Seiberg-Witten theory. (English) Zbl 0881.57032

Eichhorn, Jürgen; Friedrich, Thomas

Budzyński, Robert (ed.) et al., Symplectic singularities and geometry of gauge fields. Proceedings of the Banach Center symposium on differential geometry and mathematical physics in Spring 1995, Warsaw, Poland. Warsaw: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 39, 231-267 (1997).

Based on lectures given in January 1995 and May 1995, this is an expository article describing Seiberg-Witten theory for smooth 4-manifolds. After giving some background material on \(\text{spin}^c\) structures, the authors describe the Seiberg-Witten equations, the moduli space of their solutions, and the manifold invariants constructed from them. They discuss the special case in which the four-manifold is a Kaehler surface, and show how the Seiberg-Witten invariants can be calculated in this case. They discuss applications of Seiberg-Witten theory to the proof of the Thom conjecture (by Kronheimer and Mrowka), and to symplectic topology (by Taubes).
For the entire collection see [Zbl 0863.00038].

Reviewer: Steven B.Bradlow (Urbana)

MSC:

57R57	Applications of global analysis to structures on manifolds
58J20	Index theory and related fixed-point theorems on manifolds
58J05	Elliptic equations on manifolds, general theory
57R15	Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
53C07	Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53B35	Local differential geometry of Hermitian and Kählerian structures

Keywords:

Seiberg-Witten equations; \(\text{spin}^ c\) structures; Thom conjecture