Applied Differential Geometry: A Modern Introduction

Preface......Page 8
Glossary of Frequently Used Symbols......Page 12
1.1 Manifolds and Related Geometrical Structures......Page 36
1.1.1 Geometrical Atlas......Page 41
1.1.2 Topological Manifolds......Page 43
1.1.2.3 Properties of topological manifolds......Page 45
1.1.3 Differentiable Manifolds......Page 47
1.1.4.1 Tangent Bundle of a Smooth Manifold......Page 49
1.1.4.3 Fibre--, Tensor--, and Jet--Bundles......Page 50
1.1.5 Riemannian Manifolds: Configuration Spaces for Lagrangian Mechanics......Page 51
1.1.5.1 Riemann Surfaces......Page 52
1.1.5.2 Riemannian Geometry......Page 54
1.1.5.3 Application: Lagrangian Mechanics......Page 57
1.1.6 Symplectic Manifolds: Phase--Spaces for Hamiltonian Mechanics......Page 60
1.1.7 Lie Groups......Page 63
1.1.7.1 Application: Physical Examples of Lie Groups......Page 65
1.1.8.1 Three Pillars of 20th Century Physics......Page 66
1.1.8.2 String Theory in `Plain English\'......Page 68
1.2 Application: Paradigm of Differential--Geometric Modelling of Dynamical Systems......Page 81
2.1.1 Transformation of Coordinates and Elementary Tensors......Page 86
2.1.1.1 Transformation of Coordinates......Page 87
2.1.1.3 Vectors and Covectors......Page 88
2.1.1.4 Second--Order Tensors......Page 89
2.1.1.5 Higher--Order Tensors......Page 91
2.1.1.6 Tensor Symmetry......Page 92
2.1.2.1 Basis Vectors and the Metric Tensor in Rn......Page 93
2.1.2.2 Tensor Products in Rn......Page 94
2.1.3.1 Christoffel\'s Symbols......Page 95
2.1.3.3 Covariant Derivative......Page 96
2.1.3.4 Covariant Form of Differential Operators......Page 97
2.1.3.5 Absolute Derivative......Page 98
2.1.3.7 Mechanical Acceleration and Force......Page 99
2.1.4 Application: Covariant Mechanics......Page 100
2.1.4.1 Riemannian Curvature Tensor......Page 105
2.1.4.2 Exterior Differential Forms......Page 106
2.1.4.3 The Covariant Force Law......Page 111
2.1.5.1 Continuity Equation......Page 113
2.1.5.2 Forces Acting on a Fluid......Page 115
2.1.5.3 Constitutive and Dynamical Equations......Page 116
2.1.5.4 Navier--Stokes Equations......Page 117
2.2 Actions: The Core Machinery of Modern Physics......Page 118
2.3 Functors: Global Machinery of Modern Mathematics......Page 122
2.3.1.1 Notes from Set Theory......Page 123
2.3.1.4 Algebra of Maps......Page 124
2.3.1.7 Integration and Change of Variables......Page 125
2.3.1.8 Notes from General Topology......Page 126
2.3.1.9 Topological Space......Page 127
2.3.1.10 Homotopy......Page 128
2.3.1.11 Commutative Diagrams......Page 130
2.3.1.12 Groups and Related Algebraic Structures......Page 132
2.3.2 Categories......Page 137
2.3.3 Functors......Page 140
2.3.4Natural Transformations......Page 143
2.3.4.2 Dinatural Transformations......Page 144
2.3.6 Adjunction......Page 146
2.3.7 Abelian Categorical Algebra......Page 148
2.3.8 n-Categories......Page 151
2.3.8.1 Generalization of `Small\' Categories......Page 152
2.3.8.3 Homotopy Theory and Related n-Categories......Page 156
2.3.8.4 Categorification......Page 158
2.3.9 Application: n-Categorical Framework for Higher Gauge Fields......Page 159
2.3.10 Application: Natural Geometrical Structures......Page 163
2.3.11Ultimate Conceptual Machines: Weak n-Categories......Page 167
3.1 Introduction......Page 172
3.1.1 Intuition behind Einstein\'s Geometrodynamics......Page 173
3.1.2 Einstein\'s Geometrodynamics in Brief......Page 177
