nÃo hà / Neste trabalho, apresenta-se o formalismo canÃnico de quantizaÃÃoo da gravidade, tanto em sua formulaÃÃo original, para a qual a mÃtrica à a variÃvel canÃnica, quanto na de Ashtekar, onde a conexÃo autodual assume o papel de variÃvel canÃnica. Nesta Ãltima formulaÃÃo, as equaÃÃes de vÃnculo do formalismo sÃo drasticamente simplificadas, e, fazendo uso da teoria de Chern-Simons, constrÃi-se um estado que satisfaz estas equaÃÃes no vÃcuo, constituindo uma importante soluÃÃo para a equaÃÃo de Wheeler-DeWitt. O estado de Chern-Simons tambÃm tem uma representaÃÃo em loops, que recebe este nome por ser formulada em termos dos loops de Wilson.

A passagem progressiva de estados de Landau bidimensional (2D) para estados tridimensional (3D) foi estudada em Poços Quânticos Parabólicos (PQW) largos (W = 1000 6000 Å). Utilizou-se como técnica de transporte medidas da magnetoresistência em campo magnético intenso (B = 0 15 T) e inclinado ( = 0 90°; perpendicular paralelo), a baixas temperaturas (T = 50 mK). Observou-se, através da dependência angular das oscilações de Shubnikov de Haas ( = 0 90°), em PQWs cheios, várias sub-bandas ocupadas (5 a 8), a coexistência de estados de Landau 2D e 3D, sendo o gás 3D formado pelo colapso das sub-bandas elevadas, e o gás 2D pertencendo à primeira sub-banda. Através de cálculos do alargamento dos níveis de Landau devido ao espalhamento elástico ( = /2 , onde é o tempo quântico) e de cálculos auto-consistentes da energia de separação entre sub-bandas do PQW (ij = Ej Ei; e 12=12/2), obtiveram-se as condições 2 j-1,j para as sub-bandas elevadas j = 3,4,..., corroborando com as observações experimentais da coexistência de estados de Landau 2D e 3D no poço. Em PQWs parcialmente cheios, com apenas 2 sub-bandas ocupadas, observou-se, através do efeito do anticruzamento de níveis de Landau, de medidas da dependência angular da energia de ativação no regime de efeito Hall quântico, e de comparações com resultados de cálculos da estrutura eletrônica de PQWs em campo magnético inclinado, a coexistência de estados de Landau 2D e 3D, ocorrendo somente em campos intensos e com inclinação acentuada ( = 80 90°). Esta coexistência é diferente da mencionada anteriormente, quando od estados de Landau 3D são observados já em campo perpendicular. / The gradual progress, or evolution, of the two-dimensional (2D) toward three-dimensional (3D) Landau states was studied in wide parabolic quantum Wells (W = 1000 6000 Å). As transport technique, we used measurements of the magnetoresistence in intense (B = 0 15 T) and tilted ( = 0 90°; perpendicular parallel) magnetic Field at low temperature (T = 50 Mk). We observed in PQWs with Five to eight sub-bands occupied full well the coexistence of the 2D and 3D Landau states, through the angular dependence of the Shubnikov de Hass oscillation ( = 0 90°), where the 2D states belong to the lowest sub-band and the 3D states are formed by overlap of the other sub-bands. We calculated the level broadening due to the elastic scattering rate ( = /2 , where is the quantum time), and the energy separation between sub-bands (ij = Ej Ei; e 12=12/2). We obtained 2 j-1,j to j=3,4,... . This confirms the experimental observations of the coexistence of the 2D and 3D states in the well. We also measured PQWs partially full 2 sub-bands occupied. Experiments revel anticrossing of the Landau level (LL) belonging to the lowest sub-band and the last LL belonging to the second sub-band. Such antisrossuing occurs due to a decrease of the energy of the LL with tilt angle. This observation was supported by measurements of the angular dependence of the activation energy in the quantum Hall regime. In these measurements, we also observed the coexistence of the 2D and 3D Landau states. However, the coexistence only occurs at large tilt angles ( = 80 90°). Thus, it is different from the coexistence above mentioned, when 3D Landau states are observed already in the perpendicular magnetic field.

