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Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--University of Hong Kong, 1994. / Includes bibliographical references.

Dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Master of Science. Johannesburg, 2017. / In this dissertation we tackle the question: is there an emergent Yang-Mills theory coming from the low energy description of branes and open strings? This new Yang-Mills theory has no connection to the original gauge symmetry of the CFT. We thus explore a large N but non-planar limit of the theory. This is done with new methods developed in group representation theory. A study the dilatation operator D in N = 4 SYM theory is done since its eigenvalue, the anomalous dimension, is mapped to the energy of the open string in the IIB string theory. The construction of the spherical harmonics from the harmonic expansion on the 3-sphere, S3, is done to understand the theory of the giant graviton's worldvolume. The light-front parton picture is examined, since it explains how one can \glue" single momentum modes together to obtain higher momentum modes, and we believe that this procedure is described dynamically using magnon bound states. Following from this, we work on determining the exact magnon bound state spectrum. Finally, we test our hypothesis and see if the spectrum of the bound states matches the harmonic spectrum from the harmonic expansion on the 3-sphere, S3. A non-trivial check is also performed to show that the bound state spectrum does indeed match the spectrum coming from N = 4 SYM. / LG2018

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.

A dissertation submitted in fulfillment of the requirements for the degree of Master of Science. August 2016. / The six scalar fields in N = 4 super Yang-Mills theory enjoy a global SO(6) symmetry, and large N but non-planar limits of this theory are well-described by adopting a group representation approach. Studies have shown that the one-loop dilatation operator is highly determined by the action of the su(2)=su(3) subalgebras on restricted Schur polynomials. These actions involve the traces of products of projection operators. In this dissertation, exact analytical formulae for these traces are found which in turn are used to find the exact action of these algebras on restricted Schur polynomials. The potential of the su(2) algebra to determine the one-loop dilatation operator is also explored. This is done by exploiting necessary symmetry conditions and moving to a continuum limit in order to derive a number of partial differential equations which determine the dilatation operator. The ultimate goal of this work is to provide tools to find the exact one-loop dilatation operator in the non-planar limit. / LG2017

A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. 14 May 2013 / We study the action of the dilatation operator on restricted Schur polynomials labeled by Young diagrams with p long columns or p long rows. A new version of Schur-Weyl duality provides a powerful approach to the computation and manipulation of the symmetric group operators appearing in the restricted Schur polynomials. Using this new technology, we are able to evaluate the action of the one loop dilatation operator. The result has a direct and natural connection to the Gauss Law constraint for branes with a compact world volume. Generalzing previous results, we find considerable evidence that the dilatation operator reduces to a decoupled set of harmonic oscillators. This strongly suggests that integrability in N = 4 super Yang-Mills theory is not just a feature of the planar limit, but extends to other large N but non-planar limits.

In this thesis we explore certain limits of the AdS/CFT correspondence for integrability. This is done by calculating the action of the dilatation operator on operators known as restricted Schur polynomials, which are AdS/CFT dual to D3-branes known as giant gravitons. We focus on operators in N = 4 super-Yang-Mills theory, which is dual to type IIB string theory on an AdS5×S5 background. We find that, in various cases, this theory is integrable in a large N non-planar limit.

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.

Planar maximally supersymmetric Yang-Mills theory (N = 4 SYM) is a special quantum fi eld theory. A few of its remarkable features are conformal symmetry at the quantum level, evidence of integrability and, moreover, it is a prime example of the AdS/CFT duality. Triggered by Witten's twistor string theory [1], the past 15 years have witnessed enormous progress in reformulating this theory to make as many of these special features manifest, from the choice of convenient variables to recursion relations that allowed new mathematical structures to appear, like the Grassmannian [2]. These methods are collectively referred to as on-shell methods. The ultimate hope is that, by understanding N = 4 SYM in depth, one can learn about other, more realistic quantum fi eld theories. The overarching theme of this thesis is the investigation of how on-shell methods can aid the computation of quantities other than scattering amplitudes. In this spirit we study form factors and correlation functions, said to be partially and completely off-shell quantities, respectively. More explicitly, we compute form factors of half-BPS operators up to two loops, and study the dilatation operator in the SO(6) and SU(2j3) sectors using techniques originally designed for amplitudes. A second part of the work is dedicated to the study of scattering amplitudes beyond the planar limit, an area of research which is still in its infancy, and not much is known about which special features of the planar theory survive in the non-planar regime. In this context, we generalise some aspects of the on-shell diagram formulation of Arkani-Hamed et al. [3] to take into account non-planar corrections.