Lectures on Seiberg-Witten Invariants

  • John Douglas Moore
Part of the Lecture Notes in Mathematics book series (LNM, volume 1629)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Pages 1-44
  3. Back Matter
    Pages 117-121

About this book

Introduction

Riemannian, symplectic and complex geometry are often studied by means ofsolutions to systems ofnonlinear differential equations, such as the equa­ tions of geodesics, minimal surfaces, pseudoholomorphic curves and Yang­ Mills connections. For studying such equations, a new unified technology has been developed, involving analysis on infinite-dimensional manifolds. A striking applications of the new technology is Donaldson's theory of "anti-self-dual" connections on SU(2)-bundles over four-manifolds, which applies the Yang-Mills equations from mathematical physics to shed light on the relationship between the classification of topological and smooth four-manifolds. This reverses the expected direction of application from topology to differential equations to mathematical physics. Even though the Yang-Mills equations are only mildly nonlinear, a prodigious amount of nonlinear analysis is necessary to fully understand the properties of the space of solutions. . At our present state of knowledge, understanding smooth structures on topological four-manifolds seems to require nonlinear as opposed to linear PDE's. It is therefore quite surprising that there is a set of PDE's which are even less nonlinear than the Yang-Mills equation, but can yield many of the most important results from Donaldson's theory. These are the Seiberg-Witte~ equations. These lecture notes stem from a graduate course given at the University of California in Santa Barbara during the spring quarter of 1995. The objective was to make the Seiberg-Witten approach to Donaldson theory accessible to second-year graduate students who had already taken basic courses in differential geometry and algebraic topology.

Keywords

Characteristic class Clifford algebras Dirac operators Hodge theory Seiberg-Witten invariants algebra four-manifolds vector bundle

Authors and affiliations

  • John Douglas Moore
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-540-40952-6
  • Copyright Information Springer Berlin Heidelberg 2001
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-540-41221-2
  • Online ISBN 978-3-540-40952-6
  • Series Print ISSN 0075-8434