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<p>The appearance in string theory of higher-dimensional objects known as D-branes has been a source of much of the interesting developements in the subject during the past ten years. A very interesting phenomenon occurs when several of these D-branes are made to coincide: The abelian gauge theory living on each brane is enhanced to a non-abelian gauge theory living on the stack of coincident branes. This gives rise to interesting effects like the natural appearance of non-commutative geometry. The theory governing the dynamics of these coincident branes is still poorly understood however and only hints of the underlying structure have been seen.</p><p>This thesis focuses on an attempt to better this understanding by writing down actions for coincident branes using so-called boundary fermions, originating in considerations of open strings, instead of matrices to describe the non-abelian fields. It is shown that by gauge-fixing and by suitably quantizing these boundary fermions the non-abelian action that is known, the Myers action, can be reproduced. Furthermore it is shown that under natural assumptions, unlike the Myers action, the action formulated using boundary fermions also posseses kappa-symmetry, the criterion for being the correct supersymmetric action for coincident D-branes.</p><p>Another aspect of string theory discussed in this thesis is that of tensionless strings. These are of great interest for example because of their possible relation to higher spin gauge theories via the AdS/CFT-correspondence. The tensionless superstring in a plane wave background, arising as a particular limit of the near-horizon geometry of a stack of D3-branes, is considered and compared to the tensile case.</p>

The appearance in string theory of higher-dimensional objects known as D-branes has been a source of much of the interesting developements in the subject during the past ten years. A very interesting phenomenon occurs when several of these D-branes are made to coincide: The abelian gauge theory living on each brane is enhanced to a non-abelian gauge theory living on the stack of coincident branes. This gives rise to interesting effects like the natural appearance of non-commutative geometry. The theory governing the dynamics of these coincident branes is still poorly understood however and only hints of the underlying structure have been seen. This thesis focuses on an attempt to better this understanding by writing down actions for coincident branes using so-called boundary fermions, originating in considerations of open strings, instead of matrices to describe the non-abelian fields. It is shown that by gauge-fixing and by suitably quantizing these boundary fermions the non-abelian action that is known, the Myers action, can be reproduced. Furthermore it is shown that under natural assumptions, unlike the Myers action, the action formulated using boundary fermions also posseses kappa-symmetry, the criterion for being the correct supersymmetric action for coincident D-branes. Another aspect of string theory discussed in this thesis is that of tensionless strings. These are of great interest for example because of their possible relation to higher spin gauge theories via the AdS/CFT-correspondence. The tensionless superstring in a plane wave background, arising as a particular limit of the near-horizon geometry of a stack of D3-branes, is considered and compared to the tensile case.

O caso finito de teorias topológicas de 1-gauge, quando nenhuma simetria global está presente, é bastante bem compreendido e classificado. Nos últimos anos, as tentativas de generalizar as teorias de 1-gauge através das chamadas teorias de 2-gauge abriram a porta para novos modelos interessantes e novas fases topológicas, as quais não são descritas pelos esquemas de classificação anteriores. Nesta tese, vamos além da construção de 2-gauge, e consideramos uma classe de modelos que vivem em maiores dimensões. Esses modelos estão inseridos em uma estrutura de complexos de cadeia de grupos abelianos, forçando a generalizar o conceito usual de configurações de gauge. A vantagem de tal abordagem é que, a ordem topológica fica manifestamente explcita. Isto é feito em ter- mos de uma cohomologia com coeficientes em um complexo de cadeia finita. Além disso, mostramos que a degenerescência do estado fundamental suporta um conjunto conve- niente de números quânticos que indexam os estados e que, além, foram completamente caracterizados. Consequentemente, nós também mostramos que muitos dos exemplos abelianos de teorias de 1 -gauge 2-gauge são recuperados como casos especiais desta construção. / The finite case of 1-gauge topological theories, when no global symmetries are present, is fairly well understood and classified. In recent years, attempts to generalize the latter situation through the so called 2-gauge theories have opened the door to interesting new models and new topological phases, not described by the previous schemes of classifica- tion. In this paper we go even beyond the 2-gauge construction by considering a class of models that live in arbitrary higher dimensions. These models are embedded in a structure of chain complexes of abelian groups, forcing to generalize the usual notion of gauge configurations. The advantage of such an approach is that, the topological order is explicitly manifest when the ground state space of these models is described. This is done in terms of a cohomology with coefficients in a finite chain complex. Furthermore, we show that the ground state degeneracy underpins a convenient set of quantum num- bers that label the states and that have been completely characterized. We also show that abelian examples of 1-gauge 2-gauge theories are recovered as special cases of this construction.

