Classical Mirror Symmetry

This book furnishes a brief introduction to classical mirror symmetry, a term that denotes the process of computing Gromov–Witten invariants of a Calabi–Yau threefold by using the Picard–Fuchs differential equation of period integrals of its mirror Calabi–Yau threefold. The book concentrates on the best-known example, the quintic hypersurface in 4-dimensional projective space, and its mirror manifold. First, there is a brief review of the process of discovery of mirror symmetry and the striking result proposed in the celebrated paper by Candelas and his collaborators. Next, some elementary results of complex manifolds and Chern classes needed for study of mirror symmetry are explained. Then the topological sigma models, the A-model and the B-model, are introduced. The classical mirror symmetry hypothesis is explained as the equivalence between the correlation function of the A-model of a quintic hyper-surface and that of the B-model of its mirror manifold. On the B-model side, the process of construction of a pair of mirror Calabi–Yau threefold using toric geometry is briefly explained. Also given are detailed explanations of the derivation of the Picard–Fuchs differential equation of the period integrals and on the process of deriving the instanton expansion of the A-model Yukawa coupling based on the mirror symmetry hypothesis. On the A-model side, the moduli space of degree d quasimaps from CP^1 with two marked points to CP^4 is introduced, with reconstruction of the period integrals used in the B-model side as generating functions of the intersection numbers of the moduli space. Lastly, a mathematical justification for the process of the B-model computation from the point of view of the geometry of the moduli space of quasimaps is given. The style of description is between that of mathematics and physics, with the assumption that readers have standard graduate student backgrounds in both disciplines. Preface 6 Contents 8 1 Brief History of Classical Mirror Symmetry 10 1.1 Grand Unified Theory and Superstring Theory 10 1.2 Compactification of Heterotic String Theory 14 1.3 Discovery of Mirror Symmetry of N=2 Superconformal Field Theory 21 1.4 First Striking Prediction of Mirror Symmetry 29 References 35 2 Basics of Geometry of Complex Manifolds 36 2.1 Complex Manifold 36 2.2 Vector Bundles of Complex Manifold 39 2.2.1 Definition of Holomorphic Vector Bundles, Covariant Derivatives, Connections and Curvatures 39 2.3 Chern Classes 45 2.3.1 Calculus of Holomorphic Vector Bundles and Chern Classes 49 2.4 Khler Manifolds and Projective Spaces 53 2.4.1 Definition of Khler Manifolds 53 2.4.2 Dolbeault Cohomology of Khler Manifolds 56 2.5 Complex Projective Space as an Example of Compact Khler Manifold 58 Reference 62 3 Topological Sigma Models 63 3.1 N=2 Supersymmetric Sigma Model 63 3.2 Topological Sigma Model (A-Model) 65 3.2.1 Lagrangian and Saddle Point Approximation 65 3.2.2 Observable Satisfying {Q,calO}=0 and Topological Selection Rule 71 3.2.3 Geometrical Interpretation of Degree d Correlation Function 73 3.2.4 Degree d Correlation Function of Degree 5 Hypersurface in CP4 75 3.3 Topological Sigma Model (B-Model) 83 3.3.1 Lagrangian 83 3.3.2 Observable that Satisfies {Q,calO}=0 and Topological Selection Rule 86 3.3.3 Correlation Function 87 References 89 4 Details of B-Model Computation 90 4.1 Toric Geometry 90 4.1.1 Outline of Toric Geometry 90 4.1.2 An Example: Complex Projective Space 91 4.2 Formulation of Mirror Symmetry by Toric Geometry 97 4.2.1 An Example: Quintic Hypersurface in CP4 97 4.2.2 Blow-Up (Resolution of Singularity) and Hodge Number 101 4.3 Details of B-Model Computation 106 4.3.1 Derivation of Differential Equations Satisfied by Period Integrals 106 4.3.2 Derivation of B-Model Yukawa Coupling 110 4.3.3 Instanton Expansion of A-Model Yukawa Couplings 112 References 115 5 Reconstruction of Mirror Symmetry Hypothesis from a Geometrical Point of View 116 5.1 Simple Compactification of Holomorphic Maps from CP1 to CP4 116 5.1.1 A-Model Correlation Functions as Intersection Numbers 116 5.1.2 Evaluation of Yukawa Coupling of Quintic Hypersurface in CP4 by Using Simple Compactification of the Moduli Space 119 5.2 Toric Compactification of the Moduli Space of Degree d Quasi Maps with Two Marked Points 123 5.3 Construction of Two Point Intersection Numbers on overlineMp0,2(CP4,d) 127 5.4 Fixed Point Theorem and Computation of w(calOhacalOhb)0,d 129 5.4.1 Fixed Point Theorem 129 5.4.2 Computation of w(calOhacalOhb)0,d 132 5.5 Reconstruction of Mirror Symmetry Computation 137 References 147 Front Matter ....Pages i-viii Brief History of Classical Mirror Symmetry (Masao Jinzenji)....Pages 1-26 Basics of Geometry of Complex Manifolds (Masao Jinzenji)....Pages 27-53 Topological Sigma Models (Masao Jinzenji)....Pages 55-81 Details of B-Model Computation (Masao Jinzenji)....Pages 83-108 Reconstruction of Mirror Symmetry Hypothesis from a Geometrical Point of View (Masao Jinzenji)....Pages 109-140