Pietro Fre' SISSA-Trieste. Paolo Soriani University degli Studi di Milano. From Calabi-Yau manifolds to topological field theories

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1 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

2 CONTENTS 1 AN INTRODUCTION TO THE SUBJECT The Remarkable Interplay Supergravity and Kahler geometry Special Kahler geometry ; Moduli and Criticality Landau Ginzburg critical models and the moduli N=2 superconformal field theories Moduli and Algebraic Varieties The chiral ring in N=2 superconformal theories The vanishing locus of the superpotential as a Calabi-Yau manifold The Griffiths residue map and the Hodge ring The Art of Quantizing Zero Mirror Maps Bibliographical Note 23 2 A BIT OF GEOMETRY AND TOPOLOGY Introduction Fibre Bundles Definition of a fibre bundle Sheaves and Cech cohomology Sections of a fibre bundle Bundle maps Equivalent bundles Pull-back bundles Vector Bundles, Connections and Curvatures Fibre metrics Product bundle Whitney sum Tensor product bundle Principal fibre bundles Connections on a vector bundle 36 ix

3 x CONTENTS 2.4 Complex Structures on 2n-Dimensional Manifolds Metric and Connections on Holomorphic Vector Bundles Kahler Metrics Characteristic Classes and Elliptic Complexes Hodge Manifolds and Chern Classes Bibliographical Note 64 3 SUPERGRAVITY AND KAHLER GEOMETRY Introduction The Geometric Structure of Standard N=l Supergravity Holomorphic Killing vectors on the scalar manifold and the momentum map The momentum map and the complete bosonic lagrangian of N=l matter-coupled supergravity Extrema of the potential and Kahler quotients Effective N=l supergravities obtained from Calabi-Yau compactifications Special Kahler Geometry Special Kahler manifolds with special Killing vectors Special geometry and N=2, D=4 supergravity Bibliographical Note 91 4 COMPACTIFICATIONS ON CALABI-YAU MANIFOLDS Introduction to Calabi Yau Compactifications D=10, N=l matter-coupled supergravity Killing spinors and SU(3) holonomy The plan of this chapter D=10 Anomaly-Free Supergravity The role of anomaly-free supergravity in the derivation of Calabi- Yau compactifications Strategy to derive anomaly-free supergravity The free differential algebra {step 1) Parametrization of the super-poincare curvatures (step 2) Cohomology of superforms (step 3) Discussion of the homogeneous i/-bianchi (step 4) The BPT-theorem (step 5) Ill Construction of the 3-form X (step 6) Field equations of MAFS (step 7) Calabi-Yau compactifications as exact solutions of minimal anomalyfree supergravity Properties of Calabi-Yau Manifolds Ricci-flatness and SU(n) holonomy Harmonic forms and spinors 117

4 CONTENTS xi The covariantly constant spinor The holomorphic n-form The Hodge diamond of Calabi-Yau 3-folds Kaluza-Klein zero-modes and Yukawa Couplings Analysis of the gauge sector Analysis of the gravitational sector Yukawa couplings Complete Intersection Calabi-Yau Manifolds Bibliographical Note N=2 FIELD THEORIES IN TWO DIMENSIONS Introduction Abstract N=2 Superconformal Theories N=2 Minimal Models The Rheonomy Framework for N=2 Field Theories N=2 2D supergravity and the super-poincare algebra Chiral multiplets in curved superspace An N=2 Gauge Theory and Its Two Phases The N=2 abelian gauge multiplet N=2 Landau-Ginzburg models with an abelian gauge symmetry Structure of the scalar potential Extension to non abelian gauge symmetry R-symmetries and the rigid Landau-Ginzburg model N=2 sigma models Extrema of the N=2 scalar potential, phases of the gauge theory and reconstruction of the effective N=2 tr-model N=2 Landau-Ginzburg Models and N=2 Superconformal Theories Landau-Ginzburg Models and Calabi-Yau Manifolds Landau-Ginzburg Potentials and Pseudo-Ghost First Order Systems The Griffiths Residue Mapping and the Chiral Ring Rational meromorphic (n + l)-forms and the Hodge filtration Interpretation of the residue map in N=2 conformal field theory Explicit construction of the harmonic (n k, fc)-forms and the realization of the chiral ring on the Hodge filtration Bibliographical Note MODULI SPACES AND SPECIAL GEOMETRY Introduction The Special Geometry of (2,1)-Forms The Special Geometry of (1,1 )-Forms Special Geometry from N=2 World Sheet Supersymmetry Concluding Remarks Bibliographical Note 256

5 xii CONTENTS 7 TOPOLOGICAL FIELD THEORIES Introduction The Geometric Formulation of BRST Symmetry Topological Yang-Mills Theories Topological Sigma Models The A and B Topological Twists of an N=2 Field Theory Twists of the Two-Phase N=2 Gauge Theory The topological BRST algebra Interpretation of the A-model and topological cr-models Interpretation of the B-model and topological Landau-Ginzburg theories Correlators of the Topological Sigma Model The topological er-model or A-twist case Topological cr-models on Calabi-Yau 3-folds The B-twist case and the Hodge structure deformations Topological Conformal Field Theories Correlators of the Topological Landau-Ginzburg Model Applications of the residue pairing formula Topological Observables in the Two-Phase Theory Bibliographical Note PICARD-FUCHS EQUATIONS AND MIRROR MAPS Introduction to Mirror Symmetry The mirror quintic The issue of flat coordinates and Picard Fuchs equation Picard-Fuchs Equations for the Period Matrix Picard Fuchs equations for the cubic torus Picard-Fuchs equation for the one-modulus MCJP p -i(p) hypersurfaces, and its singularity structure Perspective Picard Fuchs Equations and Special Geometry Introduction and summary Differential equations and W-generators Associated first order linear systems The flat holomorphic connection of special Kahler manifolds Holomorphic Picard Fuchs equations for n-dimensional special manifolds The non-holomorphic Picard Fuchs equations of special manifolds Monodromy and Duality Groups Introduction The duality group Fw of MCP p _i(p) hypersurfaces Monodromy group of the cubic torus 416

6 CONTENTS xiii Barne's integral transform and the calculation of the monodromy matrix T o The Mirror Map and the Sum over Instantons Yukawa coupling as the fusion coefficient of the chiral ring General strategy for the evaluation of the Yukawa coupling of the mirror quintic Logarithmic behaviour of the solutions in the neighbourhood of? 4> = oo The instanton expansion of the Yukawa coupling and the prediction of the number of rational curves on the quintic 3-fold The special Kahlerian metric of the moduli space of Kahler class deformations for the quintic 3-fold Summary and conclusion Bibliographical Note FAREWELL 449 BIBLIOGRAPHY 450