Η παρούσα διπλωματική εργασία πραγματοποιήθηκε στο τμήμα Σωματιδιακής Φυσικής του Πανεπιστημίου Santiago de Compostela της Ισπανίας και αποτελεί τη μελέτη της υπερσυμμετρίας στις τρεις χωροχρονικές διαστάσεις. Έμφαση δίνεται σε θεωρίες που περιέχουν τον όρο Chern-Simons που παιζεί συμαντικό ρόλο στους τομείς έρευνας της θεωρητικής φυσικής. Αρχικά, εισάγουμε τις εισαγωγικές ένοιες της υπερσυμμετρίας στις τρεις διαστάσεις και ακολούθως μελέτουμε την Ν=1 ελάχιστη θεωρία με διάφορες φυσικές ποσότες που περιέχουν τον όρο Chern-Simons. Στην συνέχεια, μελετάμε τις ABJM θεωρίες και αποδεικνύουμε ότι είναι αναλλοίωτες κάτω από μετασχηματισμούς βαθμίδας. Τέλος υπολογίζουμε τις κβαντικές διορθώσεις στην διαταρακτική θεωρία Chern-Simons. / The present thesis took part in Department of Particle Physics of University of Santiago de Compostela, Spain, and is the study of supersymmetry in three spacetime dimensions. Emphasis is given to theories containing the Chern-Simons term that plays an important role in the research areas of theoretical physics. First, we introduce the notion of supersymmetry in three dimensions and then we study the N = 1 minimal theory with various physical quantitative containing the term Chern-Simons. Then, we study the ABJM theories and prove that they are invariant under gauge transformations. Finally we calculate the quantum corrections to the perturbative Chern-Simons theory.

Submitted by Vasti Diniz (vastijpa@hotmail.com) on 2017-09-13T14:26:30Z No. of bitstreams: 1 arquivototal.pdf: 2274734 bytes, checksum: 20f744b4f2279525b9a574a0a0de7838 (MD5) / Made available in DSpace on 2017-09-13T14:26:30Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 2274734 bytes, checksum: 20f744b4f2279525b9a574a0a0de7838 (MD5) Previous issue date: 2017-02-22 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this thesis we deal with G odel-type Universes in the context of modi ed gravity, in particular, Chern-Simons modi ed gravity and Brans-Dicke theory with cosmological constant (BD- ). The G odel-type metrics have been intensively discussed in the General Relativity (GR) over years. It is known that such a metrics present Closed Time-like Curves (CTC's), in other words, the G odel-type metrics are themselves an example of the causality violation. Our goal is verify the consistency of the G odel-type metrics within the Chern-Simons modi ed gravity in both: non-dynamical and dynamical formulations. In the non-dynamical framework, we show that is possible a vacuum solution in contrast to GR. Another essentially new result that we get is the presence of causal solutions for a well-motivated matter source, in general, the solutions have no analogue in GR. Moreover, the vacuum solution represents the limiting case separating the completely causal and non-causal regions, such a property re ects the topological features of the Chern-Simons theory. The primordial distinguishing feature between the Chern-Simons modi ed gravity and GR solutions is the presence of the breaking Lorentz symmetry. It turns out this breaking opens up a range of new solutions. We show that the non-trivial Chern-Simons solutions in the non-dynamical framework is accompanied by rst-order corrections of the Lorentz-violating parameter. Furthermore, in the dynamical framework the geometric parameters are also a ected by second-order corrections of the Lorentzviolating parameter. We also investigated the G odel-type metrics in BD- model. We obtain a vacuum solution which is completely causal, m2 = 4!2, where for ~! ! 