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A modern and straight forward summary of the necessary tools andconcepts needed to understand and work with gauge theory in a fibre bundle formalism. Due to the aim of being a quick but thorough introductionfull derivations are rarely included, but references to such are given wherethey have been omitted. General Relativity, although being a geometrical theory, in the sense that the gravitational force is described by the curvature of space-time, may not be derived from geometry like the other fundamental forces as in Yang-Mills theory. Thus, a possibility of unification lies in a geometrical derivation of gravity from gauge principles. By applying the presented formalism to the case of Gravity such a derivationis pursued along the lines of nonlinear realizations of the gauge group.

Thesis (Ph.D.)--Boston University / In this dissertation I use lattice gauge theory to study models of electroweak symmetry breaking that involve new strong dynamics. Electroweak symmetry breaking (EWSB) is the process by which elementary particles acquire mass. First proposed in the 1960s, this process has been clearly established by experiments, and can now be considered a law of nature. However, the physics underlying EWSB is still unknown, and understanding it remains a central challenge in particle physics today. A natural possibility is that EWSB is driven by the dynamics of some new, strongly-interacting force. Strong interactions invalidate the standard analytical approach of perturbation theory, making these models difficult to study. Lattice gauge theory is the premier method for obtaining quantitatively-reliable, nonperturbative predictions from strongly-interacting theories. In this approach, we replace spacetime by a regular, finite grid of discrete sites connected by links. The fields and interactions described by the theory are likewise discretized, and defined on the lattice so that we recover the original theory in continuous spacetime on an infinitely large lattice with sites infinitesimally close together. The finite number of degrees of freedom in the discretized system lets us simulate the lattice theory using high-performance computing. Lattice gauge theory has long been applied to quantum chromodynamics, the theory of strong nuclear interactions. Using lattice gauge theory to study dynamical EWSB, as I do in this dissertation, is a new and exciting application of these methods. Of particular interest is non-perturbative lattice calculation of the electroweak S parameter. Experimentally S ~ -0.15(10), which tightly constrains dynamical EWSB. On the lattice, I extract S from the momentum-dependence of vector and axial-vector current correlators. I created and applied computer programs to calculate these correlators and analyze them to determine S. I also calculated the masses and other properties of the new particles predicted by these theories. I find S > 0.1 in the specific theories I study. Although this result still disagrees with experiment, it is much closer to the experimental value than is the conventional wisdom S > 0.3. These results encourage further lattice studies to search for experimentally viable strongly-interacting theories of EWSB.

This thesis is devoted to a discussion of various aspects of supersymmetric quantum field theories in four and two dimensions. In four dimensions, 𝒩 = 1 supersymmetric quantum gauge theories on various four-manifolds are constructed. Many of their properties, some of which are distinct to the theories on flat spacetime, are analyzed. In two dimensions, general 𝒩 = (2, 2) nonlinear sigma models on S² are constructed, both for chiral multiplets and twisted chiral multiplets. The explicit curvature coupling terms and their effects are discussed. Finally, 𝒩 = (0, 2) gauged linear sigma models with nonabelian gauge groups are analyzed. In particular, various dualities between these nonabelian gauge theories are discussed in a geometric content, based on their Higgs branch structure. / Ph. D.

Quantum Chromo Dynamics (QCD) is a relativistic field theory of a non-abelian gauge field coupled to several flavors of fermions. Two dimensional (one space and one time) QCD serves as an interesting toy model that shares several features with the four dimensional physically relevant theory. The main aim of the research is to study two dimensional QCD using the lattice regularization. Two dimensional QCD without any fermion content is solved analytically using lattice regularization. Explicit expressions for the expectation values of Wilson loops and the correlation of two Polyakov loops oriented in two different directions are obtained. Physics of the QCD vacuum is explained using these results. The Hamiltonian formalism of lattice QCD with fermion content serves as an approach to study quark excitations out of the vacuum. The formalism is first developed and techniques to numerically evaluate the spectrum of physical particles, namely, meson and baryons are described. The Hybrid Monte Carlo technique was used to numerically extract the lowest meson and baryon masses as a function of the quark masses. It is shown that neither the lowest meson mass nor the lowest baryon mass goes to zero as the quark mass is taken to zero. This numerically establishes the presence of a mass gap in two dimensional QCD.

