Lectures on Seiberg-Witten Invariants

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April 2001



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.


1. Preliminaries 1.1. Introduction 1.2. What is a vector bundle? 1.3. What is a connection? 1.4. The curvature of a connection 1.5. Characteristic classes 1.6. The Thom form 1.7. The universal bundle 1.8. Classification of connections 1.9. Hodge theory 2. Spin geometry on four-manifolds 2.1. Euclidean geometry and the spin groups 2.2. What is a spin structure? 2.3. Almost complex and spin-c structures 2.4. Clifford algebras 2.5. The spin connection 2.6. The Dirac operator 2.7. The Atiyah-Singer index theorem 3. Global analysis 3.1. The Seiberg-Witten equations 3.2. The moduli space 3.3. Compactness of the moduli space 3.4. Transversality 3.5. The intersection form 3.6. Donaldson's theorem 3.7. Seiberg-Witten invariants 3.8. Dirac operators on Kaehler surfaces 3.9. Invariants of Kaehler surfaces Bibliography Index
EAN: 9783540412212
ISBN: 3540412212
Untertitel: 2nd ed. 2001. Book. Sprache: Englisch.
Verlag: Springer
Erscheinungsdatum: April 2001
Seitenanzahl: 136 Seiten
Format: kartoniert
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