3.2 Intuition Behind the Manifold Concept......Page 178
3.3 Definition of a Differentiable Manifold......Page 180
3.4 Smooth Maps between Smooth Manifolds......Page 182
3.4.1 Intuition behind Topological Invariants of Manifolds......Page 183
3.5.1.2 Definition of a Tangent Bundle......Page 185
3.5.2.1 Definition of a Cotangent Bundle......Page 188
3.5.3 Application: Command/Control in Human-- Robot Interactions......Page 189
3.5.4 Application: Generalized Bidirectional Associative Memory......Page 192
3.6.1 Tensor Bundle......Page 198
3.6.1.1 Pull--Back and Push--Forward......Page 199
3.6.1.2 Dynamical Evolution and Flow......Page 200
3.6.1.4 Vector--Fields on M......Page 202
3.6.1.5 Integral Curves as Dynamical Trajectories......Page 203
3.6.1.6 Dynamical Flows on M......Page 207
3.6.1.7 Categories of ODEs......Page 208
3.6.2.1 1-Forms on M......Page 209
3.6.2.2 k-Forms on M......Page 211
3.6.2.3 Exterior Differential Systems......Page 214
3.6.3 Exterior Derivative and (Co)Homology......Page 215
3.6.3.1 Intuition behind Cohomology......Page 217
3.6.3.2 Intuition behind Homology......Page 218
3.6.3.3 De Rham Complex and Homotopy Operators......Page 220
3.6.4.4 Stokes Theorem and de Rham Cohomology......Page 221
3.6.4.6 Duality of Chains and Forms on M......Page 223
3.6.4.7 Hodge Star Operator and Harmonic Forms......Page 225
3.7.1 Lie Derivative Operating on Functions......Page 227
3.7.2 Lie Derivative of Vector Fields......Page 229
3.7.4 Lie Derivative of Differential Forms......Page 232
3.7.5 Lie Derivative of Various Tensor Fields......Page 233
3.7.6 Application: Lie–Derivative Neurodynamics......Page 235
3.8 Lie Groups and Associated Lie Algebras......Page 237
3.8.1 Definition of a Lie Group......Page 238
3.8.2 Actions of Lie Groups on Smooth Manifolds......Page 242
3.8.3.1 Galilei Group......Page 245
3.8.3.2 General Linear Group......Page 246
3.8.4.1 Lie Groups of Joint Rotations......Page 247
3.8.4.2 Euclidean Groups of Total Joint Motions......Page 251
3.8.4.3 Group Structure of Biodynamical Manifold......Page 257
3.8.5.1 Configuration Models for Planar Vehicles......Page 262
3.8.5.2 Two–Vehicles Conflict Resolution Manoeuvres......Page 263
3.8.5.3 Symplectic Reduction and Dynamical Games on SE(2)......Page 265
3.8.5.4 Nash Solutions for Multi–Vehicle Manoeuvres......Page 268
3.8.6 Classical Lie Theory......Page 270
3.8.6.1 Basic Tables of Lie Groups and their Lie Algebras......Page 271
3.8.6.2 Representations of Lie groups......Page 274
3.8.6.3 Root Systems and Dynkin Diagrams......Page 275
3.8.6.4 Simple and Semisimple Lie Groups and Algebras......Page 280
3.9.1.1 Exponentiation of Vector Fields on M......Page 282
3.9.1.2 Lie Symmetry Groups and General DEs......Page 284
3.9.2.1 Prolongations of Functions......Page 285
3.9.2.2 Prolongations of Differential Equations......Page 286
3.9.2.3 Prolongations of Group Actions......Page 287
3.9.2.4 Prolongations of Vector Fields......Page 288
3.9.2.5 General Prolongation Formula......Page 289
3.9.3 Generalized Lie Symmetries......Page 291
3.9.3.1 Noether Symmetries......Page 292
3.9.4.1 The Heat Equation......Page 296
3.9.5.1 Robot Kinematics......Page 297
3.9.5.2 Maurer–Cartan 1–Forms......Page 299
3.9.5.3 General Structure of Integrable One–Forms......Page 300
3.9.5.4 Lax Integrable Dynamical Systems......Page 302
3.9.5.5 Application: Burgers Dynamical System......Page 303
