Introduction: the Euler-Gauss Hypergeometric Function / 1: |
?-Function / 1.1: |
Infinite-Product Representation Due to Euler / 1.1.1: |
?-Function as Meromorphic Function / 1.1.2: |
Connection Formula / 1.1.3: |
Power Series and Higher Logarithmic Expansion / 1.2: |
Hypergeometric Series / 1.2.1: |
Gauss' Differential Equation / 1.2.2: |
First-Order Fuchsian Equation / 1.2.3: |
Logarithmic Connection / 1.2.4: |
Higher Logarithmic Expansion / 1.2.5: |
D-Module / 1.2.6: |
Integral Representation Due to Euler and Riemann / 1.3: |
Kummer's Method / 1.3.1: |
Gauss' Contiguous Relations and Continued Fraction Expansion / 1.4: |
Gauss' Contiguous Relation / 1.4.1: |
Continued Fraction Expansion / 1.4.2: |
Convergence / 1.4.3: |
The Mellin-Barnes Integral / 1.5: |
Summation over a Lattice / 1.5.1: |
Barnes' Integral Representation / 1.5.2: |
Mellin's Differential Equation / 1.5.3: |
Plan from Chapter 2 / 1.6: |
Representation of Complex Integrals and Twisted de Rham Cohomologies / 2: |
Formulation of the Problem and Intuitive Explanation of the Twisted de Rham Theory / 2.1: |
Concept of Twist / 2.1.1: |
Intuitive Explanation / 2.1.2: |
One-Dimensional Case / 2.1.3: |
Two-Dimensional Case / 2.1.4: |
Higher-Dimensional Generalization / 2.1.5: |
Twisted Homology Group / 2.1.6: |
Locally Finite Twisted Homology Group / 2.1.7: |
Review of the de Rham Theory and the Twisted de Rham Theory / 2.2: |
Preliminary from Homological Algebra / 2.2.1: |
Current / 2.2.2: |
Current with Compact Support / 2.2.3: |
Sheaf Cohomology / 2.2.4: |
The Case of Compact Support / 2.2.5: |
De Rham's Theorem / 2.2.6: |
Duality / 2.2.7: |
Integration over a Simplex / 2.2.8: |
Twisted Chain / 2.2.9: |
Twisted Version of § 2.2.4 / 2.2.10: |
Poincaré Duality / 2.2.11: |
Reformulation / 2.2.12: |
Comparison of Cohomologies / 2.2.13: |
Computation of the Euler Characteristic / 2.2.14: |
Construction of Twisted Cycles (1): One-Dimensional Case / 2.3: |
Twisted Cycle Around One Point / 2.3.1: |
Construction of Twisted Cycles / 2.3.2: |
Intersection Number (i) / 2.3.3: |
Comparison Theorem / 2.4: |
Algebraic de Rham Complex / 2.4.1: |
Cech Cohomology / 2.4.2: |
Hypercohomology / 2.4.3: |
Spectral Sequence / 2.4.4: |
Algebraic de Rham Cohomology / 2.4.5: |
Analytic de Rham Cohomology / 2.4.6: |
de Rham-Saito Lemma and Representation of Logarithmic Differential Forms / 2.4.7: |
Logarithmic Differential Forms / 2.5.1: |
de Rham-Saito Lemma / 2.5.2: |
Representation of Logarithmic Differential Forms (i) / 2.5.3: |
Vanishing of Twisted Cohomology for Homogeneous Case / 2.6: |
Basic Operators / 2.6.1: |
Homotopy Formula / 2.6.2: |
Eigenspace Decomposition / 2.6.3: |
Vanishing Theorem (i) / 2.6.4: |
Filtration of Logarithmic Complex / 2.7: |
Filtration / 2.7.1: |
Comparison with Homogeneous Case / 2.7.2: |
Isomorphism / 2.7.3: |
Vanishing Theorem of the Twisted Rational de Rham Cohomology / 2.8: |
Vanishing of Logarithmic de Rham Cohomology / 2.8.1: |
