Topological Quantum Field Theory and Four Manifolds

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