Lattice simulations are a popular tool for studying the non-perturbative physics of nonlinear field theories. To perform accurate lattice simulations, a careful account of the discretization error is necessary. Spatial discretization error as a result of lattice spacing dependence in Langevin simulations of anisotropic (2 + 1)-dimensional classical scalar field theories is studied. A transfer integral operator (TIO) method and a one-loop renormalization (1LR) procedure are used to formulate effective potentials. The effective potentials contain counterterms which are intended to suppress the lattice spacing dependence. The two effective potentials were tested numerically in the case of a phi-4 model. A high accuracy modified Euler method was used to evolve a phenomenological Langevin equation. Large scale Langevin simulations were performed in parameter ranges determined to be appropriate. Attempts at extracting correlation lengths as a means of determining effectiveness of each method were not successful. Lattice sizes used in this study were not of a sufficient size to obtain an accurate representation of thermal equilibrium. As an alternative, the initial behaviour of the ensemble field average was observed. Results for the TIO method showed that it was successful at suppressing lattice spacing dependence in a mean field limit. Results for the 1LR method showed that it performed poorly.

The connection discovered by M. Sato, T. Miwa and M. Jimbo (SMJ) between the monodromy-preserving deformation theory of the two-dimensional Euclidean Dirac operator and quantum fields is rigorously established for the case of nonreal S¹ monodromy parameters. This connection involves the expression of the associated n-point functions in terms of solutions to deformation equations which arise as necessary conditions for the monodromy exhibited by a class of multivalued solutions of the Euclidean Dirac equation to be preserved under perturbations of branch points. Our approach utilizes recent results involving infinite-dimensional group representations. A lattice version of the n-point function is introduced as a section of a determinant bundle defined over an infinite dimensional Grassmannian. A trivialization for this bundle is singled out so that the corresponding n-point functions behave like Ising correlations in the massive scaling regime. Then the SMJ n-point functions are recovered as the scaled functions. A parallel scaling analysis is carried out with lattice analogues of the Euclidean Dirac wave functions which scale to square-integrable multivalued solutions of the Euclidean Dirac equation and the connection between the SMJ deformation theory and the n-point functions is rigorously established in terms of local Fourier expansion coefficients of these wave functions. These results are presented in detail for two-point functions with the same monodromy associated to each site.

In chapter one, we explain briefly the continuum limit, scaling, and high temperature expansion of critical phenomena, Monte Carlo algorithms and fitting. In chapter two, different continuum limits of the Ising model in dimensions (D) 2, 3 and 4 are investigated numerically. The data indicate that triviality occurs for D = 4 and fails for D < 4 in each limit. In chapter three, a relation between the critical exponents of the leading and confluent scaling terms is derived using the finite size scaling argument. We also determine the new scaling variable of the 4D Ising model based on a new Monte Carlo simulation data. In chapter four, a Monte Carlo study of two dimensional diluted Ising systems is reported. It is shown that regular dilution does not affect critical exponents, while a random one does, with critical exponents varying continuously with impurity concentration. The importance of fluctuations in producing such effects is emphasized. In chapter five, a different point of view regarding the critical exponent of the specific heat of the 3D Ising model is presented. Based on the analysis of high temperature expansion, finite size scaling and Monte Carlo data in the symmetric phase of the 3D Ising model, it is shown that logarithmic scaling behavior of specific heat is more consistent than power scaling behavior.

We investigate semiclassical contributions to correlation functions in N = 1 supersymmetric gauge theories. Our principal example is the gluino condensate, which signals the breaking of chiral symmetry, and should be exactly calculable, according to a persymmetric non-renormalisation theorem. However, the two calculational approaches previously employed, SCI and WCI methods, yield different values of the gluino condensate. We describe work undertaken to resolve this discrepancy, involving a new type of calculation in which the space is changed from R(^4) to the cylinder R(3) x S(1) This brings control over the coupling, and supersymmetry ensures that we are able to continue to large radii and extract answers relevant to R(^4). The dominant semiclassical configurations on the cylinder are all possible combinations of various types of fundamental monopoles. One specific combination is a periodic instanton, so monopoles are the analogue of the instanton partons that have been conjectured to be important at strong coupling. Other combinations provide significant contributions that are neglected in the SCI approach. Monopoles are shown to generate a superpotential that determines the quantum vacuum, where the theory is confining. The gluino condensate is calculated by summing the direct contributions from all fundamental monopoles. It is found to be in agreement with the WCI result for any classical gauge group, whereas the values for the exceptional groups have not been calculated before. The ADS superpotential, which describes the low energy dynamics of matter in a supersymmetric gauge theory, is derived using monopoles for all cases where instantons do not contribute. We report on progress made towards a two monopole calculation, in an attempt to quantify the missed contributions of the SCI method. Unfortunately, this eventually proved too complicated to be feasible.