3.10.1 Local Riemannian Geometry......Page 306
3.10.1.1 Riemannian Metric on M......Page 307
3.10.1.2 Geodesics on M......Page 312
3.10.1.3 Riemannian Curvature on M......Page 313
3.10.2.1 The Second Variation Formula......Page 316
3.10.2.2 Gauss–Bonnet Formula......Page 319
3.10.2.3 Ricci Flow on M......Page 320
3.10.2.4 Structure Equations on M......Page 322
3.10.3.1 Basis of Lagrangian Dynamics......Page 324
3.10.3.2 Lagrange–Poincaré Dynamics......Page 326
3.10.4.1 What is Quantum Gravity?......Page 327
3.10.4.2 Main Approaches to Quantum Gravity......Page 328
3.10.4.3 Traditional Approaches to Quantum Gravity......Page 335
3.10.4.4 New Approaches to Quantum Gravity......Page 339
3.10.4.5 Black Hole Entropy......Page 345
3.10.5.1 Morse Theory on Smooth Manifolds......Page 346
3.10.5.2 (Co)Bordism Theory on Smooth Manifolds......Page 349
3.11.1 Definition of a Finsler Manifold......Page 351
3.11.2 Energy Functional, Variations and Extrema......Page 352
3.11.3 Application: Finsler--Lagrangian Field Theory......Page 356
3.11.4.1 Model Specification and Parameter Estimation......Page 358
3.11.4.3 Quantitative Criteria......Page 359
3.11.4.4 Selection Among Different Models......Page 362
3.11.4.5 Riemannian Geometry of Minimum Description Length......Page 365
3.11.4.6 Finsler Approach to Information Geometry......Page 368
3.12.1 Symplectic Algebra......Page 370
3.12.2 Symplectic Geometry......Page 371
3.12.3.1 Basics of Hamiltonian Mechanics......Page 373
3.12.3.2 Library of Basic Hamiltonian Systems......Page 386
3.12.3.3 Hamilton–Poisson Mechanics......Page 396
3.12.3.4 Completely Integrable Hamiltonian Systems......Page 398
3.12.3.5 Momentum Map and Symplectic Reduction......Page 407
3.12.4 Multisymplectic Geometry......Page 409
3.13 Application: Biodynamics--Robotics......Page 410
3.13.1 Muscle--Driven Hamiltonian Biodynamics......Page 411
3.13.2 Hamiltonian–Poisson Biodynamical Systems......Page 414
3.13.3 Lie–Poisson Neurodynamics Classifier......Page 418
3.13.4.1 The Covariant Force Functor......Page 419
3.13.4.2 Lie–Lagrangian Biodynamical Functor......Page 420
3.13.5.1 (Co)Chain Complexes in Biodynamics......Page 436
3.13.5.2 Morse Theory in Biodynamics......Page 440
3.13.5.3 Hodge–De Rham Theory in Biodynamics......Page 450
3.13.5.4 Lagrangian–Hamiltonian Duality in Biodynamics......Page 454
3.14 Complex and K¨ahler Manifolds and Their Applications......Page 463
3.14.1 Complex Metrics: Hermitian and K¨ahler......Page 466
3.14.2 Calabi–Yau Manifolds......Page 471
3.14.3 Special Lagrangian Submanifolds......Page 472
3.14.4 Dolbeault Cohomology and Hodge Numbers......Page 473
3.15 Conformal Killing–Riemannian Geometry......Page 476
3.15.1 Conformal Killing Vector–Fields and Forms on M......Page 477
3.15.2 Conformal Killing Tensors and Laplacian Symmetry......Page 478
3.15.3 Application: Killing Vector and Tensor Fields in Mechanics......Page 480
3.16 Application: Lax–Pair Tensors in Gravitation......Page 483
3.16.1 Lax–Pair Tensors......Page 485
3.16.2 Geometrization of the 3–Particle Open Toda Lattice......Page 487
3.16.2.1 Tensorial Lax Representation......Page 488
3.16.3.1 Case I......Page 491
3.16.3.3 Energy–Momentum Tensors......Page 492
3.17.1.1 Moyal Product and Noncommutative Algebra......Page 493
3.17.1.2 Noncommutative Space–Time Manifolds......Page 494
3.17.1.3 Symmetries and Di eomorphisms on Deformed Spaces......Page 497
