A dissertation presented to The Faculty of Science University of the Witwatersrand Johannesburg in ful lment of the requirements for the degree of Master of Science June 2015 / At nite N, the number of restricted Schur polynomials is greater than, or equal to the number of generalized restricted Schur polynomials. In this dissertation we study this dis- crepancy and explain its origin. We conclude that, for quiver gauge theories, in general, the generalized restricted Shur polynomials correctly account for the complete set of nite N constraints and they provide a basis, while the restricted Schur polynomials only account for a subset of the nite N constraints and are thus overcomplete. We identify several situations in which the restricted Schur polynomials do in fact account for the complete set of nite N constraints. In these situations the restricted Schur polynomials and the gen- eralized restricted Schur polynomials both provide good bases for the quiver gauge theory. Further, we demonstrate situations in which the generalized restricted Schur polynomials reduce to the restricted Schur polynomials and use these results to study the anomalous dimensions for scalar operators in ABJM theory in the SU(2) sector. The operators we consider have a classical dimension that grows as N in the large N limit. Consequently, the large N limit is not captured by summing planar diagrams { non-planar contributions have to be included. We nd that the mixing matrix at two-loop order is diagonalized using a double coset ansatz, reducing it to the Hamiltonian of a set of decoupled oscilla- tors. The spectrum of anomalous dimensions, when interpreted in the dual gravity theory, shows that the energy of the uctuations of the corresponding giant graviton is dependent on the size of the giant. The rst subleading corrections to the large N limit are also considered. These subleading corrections to the dilatation operator do not commute with the leading terms, indicating that integrability probably does not survive beyond the large N limit.

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 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]].