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1 Introduction 1

1.1 What is Gauss-Manin connection in disguise? 3

1.2 Why mirror quintic Calabi-Yau threefold? 4

1.3 How to read the text? 5

1.4 Why differential Calabi-Yau modular form? 5

2 Summary of results and computations 7

2.1 Mirror quintic Calabi-Yau threefolds 7

2.2 Ramanujan differential equation 8

2.3 Modular vector fields 9

2.4 Geometric differential Calabi-Yau modular forms 11

2.5 Eisenstein series 12

2.6 Elliptic integrals and modular forms 14

2.7 Periods and differential Calabi-Yau modular forms,Ⅰ 15

2.8 Integrality of Fourier coefficients 17

2.9 Quasi-or differential modular forms 18

2.10 Functional equations 19

2.11 Conifold singularity 21

2.12 The Lie algebra sl2 22

2.13 BCOV holomorphic anomaly equation,Ⅰ 23

2.14 Gromov-Witten invariants 24

2.15 Periods and differential Calabi-Yau modular forms,Ⅱ 25

2.16 BCOV holomorphic anomaly equation,Ⅱ 28

2.17 The polynomial structure of partition functions 29

2.18 Future developments 30

3 Moduli of enhanced mirror quintics 31

3.1 What is mirror quintic？ 31

3.2 Moduli space,Ⅰ 32

3.3 Gauss-Manin connection,Ⅰ 33

3.4 Intersection form and Hodge filtration 34

3.5 A vector field on S 35

3.6 Moduli space,Ⅱ 35

3.7 The Picard-Fuchs equation 36

3.8 Gauss-Manin connection,Ⅱ 37

3.9 Proof of Theorem 2 39

3.10 Algebraic group 39

3.11 Another vector field 41

3.12 Weights 42

3.13 A Lie algebra 43

4 Topology and periods 45

4.1 Period map 45

4.2 τ-locus 46

4.3 Positivity conditions 48

4.4 Generalized period domain 49

4.5 The algebraic group and τ-locus 50

4.6 Monodromy covering 51

4.7 A particular solution 52

4.8 Action of the monodromy 52

4.9 The solution in terms of periods 54

4.10 Computing periods 55

4.11 Algebraically independent periods 57

4.12 θ-locus 58

4.13 The algebraic group and the θ-locus 60

4.14 Comparing τ and θ-loci 61

4.15 All solutions of R0,？0 62

4.16 Around the elliptic point 62

4.17 Halphen property 63

4.18 Differential Calabi-Yau modular forms around the conifold 64

4.19 Logarithmic mirror map aroundthe conifold 65

4.20 Holomorphic mirror map 67

5 Formal power series solutions 69

5.1 Singularities of modular differential equations 69

5.2 q-expansion around maximal unipotent cusp 70

53 Another q-expansion 71

5.4 q-expansion around conifold 72

5.5 New coordinates 73

5.6 Holomorphic foliations 74

6 Topological string partition functions 75

6.1 Yamaguchi-Yau's elements 75

6.2 Proof of Theorem 8 76

6.3 Genus 1 topological partition function 77

6.4 Holomorphic anomaly equation 78

6.5 Proof of Proposition 1 80

6.6 The ambiguity of Falg g 81

6.7 Topological partition functions Falg g,g=2,3 82

6.8 Topological partition functions for elliptic curves 82

7 Holomorphic differential Calabi-Yau modular forms 85

7.1 Fourth-order differential equations 85

7.2 Hypergeometric differential equations 86

7.3 Picard-Fuchs equations 87

7.4 Intersection form 89

7.5 Maximal unipotent monodromy 91

7.6 The field of differential Calabi-Yau modular forms 92

7.7 The derivation 94

7.8 Yukawa coupling 94

7.9 q-expansion 95

8 Non-holomorphic differential Calabi-Yau modular forms 97

8.1 The differential field 97

8.2 Anti-holomorphic derivation 99

8.3 A new basis 100

8.4 Yamaguchi-Yau elements 101

8.5 Hypergeometric cases 103

9 BCOV holomorphic anomaly equation 105

9.1 Genus 1 topological partition function 105

9.2 The covariant derivative 106

9.3 Holomorphic anomaly equation 108

9.4 Master anomaly equation 108

9.5 Algebraic anomaly equation 109

9.6 Proof of Theorem 9 110

9.7 A kind of Gauss-Manin connection 111

9.8 Seven vector fields 112

9.9 Comparison of algebraic and holomorphic anomaly equations 113

9.10 Feynman rules 114

9.11 Structure of the ambiguity 116

10 Calabi-Yau modular forms 119

10.1 Classical modular forms 119

10.2 A general setting 120

10.3 The algebra of Calabi-Yau modular forms 121

11 Problems 125

11.1 Vanishing of periods 125

11.2 Hecke operators 126

11.3 Maximal Hodge structure 127

11.4 Monodromy 129

11.5 Torelli problem 129

11.6 Monstrous moonshine conjecture 130

11.7 Integrality of instanton numbers 131

11.8 Some product formulas 132

11.9 A new mirror map 133

11.10 Yet another coordinate 135

11.11 Gap condition 136

11.12 Algebraic gap condition 137

11.13 Arithmetic modularity 140

A Second-order linear differential equations 143

A.1 Holomorphic and non-holomorphic quasi-modular forms 143

A.2 Full quasi-modular forms 145

B Metric 147

B.1 Poincaré metric 148

B.2 K？hler metric for moduli of mirror quintics 149

C Integrality properties 151

HOSSEIN MOVASATI,KHOSRO M.SHOKRI 151

C.1 Introduction 151

C.2 Dwork map 155

C.3 Dwork lemma and theorem on hypergeometric functions 156

C.4 Consequences of Dwork's theorem 157

C.5 Proof of Theorem 13,Part 1 158

C.6 A problem in computational commutative algebra 158

C.7 The case n=2 159

C.8 The symmetry 160

C.9 Proof of Theorem 13,Part 2 160

C.10 Computational evidence for Conjecture 161

C.11 Proof of Corollary 1 162

D Kontsevich's formula 163

CARLOS MATHEUS 163

D.1 Examples of variations of Hodge structures of weight k 164

D.2 Lyapunov exponents 166

D.3 Kontsevich's formula in the classical setting 168

D.4 Kontsevich's formula in Calabi-Yau 3-folds setting 170

D.5 Simplicity of Lyapunov exponents of mirror quintics 173

References 175