Noncommutative Geometry Quantum Fields and Motives Alain Connes Matilde Marcolli

808 Pages·2007·6.46 MB·English

American Mathematical Society
Colloquium Publications
Volume 55
Noncommutative Geometry,
Quantum Fields and Motives
Alain Connes
Matilde Marcolli
Editorial Board
Paul J. Sally, Jr., Chair
Yuri Manin
Peter Sarnak
2000 Mathematics Subject Classification. Primary 58B34, 11G35,11M06,11M26, 11G09,
81T15, 14G35, 14F42, 34M50, 81V25.
This edition is published by the American Mathematical Society
under license from Hindustan Book Agency.
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Library of Congress Cataloging-in-Publication Data
Connes,Alain.
Noncommutativegeometry,quantumfieldsandmotives/AlainConnes,MatildeMarcolli.
p. cm. — (Colloquium publications (American Mathematical Society), ISSN 0065-9258 ;
v.55)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-4210-2(alk.paper)
1. Noncommutative differential geometry. 2. Quantum field theory, I. Marcolli, Matilde.
II.Title.
QC20.7.D52C66 2007
512(cid:1).55—dc22 2007060843
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Contents
Preface xiii
Chapter 1. Quantum fields, noncommutative spaces, and motives 1
1. Introduction 1
2. Basics of perturbative QFT 7
2.1. Lagrangian and Hamiltonian formalisms 8
2.2. Lagrangian and the Feynman integral 10
2.3. The Hamiltonian and canonical quantization 11
2.4. The simplest example 13
2.5. Green’s functions 17
2.6. Wick rotation and Euclidean Green’s functions 18
3. Feynman diagrams 22
3.1. The simplest case 23
3.2. The origins of renormalization 27
3.3. Feynman graphs and rules 31
3.4. Connected Green’s functions 35
3.5. The effective action and one-particle irreducible graphs 37
3.6. Physically observable parameters 41
3.7. The physics idea of renormalization 43
4. Dimensional regularization 46
5. The graph by graph method of Bogoliubov–Parasiuk–Hepp–
Zimmermann 52
5.1. The simplest example of subdivergence 54
5.2. Superficial degree of divergence 58
5.3. Subdivergences and preparation 59
6. The Connes–Kreimer theory of perturbative renormalization 66
6.1. Commutative Hopf algebras and affine group schemes 67
6.2. The Hopf algebra of Feynman graphs: discrete part 71
6.3. The Hopf algebra of Feynman graphs: full structure 78
6.4. BPHZ as a Birkhoff factorization 81
6.5. Diffeographisms and diffeomorphisms 88
6.6. The renormalization group 89
7. Renormalization and the Riemann–Hilbert correspondence 95
7.1. Counterterms and time-ordered exponentials 96
7.2. Flat equisingular connections 103
7.3. Equivariant principal bundles and the group G∗ = G(cid:1)G 114
m
v
vi CONTENTS
7.4. Tannakian categories and affine group schemes 119
7.5. Differential Galois theory and the local Riemann–Hilbert
correspondence 123
7.6. Universal Hopf algebra and the Riemann–Hilbert
correspondence 128
8. Motives in a nutshell 137
8.1. Algebraic varieties and motives 137
8.2. Pure motives 146
8.3. Mixed motives 151
8.4. Mixed Hodge structures 156
8.5. Tate motives, periods, and quantum fields 159
9. The Standard Model of elementary particles 160
9.1. Particles and interactions 162
9.2. Symmetries 163
9.3. Quark mixing: the CKM matrix 166
9.4. The Standard Model Lagrangian 166
9.5. Quantum level: anomalies, ghosts, gauge fixing 170
9.6. Massive neutrinos 174
9.7. The Standard Model minimally coupled to gravity 179
9.8. Higher derivative terms in gravity 183
9.9. Symmetries as diffeomorphisms 184
10. The framework of (metric) noncommutative geometry 186
10.1. Spectral geometry 187
10.2. Spectral triples 190
10.3. The real part of a real spectral triple 192
10.4. Hochschild and cyclic cohomology 193
10.5. The local index cocycle 198
10.6. Positivity in Hochschild cohomology and Yang-Mills action 201
10.7. Cyclic cohomology and Chern-Simons action 202
10.8. Inner fluctuations of the metric 203
11. The spectral action principle 206
11.1. Terms in Λ2 in the spectral action and scalar curvature 210
11.2. Seeley–DeWitt coefficients and Gilkey’s theorem 216
11.3. The generalized Lichnerowicz formula 217
11.4. The Einstein–Yang–Mills system 218
