Gauge Theory and the Topology of Four-Manifolds
About this Title
Robert Friedman, Columbia University, New York, NY and John W. Morgan, Columbia University, New York, NY, Editors
Publication: IAS/Park City Mathematics Series
Publication Year: 1998; Volume 4
ISBNs: 978-0-8218-0591-6 (print); 978-1-4704-3903-3 (online)
MathSciNet review: MR1611444
MSC: Primary 57-06; Secondary 58-06
The lectures in this volume provide a perspective on how 4-manifold theory was studied before the discovery of modern-day Seiberg-Witten theory. One reason the progress using the Seiberg-Witten invariants was so spectacular was that those studying $SU(2)$-gauge theory had more than ten years' experience with the subject. The tools had been honed, the correct questions formulated, and the basic strategies well understood. The knowledge immediately bore fruit in the technically simpler environment of the Seiberg-Witten theory.
Gauge theory long predates Donaldson's applications of the subject to 4-manifold topology, where the central concern was the geometry of the moduli space. One reason for the interest in this study is the connection between the gauge theory moduli spaces of a Kähler manifold and the algebro-geometric moduli space of stable holomorphic bundles over the manifold. The extra geometric richness of the $SU(2)$-moduli spaces may one day be important for purposes beyond the algebraic invariants that have been studied to date. It is for this reason that the results presented in this volume will be essential.
Graduate students and research mathematicians working in algebraic geometry.
Table of Contents
- Geometric invariant theory and the moduli of bundles
- Anti-self-dual connections and stable vector bundles
- An introduction to gauge theory
- Computing Donaldson invariants
- Donaldson-Floer theory