The study of supersymmetric gauge theories has been a fruitful area of research for both Physics and Mathematics. Better understanding of these theories on their own or embedded in superstring theory is useful in obtaining realistic breaking of supersymmetry. On the other hand, there theories when embedded into superstring theory are dual to various geometries and provide an interesting arena in which to study properties of these geometries. For example, the surprising connection between N=2 gauge theories and Calabi-Yau spaces has been used to calculate the all genus Gromov-Witten partitiion function from the gauge theory.

The aim of this course is to provide a thorough treatment of N=4,2,1 supersymmetric gauge theories from a physical and geometric point of view by embedding these theories in superstring theory.

Topics:

1. Classical and Quantum Yang-Mills Theory
   • Basic Definitions
   • Relation with Electromagnetism
   • Instantons and Monopoles
   • Different Phases of guage theory
2. Supersymmetric N=4 Guage Theory
   • Classical Theory
   • Quantum Theory and Conformal Invariance
   • Using Conformal Field Theory Ideas
   • Duality (Modular invariance)
   • Embedding in superstring theory
   • Geometric Nature of Duality
3. Nature of Duality
   • T-duality
   • Bosonization as Duality
   • Three Dimensional N = 2 Abelian Duality
4. Supersymmetric Breaking N=4 →N=2
   • Quantum Moduli Spaces
   • Instantons
   • Gauge theory from D-Branes

Expected attendance: Mathematics graduate students with an interest in geometry related to string theory and Mathematical physics and Physics graduate students with an interest in string theory will form the core constituency for this course.

Prerequisites: The course will be accessible to students who have had either of the following:

1. Two quarters of Quantum Field Theory course
2. Introductory string theory course.
3. Introductory supersymmetry course.