs où les p-formes permettent de modeler la géométrie de l’horizon. Nous exposerons ensuite l’extension la plus générale de la théorie d’Einstein en dimension quelconque qui génère des équations du second ordre en la métrique : la théorie de Lovelock. Nous déterminerons dans ce contexte une large classe de solutions en dimension 6 pour laquelle la théorie se réduit à celle d’Einstein-Gauss-Bonnet avec toujours la présence de p-formes. Enfin, nous étudierons une généralisation de la Relativité Générale en dimension 4 dont la modification est induite par un champ scalaire couplé conformément à la gravitation. Nous exhiberons notamment une nouvelle solution de trou noir avec un horizon plat dans cette théorie en présence de champs axioniques. Pour clore cette thèse, l’aspect thermodynamique de ces théories gravitationnelles sera étudié ; ce qui permettra de déterminer la masse et les charges de ces nouvelles solutions et d’étudier des phénomènes de transitions de phase en présence d’un champ scalaire conforme. / The main interest of the work exposed in this thesis is to explore hairy black holes in a more general framework than General Relativity by taking into account the presence of a cosmological constant, of higher dimensions, of exotic matter fields or of higher curvature terms. These extensions to General Relativity can be derived in the context of String Theory. It is also by studying natural extensions to General Relativity that we can more deeply understand the theory of Einstein. Firstly, we will display the theory of General Relativity with its building blocks in particular and we will give the mathematical tools that we need afterwards. Then, a first extension will be detailed with the introduction of higher dimensions and p-form fields which constitute the natural generalization of the electromagnetic interaction. We will build in this framework new static black hole solutions where p-form fields allow to shape the geometry of the horizon. Secondly, we will present the general extension of Einstein theory in any dimension which produces second order field equations: Lovelock theory. We will determine in this context a large class of solutions in dimension 6 for which the theory is reduced to Einstein-Gauss-Bonnet theory with the presence of p-form fields. Thirdly, we will study a generalization of General Relativity in dimension 4 whose modification is induced by a conformally coupled scalar field. We will namely exhibit a new black hole solution with a flat horizon in the presence of axionic fields. To conclude this thesis, thermodynamical aspects of these gravitational theories will be studied. In this way, we will be able to determine the mass and the charges of these new solutions and we will examine phase transition phenomena in the presence of a conformally scalar field.

This D.Sc. thesis proposes new perspectives for modeling drill string torsional dynamics under uncertainties. This work develops a novel stochastic hysteretic (nonreversible) bit-rock interaction model. Firstly, a new nominal interaction model, which depends not only on the bit speed, but also on the bit acceleration is developed. Then, a new stochastic model for the bit-rock interaction, taking into account the inherent fluctuations during the drilling, is also proposed. Furthermore, here a new test-rig is proposed to analyze drill string dynamics and bit-rock interaction, which is able to reproduce stick-slip phenomena while drilling a rock sample using standard masonry bits, as well as to validate bit-rock interaction models. An original strategy for modeling uncertainties globally, based on terms of the nonparametric probabilistic approach, considering a simple torsional model for a drill string, is also proposed herein. This strategy allows to control the dispersion level of each interior and interface DOFs of each drill string substructure independently, which can provide more information to improve the operational safety. / This D.Sc. thesis proposes new perspectives for modeling drill string torsional dynamics under uncertainties. This work develops a novel stochastic hysteretic (nonreversible) bit-rock interaction model. Firstly, a new nominal interaction model, which depends not only on the bit speed, but also on the bit acceleration is developed. Then, a new stochastic model for the bit-rock interaction, taking into account the inherent fluctuations during the drilling, is also proposed. Furthermore, here a new test-rig is proposed to analyze drill string dynamics and bit-rock interaction, which is able to reproduce stick-slip phenomena while drilling a rock sample using standard masonry bits, as well as to validate bit-rock interaction models. An original strategy for modeling uncertainties globally, based on terms of the nonparametric probabilistic approach, considering a simple torsional model for a drill string, is also proposed herein. This strategy allows to control the dispersion level of each interior and interface DOFs of each drill string substructure independently, which can provide more information to improve the operational safety

