Topological Quantum Field Theory and Four Manifolds

The present book is the first of its kind in dealing with topological quantum field theories and their applications to topological aspects of four manifolds. It is not only unique for this reason but also because it contains sufficient introductory material that it can be read by mathematicians and theoretical physicists. On the one hand, it contains a chapter dealing with topological aspects of four manifolds, on the other hand it provides a full introduction to supersymmetry. The book constitutes an essential tool for researchers interested in the basics of topological quantum field theory, since these theories are introduced in detail from a general point of view. In addition, the book describes Donaldson theory and Seiberg-Witten theory, and provides all the details that have led to the connection between these theories using topological quantum field theory. It provides a full account of Witten's magic formula relating Donaldson and Seiberg-Witten invariants. Furthermore, the book presents some of the recent developments that have led to important applications in the context of the topology of four manifolds.

Texte du rabat

The emergence of topological quantum ?eld theory has been one of the most important breakthroughs which have occurred in the context of ma- ematical physics in the last century, a century characterizedbyindependent developments of the main ideas in both disciplines, physics and mathematics, which has concluded with two decades of strong interaction between them, where physics, as in previous centuries, has acted as a source of new mat- matics. Topological quantum ?eld theories constitute the core of these p- nomena, although the main drivingforce behind it has been the enormous e?ort made in theoretical particle physics to understand string theory as a theory able to unify the four fundamental interactions observed in nature. These theories set up a new realm where both disciplines pro?t from each other. Although the most striking results have appeared on the mathema- calside,theoreticalphysicshasclearlyalsobene?tted,sincethecorresponding developments have helped better to understand aspects of the fundamentals of ?eld and string theory.

Contenu

Table of Contents Preface vii 1. Topological Aspects of Four-Manifolds 1 1.1. Homology and cohomology 1 1.2. The intersection form 2 1.3. Self-dual and anti-self-dual forms 4 1.4. Characteristic classes 5 1.5. Examples of four-manifolds. Complex surfaces 6 1.6. Spin and Spinc-structures on four-manifolds. 9 2. The Theory of Donaldson Invariants 12 2.1. Yang-Mills theory on a four-manifold 12 2.2. SU(2) and SO(3) bundles 14 2.3. ASD connections 16 2.4. Reducible connections 18 2.5. A local model for the moduli space 19 2.6. Donaldson invariants 22 2.7. Metric dependence 27 3. The Theory of Seiberg-Witten Invariants 31 3.1. The Seiberg-Witten equations 31 3.2. The Seiberg-Witten invariants 32 3.3. Metric dependence 36 4. Supersymmetry in Four Dimensions 39 4.1. The supersymmetry algebra 39 4.2. N = 1 superspace and super.elds 40 4.3. N = 1 supersymmetric Yang-Mills theories 45 4.4. N = 2 supersymmetric Yang-Mills theories 50 4.5. N = 2 supersymmetric hypermultiplets 53 4.6. N = 2 supersymmetric Yang-Mills theories with matter 55 5. Topological Quantum Field Theories in Four Dimensions 58 5.1. Basic properties of topological quantum .eld theories 58 5.2. Twist of N = 2 supersymmetry 61 5.3. Donaldson-Witten theory 64 5.4. Twisted N = 2 supersymmetric hypermultiplet 71 5.5. Extensions of Donaldson-Witten theory 72 5.6. Monopole equations 74 6. The Mathai-Quillen Formalism 78 6.1. Equivariant cohomology 79 6.2. The .nite-dimensional case 82 6.3. A detailed example 88 6.4. Mathai-Quillen formalism: In.nite-dimensional case 93 6.5. The Mathai-Quillen formalism fortheories with gauge symmetry 102 6.6. Donaldson-Witten theory in the Mathai-Quillen formalism 105 6.7. Abelian monopoles in the Mathai-Quillen formalism 107 7. The Seiberg-Witten Solution of N = 2 SUSY Yang-Mills Theory 110 7.1. Low energy e.ective action: semi-classical aspects 110 7.2. Sl(2,Z) duality of the e.ective action 116 7.3. Elliptic curves 120 7.4. The exact solution of Seiberg and Witten 123 7.5. The Seiberg-Witten solution in terms of modular forms 129 8. The u-plane Integral 133 8.1. The basic principle (or, 'Coulomb + Higgs=Donaldson') 133 8.2. E.ective topological quantum .eld theory on the u-plane 134 8.3. Zero modes 140 8.4. Final form for the u-plane integral 144 8.5. Behavior under monodromy and duality 149 9. Some Applications of the u-plane Integral 154 9.1. Wall crossing 154 9.2. The Seiberg-Witten contribution 157 9.3. The blow-up formula. 165 10. Further Developments in Donaldson-Witten Theory 170 10.1. More formulae for Donaldson invariants 170 10.2. Applications to the geography of four-manifolds 177 10.3. Extensions to higher rank gauge groups 188 Appendix A. Spinors in Four Dimensions 204 Appendix B. Elliptic Functions and Modular Forms 209 Bibliography 213