A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. May 23, 2017. / After studying matrix model quantum mechanics for one and two matrices and the eigenvalue dynamics of the one matrix model, evidence is found that there is a sector of the two matrix model that can be reduced to eigenvalue dynamics. This sector is defined by the SU(2) sector of N = 4 super Yang-Mills theory. Evidence is found by doing explicit computations of correlation functions using an interesting generalization of the usual Van der Monde determinant. The observables we study are the BPS operators of the SU(2) sector and include traces of products of both matrices, which are genuine multi matrix observables. These operators are associated to supergravity solutions of string theory. / XL2018

We investigate nonperturbative effects due to instantons in N = 2 supersymmetric SU(N) Yang-Mills models, with the aim of testing the exact results predicted for these models. In two separate semiclassical calculations we obtain the one-instanton contribution to the Higgs condensate u(_3) = (TrA(^3)) and to the prepotential F. Comparing our results with the exact predictions, we find complete agreement except when the number of flavours of fundamental matter hypermultiplets, N(_f), takes certain values. The source of the u(_3) discrepancy is an ambiguity in the parameterization of the hyperelliptic curves from which the exact predictions are derived when N(_f) ≥ N. This ambiguity can easily be fixed using the results of instanton calculations. The discrepancy associated with T appears in the finite N(_f) = 2N models. For these models we are unable to modify the curves to agree with the instanton calculations when N > 3. Our one-instanton calculation of the prepotential is facilitated by a multi-instanton calculus which we construct, starting from the general solution of Atiyah, Drinfeld, Hitchin and Manin. Our calculus comprises: (i) the super-multi-instanton background, (ii) the su persymmetric multi-instanton action and (iii) the supersymmetric semiclassical collective coordinate measure. Our calculus has application to supersymmetric Yang-Mills theory with gauge group U(N) or SU(_N). We employ our instanton calculus to derive results at arbitrary k-instanton levels. In N =2 supersymmetric SU(N) Yang-Mills theory, we derive a closed form expression for the A;-instanton contribution to the prepotential. This amounts to a solution, in quadratures, of the low-energy physics of the theory, obtained from first principles. In supersymmetric SU(2) Yang-Mills theory, we use our calculus to investigate multi-instanton contributions to higher-derivative terms in the Wilsonian effective action. Using a scaling argument, based on general properties of the SU(2) k-instanton action and measure, we show that in the finite, massless N = 2 and N = 4 models, all k-instanton contributions to the next-to- leading higher-derivative terms vanish. This confirms a nonperturbative nonrenormalization theorem due to Dine and Seiberg.

A Canonical Basis m ethod is presented for computing the cut-constructible parts of loop amplitudes in massless Yang-Mills theory by generalised Unitarity. The method relies upon constructing a canonical basis of general cut solutions and constructing the amplitude by identifying the integral coefficients with elements of the canonical basis, thereby avoiding repeated integration of similar integrand structures. The method provides closed, rational, fully analytic expressions for the integral coefficients. As an application the m ethod is applied to compute the previously unknown NMHV partial amplitudes of the 7-gluon Af = 1 chiral multiplet loop, and is then extended to rederive the cut-constructible terms of the NMHV 6-gluon partial amplitudes with a complex scalar loop.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science. April 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. 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.

published_or_final_version / Physics / Doctoral / Doctor of Philosophy

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A thesis submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Doctor of Philosophy. 3rd October 2013. / A complete understanding of quantum gravity remains an open problem. However, the AdS/CFT correspondence which relates quantum eld theories that enjoy conformal symmetry to theories of (quantum) gravity is proving to be a useful tool in shedding light on this formidable problem. Recently developed group representation theoretic methods have proved useful in understanding the large N; but non-planar limit of N = 4 supersymmetric Yang-Mills theory. In this work, we study operators that are dual to excited giant gravitons, which corresponds to a sector of N = 4 super Yang-Mills theory that is described by a large N; but non-planar limit. After a brief review of the work done in the su (2) sector, we compute the spectrum of anomalous dimensions in the su (2) sector of the Leigh-Strassler deformed theory. The result resembles the spectrum of a shifted harmonic oscillator. We then explain how to construct restricted Schur polynomials built using both fermionic and bosonic elds which transform in the adjoint of the gauge group U (N) : We show that these operators diagonalise the free eld two point function to all orders in 1=N: As an application of our new operators, we study the action of the one-loop dilatation operator in the su (2,3) sector in a large N; but non-planar limit of N = 4 super Yang-Mills theory. As in the su (2) case, the resulting spectrum matches the spectrum of a set of decoupled oscillators. Finally, in an appendix, we study the action of the one-loop dilatation operator in an sl (2) sector of N = 4 super Yang-Mills theory. Again, the resulting spectrum matches that of a set of harmonic oscillators. In all these cases, we nd that the action of the dilatation operator is diagonalised by a double coset ansatz.