Esta tese visa realizar um estudo acerca das freqüências quasinormais das p-branas negras utilizando um campo escalar de teste. No contexto da correspondência AdS/CFT estudamos alguns aspectos da formação de estados supercondutores na teoria de campos definida na borda do espaço-tempo de Gauss-Bonnet AdS d-dimensional. Apresentamos a derivação das soluções das p-branas negras, bem como a análise de sua estrutura causal. Revisamos o regime perturbativo da Relatividade Geral e a interpretação dos modos quasinormais de um buraco negro em termos dos elementos que compõem a correspondência AdS/CFT. Estudamos a evolução de um campo escalar sem massa na geometria das p-branas negras em detalhe, sendo que utilizamos o método semianalítico WKB e a integração característica (Problema de Goursat) para calcular as freqüências quasinormais correspondentes. Por fim, apresentamos uma expressão aproximada de um estado supercondutor na teoria de campos definida na borda do buraco negro de Gauss-Bonnet AdS através do dicionário AdS/CFT. / This thesis aims at studying the quasinormal frequencies of black p-branes using a scalar field in the probe limit. In the context of AdS/CFT correspondence we study some aspects regarding the formation of superconductor states in the field theory set on the AdS boundary of the Gauss-Bonnet AdS spacetime. We present the derivation of black p-brane solutions, as well as the analysis of its causal structure. We review the perturbative regime of General Relativity and the interpretation of the black hole quasinormal modes in terms of the correspondence AdS/CFT. We study the evolution of a massless scalar field in the geometry of black p-branes in detail, and we use the WKB method and the characteristic integration (Goursats problem) in order to calculate the corresponding quasinormal frequencies. Ultimately, we present an approximate expression for a superconductor state in the field theory defined in the AdS boundary of Gauss-Bonnet AdS spacetime.

Este é um trabalho de revisão. Os principais formalismos no desenvolvimento da teoria de cordas são introduzidos e discutidos: a corda bosônica e as supercordas no formalismo de Ramond-Neveu-Schwarz e no formalismo de Green-Schwarz. São introduzidas também as açôes efetivas no limite de baixas energias das D-branas (ação DBI) e das teorias de supergravidade (D=10,11). Por fim, são analisados alguns aspectos da correspondência AdS/CFT, enunciando a conjectura e introduzindo algumas das entradas do dicionário. Os apêndices contêm tópicos de grande relevância para o texto e podem ajudar a esclarecer vários raciocínios e passagens ao longo do mesmo. / This is a review work. The main formalisms on the development of string theory are introduced and discussed: the bosonic string and the superstrings formalisms of Ramond-Never-Schwarz and Green-Schwarz. The effective actions in the low energy limit of the D-branes (DBI action) and supergravity theories (D=10,11) are introduced as well. Finally, some aspects of the AdS/CFT correspondence are analised, enunciating the conjecture and introducing some entries of the dictionary. The appendices contains some topics that are highly relevant for the whole text and may help clarify several ideas and arguments through it.

Nesta tese investigamos a integrabilidade quântica do modelo de Alday-Arutyunov-Frolov (AAF) através da diagonalização da Hamiltoniana do modelo transformado de AAF. O modelo transformado de AAF é obtido do modelo original após uma redefinição nos campos que tem o objetivo de transformar os parênteses de Dirac na estrutura canônicos padrão entre dois fêrmions. Para diagonalização da Hamiltoniana do modelo transformado nós usamos um método desenvolvido para o modelo de Landau-Lifshitz. Este método exige regularização do produto de operadores definidos no mesmo ponto bem como a construção de extensões auto-adjuntas. Mostramos que a função de onda e os estados do modelo transformado de AAF apresentam descontinuidades que devem ser tratadas cuidadosamente. Nós também investigamos a equivalência entre as matrizes S das duas formulações do modelo de AAF através do cálculo direto da matriz S em ambas as teorias. Também consideramos o teorema da equivalência para o modelo de AAF, para mostrar que a matriz S é invariante por redefinições dos campos da teoria. / In this thesis we investigate the quantum integrability of the Alday-Arutyunov-Frolov model (AAF) by diagonalization of the Hamiltonian of the AAF transformed model. The AAF transformed model is obtained from original model after fields redefinition whose purpose is to transform the Dirac brackets in the standard canonical structure between two fermions. For the diagonalization of the Hamiltonian of the transformed model we use the method developed for the Landau-Lifshitz model. This method requires regularization of the singular product of operators at the same point as well as construction of the selfadjoint extensions. We show that the wave function and the states of the transformed AAF exhibit discontinuities. We also investigated the equivalence of the matrices S of the two formulations of the AAF model through direct calculation of the S-matrix in both theories. We also consider the equivalence theorem applied to AAF model. To shows that the S matrix is invariant under field redefinitions.