3.17.1.4 Deformed Di eomorphisms......Page 500
3.17.1.5 Noncommutative Space–Time Geometry......Page 502
3.17.1.6 Star–Products and Expanded Einstein–Hilbert Action......Page 505
3.17.2 Synthetic Di erential Geometry......Page 508
3.17.2.1 Distributions......Page 509
3.17.2.2 Synthetic Calculus in Euclidean Spaces......Page 511
3.17.2.3 Spheres and Balls as Distributions......Page 513
3.17.2.4 Stokes Theorem for Unit Sphere......Page 515
3.17.2.5 Time Derivatives of Expanding Spheres......Page 516
3.17.2.6 The Wave Equation......Page 517
4.1 Intuition Behind a Fibre Bundle......Page 520
4.2 Definition of a Fibre Bundle......Page 521
4.3 Vector and Affine Bundles......Page 526
4.3.1 The Second Vector Bundle of the Manifold M......Page 530
4.3.2 The Natural Vector Bundle......Page 531
4.3.3.1 Tangent and Cotangent Bundles Revisited......Page 533
4.3.4 Affine Bundles......Page 535
4.4.1 Vector–Fields and Connections......Page 536
4.4.2 Hamiltonian Structures on the Tangent Bundle......Page 538
4.5.1 Topological K-Theory......Page 543
4.5.1.1 Bott Periodicity Theorem......Page 544
4.5.2 Algebraic K-Theory......Page 545
4.5.3 Chern Classes and Chern Character......Page 546
4.5.4 Atiyah’s View on K-Theory......Page 550
4.5.5 Atiyah–Singer Index Theorem......Page 553
4.5.6 The Infinite–Order Case......Page 555
4.5.7 Twisted K-Theory and the Verlinde Algebra......Page 558
4.5.8.1 Classification of Ramond–Ramond Fluxes......Page 561
4.5.8.2 Classification of D-Branes......Page 563
4.6 Principal Bundles......Page 564
4.7 Distributions and Foliations on Manifolds......Page 568
4.8 Application: Nonholonomic Mechanics......Page 569
4.9.1 Introduction to Geometrical Nonlinear Control......Page 572
4.9.2 Feedback Linearization......Page 574
4.9.3 Nonlinear Controllability......Page 582
4.9.4.1 Abstract Control System......Page 589
4.9.4.3 Local Controllability of Affine Control Systems......Page 590
4.9.4.4 Lagrangian Control Systems......Page 591
4.9.4.5 Lie–Adaptive Control......Page 601
4.9.5.1 Hamiltonian Control Systems......Page 602
4.9.5.2 Pontryagin’s Maximum Principle......Page 605
4.9.5.3 Affine Control Systems......Page 606
4.9.6 Brain–Like Control Functor in Biodynamics......Page 608
4.9.6.1 Functor Control Machine......Page 609
4.9.6.2 Spinal Control Level......Page 611
4.9.6.3 Cerebellar Control Level......Page 616
4.9.6.4 Cortical Control Level......Page 619
4.9.6.5 Open Liouville Neurodynamics and Biodynamical Self– Similarity......Page 622
4.9.7.1 Neurodynamical 2-Functor......Page 629
4.9.7.2 Solitary ‘Thought Nets’ and ‘Emerging Mind’......Page 632
4.9.8 Geometrodynamics of Human Crowd......Page 637
4.9.8.2 Geometrodynamics of Individual Agents......Page 638
4.9.8.3 Collective Crowd Geometrodynamics......Page 640
4.10 Multivector–Fields and Tangent–Valued Forms......Page 641
4.11.1 Quantization of Hamiltonian Mechanics......Page 649
4.11.2 Quantization of Relativistic Hamiltonian Mechanics......Page 652
4.12 Symplectic Structures on Fiber Bundles......Page 659
4.12.1.1 Characterizing Hamiltonian Bundles......Page 660
4.12.1.2 Hamiltonian Structures......Page 661
4.12.1.3 Marked Hamiltonian Structures......Page 665
4.12.1.5 Cohomological Splitting......Page 667
4.12.1.6 Homological Action of Ham(M) on M......Page 669
4.12.1.7 General Symplectic Bundles......Page 671
4.12.1.8 Existence of Hamiltonian Structures......Page 672
