Cours SpécialisésCollection SMF 2012; 191 pp; softcover Number: 18 ISBN10: 2856293220 ISBN13: 9782856293225 List Price: US$90 Member Price: US$72 Order Code: COSP/18
 The fundamental idea of mathematical gauge theory is to study the moduli spaces of solutions of certain systems of partial differential equations on a differentiable manifold and to obtain information about this manifold (for instance, information on its diffeomorphism type) using them. This idea brought the first spectacular results in 4dimensional differential topology:  The ability to show that the intersection form of a compact, oriented, differentiable 4manifold is standard over \(\mathbb {Z}\) whenever it is (positively or negatively) defined. By Freedman's results on the classification of topological 4manifolds, the analogue statement is definitely false in the topological framework.
 The ability to introduce and compute explicitly the first \({\mathcal C}^\infty\)invariants in dimension 4, which, in turn, were used to discover the first exotic pairs (i.e. homeomorphic but not diffeomorphic pairs of differentiable 4manifolds).
The goal of these lecture notes is to give a solid introduction to mathematical gauge theory and to explain in detail some of its important applications in 4dimensional differential topology, e.g., the Donaldson theorem concerning the intersection form of differentiable 4manifolds and the Van de Ven conjecture concerning the differential topological classification of complex surfaces. This book deals essentially with SeibergWitten theory, which is easily accessible to students, but also contains elements of Donaldson theory: the gauge group of a principal fiberbundle, YangMills equations, ASDequations, and examples of moduli spaces of YangMills equations. These lecture notes are fully accessible to students who have attended lectures on differentiable geometry and algebraic topology and have a basic background in modern analysis (Sobolev spaces, distributions, and differential operators). A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in Gauge theory, SeibergWitten theory, and Donaldson theory. Table of Contents  Introduction
 Théorie de Hodge sur les variétés compactes
 Connexions linéaires et courbure
 Fibrés principaux et connexions sur les fibrés principaux
 Connexions de YangMills et connexions antiautoduales
 Structures Spin et Spinc\(^c\), opérateurs de Dirac, la formule de Weitzenböck
 Espaces de modules de monopoles de SeibergWitten. Le théorème de Donaldson sur la forme d'intersection d'une 4variété
 Les invariants de SeibergWitten
 Monopoles sur les surfaces kähleriennes
 Exemples et applications
 Appendices
 Bibliographie
