From Calabi-Yau manifolds to topological field theories Pietro Fre' SISSA-Trieste Paolo Soriani University degli Studi di Milano World Scientific Singapore New Jersey London Hong Kong
CONTENTS 1 AN INTRODUCTION TO THE SUBJECT 1 1.1 The Remarkable Interplay 2 1.1.1 Supergravity and Kahler geometry 2 1.1.2 Special Kahler geometry ; 4 1.2 Moduli and Criticality 4 1.2.1 Landau Ginzburg critical models and the moduli 4 1.2.2 N=2 superconformal field theories 11 1.3 Moduli and Algebraic Varieties 12 1.3.1 The chiral ring in N=2 superconformal theories 14 1.3.2 The vanishing locus of the superpotential as a Calabi-Yau manifold 15 1.3.3 The Griffiths residue map and the Hodge ring 15 1.4 The Art of Quantizing Zero 17 1.5 Mirror Maps 22 1.6 Bibliographical Note 23 2 A BIT OF GEOMETRY AND TOPOLOGY 25 2.1 Introduction 25 2.2 Fibre Bundles 25 2.2.1 Definition of a fibre bundle 25 2.2.2 Sheaves and Cech cohomology 27 2.2.3 Sections of a fibre bundle 30 2.2.4 Bundle maps 30 2.2.5 Equivalent bundles 31 2.2.6 Pull-back bundles 31 2.3 Vector Bundles, Connections and Curvatures 33 2.3.1 Fibre metrics 33 2.3.2 Product bundle 34 2.3.3 Whitney sum 34 2.3.4 Tensor product bundle 35 2.3.5 Principal fibre bundles 35 2.3.6 Connections on a vector bundle 36 ix
x CONTENTS 2.4 Complex Structures on 2n-Dimensional Manifolds 37 2.5 Metric and Connections on Holomorphic Vector Bundles 41 2.6 Kahler Metrics 44 2.7 Characteristic Classes and Elliptic Complexes 46 2.8 Hodge Manifolds and Chern Classes 57 2.9 Bibliographical Note 64 3 SUPERGRAVITY AND KAHLER GEOMETRY 65 3.1 Introduction 65 3.2 The Geometric Structure of Standard N=l Supergravity 66 3.2.1 Holomorphic Killing vectors on the scalar manifold and the momentum map 71 3.2.2 The momentum map and the complete bosonic lagrangian of N=l matter-coupled supergravity 74 3.2.3 Extrema of the potential and Kahler quotients 76 3.2.4 Effective N=l supergravities obtained from Calabi-Yau compactifications 79 3.3 Special Kahler Geometry 81 3.3.1 Special Kahler manifolds with special Killing vectors 81 3.3.2 Special geometry and N=2, D=4 supergravity 87 3.4 Bibliographical Note 91 4 COMPACTIFICATIONS ON CALABI-YAU MANIFOLDS 93 4.1 Introduction to Calabi Yau Compactifications 93 4.1.1 D=10, N=l matter-coupled supergravity 94 4.1.2 Killing spinors and SU(3) holonomy 96 4.1.3 The plan of this chapter 99 4.2 D=10 Anomaly-Free Supergravity 101 4.2.1 The role of anomaly-free supergravity in the derivation of Calabi- Yau compactifications 101 4.2.2 Strategy to derive anomaly-free supergravity 103 4.2.3 The free differential algebra {step 1) 107 4.2.4 Parametrization of the super-poincare curvatures (step 2) 108 4.2.5 Cohomology of superforms (step 3) 109 4.2.6 Discussion of the homogeneous i/-bianchi (step 4) 110 4.2.7 The BPT-theorem (step 5) Ill 4.2.8 Construction of the 3-form X (step 6) 112 4.2.9 Field equations of MAFS (step 7) 113 4.2.10 Calabi-Yau compactifications as exact solutions of minimal anomalyfree supergravity 115 4.3 Properties of Calabi-Yau Manifolds 115 4.3.1 Ricci-flatness and SU(n) holonomy 116 4.3.2 Harmonic forms and spinors 117
CONTENTS xi 4.3.3 The covariantly constant spinor 119 4.3.4 The holomorphic n-form 119 4.3.5 The Hodge diamond of Calabi-Yau 3-folds 122 4.4 Kaluza-Klein zero-modes and Yukawa Couplings 127 4.4.1 Analysis of the gauge sector 128 4.4.2 Analysis of the gravitational sector 131 4.4.3 Yukawa couplings 135 4.5 Complete Intersection Calabi-Yau Manifolds 136 4.6 Bibliographical Note 141 5 N=2 FIELD THEORIES IN TWO DIMENSIONS 143 5.1 Introduction 143 5.2 Abstract N=2 Superconformal Theories 146 5.3 N=2 Minimal Models 153 5.4 The Rheonomy Framework for N=2 Field Theories 158 5.4.1 N=2 2D supergravity and the super-poincare algebra 158 5.4.2 Chiral multiplets in curved superspace 165 5.5 An N=2 Gauge Theory and Its Two Phases 169 5.5.1 The N=2 abelian gauge multiplet 170 5.5.2 N=2 Landau-Ginzburg models with an abelian gauge symmetry. 