In this work we review the basic principles of the theory of the relativistic bosonic string through the study of the action functionals of Nambu-Goto and Polyakov and the techniques required for their canonical, light-cone, and path-integral quantisation. For this purpose, we briefly review the main properties of the gauge symmetries and conformal field theory involved in the techniques studied. / Neste trabalho fazemos uma revisão dos princípios básicos da teoria da corda bosônica relativística através do estudo dos funcionais ação de Nambu-Goto e de Polyakov e das técnicas necessárias para sua quantização canônica, no cone de luz e usando integrais de trajetória. Para tanto apresentamos uma pequena revisão das principais propriedades das simetrias de calibre a da teoria de campos conforme envolvidas nas técnicas estudadas.

Wong Hiu Lam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 104-109). / Abstracts in English and Chinese. / Abstract --- p.iv / Abstract (Chinese Translation) --- p.v / Acknowledgements --- p.vi / Chapter Chapter 1 --- Introduction --- p.1 / Overview of Bartok's six quartets --- p.1 / Literature review --- p.4 / Objectives --- p.9 / Chapter Chapter 2 --- "Karpati, Antokoletz and Wilson's Analytical Approaches" --- p.11 / Karpati's idea of mistiming --- p.11 / Antokoletz's interval cycle and interval ratio --- p.13 / Wilson's interest in structural hierarchy --- p.18 / Chapter Chapter 3 --- "The Fifth Quartet, First and Fifth Movements" --- p.23 / "The Fifth Quartet, first movement" --- p.24 / Form --- p.24 / Pitch centres --- p.26 / Fourths --- p.28 / "The Fifth Quartet, fifth movement" --- p.35 / Form --- p.35 / Pitch centres --- p.37 / Fourths --- p.37 / Symmetries --- p.41 / Chapter Chapter 4 --- "The Fifth Quartet, Second and Fourth Movements" --- p.45 / Bartok's analysis of the Fifth Quartet --- p.45 / Other analyses --- p.46 / Bartok's formal analysis of the second and fourth movements --- p.47 / An extended introduction --- p.48 / Similarities between the second and fourth movements --- p.49 / Prominent intervals and pc sets --- p.49 / Instrumental sequence --- p.53 / Juxtaposition and superimposition of(0123)s and (0167)s --- p.53 / Juxtaposition and superimposition: melody and chordal accompaniment --- p.54 / Thematic kinship --- p.55 / Texture --- p.57 / Chapter Chapter 5 --- "Figuring Thirds in the Fifth Quartet, Third Movement" --- p.59 / Formal structure of the whole quartet and the third movement --- p.59 / Motive a and third-structure --- p.60 / "Definition of ""third-structure""" --- p.61 / Arch-shaped melodies formed by thirds --- p.61 / Third-structures and the diatonic collection --- p.62 / Third-structures and the octatonic collection --- p.62 / Third-structures and the chromatic collection --- p.67 / Referential collections and formal structure --- p.67 / Chapter Chapter 6 --- "The Sixth Quartet, First and Fourth Movements" --- p.69 / Existing Literature --- p.69 / Formal structure --- p.71 / Mesto themes of the two movements --- p.73 / An unusual finale --- p.75 / Fourths --- p.79 / Chapter Chapter 7 --- "Duality in the Sixth Quartet, Second Movement" --- p.84 / Karpati and dual thirds --- p.87 / Dual centres B and G# --- p.88 / (0347)s --- p.90 / Relationship between duality and form --- p.92 / Chapter Chapter 8 --- "The Sixth Quartet, Third Movement" --- p.94 / Chapter Chapter 9 --- Conclusion --- p.100 / Selected Bibliography --- p.104

Information about the very early universe can be accessed from observations of the cosmic microwave background (CMB) radiation and the later formation of large-scale structure (LSS) that are produced from cosmological perturbations of the early universe. The most developed theoretical explanation for the origin of these perturbations is the theory of inflation, in which the early universe undergoes a period of accelerated expansion, amplifying quantum fluctuations to macroscopic size, which act as the seeds for the CMB anisotropies and the cosmic web of the LSS. The work in this thesis aims to connect the theory of inflation to properties of these observables in a highly detailed way, suitable for future high-precision astronomical surveys. After some introductory review chapters, we begin with new research on a study of inflation from string theory, deriving an observably-large value of the tensor-to-scalar ratio, which had been previously difficult to achieve theoretically. The next study investigates the link between the observed CMB power asymmetry and non-Gaussianity, including a novel non-zero value for the trispectrum. Next we study soft limits of non-Gaussian inflationary correlation functions, focussing first on the squeezed limit of the bispectrum and then generalizing to soft limits of higher-point correlation functions, giving results valid for multi-fi eld models of inflation.