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.

In the paper we study the possibility to represent the index formula for spectral boundary value problems as a sum of two terms, the first one being homotopy invariant of the principal symbol, while the second depends on the conormal symbol of the problem only. The answer is given in analytical, as well as in topological terms.

This thesis studies U(1) Chern-Simons theory and its relation to the results of Chris Beasley and Edward Witten (2005). Using the partition function formalism, we are led to compare U(1) Chern-Simons theory as constructed by Manoliu (1998) to the results of Beasley and Witten (2005). This leads to an explicit calculation of the U(1) Chern Simons partition function on a closed Sasakian three-manifold and opens the door to studying rigorous extensions of this theory to more general gauge groups and three-manifold geometries.

In this thesis, we considered electrically charged vortex solutions of Born- Infeld Chern-Simons gauge theory in 2+1 dimensions, with a sixth order charged scalar eld potential. For this purpose, rst Nielsen-Olesen vortex solutions are extensively reviewed. Then, Born-Infeld and Chern-Simons theories are summarized. Finally, vortex solutions are obtained for the Born-Infeld-Higgs system with a Chern-Simons term. These solutions are analyzed numerically, comparing their properties with Nielsen-Olesen vortices.

This thesis studies U(1) Chern-Simons theory and its relation to the results of Chris Beasley and Edward Witten (2005). Using the partition function formalism, we are led to compare U(1) Chern-Simons theory as constructed by Manoliu (1998) to the results of Beasley and Witten (2005). This leads to an explicit calculation of the U(1) Chern Simons partition function on a closed Sasakian three-manifold and opens the door to studying rigorous extensions of this theory to more general gauge groups and three-manifold geometries.

This thesis is a study of three categories of problems in fermionic systems for which topology plays an important role: (i) The properties of zero modes arising in systems of fermions interacting with a bosonic background, with a special focus on Majorana modes arising in the superconductor state. We propose a method for counting Majorana modes and we study a mechanism for controlling their number parity in lattice systems, two questions that are of relevance to the protection of quantum bits. (ii) The study of dispersionless bands in two dimensions as a platform for correlated physics, where it is shown the possibility of stabilizing the fractional quantum Hall effect in a flat band with Chern number. (iii) The extension of the hierarchy of quantum Hall fluids to the case of time-reversal symmetric incompressible ground states describing a phase of strongly interacting topological insulators in two dimensions. / Physics

Gauge theories are one of the corner stones of modern theoretical physics. They describe the nature of all fundamental interactions and have been applied in multiple branches of physics. The most challenging problem of gauge theories, which has not been solved yet, is their strong coupling dynamics. A class of gauge theories that admits simplifications allowing to deal with the strong coupling regime are supersymmetric ones. For example, recently proposed method of supersymmetric localization allows to reduce expectation values of supersymmetric observables, expressed through the path integral, to finite-dimensional matrix integral. The last one is usually easier to deal with compared to the original infinite-dimensional integral. This thesis deals with the matrix models obtained from the localization of different 5D gauge theories. The focus of our study is N=1 super Yang-Mills theory with different matter content as well as N=1 Chern-Simons-Matter theory with adjoint hypermultiplets. Both theories are considered on the five-spheres. We make use of the saddle-point approximation of the matrix integrals, obtained from localization, to evaluate expectation values of different observables in these theories. This approximation corresponds to the large-N limit of the localized gauge theory. We derive <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?N%5E%7B3%7D" /> behavior for the free energy of 5D N=1* super Yang-Mills theory at strong coupling. This result is important in light of the relation between 5D theory and the world-volume theories of M5-branes, playing a significant role in string theory. We have also explored rich phase structure of 5D SU(N) N=1 super Yang-Mills theory coupled to massive matter in different representations of the gauge group. We have shown that in the case of the massive adjoint hypermultiplet theory undergoes infinite chain of the third order phase transitions while interpolating between weak and strong coupling in the decompactification limit. Finally, we obtain several interesting results for 5D Chern-Simons theory, suggesting existence of the holographic duals to this theory. In particular, we derive <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?N%5E%7B5/2%7D" /> behavior of the free energy of this theory, which reproduces the behavior of the free energy for 5D theories with known  holographic duals.

MORAES, Jason Roberto Alves de. Gravitação quântica canônica. 2016. 72 f. Dissertação (Mestrado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2016. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2016-07-18T13:04:07Z No. of bitstreams: 1 2016_dis_jramoraes.pdf: 629993 bytes, checksum: 4192a98f10583b124eae37f6fd751afe (MD5) / Approved for entry into archive by Edvander Pires (edvanderpires@gmail.com) on 2016-07-18T13:04:46Z (GMT) No. of bitstreams: 1 2016_dis_jramoraes.pdf: 629993 bytes, checksum: 4192a98f10583b124eae37f6fd751afe (MD5) / Made available in DSpace on 2016-07-18T13:04:46Z (GMT). No. of bitstreams: 1 2016_dis_jramoraes.pdf: 629993 bytes, checksum: 4192a98f10583b124eae37f6fd751afe (MD5) Previous issue date: 2016 / Neste trabalho, apresenta-se o formalismo canônico de quantização da gravidade, tanto em sua formulação original, para a qual a métrica é a variável canônica, quanto na de Ashtekar, onde a conexão autodual assume o papel de variável canônica. Nesta última formulação, as equações de vínculo do formalismo são drasticamente simplificadas, e, fazendo uso da teoria de Chern-Simons, constrói-se um estado que satisfaz estas equações no vácuo, constituindo uma importante solução para a equação de Wheeler-DeWitt. O estado de Chern-Simons também tem uma representação em loops, que recebe este nome por ser formulada em termos dos loops de Wilson.