Some Aspects Of Dualities In Field Theories And String Theories

In this thesis we study examples of the dualities that have played very important roles in modern quantum field theories and string theories. In chapter 1, we start with an example of strong-weak coupling duality, the electromagnetic duality of d = 4 N = 4 super Yang-Mills (SYM4) theory. The low energy effective action of SYM4 is believed to have a selfdual description. We use the formulation of N = 2 superspace. In chapter 2 we study the partial breaking of N = 2 supersymmetry of the so called O(2) projective multiplet. We obtained the N = 1 action of the real linear multiplet by imposing the nilpotency constraint to N = 1 antichiral superfield. D-branes are topological defects on which open strings can end. Maldacena made a conjecture: The weaker form of the conjecture states that IIB supergravity on AdS5 x S5 is holographically dual to large N SYM4 theory and the stronger form states that IIB string theory on AdS5 x S5 is dual to SYM 4 for any value of gs and N. In chapter 3, we address the question of why the weaker form of the conjecture should be true taking a viewpoint that there are two different but dual descriptions of D- branes. In chapter 4, we discuss a check of the weaker form of AdS/CFT conjecture. All CPOs at large N are independent of l≡Ng2YM . We present a N = 1 superspace computation of g2YM corrections to certain two and three point functions of gauge invariant operators. String theory provides a very powerful setting for the study of gauge theories. The hyperkähler quotient introduced mathematically refined. Remarkably, the Higgs branch of a quiver gauge theory based on the extended E6 Dynkin diagram turned out to have the same form as the curve with E6 global symmetry obtained by Minahan and Nemeschansky. In chapter 5, we work out the full curve corresponding to the E7 extended Dynkin diagram. We understand the origin of these phenomena through mirror symmetry and F-theory compactifications. D3 branes are used to probe the singularities of the backgrounds under consideration. We conjecture that mirror symmetry in these four dimensional models acts in such a way that it is the generalized Coulomb branch of the original gauge theory that gets interchanged with the Higgs branch of the mirror dual theory. (Abstract shortened by UMI.