4.12.1.9 Classification of Hamiltonian Structures......Page 677
4.12.2.1 Stability......Page 680
4.12.2.2 Functorial Properties......Page 683
4.12.2.3 Splitting of Rational Cohomology......Page 685
4.12.2.4 Hamiltonian Bundles and Gromov–Witten Invariants......Page 689
4.12.2.5 Homotopy Reasons for Splitting......Page 694
4.12.2.6 Action of the Homology of (M) on H (M)......Page 696
4.12.2.7 Cohomology of General Symplectic Bundles......Page 699
4.13.1 Clifford Algebras and Modules......Page 701
4.13.1.1 The Exterior Algebra......Page 704
4.13.1.3 4D Case......Page 707
4.13.2.1 Basic Properties......Page 710
4.13.2.2 4D Case A......Page 712
4.13.2.3 (Anti) Self Duality......Page 716
4.13.2.4 Hermitian Structure on the Spinors......Page 721
4.13.2.5 Symplectic Structure on the Spinors......Page 724
4.13.3.1 Penrose Index Formalism......Page 726
4.13.3.2 Twistor Calculus......Page 733
4.13.4.1 Introduction to Loop Quantum Gravity......Page 736
4.13.4.2 Formalism of Loop Quantum Gravity......Page 743
4.13.4.3 Loop Algebra......Page 744
4.13.4.4 Loop Quantum Gravity......Page 746
4.13.4.5 Loop States and Spin Network States......Page 747
4.13.4.6 Diagrammatic Representation of the States......Page 750
4.13.4.7 Quantum Operators......Page 751
4.13.4.8 Loop v.s. Connection Representation......Page 752
4.14 Application: Seiberg–Witten Monopole Field Theory......Page 753
4.14.1.2 N = 2 Super–Action......Page 756
4.14.1.3 Spontaneous Symmetry–Breaking......Page 758
4.14.1.5 The SW Prepotential......Page 759
4.14.2 Clifford Actions, Dirac Operators and Spinor Bundles......Page 760
4.14.2.1 Clifford Algebras and Dirac Operators......Page 762
4.14.2.2 Spin and Spinc Structures......Page 764
4.14.2.3 Spinor Bundles......Page 765
4.14.2.4 The Gauge Group and Its Equations......Page 766
4.14.3 Original SW Low Energy Effective Field Action......Page 767
4.14.4 QED With Matter......Page 770
4.14.5 QCD With Matter......Page 772
4.14.6 Duality......Page 773
4.14.6.1 Witten’s Formalism......Page 775
4.14.7.1 Singularity at Infinity......Page 781
4.14.7.2 Singularities at Strong Coupling......Page 782
4.14.7.3 Effects of a Massless Monopole......Page 783
4.14.7.4 The Third Singularity......Page 784
4.14.7.5 Monopole Condensation and Confinement......Page 785
4.14.8 Masses and Periods......Page 787
4.14.9 Residues......Page 789
4.14.10 SW Monopole Equations and Donaldson Theory......Page 792
4.14.10.1 Topological Invariance......Page 795
4.14.10.2 Vanishing Theorems......Page 797
4.14.10.3 Computation on Kahler Manifolds......Page 800
4.14.11 SW Theory and Integrable Systems......Page 804
4.14.11.1 SU(N) Elliptic CM System......Page 806
4.14.11.2 CM Systems Defined by Lie Algebras......Page 807
4.14.11.3 Twisted CM–Systems Defined by Lie Algebras......Page 808
4.14.11.4 Scaling Limits of CM–Systems......Page 809
4.14.11.5 Lax Pairs for CM–Systems......Page 811
4.14.11.6 CM and SW Theory for SU(N)......Page 814
4.14.11.7 CM and SW Theory for General Lie Algebra......Page 817
4.14.12.1 WDVV Equations......Page 819
4.14.12.2 Perturbative SW Prepotentials......Page 822
4.14.12.3 Associativity Conditions......Page 824
4.14.12.4 SW Theories and Integrable Systems......Page 825
4.14.12.5 WDVV Equations in SW Theories......Page 828
5.1 Intuition Behind a Jet Space......Page 832
5.2 Definition of a 1–Jet Space......Page 836
5.3 Connections as Jet Fields......Page 841
5.3.1 Principal Connections......Page 850
