Random matrix models have found numerous applications in both Theoretical Physics and Mathematics. In the gauge-gravity duality, for example, the dynamics of the half- BPS sector can be fully described by the holomorphic sector of a single complex matrix model. In this thesis, we study the large-N limit of multi-matrix models at strong-coupling. In particular, we explore the significance of rescaling the matrix fields. In order to investigate this, we consider the matrix quantum mechanics of a single Hermitian system with a quartic interaction. We “compactify” this system on a circle and compute the first-order perturbation theory correction to the ground-state energy. The exact ground-state energy is obtained using the Das-Jevicki-Sakita Collective Field Theory approach. We then discuss the multi-matrix model that results from the compactification of the Higgs sector of N = 4 SYM on S4 (or T S3). For the radial subsector, the saddle-point equations are solved exactly and hence the radial density of eigenvalues for an arbitrary number of even Hermitian matrices is obtained. The single complex matrix model is parametrized in terms of the matrix valued polar coordinates and the first-order perturbation theory density of eigenstates is obtained. We make use of the Harish-Chandra- Itzykson-Zuber (HCIZ) formula to write down the exact saddle-point equations. We then give a complementary approach - based on the Dyson-Schwinger (loop) equations formalism - to the saddle-point method. We reproduce the results obtained for the radial (single matrix) subsector. The two-matrix integral does not close on the original set of variables and thus we map the system onto an auxiliary Penner-type two matrix model. In the absence of a logarithmic potential we derive a radial hemispherical density of eigenvalues. The system is regulated with a logarithm potential, and the Dobroliubov-Makeenko-Semenoff (DMS) loop equations yield an equation of third degree that is satisfied by the generating function. This equation is solved at strong coupling and, accordingly, we obtain the radial density of eigenvalues.

Massive QED (Schwinger model) for one and two fermion species in 1+1 dimensions is studied using Hamiltonian lattice techniques. Bound-state masses are calculated as strong-coupling expansions in inverse powers of the dimensionless coupling constant. Various Pade approximant methods for extracting continuum predictions from these are compared. The non-relativistic limit of both lattice theories is the lattice linear potential model. This can be solved exactly. It is used to test convergence of the sequence of Pade approximants. The investigation is continued for the ordinary Schwinger model. At all coupling strengths, the best continuum estimates for bound-state masses come from values of the Pade approximants at non-zero lattice spacing. Two different lattice formulations of the two-species Schwinger model are studied. Both have a restoration of chiral SU(2) symmetry as the fermion mass vanishes. The corresponding symmetric vacuum is too complicated to do a perturbative calculation beyond second order, where the low-lying states are those of a Heisenberg antiferromagnetic chain, in qualitative agreement with the continuum theory. Strong-coupling expansions are carried out to high orders about the unsymmetric vacua of the massive theories. Continuum estimates for bound-state masses are compared. For weak coupling their convergence is understood in terms of the linear potential model. But for strong coupling convergence is slow; neither lattice can account for the whole particle spectrum, though each treats part of it well. Matrix methods are studied in an attempt to obtain better convergence from low-order calculations. Strong-coupling expansions for the Hamiltonian matrix in a non-degenerate subspace are extrapolated to zero lattice spacing using matrix Pade approximants. Improved continuum estimates ore obtained from the scalar mass matrix of the ordinary Schwinger model, but not from the pseudoscalar mass matrix.

Thesis (PhD)--Stellenbosch University, 2001. / ENGLISH ABSTRACT: In this dissertation, we give an overview of disordered systems, where we concentrate on the theoretical calculation techniques used in this field. We first discuss the general properties of disordered systems and the different models and quantities used in the study of these systems, before describing calculation techniques used to investigate the quantities introduced. These calculation techniques include the phase formalism method used one dimension, as well as the scaling approach and field theoretic approaches leading to non-linear c-models in higher dimensions. We then introduce a complementary effective field theoretic approach based on the logarithmic derivative of the wave-function, and show how the quantities of interest are calculated using this method. As an example, the effective field theory is applied to one dimensional systems with Gaussian disorder. The average density of states, the average 2-point correlator and the conductivity are calculated in a weak disorder saddle-point approximation and in strong disorder duality approximation. These results are then calculated numerically and in the case of the density of states compared to the exact result. / AFRIKAANSE OPSOMMING: In hierdie tesis, gee ons 'n oorsig van sisteme met wanorde, waar ons konsentreer op teoretiese berekeningsmetodes wat in die veld gebruik word. Eerstens bespreek ons die algemene eienskappe van sisteme met wanorde en verskillende modelle en hoeveelhede wat gebruik word in die studie van hierdie sisteme, voordat ons die berekeningsmetodes beskryf wat gebruik word om die bogenoemde hoeveelhede te ondersoek. Hierdie berekeningstegnieke sluit in die fase formalisme wat in een dimensie gebruik word, asook die skalingsbenadering en veldteoretiese metodes wat lei tot nie-lineêr u-modelle in hoër dimensies. Ons voer in 'n komplementere effektiewe veldeteorie gebaseer op die logaritmiese afgeleide van die golffunksie, en wys hoe hoeveelhede van belang met hierdie metode bereken word. As 'n voorbeeld, word die effektiewe veldetoerie toegepas op 'n een dimensionele sisteem met 'n Gauss verdeling. Die gemiddelde digtheid van toestande, die gemiddelde 2-punt korrelator en die gemiddelde geleidingsvermoë word bereken in 'n swak wanorde saalpunt benadering en in 'n sterk wanorde duale benadering. Hierdie resultate word dan numeries bereken, en in die geval van die digtheid van toestande vergelyk met die eksakte resultaat.