Vanishing of Algebraic de Rham Cohomology / 2.8.2: |
Example / 2.8.3: |
Arrangement of Hyperplanes in General Position / 2.9: |
Vanishing Theorem (ii) / 2.9.1: |
Representation of Logarithmic Differential Forms (ii) / 2.9.2: |
Reduction of Poles / 2.9.3: |
Basis of Cohomology / 2.9.4: |
Arrangement of Hyperplanes and Hypergeometric Functions over Grassmannians / 3: |
Classical Hypergeometric Series and Their Generalizations, in Particular, Hypergeometric Series of Type (n + 1, m + 1) / 3.1: |
Definition / 3.1.1: |
Simple Examples / 3.1.2: |
Hypergeometric Series of Type (n + 1, m + 1) / 3.1.3: |
Appell-Lauricella Hypergeometric Functions (i) / 3.1.4: |
Appell-Lauricella Hypergeometric Functions (ii) / 3.1.5: |
Restriction to a Sublattice / 3.1.6: |
Examples / 3.1.7: |
Appell-Lauricella Hypergeometric Functions (iii) / 3.1.8: |
Horn's Hypergeometric Functions / 3.1.9: |
Construction of Twisted Cycles (2): For an Arrangement of Hyperplanes in General Positiion / 3.2: |
Bounded Chambers / 3.2.1: |
Basis of Locally Finite Homology / 3.2.3: |
Regularization of Integrals / 3.2.4: |
Kummer's Method for Integral Representations and Its Modernization via the Twisted de Rham Theory: Integral Representations of Hypergeometric Series of Type (n + 1, m +1) / 3.3: |
Higher-Dimensional Case / 3.3.1: |
Elementary Integral Representations / 3.3.4: |
Hypergeometric Function of Type (3,6) / 3.3.5: |
Hypergeometric Functions of Type (n + 1, m + 1) / 3.3.6: |
Horn's Cases / 3.3.7: |
System of Hypergeometric Differential Equations E(n + 1, m + 1; ?) / 3.4: |
Hypergeometric Integral of Type (n + 1, m + 1; ?) / 3.4.1: |
Differential Equation E(n + 1, m + 1; ?) / 3.4.2: |
Equivalent System / 3.4.3: |
Integral Solutions of E(n + 1, m + 1; ?) and Wronskian / 3.5: |
Hypergeometric Integrals as a Basis / 3.5.1: |
Gauss' Equation E'(2, 4; ?') / 3.5.2: |
Appell-Lauricella Hypergeometric Differential Equation E'(2, m + 1; ?') / 3.5.3: |
Equation E'(3.6; ?') / 3.5.4: |
Equation E'(4, 8; ?') / 3.5.5: |
General Cases / 3.5.6: |
Wronskian / 3.5.7: |
Varchenko's Formula / 3.5.8: |
Intersection Number (ii) / 3.5.9: |
Twisted Riemann's Period Relations and Quadratic Relations of Hypergeometric Functions / 3.5.10: |
Determination of the Rank of E(n + 1, m + 1; ?) / 3.6: |
Equation E'(n + 1, m + 1; ?') / 3.6.1: |
Equation E'(2,4; ?') / 3.6.2: |
Equation E'(2, m + 1; ?') / 3.6.3: |
Equation E'(3, 6; ?') / 3.6.4: |
Duality of E(n + 1, m + 1; ?) / 3.6.5: |
Duality of Equations / 3.7.1: |
Duality of Grassmannians / 3.7.2: |
Duality of Hypergeometric Functions / 3.7.3: |
Duality of Integral Representations / 3.7.4: |
Logarithmic Gauss-Manin Connection Associated to an Arrangement of Hyperplanes in General Position / 3.7.5: |
Review of Notation / 3.8.1: |
Variational Formula / 3.8.2: |
Partial Fraction Expansion / 3.8.3: |
Logarithmic Gauss-Manin Connection / 3.8.4: |
Holonomic Difference Equations and Asymptotic Expansion / 4: |
