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The Geometry and Physics of the Seiberg—Witten Equations

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Geometric Analysis and Applications to Quantum Field Theory

Part of the book series: Progress in Mathematics ((PM,volume 205))

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.

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  1. Department of Pure Mathematics, University of Adelaide, Adelaide, SA, 5005, Australia

    Siye Wu

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  1. Department of Physics and Mathematical Physics, Adelaide, SA, 5005, Australia

    Peter Bouwknegt

  2. Department of Pure Mathematics, University of Adelaide, Adelaide, SA, 5005, Australia

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  3. Department of Mathematics, University of Colorado, Boulder, CO, 80309-0395, USA

    Siye Wu

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Wu, S. (2002). The Geometry and Physics of the Seiberg—Witten Equations. In: Bouwknegt, P., Wu, S. (eds) Geometric Analysis and Applications to Quantum Field Theory. Progress in Mathematics, vol 205. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0067-3_7

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