Gauge theories is one of the most pervasive and important subject of modern theoretical physics, and there are still many things about them we do not understand. In particular dealing with strongly coupled theories where normal perturbative techniques do not apply is a fundamental open problem. In this thesis, we study a particular class of toy-models that have supersymmetry, which makes them much easier to deal with. We employ the mathematical technique of localization, which for supersymmetric theories lets us evaluate certain path integrals exactly and for any value of the coupling. This is used to study the 5d N=1 theories placed on toric Sasaki-Einstein manifolds and compute their partition functions, finding that they factorize into a product of contributions from each closed Reeb orbit of the manifold. This computation leads us to define two new hierarchies of special functions associated to these manifolds, and we study their properties. Finally, we use the 5d N=1 theories to construct new 4d N=2 theories on a large class of curved backgrounds. These theories have some interesting features, such as supporting both instantons and anti-instantons, and having a position-dependent complexified coupling constant.

Calculamos as funções de Green térmicas de um e dois grávitons em um \"loop\" no \"gauge\" temporal. Para tratar os polos extras que surgem neste \"gauge\" utilizamos uma técnica livre de ambiguidades no formalismo de tempo imaginário. Foram calculadas, para temperaturas altas comparadas com o momento externo e menores que a escala de Plank, as contribuições dominantes T4 e sub-dominantes T2 e log(T) para a auto-energia do gráviton. Mostramos que as contribuições dos polos extras não modificam o comportamento das contribuições dominantes e sub-dominantes em altas temperaturas. Verificamos que os termos dominantes T4 não dependem do parâmetro de \"gauge\". Verificamos também que as identidades de \' t Hooft são satisfeitas pelos termos T2 e mostramos que o termos log(T) tem a mesma estrutura que o resíduo do polo ultravioleta da auto-energia a temperatura zero. Calculamos as relações de dispersão no plasma de grávitons até a ordem GT2 e discutimos a dependência de \"gauge\" induzida pelos termos T 2 da auto-energia. Calculamos também a função de um gráviton em dois \"loops\" em duas classes distintas de \"gauge\". / We have computed the thermal one-graviton function and the self-energy in oneloop using a temporal gauge. In order to deal with the extra poles which are present in the temporal gauge, we employ an ambiguity-free technique in the imaginary-time formalismo We obtain, for temperatures T high compared with the externaI momentum and well below the Plank scale, the leading T4 as well as the sub-Ieading T2 and log(T) contributions to the graviton self-energy. We show that the extra pole contributions do not change the behaviour of the leading and sub-Ieading contributions from hard thermal loop region. We verify that the leading contributions are gauge independent. We also verify the t Hooft identities for the sub-Ieading T 2 terms and show that the logarithmic part has the same structure as the residue of ultraviolet pole of zero temperature graviton self-energy. We compute the dispersion relations up to GT2 order and we discuss the gauge dependence induced by sub-Ieading contributions T 2. We also compute the thermal one-graviton Green\'s function in the two-Ioop approximation for two distinct gauge classes.

Calculamos as funções de Green térmicas de um e dois grávitons em um \"loop\" no \"gauge\" temporal. Para tratar os polos extras que surgem neste \"gauge\" utilizamos uma técnica livre de ambiguidades no formalismo de tempo imaginário. Foram calculadas, para temperaturas altas comparadas com o momento externo e menores que a escala de Plank, as contribuições dominantes T4 e sub-dominantes T2 e log(T) para a auto-energia do gráviton. Mostramos que as contribuições dos polos extras não modificam o comportamento das contribuições dominantes e sub-dominantes em altas temperaturas. Verificamos que os termos dominantes T4 não dependem do parâmetro de \"gauge\". Verificamos também que as identidades de \' t Hooft são satisfeitas pelos termos T2 e mostramos que o termos log(T) tem a mesma estrutura que o resíduo do polo ultravioleta da auto-energia a temperatura zero. Calculamos as relações de dispersão no plasma de grávitons até a ordem GT2 e discutimos a dependência de \"gauge\" induzida pelos termos T 2 da auto-energia. Calculamos também a função de um gráviton em dois \"loops\" em duas classes distintas de \"gauge\". / We have computed the thermal one-graviton function and the self-energy in oneloop using a temporal gauge. In order to deal with the extra poles which are present in the temporal gauge, we employ an ambiguity-free technique in the imaginary-time formalismo We obtain, for temperatures T high compared with the externaI momentum and well below the Plank scale, the leading T4 as well as the sub-Ieading T2 and log(T) contributions to the graviton self-energy. We show that the extra pole contributions do not change the behaviour of the leading and sub-Ieading contributions from hard thermal loop region. We verify that the leading contributions are gauge independent. We also verify the t Hooft identities for the sub-Ieading T 2 terms and show that the logarithmic part has the same structure as the residue of ultraviolet pole of zero temperature graviton self-energy. We compute the dispersion relations up to GT2 order and we discuss the gauge dependence induced by sub-Ieading contributions T 2. We also compute the thermal one-graviton Green\'s function in the two-Ioop approximation for two distinct gauge classes.

