TOPOLOGICAL QUANTUM FIELD THEORY AND FOUR MANIFOLDS

By DAVID WHITE

1 1 HOMOLOGY AND COHOMOLOGY 1 2 THE INTERSECTION FORM 1 3 SELF-DUAL AND ANTI-SELF-DUAL FORMS 1 4 CHARACTERISTIC CLASSES 1 5 EXAMPLES OF FOUR-MANIFOLDS COMPLEX SURFACES 1 6 SPIN AND SPINC-STRUCTURES ON FOUR-MANIFOLDS 2 THE THEORY OF DONALDSON INVARIANTS 2 1 YANG–MILLS THEORY ON A FOUR-MANIFOLD 2 2 SU(2) AND SO(3) BUNDLES 2 3 ASD CONNECTIONS 2 4 REDUCIBLE CONNECTIONS 2 5 A LOCAL MODEL FOR THE MODULI SPACE 2 6 DONALDSON INVARIANTS 2 7 METRIC DEPENDENCE 3 THE THEORY OF SEIBERG–WITTEN INVARIANTS 3 1 THE SEIBERG–WITTEN EQUATIONS 3 2 THE SEIBERG–WITTEN INVARIANTS 3 3 METRIC DEPENDENCE 4 SUPERSYMMETRY IN FOUR DIMENSIONS 4 1 THE SUPERSYMMETRY ALGEBRA 4 2 N = 1 SUPERSPACE AND SUPER?ELDS 4 3 N = 1 SUPERSYMMETRIC YANG–MILLS THEORIES 4 4 N = 2 SUPERSYMMETRIC YANG–MILLS THEORIES 4 5 N = 2 SUPERSYMMETRIC HYPERMULTIPLETS 4 6 N = 2 SUPERSYMMETRIC YANG–MILLS THEORIES WITH MATTER 5 TOPOLOGICAL QUANTUM FIELD THEORIES IN FOUR DIMENSIONS 5 1 BASIC PROPERTIES OF TOPOLOGICAL QUANTUM ?ELD THEORIES 5 2 TWIST OF N = 2 SUPERSYMMETRY 5 3 DONALDSON–WITTEN THEORY 5 4 TWISTED N = 2 SUPERSYMMETRIC HYPERMULTIPLET 5 5 EXTENSIONS OF DONALDSON–WITTEN THEORY 5 6 MONOPOLE EQUATIONS 6 THE MATHAI–QUILLEN FORMALISM 6 1 EQUIVARIANT COHOMOLOGY 6 2 THE ?NITE-DIMENSIONAL CASE 6 3 A DETAILED EXAMPLE 6 4 MATHAI–QUILLEN FORMALISM: IN?NITE-DIMENSIONAL CASE 6 5 THE MATHAI–QUILLEN FORMALISM FOR THEORIES WITH GAUGE SYMMETRY 6 6 DONALDSON–WITTEN THEORY IN THE MATHAI–QUILLEN FORMALISM 6 7 ABELIAN MONOPOLES IN THE MATHAI–QUILLEN FORMALISM 7 THE SEIBERG–WITTEN SOLUTION OF N = 2 SUSY YANG–MILLS THEORY 7 1 LOW ENERGY E?ECTIVE ACTION: SEMI-CLASSICAL ASPECTS 7 2 SL(2, Z) DUALITY OF THE E?ECTIVE ACTION 7 3 ELLIPTIC CURVES 7 4 THE EXACT SOLUTION OF SEIBERG AND WITTEN 7 5 THE SEIBERG–WITTEN SOLUTION IN TERMS OF MODULAR FORMS 8 THE U-PLANE INTEGRAL 8 1 THE BASIC PRINCIPLE (OR, ‘COULOMB + HIGGS=DONALDSON’) 8 2 E?ECTIVE TOPOLOGICAL QUANTUM ?ELD THEORY ON THE U-PLANE 8 3 ZERO MODES 8 4 FINAL FORM FOR THE U-PLANE INTEGRAL 8 5 BEHAVIOR UNDER MONODROMY AND DUALITY 9 SOME APPLICATIONS OF THE U-PLANE INTEGRAL 9 1 WALL CROSSING 9 2 THE SEIBERG–WITTEN CONTRIBUTION 9 3 THE BLOW-UP FORMULA 10 FURTHER DEVELOPMENTS IN DONALDSON–WITTEN THEORY 10 1 MORE FORMULAE FOR DONALDSON INVARIANTS 10 2 APPLICATIONS TO THE GEOGRAPHY OF FOUR-MANIFOLDS 10 3 EXTENSIONS TO HIGHER RANK GAUGE GROUPS