172 5.5.3 Structure of the scalar potential 174 5.5.4 Extension to non abelian gauge symmetry 175 5.5.5 R-symmetries and the rigid Landau-Ginzburg model 178 5.5.6 N=2 sigma models 182 5.5.7 Extrema of the N=2 scalar potential, phases of the gauge theory and reconstruction of the effective N=2 tr-model 186 5.6 N=2 Landau-Ginzburg Models and N=2 Superconformal Theories... 196 5.7 Landau-Ginzburg Models and Calabi-Yau Manifolds 203 5.8 Landau-Ginzburg Potentials and Pseudo-Ghost First Order Systems.. 206 5.9 The Griffiths Residue Mapping and the Chiral Ring 215 5.9.1 Rational meromorphic (n + l)-forms and the Hodge filtration.. 216 5.9.2 Interpretation of the residue map in N=2 conformal field theory. 220 5.9.3 Explicit construction of the harmonic (n k, fc)-forms and the realization of the chiral ring on the Hodge filtration 221 5.10 Bibliographical Note 226 6 MODULI SPACES AND SPECIAL GEOMETRY 229 6.1 Introduction 229 6.2 The Special Geometry of (2,1)-Forms 232 6.3 The Special Geometry of (1,1 )-Forms 238 6.4 Special Geometry from N=2 World Sheet Supersymmetry 240 6.5 Concluding Remarks 256 6.6 Bibliographical Note 256
xii CONTENTS 7 TOPOLOGICAL FIELD THEORIES 259 7.1 Introduction 259 7.2 The Geometric Formulation of BRST Symmetry 265 7.3 Topological Yang-Mills Theories 274 7.4 Topological Sigma Models 283 7.5 The A and B Topological Twists of an N=2 Field Theory 289 7.6 Twists of the Two-Phase N=2 Gauge Theory 293 7.6.1 The topological BRST algebra 293 7.6.2 Interpretation of the A-model and topological cr-models 297 7.6.3 Interpretation of the B-model and topological Landau-Ginzburg theories 301 7.7 Correlators of the Topological Sigma Model 305 7.7.1 The topological er-model or A-twist case 307 7.7.2 Topological cr-models on Calabi-Yau 3-folds 319 7.7.3 The B-twist case and the Hodge structure deformations 325 7.8 Topological Conformal Field Theories 334 7.9 Correlators of the Topological Landau-Ginzburg Model 341 7.9.1 Applications of the residue pairing formula 345 7.10 Topological Observables in the Two-Phase Theory 352 7.11 Bibliographical Note 360 8 PICARD-FUCHS EQUATIONS AND MIRROR MAPS 363 8.1 Introduction to Mirror Symmetry 363 8.1.1 The mirror quintic 368 8.1.2 The issue of flat coordinates and Picard Fuchs equation 375 8.2 Picard-Fuchs Equations for the Period Matrix 377 8.2.1 Picard Fuchs equations for the cubic torus 380 8.2.2 Picard-Fuchs equation for the one-modulus MCJP p -i(p) hypersurfaces, and its singularity structure 383 8.2.3 Perspective 386 8.3 Picard Fuchs Equations and Special Geometry 387 8.3.1 Introduction and summary 387 8.3.2 Differential equations and W-generators 391 8.3.3 Associated first order linear systems 395 8.3.4 The flat holomorphic connection of special Kahler manifolds... 398 8.3.5 Holomorphic Picard Fuchs equations for n-dimensional special manifolds 403 8.3.6 The non-holomorphic Picard Fuchs equations of special manifolds 408 8.4 Monodromy and Duality Groups 410 8.4.1 Introduction 410 8.4.2 The duality group Fw of MCP p _i(p) hypersurfaces 411 8.4.3 Monodromy group of the cubic torus 416
CONTENTS xiii 8.4.4 Barne's integral transform and the calculation of the monodromy matrix T o 419 8.5 The Mirror Map and the Sum over Instantons 429 8.5.1 Yukawa coupling as the fusion coefficient of the chiral ring... 431 8.5.2 General strategy for the evaluation of the Yukawa coupling of the mirror quintic 434 8.5.3 Logarithmic behaviour of the solutions in the neighbourhood of? 4> = oo 438 8.5.4 The instanton expansion of the Yukawa coupling and the prediction of the number of rational curves on the quintic 3-fold 440 8.5.5 The special Kahlerian metric of the moduli space of Kahler class deformations for the quintic 3-fold 443 8.5.6 Summary and conclusion 446 8.6 Bibliographical Note 447 9 FAREWELL 449 BIBLIOGRAPHY 450