伪装的Gauss-Manin联络2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

伪装的Gauss-Manin联络PDF格式电子书版下载

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

1 Introduction1

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 computations7

2.1 Mirror quintic Calabi-Yau threefolds7

2.2 Ramanujan differential equation8

2.3 Modular vector fields9

2.4 Geometric differential Calabi-Yau modular forms11

2.5 Eisenstein series12

2.6 Elliptic integrals and modular forms14

2.7 Periods and differential Calabi-Yau modular forms,Ⅰ15

2.8 Integrality of Fourier coefficients17

2.9 Quasi-or differential modular forms18

2.10 Functional equations19

2.11 Conifold singularity21

2.12 The Lie algebra sl222

2.13 BCOV holomorphic anomaly equation,Ⅰ23

2.14 Gromov-Witten invariants24

2.15 Periods and differential Calabi-Yau modular forms,Ⅱ25

2.16 BCOV holomorphic anomaly equation,Ⅱ28

2.17 The polynomial structure of partition functions29

2.18 Future developments30

3 Moduli of enhanced mirror quintics31

3.1 What is mirror quintic？31

3.2 Moduli space,Ⅰ32

3.3 Gauss-Manin connection,Ⅰ33

3.4 Intersection form and Hodge filtration34

3.5 A vector field on S35

3.6 Moduli space,Ⅱ35

3.7 The Picard-Fuchs equation36

3.8 Gauss-Manin connection,Ⅱ37

3.9 Proof of Theorem 239

3.10 Algebraic group39

3.11 Another vector field41

3.12 Weights42

3.13 A Lie algebra43

4 Topology and periods45

4.1 Period map45

4.2 τ-locus46

4.3 Positivity conditions48

4.4 Generalized period domain49

4.5 The algebraic group and τ-locus50

4.6 Monodromy covering51

4.7 A particular solution52

4.8 Action of the monodromy52

4.9 The solution in terms of periods54

4.10 Computing periods55

4.11 Algebraically independent periods57

4.12 θ-locus58

4.13 The algebraic group and the θ-locus60

4.14 Comparing τ and θ-loci61

4.15 All solutions of R0,？062

4.16 Around the elliptic point62

4.17 Halphen property63

4.18 Differential Calabi-Yau modular forms around the conifold64

4.19 Logarithmic mirror map aroundthe conifold65

4.20 Holomorphic mirror map67

5 Formal power series solutions69

5.1 Singularities of modular differential equations69

5.2 q-expansion around maximal unipotent cusp70

53 Another q-expansion71

5.4 q-expansion around conifold72

5.5 New coordinates73

5.6 Holomorphic foliations74

6 Topological string partition functions75

6.1 Yamaguchi-Yau's elements75

6.2 Proof of Theorem 876

6.3 Genus 1 topological partition function77

6.4 Holomorphic anomaly equation78

6.5 Proof of Proposition 180

6.6 The ambiguity of Falg g81

6.7 Topological partition functions Falg g,g=2,382

6.8 Topological partition functions for elliptic curves82

7 Holomorphic differential Calabi-Yau modular forms85

7.1 Fourth-order differential equations85

7.2 Hypergeometric differential equations86

7.3 Picard-Fuchs equations87

7.4 Intersection form89

7.5 Maximal unipotent monodromy91

7.6 The field of differential Calabi-Yau modular forms92

7.7 The derivation94

7.8 Yukawa coupling94

7.9 q-expansion95

8 Non-holomorphic differential Calabi-Yau modular forms97

8.1 The differential field97

8.2 Anti-holomorphic derivation99

8.3 A new basis100

8.4 Yamaguchi-Yau elements101

8.5 Hypergeometric cases103

9 BCOV holomorphic anomaly equation105

9.1 Genus 1 topological partition function105

9.2 The covariant derivative106

9.3 Holomorphic anomaly equation108

9.4 Master anomaly equation108

9.5 Algebraic anomaly equation109

9.6 Proof of Theorem 9110

9.7 A kind of Gauss-Manin connection111

9.8 Seven vector fields112

9.9 Comparison of algebraic and holomorphic anomaly equations113

9.10 Feynman rules114

9.11 Structure of the ambiguity116

10 Calabi-Yau modular forms119

10.1 Classical modular forms119

10.2 A general setting120

10.3 The algebra of Calabi-Yau modular forms121

11 Problems125

11.1 Vanishing of periods125

11.2 Hecke operators126

11.3 Maximal Hodge structure127

11.4 Monodromy129

11.5 Torelli problem129

11.6 Monstrous moonshine conjecture130

11.7 Integrality of instanton numbers131

11.8 Some product formulas132

11.9 A new mirror map133

11.10 Yet another coordinate135

11.11 Gap condition136

11.12 Algebraic gap condition137

11.13 Arithmetic modularity140

A Second-order linear differential equations143

A.1 Holomorphic and non-holomorphic quasi-modular forms143

A.2 Full quasi-modular forms145

B Metric147

B.1 Poincaré metric148

B.2 K？hler metric for moduli of mirror quintics149

C Integrality properties151

HOSSEIN MOVASATI,KHOSRO M.SHOKRI151

C.1 Introduction151

C.2 Dwork map155

C.3 Dwork lemma and theorem on hypergeometric functions156

C.4 Consequences of Dwork's theorem157

C.5 Proof of Theorem 13,Part 1158

C.6 A problem in computational commutative algebra158

C.7 The case n=2159

C.8 The symmetry160

C.9 Proof of Theorem 13,Part 2160

C.10 Computational evidence for Conjecture161

C.11 Proof of Corollary 1162

D Kontsevich's formula163

CARLOS MATHEUS163

D.1 Examples of variations of Hodge structures of weight k164

D.2 Lyapunov exponents166

D.3 Kontsevich's formula in the classical setting168

D.4 Kontsevich's formula in Calabi-Yau 3-folds setting170

D.5 Simplicity of Lyapunov exponents of mirror quintics173

References175