On extended supersymmetry in two and four dimensions.

University of Minnesota Ph.D. dissertation. August 2012. Major: Physics. Advisor: Prof. Arkady Vainshtein. 1 computer file (PDF); viii, 172 pages, appendices A-B.We study the relationship between gauge theories in two and four dimensions with N = 2 supersymmetry. This includes the duality between their moduli spaces, comparison of Bogomolny-Prassad-Somerfield (BPS) spectra, study of instanton configurations, and other aspects. On they way we use various methods of integrability, conformal filed theories and string theory to achieve our goals. We start with describing physics of two dimensional N = 2 sigma models, geometry of their target spaces, their BPS spectra, and how they can de derived from four dimensional theories via BPS vortex construction. Two different approaches { gauge and geometric, as tools to study 2d theories, are compared in the light of perturbative as well as nonperturbative aspects of the theories in question. Then we discuss four dimensional supersymmetric gauge theories in presence of an Omega background { a special deformation used in localization of path integrals of supersymmetric theories. Instead of performing the localization we treat the Omega background physically and study BPS solitons for such theories, albeit the latter already possess less supersymmetry. Theories with N = 2 SUSY in Omega background are conjectured (and proven in special cases) to be dual to nonsupersymmetric conformal field theories in two dimensions by Alday, Gaiotto and Tachikawa (AGT duality). Employing the machinery of the 4d/2d duality combined with powerful methods of integrability we provide a proof of the AGT relation (in the limit where the 2d model in question exists). In the end we regard heterotic N = (0; 1) and N = (0; 2) sigma models in two dimensions. With fewer supersymmetry one has less control on the nonperturbative dynamics of the theory, however, we get some nice physical understanding of these models at strong coupling by means of the large number of colors approximation