A thesis submitted to the Faculty of Science University of the Witwatersrand Johannesburg in fulfillment of the requirements for the Master of Science Johannesburg 1997

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, 2015. / The AdS/CFT correspondence asserts a duality between non-Abelian gauge theories and quantum theories of gravity, established by the value of the gauge coupling . Gerard t'Hooft found that the large N0 limit in non-Abelian Yang-Mills gauge theories results in a planar diagram simpli cation of the topological expansion. In this dissertation, SU(2) gauge theories are written in terms of vector models (making use of collective eld theory to obtain an expression for the Jacobian), a saddle point analysis is performed, and the large N limit taken. Initially this procedure is done for gauge theories dimensionally reduced on T4 and R T3, and then attempted for the full eld theory (without dimensional reduction). In each case this results in an expression for the non-perturbative propagator. A nite volume must be imposed to obtain a gap equation for the full eld theory; directives for possible solutions to this di culty are discussed.

A dissertation submitted to the University of the Witwatersrand, Faculty of Science in ful lment of the academic requirements of the degree of Master of Science. Johannesburg, 2015. / The gauge/gravity correspondence is a conjectured exact duality between quantum eld theories and theories of quantum gravity. A very simple gauge/string duality, claims an equivalence between the Gaussian matrix model and the topological A-model string theory on P1. In this dissertation we study this duality, proposing concrete operators in the matrix model that are dual to gravitational descendants of the puncture operator of the topological string theory. We test our proposal by showing that a large number of matrix model correlators are in complete agreement with correlators in the dual topological string theory. Contact term interactions, as proposed by Gopakumar and Pius, play an interesting and non-trivial role in the duality.

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, 5 June 2017. / The holographic computation of extremal correlators is often frustrated by divergences. The interpretation of these divergences is incomplete. The primary goal of this study is to develop a better understanding of these divergences. Towards this end, working within the AdS/CFT correspondence we review the computation of correlators. In the field theory we review well known matrix model techniques useful to study the planar limit, as well as methods exploiting group representation theory that are useful for the computation of correlators in large N but non-planar limits. On the gravity side of the correspondence, we describe in detail the computation of two- and three point functions of a scalar field coupled to gravity on the Euclidian AdSd+1 space, three-point functions of two giant gravitons and one pointlike graviton as well as correlators of Kaluza-Klein gravitons. A key observation of this study is that extremal correlators are mapped to scattering amplitudes of particles with parallel momenta. These are naturally accompanied by involve collinear divergences. Therefore, we suggest that the divergences in the computation of extremal correlators are linked to collinear divergences. A lot more work is needed to establish this connection. / LG2017

The double-copy construction enables the calculation of scattering amplitudes in theories of gravity by combining amplitudes from gauge theories. It relies on obtaining numerators that obey a duality between color and numerator factors, called color-kinematics duality. This construction is reviewed, along with the spinor-helicity formalism for onshell states and supersymmetry in amplitudes. Using generalized unitarity, a one-loop amplitude is verified from literature for a N = 2 theory obeying color-kinematics duality. This amplitude, along with a one-loop amplitude for a N = 0 theory are combined with the double copy in order to produce one-loop amplitudes from homogeneous supergravities. The one-loop divergence is studied with the methods of counterterm analysis, that is, operators necessary to cancel the on-shell matrix element of the divergence are identified for the amplitudes studied. It is interesting to note that all vectors produced from the double copy behave in the same way, that is, have the same divergence, for the four special cases of the magical supergravities. Furthermore, one of the counterterms vanishes for these four special cases, which is likely related to the enhanced symmetry that these theories posses.

In this thesis, we investigate theoretically a new model of charge regulation of a single charged planar surface immersed in an aqueous electrolyte solution. Assuming that the adsorbed ions are mobile in the charged plane, we formulate a field theory of charge regulation where the numbers of adsorbed ions can be determined consistently by equating the chemical potentials of the adsorbed ions to that of the ions in the bulk. We analyze the mean-field treatment of the model for electrolyte of arbitrary valences, and then beyond, where correlation effects are systematically taken into account in a loop expansion. In particular, we compute exactly various one-loop quantities, including electrostatic potentials, ion distributions, and chemical potentials, not only for symmetric (1, 1) electrolyte but also for asymmetric (2, 1) electrolyte, and make use of these quantities to address charge regulation at the one-loop level. We find that correlation effects give rise to various phase transitions in the adsorption of ions, and present phase diagrams for (1, 1) and (2, 1) electrolytes, whose distinct behaviors suggest that charge regulation, at the one-loop level, is no longer universal but depends crucially on the valency of the ions. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2016. / FAU Electronic Theses and Dissertations Collection