1 one recovers the GR with a scalar eld and cosmological constant. It is worth calling attention to the role of the cosmological constant that is fundamental in this context. / Nesta tese tratamos os Universos tipo Code' no contexto da gravidade modificada, em particular, na gravidade modificada de Chern-Simons e na teoria de Brans-Dicke com constante cosmolOgica (BD-A). As metricas tipo Code' vem sendo intensamente discuti­das na Relatividade Geral (RG) ao longo dos anon. Sabe-se que tail metricas apresentam Curvas tipo Tempo Fechadas (CTC's), ou seja, as prOprias metricas tipo Code' sao um exemplo da violagao de causalidade. Nosso objetivo é verificar a consistencia das metricas do tipo Code' dentro da gravidade modificada de Chern-Simons em ambas as formula-goes: nao dinamica e dinamica. Na formulagao nao-dinamica, mostramos que é possivel uma solugao de vacuo diferentemente da RG. Outro resultado essencialmente novo que obtemos é a presenga de solugoes causais para uma fonte de materia bem motivada, em geral, as solugoes nao tem analog° na RG. Alem disso, a solugao de vacuo representa o caso limite que separa as regioes completamente causal e nao causal, tal propriedade re­flete as caracteristicas topolOgicas da teoria de Chern-Simons. A caracteristica que difere fundamentalmente as solugoes da gravidade modificada de Chern-Simons e as da RG é a presenga da quebra da simetria de Lorentz. Acontece que essa quebra abre um leque de novas solugoes. Mostramos que as solugoes nao triviais de Chern-Simons na formulagao nao dinamica sao acompanhadas por corregoes de primeira ordem do parametro de viola­gao de Lorentz. Alem disso, na formulagao dinamica os parametros geometricos tambem sao afetados por corregoes de segunda ordem do parametro de violagao de Lorentz. Investigamos tambem as metricas tipo Code' no modelo BD-A. Obtemos uma solugao de vacuo completamente causal, m2 = 4w2, onde para cD -+ oo recupera-se a GR com um campo escalar e constante cosmolOgica. Vale a pena chamar a atengao para o papel da constante cosmolOgica que é fundamental neste contexto.

Maxim Kontsevich a défini un invariant $Z$ des sphères d'homologie rationnelle orientées de dimension $3$ en 1992, en poursuivant l'étude initiée par Edward Witten du développement perturbatif de la théorie de Chern-Simons.L'invariant $Z$ de Kontsevich est gradué. Il s'écrit $Z=(Z_n)_{nin NN }$, où $Z_n$ prend ses valeurs dans un espace $CA_n$ engendré par des diagrammes trivalents à $2n$ sommets appelésdiagrammes de Feynman-Jacobi de degré $n$.L'invariant $Z$ apparait d'abord comme un invariant $Z(M,tau)$ des sphères d'homologie rationnelle $M$ de dimension $3$ munies d'une parallélisation $tau$.Il est l'exponentielle d'un invariant $z(M,tau)=(z_n(M,tau))_{nin NN }$dont la partie de degré $n$ compte algébriquement les plongements des diagrammes de Feynman-Jacobi connexes à $2n$ sommets assujettis à vérifier certaines conditions.On peut associer un invariant homotopique entier $p_1(tau)$ aux parallélisations $tau$ des variétés orientées de dimension $3$, et il existe un élément $beta=(beta_n)_{nin NN}$ de $CA_n$ appelé anomalie tel que$$z_n(M,tau)-p_1(tau)beta_n$$ soit indépendant de $tau$ et noté $z_n(M)$.$$Z(M)=expleft((z_n(M))_{nin NN}right).$$On sait depuis l'introduction de cette constante par Greg Kuperberg et Dylan Thurston en 1999 que $beta_n=0$ si $n$ est pair et que $beta_1 neq 0$.Cette thèse porte sur le calcul de la première valeur inconnue $beta_3$. Elle en présente des expressions très simplifiées et implémentables sur ordinateur. / The Kontsevich invariant $Z$ of rational homology $3-$ sphere was constructed by Maxim Kontsevich in 1992 using configuration space integrals.This invariant is graduated. It can be written as $Z=(Z_n)_{nin NN}$, where $Z_n$ values in the space $mathcal{A}_n$ of jacobi diagram with order $n$. A Jacobi diagram with order $n$ is a trivalent graph with $2n$ vertices. At a first point, we can see $Z$ as an invariant $Z(M,tau)$ of rational homology $3-$spheres equipped with a trivialisation $tau$ so that $Z$ is the exponential of an invariant $z(M,tau)=(z_n(M,tau))_{ninNN}$. In fact, we can say that $z_n(M,tau)$ counts the number of embeddings of connected jacobi diagrams with order $n$ with some additionnal conditions. We can associate an homotopic integer invariant $p_1(tau)$ to each trivialisation $tau$ of oriented $3-$manifolds and it exists $beta=(beta_n)_{ninNN}$, where $beta_ninmathcal{A}_n$ that is called anomaly so that $$z_n(M,tau) - p_1(tay)$$ is independant of $tau$. We name it $z_n(M)$ and $$Z(M)=exp((z_n(M)_{nin NN})).