The Geometry and Physics of the Seiberg—Witten Equations

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Abstract

These lectures are aimed at explaining the physical origin of the Seiberg—Witten equations and invariants to a mathematical audience. In the course of the exposition, we will cover several rich aspects of nonperturbative quantum field theory. Attempts have been made to reduce the prerequisites to a minimum and to provide a comprehensive bibliography. Lecture 1 explains classical and quantum pure gauge theory and its supersymmetric versions, with a digression on supersymmetry. Emphasis is on the non-perturbative aspects of field theories, such as vacuum structure, existence of mass gap, symmetries, and anomalies. Lecture 2 is about the duality conjecture in (supersymmetric) gauge theories and its consequences. It begins with the notion of duality and the role monopoles play in electric-magnetic duality. Lecture 3 reviews Donaldson invariants and topological field theory, followed by the low energy solution to theN =2 supersymmetric gauge theory by Seiberg and Witten, and its application to four-manifolds.