http://dx.doi.org/10.1090/coll/055 America n Mathematica l Societ y Colloquiu m Publication s Volum e 55
Noncommutativ e Geometry , Quantu m Field s an d Motive s
Alai n Conne s Matild e Marcoll i
»AMS AMERICAN MATHEMATICA L SOCIET Y öUöi HINDUSTA N SJU BOOKAGENC Y Editorial Boar d Paul J . Sally , Jr., Chai r Yuri Mani n Peter Sarna k
2000 Mathematics Subject Classification. Primar y 58B34 , 11G35 , 11M06 , 11M26 , 11G09 , 81T15, 14G35 , 14F42 , 34M50 , 81V25 . This editio n i s published b y th e America n Mathematica l Societ y under licens e fro m Hindusta n Boo k Agency .
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Library o f Congres s Cataloging-in-Publicatio n Dat a Connes, Alain . Noncommutative geometry , quantu m fields an d motive s / Alai n Connes , Matild e Marcolli , p. cm . — (Colloquiu m publication s (America n Mathematica l Society) , ISS N 0065-925 8 ; v. 55 ) Includes bibliographica l reference s an d index . ISBN 978-0-8218-4210- 2 (alk . paper ) 1. Noncommutativ e differentia l geometry . 2 . Quantu m field theory , I . Marcolli , Matilde . IL Title . QC20.7.D52C66 200 7 512/.55—dc22 200706084 3
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Preface xii i
Chapter 1 . Quantu m fields, noncommutative Spaces , and motive s 1 1. Introductio n 1 2. Basic s o f perturbative QF T 7 2.1. Lagrangia n an d Hamiltonia n formalism s 8 2.2. Lagrangia n an d the Feynma n integra l 1 0 2.3. Th e Hamiltonia n an d canonica l quantizatio n 1 1 2.4. Th e simples t exampl e 1 3 2.5. Green' s function s 1 7 2.6. Wic k rotation an d Euclidea n Green' s function s 1 8 3. Feynma n diagram s 2 2 3.1. Th e simples t cas e 2 3 3.2. Th e origin s o f renormalization 2 7 3.3. Feynma n graph s an d rule s 3 1 3.4. Connecte d Green' s function s 3 5 3.5. Th e effectiv e actio n an d one-particl e irreducibl e graph s 3 7 3.6. Physicall y observabl e parameters 4 1 3.7. Th e physic s ide a o f renormalization 4 3 4. Dimensiona l regularizatio n 4 6 5. Th e grap h b y grap h metho d o f Bogoliubov-Parasiuk-Hepp - Zimmermann 5 2 5.1. Th e simples t exampl e o f subdivergence 5 4 5.2. Superficia l degre e o f divergence 5 8 5.3. Subdivergence s an d preparatio n 5 9 6. Th e Connes-Kreime r theor y o f perturbative renormalizatio n 6 6 6.1. Commutativ e Hop f algebra s an d affin e grou p scheme s 6 7 6.2. Th e Hop f algebr a o f Feynman graphs : discret e part 7 1 6.3. Th e Hop f algebr a o f Feynman graphs : fül l structur e 7 8 6.4. BPH Z a s a Birkhof f factorizatio n 8 1 6.5. Diffeographism s an d diffeomorphism s 8 8 6.6. Th e renormalizatio n grou p 8 9 7. Renormalizatio n an d the Riemann-Hilbert correspondenc e 9 5 7.1. Counterterm s an d time-ordered exponential s 9 6 7.2. Fia t equisingula r connection s 10 3 7.3. Equivarian t principa l bundle s and the grou p G * = G x i Gm 11 4 CONTENTS
7.4. Tannakia n categorie s an d affin e grou p scheme s 11 9 7.5. Differentia l Galoi s theor y an d th e loca l Riemann-Hilber t correspondence 12 3 7.6. Universa l Hop f algebr a an d th e Riemann-Hilber t correspondence 12 8 8. Motive s i n a nutshell 13 7 8.1. Algebrai c varieties an d motive s 13 7 8.2. Pur e motive s 14 6 8.3. Mixe d motive s 15 1 8.4. Mixe d Hodg e structures 15 6 8.5. Tät e motives , periods, an d quantu m fields 15 9 9. Th e Standar d Mode l o f elementary particle s 16 0 9.1. Particle s an d interaction s 16 2 9.2. Symmetrie s 16 3 9.3. Quar k mixing : th e CK M matrix 16 6 9.4. Th e Standar d Mode l Lagrangian 16 6 9.5. Quantu m level : anomalies , ghosts , gauge fixing 17 0 9.6. Massiv e neutrinos 17 4 9.7. Th e Standard Mode l minimally couple d to gravity 17 9 9.8. Highe r derivativ e terms i n gravity 18 3 9.9. Symmetrie s a s diffeomorphisms 18 4 10. Th e framewor k o f (metric ) noncommutativ e geometr y 18 6 10.1. Spectra l geometr y 18 7 10.2. Spectra l triples 19 0 10.3. Th e rea l part o f a real spectral tripl e 19 2 10.4. Hochschil d an d cycli c cohomolog y 19 3 10.5. Th e loca l index cocycl e 19 8 10.6. Positivit y i n Hochschild cohomolog y an d Yang-Mill s actio n 20 1 10.7. Cycli c cohomolog y an d Chern-Simon s actio n 20 2 10.8. Inne r fluctuations o f the metri c 20 3 11. Th e spectra l actio n principl e 20 6 11.1. Term s i n A 2 i n the spectra l actio n an d scala r curvatur e 21 0 11.2. Seeley-DeWit t coefficient s an d Gilkey' s theorem 21 6 11.3. Th e generalize d Lichnerowic z formul a 21 7 11.4. Th e Einstein-Yang-Mill s Syste m 21 8 11.5. Scal e independent term s i n the spectra l actio n 22 3 11.6. Spectra l actio n wit h dilato n 22 7 12. Noncommutativ e geometr y an d the Standar d Mode l 23 0 13. Th e finite noncommutativ e geometr y 23 4 13.1. Th e subalgebr a an d the orde r on e condition 23 8 13.2. Th e bimodul e HF an d fermion s 24 0 13.3. Unimodularit y an d hypercharge s 24 3 13.4. Th e Classificatio n o f Dirac Operator s 24 6 13.5. Modul i spac e o f Dirac Operator s and Yukaw a parameters 25 2 13.6. Th e intersectio n pairin g o f the finite geometr y 25 5 CONTENTS vi i
14. Th e product geometr y 25 7 14.1. Th e rea l part o f the product geometr y 25 8 15. Boson s a s inner fluctuations 25 9 15.1. Th e loca l gauge transformations 25 9 15.2. Discret e part o f the inne r fluctuations an d the Higg s field 26 0 15.3. Power s o f D&'V 26 2 15.4. Continuou s part o f the inne r fluctuations an d gaug e bosons 26 5 15.5. Independenc e o f the boso n fields 26 9 15.6. Th e Dira c Operato r an d it s Squar e 26 9 16. Th e spectral actio n an d the Standar d Mode l Lagrangian 27 1 16.1. Th e asymptoti c expansio n o f the spectra l actio n onMxF 27 1 16.2. Fermioni c actio n an d Pfaffia n 27 5 16.3. Fermio n doubling , Pfaffia n an d Majoran a fermion s 27 7 17. Th e Standar d Mode l Lagrangian fro m th e spectra l actio n 28 0 17.1. Chang e o f variables i n the asymptotic formul a an d unification28 1 17.2. Couplin g constant s a t unificatio n 28 2 17.3. Th e couplin g o f fermions 28 4 17.4. Th e mas s relation a t unificatio n 29 2 17.5. Th e see-sa w mechanis m 29 3 17.6. Th e mas s relation an d the top quar k mas s 29 5 17.7. Th e self-interactio n o f the gaug e bosons 29 8 17.8. Th e minima l couplin g o f the Higg s field 30 0 17.9. Th e Higg s field self-interaction 30 2 17.10. Th e Higg s scattering paramete r an d the Higg s mass 30 4 17.11. Th e gravitationa l term s 30 6 17.12. Th e parameter s o f the Standar d Mode l 30 8 18. Functiona l integra l 30 9 18.1. Rea l orientation an d volum e for m 31 1 18.2. Th e reconstructio n o f spin manifold s 31 3 18.3. Irreducibl e finite geometrie s o f K O-dimension 6 31 4 18.4. Th e functiona l integra l an d ope n question s 31 6 19. Dimensiona l regularizatio n an d noncommutativ e geometr y 31 8 19.1. Chira l anomalie s 31 8 19.2. Th e space s X z 32 2 19.3. Chira l gaug e transformations 32 5 19.4. Finitenes s o f anomalous graph s an d relatio n wit h residues 32 6 19.5. Th e simples t anomalou s graph s 32 9 19.6. Anomalou s graph s i n dimension 2 and the loca l index cocycl e 335