One of the key goals of string theory is to provide a uni cation of general relativity and quantum eld theory. In the pursuit of this goal it has become clear that the di erent string theories that have been discovered so far are all in fact, partial descriptions of a single theory. At strong coupling a new theory, called M-theory, is the correct description. M-theory includes gravitons, M2-branes and M5-branes. Up to now, the correct description of the M5-brane is outstanding. In this project some proposals for this theory are studied. In particular, there is a proposal that D=5 maximally supersymmetric Yang-Mills theory can be used to provide a description of the world volume physics of the M5-brane. According to this proposal, instantons in D=5 maximally supersymmetric Yang-Mills theory are graviton excitations of the M theory. In this M.Sc dissertation the instanton solutions of D=5 maximally supersymmetric Yang-Mills theory are explored, with the goal of testing the above proposal. The dissertation begins with a review of the uses of instantons in quantum mechanics. In particular, instantons are used to account for tunneling e ects within a path integral approach to quantum mechanics. The lifting of ground state degeneracies as well as the estimation of the lifetime of unstable states using instantons is developed. The quantization of gauge theories is reviewed in detail. The relevance of instantons for a semi-classical study of Yang-Mills theory is explained. Finally, the relevance of instantons for D = 5 maximally supersymmetric Yang-Mills theory is considered.

The study of scattering amplitudes in quantum eld theories (QFTs) is equally important for high energy phenomenology and for theoretical understanding of fundamental physics. Over the last 15 years there has been an explosion of new techniques, inspired by Witten's celebrated twistor string theory [1]. The N = 4 super Yang-Mills theory (SYM) provides a playground for applying and extending these methods, heavily constrained by spacetime, internal and hidden symmetries. Recently, Cachazo, He and Yuan proposed an algebraic construction of scattering amplitudes at tree level in various QFTs, based on the solution of certain scattering equations [2]. This formula was later extended to tree-level form factors of Tr(F2 SD) in four dimensional Yang-Mills theory [3]. In this thesis we show how this result may be naturally supersymmetrised, and derived from a dual connected formulation. Moreover, we relate our results to a geometric construction of form factors via the Grassmannian [4]. Finally, we argue that ambitwistor string theory provides a natural way to lift the result to arbitrary dimensions, paving the way for loop-level results. In complementary work, it was shown that the subleading soft behaviour of tree-level amplitudes in gauge theory and gravity is universal [5{7]. This unexpected property is related to extended symmetries of the theory acting at null in nity. Moreover, the hidden structure provides additional information relevant for resummation of physical observables. In this thesis, we extend the known results to one-loop level in N = 4 SYM, arguing that IR divergences introduce anomaly terms through nite order in the regulator. We constrain these terms using dual superconformal symmetry, and derive explicit formulae in the MHV and NMHV sectors. This thesis contains documentation for two Mathematica packages, illustrating the original calculations we have performed.

The study of scattering amplitudes in the maximally supersymmetric Yang-Mills theory (N = 4 SYM) is a thriving field of research. Since the reformulation of perturbative gauge theory as a twistor string theory by Witten, this area has witnessed a flurry of activity, leading to the discovery of a multitude of novel techniques, such as recursion relations and MHV diagrams, collectively referred to as on-shell methods. In parallel, many previously hidden properties and rich mathematical structures have been found, a powerful example of such being the dual superconformal symmetry. It is natural to ask whether this understanding can be extended to phenomenologically relevant theories as well as other quantities. The goal of the present work is to apply the modern on-shell methods to calculations of form factors, with particular focus on those which are relevant for describing Higgs production in QCD from the point of view of an effective field theory. Specifically, our analysis will be carried out in supersymmetric gauge theories at two-loop level and will consist of several steps. We focus first on operators in the SU(2j3) closed subsector of N = 4 SYM, in particular two non-protected, dimension-three operators. We then move on to consider the trilinear operator Tr(F3) and a related descendant of the Konishi operator which contains Tr(F3), also in N =4 SYM. Finally, we concentrate on two-loop form factors of these two operators in theories with less-than-maximal supersymmetry. The result of our investigation shows an emergence of a small number of universal building blocks, ultimately related to the two-loop form factor of a trilinear half-BPS operator. This finding suggests that the most complicated, maximally transcendental part of Higgs plus multi-gluon amplitudes in QCD can be equivalently computed in a remarkably simple way by considering form factors of half-BPS operators in N =4 SYM.