published_or_final_version / Civil Engineering / Doctoral / Doctor of Philosophy

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. February 2015. / We explore the non-perturbative physics of giant gravitons in type IIB string theory on the AdS5 ⇥ S5 background in this thesis. The gauge theory dual is N = 4 super Yang-Mills theory with a U(N) gauge group. We diagonalise the one and two-loop dilatation operators acting on the restricted Schur polynomial basis. These operators are dual to a system of giant gravitons with strings attached. Hence, we present evidence for integrability in certain non-planar sectors of the gauge theory. In the second half of the thesis, we turn our focus to N = 4 super Yang-Mills theory with an SO(N) gauge group. In this case, the geometry of the dual gravity theory is AdS5 ⇥RP5. The non-planar physics of the SO(N) theory is distinct from that of the U(N) theory. To pursue the goal of searching for non-planar integrability in the SO(N) gauge theory, one might try to generalise the restricted Schur basis to the SO(N) case. We propose such a basis and evaluate their two-point functions exactly in the free theory. Further, we develop techniques to compute correlation functions of multi-trace operators involving two scalar fields exactly. Lastly, we extend these results to the theory with an Sp(N) gauge group.

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, 30.09.2014. / ABSTRACT In this dissertation we study the action of the one loop dilatation operator on operators with a classical dimension of order N. We consider the su(3) and su(2) sectors. The operators in the su(3) sector are constructed using three complex fields X, Y and Z, while operators in the su(2) sector are constructed from only the two complex fields Y and Z. For the operators in these sectors non-planar diagrams contribute already at the leading order in N and the planar and large N limits are distinct. Although the spectrum of anomalous dimensions in su(3) has been computed for this class of operators, previous studies have neglected certain terms which were argued to be small. After dropping these terms diagonalizing the dilatation operator reduces to diagonalizing a set of decoupled oscillators. In this dissertation we explicitly compute the terms which were neglected previously and show that diagonalizing the dilatation operator still reduces to diagonalizing a set of decoupled oscillators. In the su(2) sector the action of the one loop and the two loop dilatation operator reduces to a set of decoupled oscillators and factorizes into an action on the Z fields and an action on the Y fields. Direct computation has shown that the action on the Y fields is the same at one and two loops. In this dissertation, using the su(2) symmetry algebra as well as structural features of field theory, we give compelling evidence that the factor in the dilatation operator that acts on the Y s is given by the one loop expression, at any loop order. I hereby declare that the content of this dissertation is based on my following original works: • R. de Mello Koch, S. Graham and W. Mabanga, “Subleading corrections to the Double Coset Ansatz preserve integrability” (2013) [arXiv:1312.6230v1 [hep-th]] • R. de Mello Koch, S. Graham and I. Messamah, “Higher Loop Nonplanar Anomalous Dimensions from Symmetry” (2013) [arXiv:1312.6227v1 [hep-th]].

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

Orientador: Luiz Agostinho Ferreira / Banca: Abraham Hirsz Zimerman / Banca: Marco Aurélio Cattacin Kneipp / Resumo: Através do formalismo de sistemas integráveis em qualquer dimensão desenvolvido em [18], nos propomos a resolver um modelo integrável em 4 dimensões Euclideanas cujos campos escalares tomam valores na esfera 'S POT. 2'. A Lagrangiana é dada pelo quadrado do pull-back da 2-forma da área de 'S POT. 2'. A ação Euclideana é invariante por reescalonamento x 'seta' 'lâmbda'x, o que garante a estabilidade das soluções a la Derrick. Encontramos a equação de curvatura nula e as correntes conservadas associadas. A invariância conforme do modelo nos fornece um ansatz que permite reduzir as equações de movimento a uma equação diferencial parcial em duas variáveis. As soluções explicitas são obtidas através de programas de manipulação algébrica, Mathematica e Maple, por uma expanão em série de potências / Abstract: Using the approach to integrable theories in any dimension introduced in [18], we analyse an integrable model in four Euclidean dimensions, with fields taking values in the sphere 'S POT. 2'. The Lagrangean is given by the square of the pull-back of the area form of 'S POT. 2'. The Euclidean action is invariant under the rescaling x 'seta' 'lâmbda'x, which guarantees the estability of the solutions a la Derrick. We construct the generalized zero curvature condition and find the associated conserved currents. The four dimensional conformal invariance of the model leads to an ansatz which reduces the equations of motion to a single (linaer) partial differential equation in two variables. The solutions are obtained by through computer programs for agebraic manipulation, as Maple and Mathematica, using a method of power series expansion / Mestre

In a projective plane (PG(2, K) defined over an algebraically closed field K of characteristic p = 0, we give a complete classification of 3-nets realizing a finite group. The known infinite family, due to Yuzvinsky, arised from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky, comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family and list all possible sporadic examples. If p is larger than the order of the group, the same classification holds true apart from three possible exceptions: Alt4, Sym4 and Alt5. / by Nicola Pace. / Thesis (Ph.D.)--Florida Atlantic University, 2012. / Includes bibliography. / System requirements: Adobe Reader. / Mode of access: World Wide Web.

Spin-2 particles, or gravitons, present both virtues and vices not displayed by their lower spin peers. A massless graviton can only be described consistently by a single theory---general relativity---while mutual couplings among ``colored'' gravitons are simply not allowed. A massive graviton is also believed to admit a unique set of interactions, ones that are however pestered by superluminal perturbations and a rather limited effective field theory. And then there is the third member of the clique, the partially massless graviton, who lives in a universe with a naturally small cosmological constant, but which nonetheless seems not to exist at all. The aim of this thesis is to explore this enormously rich and tightly fettered realm of classical theories of spin-2 fields.