5.4 Definition of a 2–Jet Space......Page 853
5.5 Higher–Order Jet Spaces......Page 857
5.6 Application: Jets and Non–Autonomous Dynamics......Page 859
5.6.1 Geodesics......Page 865
5.6.2 Quadratic Dynamical Equations......Page 866
5.6.3 Equation of Free–Motion......Page 867
5.6.4 Quadratic Lagrangian and Newtonian Systems......Page 868
5.6.5 Jacobi Fields......Page 870
5.6.6 Constraints......Page 871
5.6.7 Time–Dependent Lagrangian Dynamics......Page 876
5.6.8 Time–Dependent Hamiltonian Dynamics......Page 878
5.6.9 Time–Dependent Constraints......Page 883
5.6.10 Lagrangian Constraints......Page 884
5.6.11 Quadratic Degenerate Lagrangian Systems......Page 887
5.6.12 Time–Dependent Integrable Hamiltonian Systems......Page 890
5.6.13 Time–Dependent Action–Angle Coordinates......Page 893
5.6.14 Lyapunov Stability......Page 895
5.6.15 First–Order Dynamical Equations......Page 896
5.6.16.1 Lyapunov Tensor......Page 898
5.6.16.2 Lyapunov Stability......Page 899
5.7 Application: Jets and Multi–Time Rheonomic Dynamics......Page 903
5.7.1 Relativistic Rheonomic Lagrangian Spaces......Page 905
5.7.2 Canonical Nonlinear Connections......Page 906
5.7.3 Cartan’s Canonical Connections......Page 909
5.7.4 General Nonlinear Connections......Page 911
5.8 Jets and Action Principles......Page 912
5.9 Application: Jets and Lagrangian Field Theory......Page 918
5.9.1 Lagrangian Conservation Laws......Page 923
5.9.2 General Covariance Condition......Page 928
5.10 Application: Jets and Hamiltonian Field Theory......Page 932
5.10.1 Covariant Hamiltonian Field Systems......Page 934
5.10.2 Associated Lagrangian and Hamiltonian Systems......Page 937
5.10.3 Evolution Operator......Page 939
5.10.4 Quadratic Degenerate Systems......Page 945
5.11.1 Connection Strength......Page 948
5.11.2 Associated Bundles......Page 949
5.11.3 Classical Gauge Fields......Page 950
5.11.4 Gauge Transformations......Page 952
5.11.5 Lagrangian Gauge Theory......Page 954
5.11.6 Hamiltonian Gauge Theory......Page 955
5.11.7 Gauge Conservation Laws......Page 958
5.11.8 Topological Gauge Theories......Page 959
5.12.1 Stress–Energy–Momentum Tensors......Page 963
5.12.2 Gauge Systems of Gravity and Fermion Fields......Page 990
5.12.3 Hawking–Penrose Quantum Gravity and Black Holes......Page 998
6. Geometrical Path Integrals and Their Applications......Page 1018
6.1.3 Continuous Random Variable......Page 1019
6.1.4 General Markov Stochastic Dynamics......Page 1020
6.1.5 Quantum Probability Concept......Page 1024
6.1.6 Quantum Coherent States......Page 1026
6.1.7 Dirac’s < bra | ket > Transition Amplitude......Page 1027
6.1.8 Feynman’s Sum–over–Histories......Page 1029
6.1.9 The Basic Form of a Path Integral......Page 1031
6.1.10 Application: Adaptive Path Integral......Page 1032
6.2.1 Extract from Feynman’s Nobel Lecture......Page 1033
6.2.2 Lagrangian Path Integral......Page 1037
6.2.3 Hamiltonian Path Integral......Page 1038
6.2.4 Feynman–Kac Formula......Page 1039
6.3.1 Canonical versus Path–Integral Quantization......Page 1041
6.3.2.1 Particles......Page 1046
6.3.2.2 Sources......Page 1047
6.3.2.4 Gauges......Page 1048
6.3.3 Riemannian–Symplectic Geometries......Page 1049
6.3.4 Euclidean Stochastic Path Integral......Page 1051
6.3.5 Application: Stochastic Optimal Control......Page 1055
6.3.5.1 Path–Integral Formalism......Page 1056
6.3.5.2 Monte Carlo Sampling......Page 1058
