Theory of Hypergeometric Functions

by Aomoto, Kazuhiko; Kita, Michitaka

• ISBN13: 9784431539124
• ISBN10: 4431539123
• Format: Hardcover
• Copyright: 2011-05-16
• Publisher: Springer Nature

Summary

This book presents a geometric theory of complex analytic integrals representing hypergeometric functions of several variables. Starting from an integrand which is a product of powers of polynomials, integrals are explained, in an open affine space, as a pair of twisted de Rham cohomology and its dual over the coefficients of local system. It is shown that hypergeometric integrals generally satisfy holonomic system of linear differential equations with respect to the coefficients of polynomials and also satisfy holonomic system of linear difference equations with respect to the exponents. These are basically deduced from Grothendieck-Deligne's rational de Rham cohomology on the one hand, and by multidimensional extension of Birkhoff's classical theory on analytic difference equations on the other.

Table of Contents

Introduction: the Euler-Gauss Hypergeometric Function	p. 1
¿-Function	p. 2
Infinite-Product Representation Due to Euler	p. 2
¿-Function as Meromorphic Function	p. 3
Connection Formula	p. 4
Power Series and Higher Logarithmic Expansion	p. 4
Hypergeometric Series	p. 4
Gauss' Differential Equation	p. 5
First-Order Fuchsian Equation	p. 6
Logarithmic Connection	p. 6
Higher Logarithmic Expansion	p. 7
D-Module	p. 10
Integral Representation Due to Euler and Riemann	p. 11
Kummer's Method	p. 11
Gauss' Contiguous Relations and Continued Fraction Expansion	p. 12
Gauss' Contiguous Relation	p. 12
Continued Fraction Expansion	p. 13
Convergence	p. 15
The Mellin-Barnes Integral	p. 16
Summation over a Lattice	p. 16
Barnes' Integral Representation	p. 16
Mellin's Differential Equation	p. 18
Plan from Chapter 2	p. 19
Representation of Complex Integrals and Twisted de Rham Cohomologies	p. 21
Formulation of the Problem and Intuitive Explanation of the Twisted de Rham Theory	p. 21
Concept of Twist	p. 21
Intuitive Explanation	p. 22
One-Dimensional Case	p. 23
Two-Dimensional Case	p. 24
Higher-Dimensional Generalization	p. 25
Twisted Homology Group	p. 26
Locally Finite Twisted Homology Group	p. 28
Review of the de Rham Theory and the Twisted de Rham Theory	p. 29
Preliminary from Homological Algebra	p. 29
Current	p. 31
Current with Compact Support	p. 33
Sheaf Cohomology	p. 33
The Case of Compact Support	p. 35
De Rham's Theorem	p. 35
Duality	p. 36
Integration over a Simplex	p. 36
Twisted Chain	p. 38
Twisted Version of § 2.2.4	p. 39
Poincaré Duality	p. 40
Reformulation	p. 41
Comparison of Cohomologies	p. 42
Computation of the Euler Characteristic	p. 44
Construction of Twisted Cycles (1): One-Dimensional Case	p. 48
Twisted Cycle Around One Point	p. 48
Construction of Twisted Cycles	p. 50
Intersection Number (i)	p. 52
Comparison Theorem	p. 54
Algebraic de Rham Complex	p. 54
Cech Cohomology	p. 55
Hypercohomology	p. 56
Spectral Sequence	p. 57
Algebraic de Rham Cohomology	p. 58
Analytic de Rham Cohomology	p. 58
Comparison Theorem	p. 59
Reformulation	p. 60
de Rham-Saito Lemma and Representation of Logarithmic Differential Forms	p. 60
Logarithmic Differential Forms	p. 60
