1 Introduction
1.1 What is Gauss-Manin connection in disguise?
1.2 Why mirror quintic Calabi-Yau threefold?
1.3 How to read the text?
1.4 Why differential Calabi-Yau modular form?
2 Summary of results and computations
2.1 Mirror quintic Calabi-Yau threefolds
2.2 Ramanujan differential equation
2.3 Modular vector fields
2.4 Geometric differential Calabi-Yau modular forms
2.5 Eisenstein series
2.6 Elliptic integrals and modular forms
2.7 Periods and differential Calabi-Yau modular forms, I
2.8 Integrality of Fourier coefficients
2.9 Quasi- or differential modular forms
2.10 Functional equations
2.11 Conifold singularity
2.12 The Lie algebra sl2
2.13 BCOV holomorphic anomaly equation, I
2.14 Gromov-Witten invariants
2.15 Periods and differential Calabi-Yau modular forms, II
2.16 BCOV holomorphic anomaly equation, II
2.17 The polynomial structure of partition functions
2.18 Future developments
3 Moduli of enhanced mirror quintics
3.1 What is mirror quintic?
3.2 Moduli space, I
3.3 Gauss-Manin connection, I
3.4 Intersection form and Hodge filtration
3.5 A vector field on S
3.6 Moduli space, II
3.7 The Picard-Fuchs equation
3.8 Gauss-Manin connection, II
3.9 Proof of Theorem 2
3.10 Algebraic group
3.11 Another vector field
3.12 Weights
3.13 A Lie algebra
4 Topology and periods
4.1 Period map
4.2 t-locus
4.3 Positivity conditions
4.4 Generalized period domain
4.5 The algebraic group and t-locus
4.6 Monodromy covering
4.7 A particular solution
4.8 Action of the monodromy
4.9 The solution in terms of periods
4.10 Computing periods
4.11 Algebraically independent periods
4.12 0-locus
4.13 The algebraic group and the 0-locus
4.14 Comparing t and 0-loci
4.15 All solutions of R0, R0
4.16 Around the elfiptic point
4.17 Halphen property
4.18 Differential Calabi-Yau modular forms around the conifold
4.19 Logarithmic mirror map around the conifold
4.20 Holomorphic mirror map
5 Formal power series solutions
5.1 Singularities of modular differential equations
5.2 q-expansion around maximal unipotent cusp
5.3 Another q-expansion
5.4 q-expansion around conifold
5.5 New coordinates
5.6 Holomorphic foliations
6 Topological string partition functions
6.1 Yamaguchi-Yau's elements
6.2 Proof of Theorem 8
6.3 Genus 1 topological partition function
6.4 Holomorphic anomaly equation
6.5 Proof of Proposition 1
6.6 The ambiguity of F
6.7 Topological partition functions F8 , g = 2, 3
6.8 Topological partition functions for elliptic curves
7 Holomorphie differential Calabi-Yau modular forms
7.1 Fourth-order differential equations
7.2 Hypergeometric differential equations
7.3 Picard-Fuchs equations
7.4 Intersection form
7.5 Maximal unipotent monodromy
7.6 The field of differential Calabi-Yau modular forms
7.7 The derivation
7.8 Yukawa coupling
7.9 q-expansion
8 Non-holomorphie differential Calabi-Yau modular forms
8.1 The differential field
8.2 Anti-holomorphic derivation
8.3 A new basis
8.4 Yamaguchi-Yau elements
8.5 Hypergeometric cases
9 BCOV holomorphie anomaly equation
9.1 Genus 1 topological partition function
9.2 The covariant derivative
9.3 Holomorphic anomaly equation
9.4 Master anomaly equation
9.5 Algebraic anomaly equation
9.6 Proof of Theorem 9
9.7 A kind of Gauss-Manin connection
9.8 Seven vector fields
9.9 Comparison of algebraic and holomorphic anomaly equations
9.10 Feynman rules
9.11 Structure of the ambiguity
10 Calabi-Yau modular forms
10.1 Classical modular forms
10.2 A general setting
10.3 The algebra of Calabi-Yau modular forms
11 Problems
11.1 Vanishing of periods
11.2 Hecke operators
11.3 Maximal Hodge structure
11.4 Monodromy
11.5 Torelli problem
11.6 Monstrous moonshine conjecture
11.7 Integrality of instanton numbers
11.8 Some product formulas
11.9 A new mirror map
11.10 Yet another coordinate
11.11 Gap condition
11.12 Algebraic gap condition
11.13 Arithmetic modularity
A Second-order linear differential equations
A.1 Holomorphic and non-holomorphic quasi-modular forms
A.2 Full quasi-modular forms
B Metric
B.1 Poincare metric
B.2 Kahler metric for moduli of mirror quintics
C Integrality properties
HOSSEIN MOVASATI, KHOSRO M. SHOKRI
C.1 Introduction
C.2 Dwork map
C.3 Dwork lemma and theorem on hypergeometric functions
C.4 Consequences of Dwork's theorem
C.5 Proof of Theorem 13, Part 1
C.6 A problem in computational commutative algebra
C.7 The casen = 2
C.8 The symmetry
C.9 Proof of Theorem 13, Part 2
C.10 Computational evidence for Conjecture 1
C.11 Proof of Corollary 1
D Kontsevich's formula
CARLOS MATHEUS
D.1 Examples of variations of Hodge structures of weight k
D.2 Lyapunov exponents
D.3 Kontsevich's formula in the classical setting
D.4 Kontsevich's formula in Calabi-Yau 3-folds setting
D.5 Simplicity of Lyapunov exponents of mirror quintics
References
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