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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.

Berlin, Humboldt-Univ., Diss., 2003. / Computerdatei im Fernzugriff.

Berlin, Humboldt-Univ., Diss., 2003. / Computerdatei im Fernzugriff.

Münster (Westfalen), Univ., Diss., 2004. / Erscheinungsjahr an der Haupttitelstelle: 2003. Computerdatei im Fernzugriff.

Münster (Westfalen), Univ., Diss., 2004. / Erscheinungsjahr an der Haupttitelstelle: 2003. Computerdatei im Fernzugriff.

Berlin, Humboldt-University, Diss., 2004.

Darmstadt, Techn. University, Diss., 2004.

Mathematisch-Naturwissenschaftliche Fakultät I, Humboldt-University, Berlin, Diss., 2003.