伪装的Gauss-Manin联络
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作者Hossein Movasati
出版社高等教育出版社
ISBN9787040468632
出版时间2017-02
版次1
装帧平装
开本16开
纸张胶版纸
页数180页
定价69元
上书时间2024-06-01
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基本信息
书名:伪装的Gauss-Manin联络
定价：69.00元
作者:Hossein Movasati
出版社：高等教育出版社
出版日期：2017-02-01
ISBN：9787040468632
字数：
页码：180
版次：1
装帧：平装
开本：16开
商品重量：
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内容提要
本书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。书中讨论了新理论和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，他们希望理解由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紧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。
目录
1 Introduction1.1 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 computations2.1 Mirror quintic Calabi-Yau threefolds2.2 Ramanujan differential equation2.3 Modular vector fields2.4 Geometric differential Calabi-Yau modular forms2.5 Eisenstein series2.6 Elliptic integrals and modular forms2.7 Periods and differential Calabi-Yau modular forms， I2.8 Integrality of Fourier coefficients2.9 Quasi- or differential modular forms2.10 Functional equations2.11 Conifold singularity2.12 The Lie algebra sl22.13 BCOV holomorphic anomaly equation， I2.14 Gromov-Witten invariants2.15 Periods and differential Calabi-Yau modular forms， II2.16 BCOV holomorphic anomaly equation， II2.17 The polynomial structure of partition functions2.18 Future developments3 Moduli of enhanced mirror quintics3.1 What is mirror quintic？3.2 Moduli space， I3.3 Gauss-Manin connection， I3.4 Intersection form and Hodge filtration3.5 A vector field on S3.6 Moduli space， II3.7 The Picard-Fuchs equation3.8 Gauss-Manin connection， II3.9 Proof of Theorem 23.10 Algebraic group3.11 Another vector field3.12 Weights3.13 A Lie algebra4 Topology and periods4.1 Period map4.2 t-locus4.3 Positivity conditions4.4 Generalized period domain4.5 The algebraic group and t-locus4.6 Monodromy covering4.7 A particular solution4.8 Action of the monodromy4.9 The solution in terms of periods4.10 Computing periods4.11 Algebraically independent periods4.12 0-locus4.13 The algebraic group and the 0-locus4.14 Comparing t and 0-loci4.15 All solutions of R0， R04.16 Around the elfiptic point4.17 Halphen property4.18 Differential Calabi-Yau modular forms around the conifold4.19 Logarithmic mirror map around the conifold4.20 Holomorphic mirror map5 Formal power series solutions5.1 Singularities of modular differential equations5.2 q-expansion around maximal unipotent cusp5.3 Another q-expansion5.4 q-expansion around conifold5.5 New coordinates5.6 Holomorphic foliations6 Topological string partition functions6.1 Yamaguchi-Yau's elements6.2 Proof of Theorem 86.3 Genus 1 topological partition function6.4 Holomorphic anomaly equation6.5 Proof of Proposition 16.6 The ambiguity of F6.7 Topological partition functions F8 ， g = 2， 36.8 Topological partition functions for elliptic curves7 Holomorphie differential Calabi-Yau modular forms7.1 Fourth-order differential equations7.2 Hypergeometric differential equations7.3 Picard-Fuchs equations7.4 Intersection form7.5 Maximal unipotent monodromy7.6 The field of differential Calabi-Yau modular forms7.7 The derivation7.8 Yukawa coupling7.9 q-expansion8 Non-holomorphie differential Calabi-Yau modular forms8.1 The differential field8.2 Anti-holomorphic derivation8.3 A new basis8.4 Yamaguchi-Yau elements8.5 Hypergeometric cases9 BCOV holomorphie anomaly equation9.1 Genus 1 topological partition function9.2 The covariant derivative9.3 Holomorphic anomaly equation9.4 Master anomaly equation9.5 Algebraic anomaly equation9.6 Proof of Theorem 99.7 A kind of Gauss-Manin connection9.8 Seven vector fields9.9 Comparison of algebraic and holomorphic anomaly equations9.10 Feynman rules9.11 Structure of the ambiguity10 Calabi-Yau modular forms10.1 Classical modular forms10.2 A general setting10.3 The algebra of Calabi-Yau modular forms11 Problems11.1 Vanishing of periods11.2 Hecke operators11.3 Maximal Hodge structure11.4 Monodromy11.5 Torelli problem11.6 Monstrous moonshine conjecture11.7 Integrality of instanton numbers11.8 Some product formulas11.9 A new mirror map11.10 Yet another coordinate11.11 Gap condition11.12 Algebraic gap condition11.13 Arithmetic modularityA Second-order linear differential equationsA.1 Holomorphic and non-holomorphic quasi-modular formsA.2 Full quasi-modular formsB MetricB.1 Poincare metricB.2 Kahler metric for moduli of mirror quinticsC Integrality propertiesHOSSEIN MOVASATI， KHOSRO M. SHOKRIC.1 IntroductionC.2 Dwork mapC.3 Dwork lemma and theorem on hypergeometric functionsC.4 Consequences of Dwork's theoremC.5 Proof of Theorem 13， Part 1C.6 A problem in computational commutative algebraC.7 The casen = 2C.8 The symmetryC.9 Proof of Theorem 13， Part 2C.10 Computational evidence for Conjecture 1C.11 Proof of Corollary 1D Kontsevich's formulaCARLOS MATHEUSD.1 Examples of variations of Hodge structures of weight kD.2 Lyapunov exponentsD.3 Kontsevich's formula in the classical settingD.4 Kontsevich's formula in Calabi-Yau 3-folds settingD.5 Simplicity of Lyapunov exponents of mirror quinticsReferences
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