New four dimensional N = 2 field theories can be engineered from compactifying six dimensional (2, 0) superconformal field theory on a punctured Riemann surface. Hitchin’s equation is defined on this Riemann surface and the fields in Hitchin’s equation are singular at the punctures. Four dimensional theory is entirely determined by the data at the punctures. Theory without lagrangian description can also be constructed in this way. We first construct new four dimensional generalized superconformal quiver gauge theory by putting regular singularity at the puncture. The algorithm of calculating weakly coupled gauge group in any duality frame is developed. The asymptotical free theory and Argyres-Douglas field theory can also be constructed using six dimensional method. This requires introducing irregular singularity of Hithcin’s equation. Compactify four dimensional theory down to three dimensions, the corresponding N = 4 theory has the interesting mirror symmetry. The mirror theory for the generalized superconformal quiver gauge theory can be derived using the data at the puncture too. Motivated by this construction, we study other three dimensional theories deformed from the above theory and find their mirrors. The surprising relation of above four dimensional gauge theory and two dimensional conformal field theory may have some deep implications. The S-duality of four dimensional theory and the crossing symmetry and modular invariance of two dimensional theory are naturally related.

Gauge theories are one of the corner stones of modern theoretical physics. They describe the nature of all fundamental interactions and have been applied in multiple branches of physics. The most challenging problem of gauge theories, which has not been solved yet, is their strong coupling dynamics. A class of gauge theories that admits simplifications allowing to deal with the strong coupling regime are supersymmetric ones. For example, recently proposed method of supersymmetric localization allows to reduce expectation values of supersymmetric observables, expressed through the path integral, to finite-dimensional matrix integral. The last one is usually easier to deal with compared to the original infinite-dimensional integral. This thesis deals with the matrix models obtained from the localization of different 5D gauge theories. The focus of our study is N=1 super Yang-Mills theory with different matter content as well as N=1 Chern-Simons-Matter theory with adjoint hypermultiplets. Both theories are considered on the five-spheres. We make use of the saddle-point approximation of the matrix integrals, obtained from localization, to evaluate expectation values of different observables in these theories. This approximation corresponds to the large-N limit of the localized gauge theory. We derive <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?N%5E%7B3%7D" /> behavior for the free energy of 5D N=1* super Yang-Mills theory at strong coupling. This result is important in light of the relation between 5D theory and the world-volume theories of M5-branes, playing a significant role in string theory. We have also explored rich phase structure of 5D SU(N) N=1 super Yang-Mills theory coupled to massive matter in different representations of the gauge group. We have shown that in the case of the massive adjoint hypermultiplet theory undergoes infinite chain of the third order phase transitions while interpolating between weak and strong coupling in the decompactification limit. Finally, we obtain several interesting results for 5D Chern-Simons theory, suggesting existence of the holographic duals to this theory. In particular, we derive <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?N%5E%7B5/2%7D" /> behavior of the free energy of this theory, which reproduces the behavior of the free energy for 5D theories with known  holographic duals.

Adinkras and arithmetical graphs have divergent origins. In the spirit of Feynman diagrams, adinkras encode representations of supersymmetry algebras as graphs with additional structures. Arithmetical graphs, on the other hand, arise in algebraic geometry, and give an arithmetical structure to a graph. In this thesis, we will interpret adinkras as arithmetical graphs and see what can be learned. Our work consists of three main strands. First, we investigate arithmetical structures on the underlying graph of an adinkra in the specific case where the underlying graph is a hypercube. We classify all such arithmetical structures and compute some of the corresponding volumes and linear ranks. Second, we consider the case of a reduced arithmetical graph structure on the hypercube and explore the wealth of relationships that exist between its linear rank and several notions of genus that appear in the literature on graph theory and adinkras. Third, we study modifications of the definition of an arithmetical graph that incorporate some of the properties of an adinkra, such as the vertex height assignment or the edge dashing. To this end, we introduce the directed arithmetical graph and the dashed arithmetical graph. We then explore properties of these modifications in an attempt to see if our definitions make sense, answering questions such as whether the volume is still an integer and whether there are still only finitely many arithmetical structures on a given graph.