In this thesis, we study the Fisher's zeros in lattice gauge theory. The analysis of singularities in the complex coupling plane is an important tool to understand the critical phenomena of statistical models. The Fisher's zero structure characterizes the scaling properties of the underlying models and has a strong influence on the complex renormalization group transformation flows in the region away from both the strong and weak coupling regimes. By reconstructing the density of states, we try to develop a systematical method to investigate these singularities and we apply the method to SU(2) and U(1) lattice gauge models with a Wilson action in the fundamental representation. We first take the perturbative approach. By using the saddle point approximation, we construct the series expansions of the density of states in both of the strong and weak regimes from the strong and weak coupling expansions of the free energy density. We analyze the SU(2) and U(1) models. The expansions in the strong and weak regimes for the two models indicate both possess finite radii of convergence, suggesting the existence of complex singularities. We then perform the numerical calculations. We use Monte Carlo simulations to construct the numerical density of states of the SU(2) and U(1) models. We also discuss the convergence of the Ferrenberg-Swendsen's method which we use for the SU(2) model and propose a practical method to find the initial values that improve the convergence of the iterations. The strong and weak series expansions are in good agreement with the numerical results in their respective limits. The numerical calculations also enable the discussion of the finite volume effects which are important to the weak expansion. We calculate the Fisher's zeros of the SU(2) and U(1) models at various volumes using the numerical entropy density functions. We compare different methods of locating the zeros. By the assumption of validity of the saddle point approximation, we find that the roots of the second derivative of the entropy density function have an interesting relation with the actual zeros and may possibly reveal the scaling property of the zeros. Using the analytic approximation of the numerical density of states, we are able to locate the Fisher's zeros of the SU(2) and U(1) models. The zeros of the SU(2) stabilize at a distance from the real axis, which is compatible with the scenario that a crossover instead of a phase transition is expected in the infinite volume limit. In contrast, with the precise determination of the locations of Fisher's zeros for the U(1) model at smaller lattice sizes L=4, 6 and 8, we show that the imaginary parts of the zeros decrease with a power law of L-3.07 and pinch the real axis at β= 1.01134, which agrees with results using other methods. Preliminary results at larger volumes indicate a first-order transition in the infinite volume limit.

The analysis of the error of perturbative series by comparing it to the exact solution is an important tool to understand the non-perturbative physics of statistical models. For some toy models, a new method can be used to calculate higher order weak coupling expansion and modified perturbation theory can be constructed. However, it is nontrivial to generalize the new method to understand the critical behavior of high dimensional spin and gauge models. Actually, it is a big challenge in both high energy physics and condensed matter physics to develop accurate and efficient numerical algorithms to solve these problems. In this thesis, one systematic way named tensor renormalization group method is discussed. The applications of the method to several spin and gauge models on a lattice are investigated. theoretically, the new method allows one to write an exact representation of the partition function of models with local interactions. E.g. O(N) models, Z2 gauge models and U(1) gauge models. Practically, by using controllable approximations, results in both finite volume and the thermodynamic limit can be obtained. Another advantage of the new method is that it is insensitive to sign problems for models with complex coupling and chemical potential. Through the new approach, the Fisher's zeros of the 2D O(2) model in the complex coupling plane can be calculated and the finite size scaling of the results agrees well with the Kosterlitz-Thouless assumption. Applying the method to the O(2) model with a chemical potential, new phase diagram of the models can be obtained. The structure of the tensor language may provide a new tool to understand phase transition properties in general.