$$Greg Kuperberg and Dylan Thurston introduced this constant in 1999. We already know that $beta_n=0$ if $n$ is even and $beta_1neq 0$. This thesis is about the computation of $beta_3$. It describes simplified expressions of $beta_3$, and this expressions can be compute with a computer.

A passagem progressiva de estados de Landau bidimensional (2D) para estados tridimensional (3D) foi estudada em Poços Quânticos Parabólicos (PQW) largos (W = 1000 6000 Å). Utilizou-se como técnica de transporte medidas da magnetoresistência em campo magnético intenso (B = 0 15 T) e inclinado ( = 0 90°; perpendicular paralelo), a baixas temperaturas (T = 50 mK). Observou-se, através da dependência angular das oscilações de Shubnikov de Haas ( = 0 90°), em PQWs cheios, várias sub-bandas ocupadas (5 a 8), a coexistência de estados de Landau 2D e 3D, sendo o gás 3D formado pelo colapso das sub-bandas elevadas, e o gás 2D pertencendo à primeira sub-banda. Através de cálculos do alargamento dos níveis de Landau devido ao espalhamento elástico ( = /2 , onde é o tempo quântico) e de cálculos auto-consistentes da energia de separação entre sub-bandas do PQW (ij = Ej Ei; e 12=12/2), obtiveram-se as condições 2 j-1,j para as sub-bandas elevadas j = 3,4,..., corroborando com as observações experimentais da coexistência de estados de Landau 2D e 3D no poço. Em PQWs parcialmente cheios, com apenas 2 sub-bandas ocupadas, observou-se, através do efeito do anticruzamento de níveis de Landau, de medidas da dependência angular da energia de ativação no regime de efeito Hall quântico, e de comparações com resultados de cálculos da estrutura eletrônica de PQWs em campo magnético inclinado, a coexistência de estados de Landau 2D e 3D, ocorrendo somente em campos intensos e com inclinação acentuada ( = 80 90°). Esta coexistência é diferente da mencionada anteriormente, quando od estados de Landau 3D são observados já em campo perpendicular. / The gradual progress, or evolution, of the two-dimensional (2D) toward three-dimensional (3D) Landau states was studied in wide parabolic quantum Wells (W = 1000 6000 Å). As transport technique, we used measurements of the magnetoresistence in intense (B = 0 15 T) and tilted ( = 0 90°; perpendicular parallel) magnetic Field at low temperature (T = 50 Mk). We observed in PQWs with Five to eight sub-bands occupied full well the coexistence of the 2D and 3D Landau states, through the angular dependence of the Shubnikov de Hass oscillation ( = 0 90°), where the 2D states belong to the lowest sub-band and the 3D states are formed by overlap of the other sub-bands. We calculated the level broadening due to the elastic scattering rate ( = /2 , where is the quantum time), and the energy separation between sub-bands (ij = Ej Ei; e 12=12/2). We obtained 2 j-1,j to j=3,4,... . This confirms the experimental observations of the coexistence of the 2D and 3D states in the well. We also measured PQWs partially full 2 sub-bands occupied. Experiments revel anticrossing of the Landau level (LL) belonging to the lowest sub-band and the last LL belonging to the second sub-band. Such antisrossuing occurs due to a decrease of the energy of the LL with tilt angle. This observation was supported by measurements of the angular dependence of the activation energy in the quantum Hall regime. In these measurements, we also observed the coexistence of the 2D and 3D Landau states. However, the coexistence only occurs at large tilt angles ( = 80 90°). Thus, it is different from the coexistence above mentioned, when 3D Landau states are observed already in the perpendicular magnetic field.

