伪装的Gauss-Mainin联络简介，目录书摘
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伪装的Gauss-Mainin联络
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内容简介:
H.莫瓦萨蒂所著的《伪装的Gauss-Manin联络（英文版）（精）》试图对于三阶上同调等于1的带Hodge数的Calabi-Yau三维体族构建一个模形式理论。书中讨论了新理论与定义在上半平面的模形式经典理论之间的不同和相似之处。新理论的主要例子是拓扑弦分拆函数，它们对镜像Calabi-Yau三维体的Gromov-Witten不变量进行了编码。
本书有两个主要的目标读者群：一个是那些经典模和自守形式领域的研究者，他们希望理解Calabi-Yau三维体得到物理学家所谓的q-展开，另一个是想要弄清镜面对称是如何对于Calabi-Yau三维体进行计数的致力于枚举几何学的数学家。本书也可推荐给研究自守形式及其在代数几何中的应用的数学家，特别是注意到以下问题的学者：在他们的研究中涉及的代数簇的类是有限的，例如，它不包括紧非刚性Calabi-Yau三维体。流畅地阅读本书需要复分析、微分方程、代数拓扑和代数几何的先导知识。
作者简介: H.莫瓦萨蒂，是伊朗裔巴西数学家，2006年起在位于里约热内卢的巴西纯粹与应用数学研究所（IMPA-InstitutodeMatematicaPuraeAplicada）工作。他的数学生涯起步于对复流形上的全纯叶状结构和微分方程的研究，并逐步转移到对于Hodge理论、模形式以及它们在数学物理特别是镜面对称中的应用的研究。
目录:
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