11.5. Scale independent terms in the spectral action 223
11.6. Spectral action with dilaton 227
12. Noncommutative geometry and the Standard Model 230
13. The finite noncommutative geometry 234
13.1. The subalgebra and the order one condition 238
13.2. The bimodule H and fermions 240
F
13.3. Unimodularity and hypercharges 243
13.4. The classification of Dirac operators 246
13.5. Moduli space of Dirac operators and Yukawa parameters 252
13.6. The intersection pairing of the finite geometry 255
CONTENTS vii
14. The product geometry 257
14.1. The real part of the product geometry 258
15. Bosons as inner fluctuations 259
15.1. The local gauge transformations 259
15.2. Discrete part of the inner fluctuations and the Higgs field 260
15.3. Powers of D(0,1) 262
15.4. Continuous part of the inner fluctuations and gauge
bosons 265
15.5. Independence of the boson fields 269
15.6. The Dirac operator and its square 269
16. The spectral action and the Standard Model Lagrangian 271
16.1. The asymptotic expansion of the spectral action on M ×F 271
16.2. Fermionic action and Pfaffian 275
16.3. Fermion doubling, Pfaffian and Majorana fermions 277
17. The Standard Model Lagrangian from the spectral action 280
17.1. Change of variables in the asymptotic formula and unification281
17.2. Coupling constants at unification 282
17.3. The coupling of fermions 284
17.4. The mass relation at unification 292
17.5. The see-saw mechanism 293
17.6. The mass relation and the top quark mass 295
17.7. The self-interaction of the gauge bosons 298
17.8. The minimal coupling of the Higgs field 300
17.9. The Higgs field self-interaction 302
17.10. The Higgs scattering parameter and the Higgs mass 304
17.11. The gravitational terms 306
17.12. The parameters of the Standard Model 308
18. Functional integral 309
18.1. Real orientation and volume form 311
18.2. The reconstruction of spin manifolds 313
18.3. Irreducible finite geometries of KO-dimension 6 314
18.4. The functional integral and open questions 316
19. Dimensional regularization and noncommutative geometry 318
19.1. Chiral anomalies 318
19.2. The spaces X 322
z
19.3. Chiral gauge transformations 325
19.4. Finiteness of anomalous graphs and relation with
residues 326
19.5. The simplest anomalous graphs 329
19.6. Anomalous graphs in dimension 2 and the local index cocycle335
Chapter 2. The Riemann zeta function and noncommutative geometry341
1. Introduction 341
2. Counting primes and the zeta function 345
3. Classical and quantum mechanics of zeta 351
viii CONTENTS
3.1. Spectral lines and the Riemann flow 352
3.2. Symplectic volume and the scaling Hamiltonian 354
3.3. Quantum system and prolate functions 356
4. Principal values from the local trace formula 362
4.1. Normalization of Haar measure on a modulated group 364
4.2. Principal values 366
5. Quantum states of the scaling flow 370
5.1. Quantized calculus 372
5.2. Proof of Theorem 2.18 375
6. The map E 377
6.1. Hermite–Weber approximation and Riemann’s ξ function 378
7. The ad`ele class space: finitely many degrees of freedom 381
7.1. Geometry of the semi-local ad`ele class space 383
7.2. The Hilbert space L2(X ) and the trace formula 388
S
8. Weil’s formulation of the explicit formulas 396
8.1. L-functions 396
8.2. Weil’s explicit formula 398
8.3. Fourier transform on CK 399
8.4. Computation of the principal values 400
8.5. Reformulation of the explicit formula 406
9. Spectral realization of critical zeros of L-functions 407
9.1. L-functions and homogeneous distributions on AK 409
9.2. Approximate units in the Sobolev spaces L2(CK) 414
δ
9.3. Proof of Theorem 2.47 416
10. A Lefschetz formula for Archimedean local factors 421
10.1. Archimedean local L-factors 422
10.2. Asymptotic form of the number of zeros of L-functions 423
10.3. Weil form of logarithmic derivatives of local factors 424
10.4. Lefschetz formula for complex places 427
10.5. Lefschetz formula for real places 428
10.6. The question of the spectral realization 431
10.7. Local factors for curves 434
10.8. Analogy with dimensional regularization 435