de Rham-Saito Lemma	p. 63
Representation of Logarithmic Differential Forms (i)	p. 69
Vanishing of Twisted Cohomology for Homogeneous Case	p. 74
Basic Operators	p. 74
Homotopy Formula	p. 76
Eigenspace Decomposition	p. 77
Vanishing Theorem (i)	p. 78
Filtration of Logarithmic Complex	p. 79
Filtration	p. 79
Comparison with Homogeneous Case	p. 80
Isomorphism	p. 81
Vanishing Theorem of the Twisted Rational de Rham Cohomology	p. 82
Vanishing of Logarithmic de Rham Cohomology	p. 83
Vanishing of Algebraic de Rham Cohomology	p. 83
Two-Dimensional Case	p. 85
Example	p. 86
Arrangement of Hyperplanes in General Position	p. 88
Vanishing Theorem (ii)	p. 88
Representation of Logarithmic Differential Forms (ii)	p. 89
Reduction of Poles	p. 93
Comparison Theorem	p. 96
Filtration	p. 96
Basis of Cohomology	p. 98
Arrangement of Hyperplanes and Hypergeometric Functions over Grassmannians	p. 103
Classical Hypergeometric Series and Their Generalizations, in Particular, Hypergeometric Series of Type (n + 1, m + 1)	p. 103
Definition	p. 103
Simple Examples	p. 104
Hypergeometric Series of Type (n + 1, m + 1)	p. 105
Appell-Lauricella Hypergeometric Functions (i)	p. 106
Appell-Lauricella Hypergeometric Functions (ii)	p. 106
Restriction to a Sublattice	p. 106
Examples	p. 107
Appell-Lauricella Hypergeometric Functions (iii)	p. 107
Horn's Hypergeometric Functions	p. 108
Construction of Twisted Cycles (2): For an Arrangement of Hyperplanes in General Positiion	p. 108
Twisted Homology Group	p. 108
Bounded Chambers	p. 109
Basis of Locally Finite Homology	p. 109
Construction of Twisted Cycles	p. 112
Regularization of Integrals	p. 115
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)	p. 117
Kummer's Method	p. 117
One-Dimensional Case	p. 117
Higher-Dimensional Case	p. 118
Elementary Integral Representations	p. 119
Hypergeometric Function of Type (3,6)	p. 121
Hypergeometric Functions of Type (n + 1, m + 1)	p. 123
Horn's Cases	p. 124
System of Hypergeometric Differential Equations E(n + 1, m + 1; ¿)	p. 126
Hypergeometric Integral of Type (n + 1, m + 1; ¿)	p. 126
Differential Equation E(n + 1, m + 1; ¿)	p. 128
Equivalent System	p. 133
Integral Solutions of E(n + 1, m + 1; ¿) and Wronskian	p. 135
Hypergeometric Integrals as a Basis	p. 135
Gauss' Equation E'(2, 4; ¿')	p. 137
Appell-Lauricella Hypergeometric Differential Equation E'(2, m + 1; ¿')	p. 138
Equation E'(3.6; ¿')	p. 139
Equation E'(4, 8; ¿')	p. 140
General Cases	p. 142
Wronskian	p. 144
Varchenko's Formula	p. 145
Intersection Number (ii)	p. 147
Twisted Riemann's Period Relations and Quadratic Relations of Hypergeometric Functions	p. 150
Determination of the Rank of E(n + 1, m + 1; ¿)	p. 153
Equation E'(n + 1, m + 1; ¿')	p. 153
Equation E'(2,4; ¿')	p. 154
Equation E'(2, m + 1; ¿')	p. 155
Equation E'(3, 6; ¿')	p. 157
Equation E'(n + 1, m + 1; ¿')	p. 160
Duality of E(n + 1, m + 1; ¿)	p. 165
Duality of Equations	p. 165
Duality of Grassmannians	p. 167
Duality of Hypergeometric Functions	p. 169
Duality of Integral Representations	p. 169
Example	p. 170
Logarithmic Gauss-Manin Connection Associated to an Arrangement of Hyperplanes in General Position	p. 171
Review of Notation	p. 171
Variational Formula	p. 173