O caso finito de teorias topológicas de 1-gauge, quando nenhuma simetria global está presente, é bastante bem compreendido e classificado. Nos últimos anos, as tentativas de generalizar as teorias de 1-gauge através das chamadas teorias de 2-gauge abriram a porta para novos modelos interessantes e novas fases topológicas, as quais não são descritas pelos esquemas de classificação anteriores. Nesta tese, vamos além da construção de 2-gauge, e consideramos uma classe de modelos que vivem em maiores dimensões. Esses modelos estão inseridos em uma estrutura de complexos de cadeia de grupos abelianos, forçando a generalizar o conceito usual de configurações de gauge. A vantagem de tal abordagem é que, a ordem topológica fica manifestamente explcita. Isto é feito em ter- mos de uma cohomologia com coeficientes em um complexo de cadeia finita. Além disso, mostramos que a degenerescência do estado fundamental suporta um conjunto conve- niente de números quânticos que indexam os estados e que, além, foram completamente caracterizados. Consequentemente, nós também mostramos que muitos dos exemplos abelianos de teorias de 1 -gauge 2-gauge são recuperados como casos especiais desta construção. / The finite case of 1-gauge topological theories, when no global symmetries are present, is fairly well understood and classified. In recent years, attempts to generalize the latter situation through the so called 2-gauge theories have opened the door to interesting new models and new topological phases, not described by the previous schemes of classifica- tion. In this paper we go even beyond the 2-gauge construction by considering a class of models that live in arbitrary higher dimensions. These models are embedded in a structure of chain complexes of abelian groups, forcing to generalize the usual notion of gauge configurations. The advantage of such an approach is that, the topological order is explicitly manifest when the ground state space of these models is described. This is done in terms of a cohomology with coefficients in a finite chain complex. Furthermore, we show that the ground state degeneracy underpins a convenient set of quantum num- bers that label the states and that have been completely characterized. We also show that abelian examples of 1-gauge 2-gauge theories are recovered as special cases of this construction.

The aim of this project is to understand the structure of the Standard Model of the particle physics. Therefore quantum field theories (QFT) are studied in the both cases of abelian and non-abelian gauge theories i.e. quantum electrodynamics (QED), quantum chromodynamics (QCD) and electroweak interaction are reviewed. The solution to the mass problem arising in these theories i.e. spontaneous symmetry breaking is also studied. / Syftet med detta projekt är att förstå strukturen för partikelfysikens standardmodell. Därför studeras kvantfältsteorier (QFT) i båda fallen av abelska och icke-abelska gaugeteorier, dvs kvantelektrodynamik (QED), kvantkromodynamik (QCD) och elektrosvag växelverkan granskas. Lösningen på massproblemet som uppstår i dessa teorier, dvs. spontant symmetribrott studeras också.

This thesis investigates the possibility that the gauge boson in a certain hypothetical SU(2) gauged sector can constitute all the non-baryonic dark matter. The gauge bosons acquire mass from the Higgs mechanism as in the Standard Model and scatter elastically among themselves non-gravitationally. It is expected that this self interaction of the dark gauge bosons would resolve the various discrepancies between the ΛCDM model and astrophysical observations on small (e.g. galactic or galaxy cluster) scales. Parameter space within the domain of validity of perturbation theory satisfying the constraints of dark matter abundance, the elastic self-scattering momentum transfer cross-section suggested by recent astrophysical observations, and consideration of the Big-Bang nucleosynthesis was found to be non-empty in the “forbidden” regime where the mass of the dark Higgs boson is greater than the mass of the dark gauge boson. / Graduate

