Introduction a la Theorie de Jauge

Introduction a la Theorie de Jauge

by Andrei Teleman

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Introduction a la Theorie de Jauge by Andrei Teleman

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).

Product Details

ISBN-13: 9782856293225
Publisher: American Mathematical Society
Publication date: 09/15/2012
Pages: 191
Product dimensions: 7.10(w) x 9.60(h) x 0.50(d)

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