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 4-dimensional differential topology:

• The ability to show that the intersection form of a compact, oriented, differentiable 4-manifold is standard over \(\mathbb {Z}\) whenever it is (positively or negatively) defined. By Freedman's results on the classification of topological 4-manifolds, 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 4-manifolds).

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 4-dimensional differential topology, e.g., the Donaldson theorem concerning the intersection form of differentiable 4-manifolds and the Van de Ven conjecture concerning the differential topological classification of complex surfaces.

This book deals essentially with Seiberg-Witten theory, which is easily accessible to students, but also contains elements of Donaldson theory: the gauge group of a principal fiber-bundle, Yang-Mills equations, ASD-equations, and examples of moduli spaces of Yang-Mills 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, Seiberg-Witten 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 Yang-Mills et connexions anti-autoduales
• Structures Spin et Spinc\(^c\), opérateurs de Dirac, la formule de Weitzenböck
• Espaces de modules de monopoles de Seiberg-Witten. Le théorème de Donaldson sur la forme d'intersection d'une 4-variété
• Les invariants de Seiberg-Witten
• Monopoles sur les surfaces kähleriennes
• Exemples et applications
• Appendices
• Bibliographie