Existence Theorem Due to G.D. Birkhoff and Infinite- Product Representation of Matrices / 4.1: |
Normal Form of Matrix-Valued Function / 4.1.1: |
Asymptotic Form of Solutions / 4.1.2: |
Existence Theorem (i) / 4.1.3: |
Infinite-Product Representation of Matrices / 4.1.4: |
Gauss' Decomposition / 4.1.5: |
Regularization of the Product / 4.1.6: |
Convergence of the First Column / 4.1.7: |
Asymptotic Estimate of Infinite Product / 4.1.8: |
Convergence of Lower Triangular Matrices / 4.1.9: |
Asymptotic Estimate of Lower Triangular Matrices / 4.1.10: |
Difference Equation Satisfied by Upper Triangular Matrices / 4.1.11: |
Resolution of Difference Equations / 4.1.12: |
Completion of the Proof / 4.1.13: |
Holonomic Difference Equations in Several Variables and Asymptotic Expansion / 4.2: |
Holonomic Difference Equations of First Order / 4.2.1: |
Formal Asymptotic Expansion / 4.2.2: |
Normal Form of Asymptotic Expansion / 4.2.3: |
Existence Theorem (ii) / 4.2.4: |
Connection Problem / 4.2.5: |
Remark on 1-Cocyles / 4.2.6: |
Gauss' Contiguous Relations / 4.2.8: |
Saddle Point Method and Asymptotic Expansion / 4.2.9: |
Contracting (Expanding) Twisted Cycles and Asymptotic Expansion / 4.3: |
Twisted Cohomology / 4.3.1: |
Saddle Point Method for Multi-Dimensional Case / 4.3.2: |
Complete Kähler Metric / 4.3.3: |
Gradient Vector Field / 4.3.4: |
Critical Points / 4.3.5: |
Vanishing Theorem (iii) / 4.3.6: |
Application of the Morse Theory / 4.3.7: |
n-Dimensional Lagrangian Cycles / 4.3.8: |
n-Dimensional Twisted Cycles / 4.3.9: |
Geometric Meaning of Asymptotic Expansion / 4.3.10: |
Difference Equations Satisfied by the Hypergeometric Functions of Type (n + l, m +1; ?) / 4.4: |
Derivation of Difference Equations / 4.4.1: |
Asymptotic Expansion with a Fixed Direction / 4.4.3: |
Non-Degeneracy of Period Matrix / 4.4.4: |
Connection Problem of System of Difference Equations / 4.5: |
Formulation / 4.5.1: |
The Case of Appell-Lauricella Hypergeometric Functions / 4.5.2: |
Mellin's Generalized Hypergeometric Functions / A: |
Toric Multinomial Theorem / A.1: |
Differential Equations of Mellin Type / A.4: |
b-Functions / A.6: |
Action of Algebraic Torus / A.7: |
Vector Fields of Torus Action / A.8: |
Lattice Defined by the Characters / A.9: |
G-G-Z Equation / A.10: |
The Selberg Integral and Hypergeometric Function of BC Type / A.11: |
Selberg's Integral / B.1: |
Generalization to Correlation Functions / B.2: |
Monodromy Representation of Hypergeometric Functions of Type (2, m + 1; ?) / C: |
Isotopic Deformation and Monodromy / C.1: |
KZ Equation (Toshitake Kohno) / D: |
Knizhnik-Zamolodchikov Equation / D.1: |
Review of Conformal Field Theory / D.2: |
Connection Matrices of KZ Equation / D.3: |
Iwahori-Hecke Algebra and Quasi-Hopf Algebras / D.4: |
Kontsevich Integral and Its Application / D.5: |
Integral Representation of Solutions of the KZ Equation / D.6: |
References |
Index |