Partial Fraction Expansion	p. 174
Reformulation	p. 174
Example	p. 177
Logarithmic Gauss-Manin Connection	p. 178
Holonomic Difference Equations and Asymptotic Expansion	p. 183
Existence Theorem Due to G.D. Birkhoff and Infinite- Product Representation of Matrices	p. 184
Normal Form of Matrix-Valued Function	p. 184
Asymptotic Form of Solutions	p. 187
Existence Theorem (i)	p. 188
Infinite-Product Representation of Matrices	p. 189
Gauss' Decomposition	p. 190
Regularization of the Product	p. 192
Convergence of the First Column	p. 194
Asymptotic Estimate of Infinite Product	p. 194
Convergence of Lower Triangular Matrices	p. 196
Asymptotic Estimate of Lower Triangular Matrices	p. 197
Difference Equation Satisfied by Upper Triangular Matrices	p. 199
Resolution of Difference Equations	p. 200
Completion of the Proof	p. 202
Holonomic Difference Equations in Several Variables and Asymptotic Expansion	p. 204
Holonomic Difference Equations of First Order	p. 204
Formal Asymptotic Expansion	p. 205
Normal Form of Asymptotic Expansion	p. 207
Existence Theorem (ii)	p. 209
Connection Problem	p. 210
Example	p. 211
Remark on 1-Cocyles	p. 213
Gauss' Contiguous Relations	p. 213
Convergence	p. 215
Continued Fraction Expansion	p. 216
Saddle Point Method and Asymptotic Expansion	p. 216
Contracting (Expanding) Twisted Cycles and Asymptotic Expansion	p. 221
Twisted Cohomology	p. 221
Saddle Point Method for Multi-Dimensional Case	p. 223
Complete Kähler Metric	p. 224
Gradient Vector Field	p. 226
Critical Points	p. 228
Vanishing Theorem (iii)	p. 228
Application of the Morse Theory	p. 231
n-Dimensional Lagrangian Cycles	p. 232
n-Dimensional Twisted Cycles	p. 239
Geometric Meaning of Asymptotic Expansion	p. 240
Difference Equations Satisfied by the Hypergeometric Functions of Type (n + l, m +1; ¿)	p. 243
Bounded Chambers	p. 243
Derivation of Difference Equations	p. 245
Asymptotic Expansion with a Fixed Direction	p. 250
Example	p. 251
Non-Degeneracy of Period Matrix	p. 251
Connection Problem of System of Difference Equations	p. 254
Formulation	p. 254
The Case of Appell-Lauricella Hypergeometric Functions	p. 256
Mellin's Generalized Hypergeometric Functions	p. 261
Definition	p. 261
Definition	p. 261
Kummer's Method	p. 262
Toric Multinomial Theorem	p. 264
Elementary Integral Representations	p. 266
Differential Equations of Mellin Type	p. 268
b-Functions	p. 269
Action of Algebraic Torus	p. 271
Vector Fields of Torus Action	p. 272
Lattice Defined by the Characters	p. 272
G-G-Z Equation	p. 274
Convergence	p. 276
The Selberg Integral and Hypergeometric Function of BC Type	p. 279
Selberg's Integral	p. 279
Generalization to Correlation Functions	p. 280
Monodromy Representation of Hypergeometric Functions of Type (2, m + 1; ¿)	p. 283
Isotopic Deformation and Monodromy	p. 283
KZ Equation (Toshitake Kohno)	p. 287
Knizhnik-Zamolodchikov Equation	p. 288
Review of Conformal Field Theory	p. 289
Connection Matrices of KZ Equation	p. 294
Iwahori-Hecke Algebra and Quasi-Hopf Algebras	p. 296
Kontsevich Integral and Its Application	p. 299
Integral Representation of Solutions of the KZ Equation	p. 303
References	p. 307
Index	p. 315
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