6.3.6.1 Theory and Simulations of Option Pricing......Page 1060
6.3.6.2 Option Pricing via Path Integrals......Page 1064
6.3.6.3 Continuum Limit and American Options......Page 1070
6.3.7 Application: Nonlinear Dynamics of Complex Nets......Page 1071
6.3.7.1 Continuum Limit of the Kuramoto Net......Page 1072
6.3.7.2 Path–Integral Approach to Complex Nets......Page 1073
6.3.8 Application: Dissipative Quantum Brain Model......Page 1074
6.3.9 Application: Cerebellum as a Neural Path–Integral......Page 1078
6.3.9.1 Spinal Autogenetic Reflex Control......Page 1080
6.3.9.2 Cerebellum – the Comparator......Page 1082
6.3.9.3 Hamiltonian Action and Neural Path Integral......Page 1084
6.3.10 Path Integrals via Jets: Perturbative Quantum Fields......Page 1085
6.4 Sum over Geometries and Topologies......Page 1090
6.4.1 Simplicial Quantum Geometry......Page 1092
6.4.2 Discrete Gravitational Path Integrals......Page 1094
6.4.3 Regge Calculus......Page 1096
6.4.4 Lorentzian Path Integral......Page 1099
6.4.5.1 Phase Transitions in Hamiltonian Systems......Page 1104
6.4.5.2 Geometry of the Largest Lyapunov Exponent......Page 1107
6.4.5.3 Euler Characteristics of Hamiltonian Systems......Page 1110
6.4.6 Application: Force–Field Psychodynamics......Page 1114
6.4.6.1 Motivational Cognition in the Life Space Foam......Page 1115
6.5.1 Topological Quantum Field Theory......Page 1132
6.5.2.1 SW Invariants and Monopole Equations......Page 1138
6.5.2.2 Topological Lagrangian......Page 1140
6.5.2.3 Quantum Field Theory......Page 1142
6.5.2.4 Dimensional Reduction and 3D Field Theory......Page 1147
6.5.2.5 Geometrical Interpretation......Page 1150
6.5.3 TQFTs Associated with SW–Monopoles......Page 1153
6.5.3.1 Dimensional Reduction......Page 1157
6.5.3.2 TQFTs of 3D Monopoles......Page 1159
6.5.3.3 Non–Abelian Case......Page 1170
6.5.4 Stringy Actions and Amplitudes......Page 1173
6.5.4.1 Strings......Page 1174
6.5.4.2 Interactions......Page 1175
6.5.4.3 Loop Expansion – Topology of Closed Surfaces......Page 1176
6.5.5 Transition Amplitudes for Strings......Page 1178
6.5.7 More General Actions......Page 1181
6.5.8 Transition Amplitude for a Single Point Particle......Page 1182
6.5.9 Witten’s Open String Field Theory......Page 1183
6.5.9.1 Operator Formulation of String Field Theory......Page 1184
6.5.9.2 Open Strings in Constant B-Field Background......Page 1186
6.5.9.3 Construction of Overlap Vertices......Page 1189
6.5.9.4 Transformation of String Fields......Page 1199
6.6 Application: Dynamics of Strings and Branes......Page 1203
6.6.1 A Relativistic Particle......Page 1204
6.6.2 A String......Page 1206
6.6.3 A Brane......Page 1208
6.6.4 String Dynamics......Page 1210
6.6.5 Brane Dynamics......Page 1212
6.7.1 Quantum Geometry Framework......Page 1215
6.7.2 Green–Schwarz Bosonic Strings and Branes......Page 1216
6.7.3 Calabi–Yau Manifolds, Orbifolds and Mirror Symmetry......Page 1221
6.7.4 More on Topological Field Theories......Page 1224
6.7.5 Topological Strings......Page 1239
6.7.6 Geometrical Transitions......Page 1256
6.7.7 Topological Strings and Black Hole Attractors......Page 1260
6.8.1 String Theory and Noncommutative Geometry......Page 1267
6.8.1.1 Noncommutative Gauge Theory......Page 1268
6.8.1.2 Open Strings in the Presence of Constant B-Field......Page 1270
6.8.2 K–Theory Classification of Strings......Page 1276
Bibliography......Page 1288
Index......Page 1330