Orientadores: Francisco Odair Vieira de Paiva, Augusto César Ponce / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T10:14:29Z (GMT). No. of bitstreams: 1 Presoto_AdilsonEduardo_D.pdf: 2067267 bytes, checksum: 79c3ffe06a88b7cba190920dcf512036 (MD5) Previous issue date: 2011 / Resumo: No trabalho precursor de Brezis, Marcus e Ponce [15], estudou-se problemas semilineares elípticos com uma não linearidade não decrescente, contínua e dependendo apenas da variável dependente e com medidas como dados. Os autores estavam particularmente interessados no caso em que a equação não possuía solução. Numa das técnicas estudadas, eles aproximaram a medida por funções suaves através da convolução e, sob a condição adicional de convexidade da não linearidade, mostraram que as soluções correspondentes convergiam para a solução do mesmo problema com a maior medida menor do que ou igual a medida inicial tal que o problema tinha solução. O nosso objetivo é explorar profundamente este método. Ao invés de lidar com a convolução, consideramos sequências de medidas de Radon que convergem na topologia fraca-estrela e tais que o problema tem solução para cada termo. A pergunta que se põe é: as soluções convergem? Se sim, temos que o limite satisfaz a mesma equação com uma medida, em geral, distinta do limite-fraco, logo desejamos também determinar esta medida. Quando temos uma não linearidade, como descrita no parágrafo acima, as respostas têm um alto grau de variação, conforme os exemplos dados nos trabalhos de Ponce, e são inconclusivas. A proposta da tese é estudar a convergência dessas soluções para equações e sistemas semilineares elípticos com a não linearidade sendo do tipo exponencial. No caso em que temos a equação semilinear no plano, as soluções convergem para a solução do mesmo problema com uma medida que depende apenas do limite-fraco da sequência. Também, vemos que em dimensões superiores essas asserções não se verificam mais. Por fim, o sistema que aplicamos a técnica acima é o Sistema de Chern-Simons, surgido na física teórica e que representa o modelo de Chern-Simons Abeliano relativístico envolvendo duas partículas Higgs e dois campos calibrados / Abstract: In the pioneering work of Brezis, Marcus and Ponce [15], the authors studied elliptic semilinear problems with a continuous nondecreasing nonlinearity which vanishes at origin and depends only on dependent variable, and with measures as inicial data. They were particularly interested in the case which the equation does not have a solution. One of the techniques discussed was the approach of the measure by smooth functions via convolution. Under the additional condition of convexity, they showed that the corresponding solutions converge to the solution for the same problem with the largest measure less than inicial datum such that the problem admits a solution. Our aim is to explore deeply this method. Instead of dealing with the convolution, we consider sequences of Radon measures which converge in weak-star topology and such that the problem has solution for each term. The question posted is: the solutions converge? If yes, the limit solves the same problem with, in general distinct from the weak limit, another measure, thus, we also wish to determine this measure. The purpose of the thesis is to study the convergence of solutions for equations and systems with exponential nonlinearity. If we have the equation semilinear on the plane, the solutions converge to a solution for the same problem with a measure which depends only on weak limit of the sequence. We also see that in upper dimensions the results are no longer assured. In the end, the system concerned is the Chern-Simons System that comes from theoretical physics and it represents a relativistic Abelian Chern- Simons model with two Higgs particles and two gauge fields / Doutorado / Matematica / Doutor em Matemática

It is well-known that in 2 + 1 dimensions the flux attachment transmutes the statistics of a particle.The aim of this master thesis is to study the dualities between bosons and fermions induced by Abeliantopological gauge fields in 2 + 1 dimensions. Chapter 1 and 2 are reviews of known results about thepath integral quantization of Chern-Simons theory and the regularization of the fermionic path integral.In the following chapters, we will derive the statistical transmutation and various Abelian dualities in2 + 1 dimensions.