Ultracold atomic systems have provided an ideal platform to study the physics of strongly interacting many body systems in an unprecedentedly controlled and clean environment. And, since fermions are the building blocks of visible matter, being naturally motivated we focus on the physics of ultracold fermionic systems in this thesis. There have been many recent experimental developments in these systems such as the creation of synthetic gauge fields, realization of dimensional crossover and realization of systems with synthetic dimensions. These developments pose many open theoretical questions, some of which we address in this thesis. We start the discussion by studying the spectral function of an ideal spin-12 Fermi gas in a harmonic trap in any dimensions. We discuss the performance of the local density approximation (LDA) in calculating the spectral function of the system by comparing it to exact numerical results. We show that the LDA gives better results for larger number of particles and in higher dimensions. Fermionic systems with quasi two dimensional geometry are of great importance because of their connections to the high-Tc superconducting cuprate materials. Keeping this in mind, we consider a spin-12 fermionic system in three dimensions interacting with a contact interaction and confined by a one dimensional optical potential in one direction. Using the Bogoliubov-de Gennes formalism, we show that with increasing the depth of the optical potential the three dimensional superfluid evolves into a two dimensional one by looking at the shifts in the radio-frequency spectrum of the system and the change in the binding energy of the pairs that are formed. The next topic of interest is studying the effect of synthetic gauge fields on the ultracold fermionic systems. We show that a synthetic non-Abelian Rashba type gauge field has experimentally observable signatures on the size and shape of a cloud of a system of non-interacting spin-12 Fermi system in a harmonic trap. Also, the synthetic gauge field in conjunction with the harmonic potential gives rise to ample possibilities of generating novel quantum Hamiltonians like the spherical geometry quantum Hall, magnetic monopoles etc. We then address the physics of fermions in “synthetic dimensions”. The hyperfine states of atoms loaded in a one dimensional optical lattice can be used as an extra dimension, called the synthetic dimension (SD), by using Raman coupling. This way a finite strip Hofstadter model is realized with a tunable flux per plaquette. The experimental realization of the SD system is most naturally possible in systems which also have SU(M) symmetric interactions between the fermions. The SU(M) symmetric interactions manifest as long-ranged along the synthetic dimension and is the root cause of all the novel physics in these systems. This rich physics is revealed by a mapping of the Hamiltonian of the system to a system of particles interacting via an SU(M) symmetric interaction under the influence of an SU(M) Zeeman field and a non-Abelian SU(M) gauge field. For example, this equivalence brings out the possibility of generating a non-local interaction between the particles at different sites; while the gauge filed mitigates the baryon (SU(M) singlet M-body bound states) breaking effect of the Zeeman field. As a result, the site localized SU(M) singlet baryon gets deformed and forms a “squished baryon”. Also, finite momentum dimers and resonance like states are formed in the system. Many body physics in the SD system is then studied using both analytical and numerical (Density Matrix Renormalization Group) techniques. This study reveals fascinating possibilities such as the formation of Fulde-Ferrell-Larkin-Ovchinnikov states even without any “imbalance” and the possibility to evolve a “ferromagnet” to a “superfluid” by the application of a magnetic field. Other novel fermionic phases with quasi-condensates of squished baryons are also demonstrated. In summary, the topics addressed in this thesis demonstrate the possibilities and versatilities of the ultracold fermionic systems used in conjunction with synthetic gauge fields and dimensions

Several different approaches to renormalization are studied. The Callan-Symanzik equation is derived and we study its beta functions. An effective potential for the Coleman-Weinberg model is studied to find that the beta function is positive and that spontaneous symmetry breaking will occur if we expand around the classical field. Lastly we renormalize a non-abelian gaugetheory to find that the beta function in QCD is negative.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste tese, estudamos os efeitos não perturbativos associados à presença do horizonte de Gribov e à condensação de operadores locais de dimensão dois, numa teoria de Yang-Mills euclidiana em SU(2), quantizada no calibre abeliano maximal. Estes efeitos são introduzidos de modo a preservar as propriedades de renormalizabilidade e localidade da teoria, e refletem-se diretamente no comportamento dos propagadores. A comparação com os dados da rede indicam um bom acordo qualitativo. / In this, we study the nonperturbative effects associated to the presence of the horizon and to the condensation of local dimension two operators in an Eucledean SU(2)Yang-Mills theory quantized in the maximal Abelian gauge. Such effects are introduced in a way to preserve the properties of renormalizability and locality of the theory. The comparison with the lattice data indicates a good qualitative agreement.