Geometry of Gauge Fields
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This course will be of particular interest to PhD students in geometry, mathematical physics and topology. For mathematical physics students it will provide the mathematical foundations of gauge theory, which remains the dominant paradigm for describing interactions in physics, from condensed matter theory to high energy physics. For students of geometry and topology it will introduce an interdisciplinary area of research which has provide vital inputs for geometers and topologists (including the study of knots and 4-manifolds).
- Fibre bundles and associated vector bundles, connections, curvature, characteristic classes;
- Maxwell theory as U(1) gauge theory, Dirac monopole as curvature, wave function of charged particle as section of associated line bundle;
- Chern-Simons theory and the moduli space of flat connections on a Riemann surface, Atiyah-Bott symplectic structure;
- Classical Yang-Mills theory, monopoles and instantons, self-duality equations, ADHMN construction of instantons and monopoles
- Introduction to moduli spaces of instantons and monopoles, S-duality and L2-cohomology.