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