Seiberg-Witten Invariants
Lecturer: Popelensky TheodoreAnnotation
Almost forty years ago the ideas and methods originating from gauge theories were applied in geometry and topology, resulting, in particular, in the substantial breakthrough in the theory of 4-manifolds. Among such applications, Seiberg-Witten theory is one of the most accessible and at the same time fruitful. The course starts with the review of the background material such as vector bundles with a structure group, Clifford algebras, and Spin structures.
Topics covered within the main part of the course include operators of Dirac type, Sieberg-Witten equations and corresponding moduli space, Seiberg-Witten invariants and their basic properties and applications.
Course plan
- Clifford algebras, groups Pin, Spin, and Spin^C;
- Fibrations with structure groups O(n), SO(n). Spin- and и Spin^C structures;
- Connection, curvature;
- Characteristic classes;
- Dirac operator and its properties;
- Seiberg-Witten equation and its perturbation, gauge group, moduli space;
- Compactness of moduli spaces;
- Smoothness of moduli spaces, orientability;
- SW-invariant and its properties;
- Kahler surfaces.
Literature
- J.D.Moore, Lectures on Seiberg-Witten invariants, 2001, Lecture Notes in Mathematics, 1629, Berlin: Springer-Verlag
- J.W.Morgan, The Seiberg–Witten equations and applications to the topology of smooth four-manifolds, 1996, Mathematical Notes, 44, Princeton University Press
- L.Nicolaescu, Notes on Seiberg-Witten theory, 2000, Graduate Studies in Mathematics, 28, Providence, RI: AMS
- A.Scorpan, The wild world of 4-manifolds, 2005, AMS