Chapter 3. Quantum statistical mechanics and Galois symmetries 437
1. Overview: three systems 437
2. Quantum statistical mechanics 442
2.1. Observables and time evolution 444
2.2. The KMS condition 445
2.3. Symmetries 449
2.4. Warming up and cooling down 451
2.5. Pushforward of KMS states 451
3. Q-lattices and commensurability 452
4. 1-dimensional Q-lattices 454
4.1. The Bost–Connes system 458
CONTENTS ix
4.2. Hecke algebras 459
4.3. Symmetries of the BC system 461
4.4. The arithmetic subalgebra 462
4.5. Class field theory and the Kronecker–Weber theorem 470
4.6. KMS states and class field theory 474
4.7. The class field theory problem: algebras and fields 476
4.8. The Shimura variety of G 479
m
4.9. QSM and QFT of 1-dimensional Q-lattices 481
5. 2-dimensional Q-lattices 483
5.1. Elliptic curves and Tate modules 486
5.2. Algebras and groupoids 488
5.3. Time evolution and regular representation 493
5.4. Symmetries 495
6. The modular field 501
6.1. The modular field of level N = 1 502
6.2. Modular field of level N 504
6.3. Modular functions and modular forms 511
6.4. Explicit computations for N = 2 and N = 4 513
6.5. The modular field F and Q-lattices 514
7. Arithmetic of the GL system 518
2
7.1. The arithmetic subalgebra: explicit elements 518
7.2. The arithmetic subalgebra: definition 521
7.3. Division relations in the arithmetic algebra 527
7.4. KMS states 532
7.5. Action of symmetries on KMS states 541
7.6. Low-temperature KMS states and Galois action 542
7.7. The high temperature range 543
7.8. The Shimura variety of GL 545
2
7.9. The noncommutative boundary of modular curves 546
7.10. Compatibility between the systems 549
8. KMS states and complex multiplication 551
8.1. 1-dimensional K-lattices 551
8.2. K-lattices and Q-lattices 553
8.3. Adelic description of K-lattices 554
8.4. Algebra and time evolution 556
8.5. K-lattices and ideals 558
8.6. Arithmetic subalgebra 559
8.7. Symmetries 560
8.8. Low-temperature KMS states and Galois action 563
8.9. High temperature KMS states 569
8.10. Comparison with other systems 572
9. Quantum statistical mechanics of Shimura varieties 574
Chapter 4. Endomotives, thermodynamics, and the Weil explicit
formula 577
x CONTENTS
1. Morphisms and categories of noncommutative spaces 582
1.1. The KK-category 582
1.2. The cyclic category 585
1.3. The non-unital case 588
1.4. Cyclic (co)homology 589
2. Endomotives 591
2.1. Algebraic endomotives 594
2.2. Analytic endomotives 598
2.3. Galois action 600
2.4. Uniform systems and measured endomotives 603
2.5. Compatibility of endomotives categories 604
2.6. Self-maps of algebraic varieties 606
2.7. The Bost–Connes endomotive 607
3. Motives and noncommutative spaces: higher dimensional
perspectives 610
3.1. Geometric correspondences 610
3.2. Algebraic cycles and K-theory 612
4. A thermodynamic “Frobenius” in characteristic zero 615
4.1. Tomita’s theory and the modular automorphism group 616
4.2. Regular extremal KMS states (cooling) 618
4.3. The dual system 622
4.4. Field extensions and duality of factors (an analogy) 623
4.5. Low temperature KMS states and scaling 626
4.6. The kernel of the dual trace 631
4.7. Holomorphic modules 634
4.8. The cooling morphism (distillation) 636
4.9. Distillation of the Bost–Connes endomotive 638
4.10. Spectral realization 648
5. A cohomological Lefschetz trace formula 650
5.1. The ad`ele class space of a global field 651
5.2. The cyclic module of the ad`ele class space 652
5.3. The restriction map to the id`ele class group 653
5.4. The Morita equivalence and cokernel for K = Q 654
5.5. The cokernel of ρ for general global fields 656
5.6. Trace pairing and vanishing 670
5.7. Weil’s explicit formula as a trace formula 671
5.8. Weil positivity and the Riemann Hypothesis 672
6. The Weil proof for function fields 674
6.1. Function fields and their zeta functions 675
6.2. Correspondences and divisors in C ×C 678
6.3. Frobenius correspondences and effective divisors 680
6.4. Positivity in the Weil proof 682
7. A noncommutative geometry perspective 685
7.1. Distributional trace of a flow 686
7.2. The periodic orbits of the action of CK on XK 690