In this thesis, topologically massive Yang-Mills theory is studied in the framework of geometric quantization. This theory has a mass gap that is proportional to the topological mass m. Thus, Yang-Mills contribution decays exponentially at very large distances compared to 1/m, leaving a pure Chern-Simons theory with level number k. The focus of this research is the near Chern-Simons limit of the theory, where the distance is large enough to give an almost topological theory, with a small contribution from the Yang-Mills term. It is shown that this almost topological theory consists of two copies of Chern-Simons with level number k/2, very similar to the Chern-Simons splitting of topologically massive AdS gravity model. As m approaches to infinity, the split parts add up to give the original Chern-Simons term with level k. Also, gauge invariance of the split CS theories is discussed for odd values of k. Furthermore, a relation between the observables of topologically massive Yang-Mills theory and Chern-Simons theory is obtained. It is shown that one of the two split Chern-Simons pieces is associated with Wilson loops while the other with 't Hooft loops. This allows one to use skein relations to calculate topologically massive Yang-Mills theory observables in the near Chern-Simons limit. Finally, motivated with the topologically massive AdS gravity model, Chern-Simons splitting concept is extended to pure Yang-Mills theory at large distances. It is shown that pure Yang-Mills theory acts like two Chern-Simons theories with level numbers k/2 and -k/2 at large scales. At very large scales, these two terms cancel to make the theory trivial, as required by the existence of a mass gap.

This thesis has four chapters. After a brief introduction in Chapter 1, the $AJ$-conjecture is introduced in Chapter 2. The $AJ$-conjecture for a knot $K \subset S^3$ relates the $A$-polynomial and the colored Jones polynomial of $K$. If $K$ satisfies the $AJ$-conjecture, sufficient conditions on $K$ are given for the $(r,2)$-cable knot $C$ to also satisfy the $AJ$-conjecture. If a reduced alternating diagram of $K$ has $\eta_+$ positive crossings and $\eta_-$ negative crossings, then $C$ will satisfy the $AJ$-conjecture when $(r+4\eta_-)(r-4\eta_+)>0$ and the conditions of Theorem 2.2.1 are satisfied. Chapter 3 is about quantum curves and their relation to the $AJ$ conjecture. The variables $l$ and $m$ of the $A$-polynomial are quantized to operators that act on holomorphic functions. Motivated by a heuristic definition of the Jones polynomial from quantum physics, an annihilator of the Chern-Simons section of the Chern-Simons line bundle is found. For torus knots, it is shown that the annihilator matches with that of the colored Jones polynomial. In Chapter 4, a tangle functor is defined using semicyclic representations of the quantum group $U_q(sl_2)$. The semicyclic representations are deformations of the standard representation used to define Kashaev's invariant for a knot $K$ in $S^3$. It is shown that at certain roots of unity the semicyclic tangle functor recovers Kashaev's invariant.