Chapter 2 . Th e Riemann zeta function an d noncommutative geometry34 1 1. Introductio n 34 1 2. Countin g prime s an d the zet a functio n 34 5 3. Classica l an d quantu m mechanic s o f zeta 35 1 viii CONTENT S
3.1. Spectra l line s and the Rieman n flow 35 2 3.2. Symplecti c volum e and the scalin g Hamiltonian 35 4 3.3. Quantu m Syste m and prolat e function s 35 6 4. Principa l value s fro m th e loca l trace formul a 36 2 4.1. Normalizatio n o f Haar measur e o n a modulated grou p 36 4 4.2. Principa l value s 36 6 5. Quantu m state s o f the scalin g flow 37 0 5.1. Quantize d calculu s 37 2 5.2. Proo f o f Theorem 2.1 8 37 5 6. Th e ma p < £ 37 7 6.1. Hermite-Webe r approximatio n an d Riemann' s £ function 37 8 7. Th e adel e clas s space: finitely man y degree s o f freedom 38 1 7.1. Geometr y o f the semi-loca l adel e clas s space 38 3 7.2. Th e Huber t spac e L 2(Xs) an d the trace formul a 38 8 8. Weil' s formulation o f the explici t formula s 39 6 8.1. L-function s 39 6 8.2. Weil' s explicit formul a 39 8 8.3. Fourie r transfor m o n C K 39 9 8.4. Computatio n o f the principa l value s 40 0 8.5. Reformulatio n o f the explici t formul a 40 6 9. Spectra l realizatio n o f critical zero s o f L-functions 40 7 9.1. L-function s an d homogeneou s distribution s o n AK 40 9 9.2. Approximat e unit s i n the Sobole v Space s L^(CK) 41 4 9.3. Proo f o f Theorem 2.4 7 41 6 10. A Lefschetz formul a fo r Archimedea n loca l factors 42 1 10.1. Archimedea n loca l L-factor s 42 2 10.2. Asymptoti c for m o f the numbe r o f zeros o f L-functions 42 3 10.3. Wei l form o f logarithmic derivative s o f local factors 42 4 10.4. Lefschet z formul a fo r comple x place s 42 7 10.5. Lefschet z formul a fo r rea l place s 42 8 10.6. Th e questio n o f the spectra l realizatio n 43 1 10.7. Loca l factors fo r curve s 43 4 10.8. Analog y wit h dimensiona l regularizatio n 43 5
Chapter 3 . Quantu m Statistica l mechanic s an d Galoi s symmetries 43 7 1. Overview : thre e System s 43 7 2. Quantu m Statistica l mechanic s 44 2 2.1. Observable s an d tim e evolution 44 4 2.2. Th e KM S condition 44 5 2.3. Symmetrie s 44 9 2.4. Warmin g u p an d coolin g down 45 1 2.5. Pushforwar d o f KMS states 45 1 3. Q-lattice s an d commensurabilit y 45 2 4. 1-dimensiona l Q-lattice s 45 4 4.1. Th e Bost-Conne s Syste m 45 8 CONTENTS i x
4.2. Heck e algebra s 45 9 4.3. Symmetrie s o f the B C Syste m 46 1 4.4. Th e arithmeti c subalgebr a 46 2 4.5. Clas s field theory an d the Kronecker-Webe r theore m 47 0 4.6. KM S states an d class field theory 47 4 4.7. Th e clas s field theory problem : algebra s an d fields 47 6 4.8. Th e Shimur a variet y o f Gm 47 9 4.9. QS M and QFT o f 1-dimensiona l Q-lattiee s 48 1 5. 2-dimensiona l Q-lattice s 48 3 5.1. Ellipti c curve s and Tät e module s 48 6 5.2. Algebra s an d groupoid s 48 8 5.3. Tim e evolution an d regulä r representatio n 49 3 5.4. Symmetrie s 49 5 6. Th e modula r field 50 1 6.1. Th e modula r field o f level N = 1 50 2 6.2. Modula r field o f leve l N 50 4 6.3. Modula r function s an d modula r form s 51 1 6.4. Explici t computation s fo r N = 2 and N = 4 51 3 6.5. Th e modula r field F an d Q-lattice s 51 4 7. Arithmeti c o f the GL 2 System 51 8 7.1. Th e arithmeti c subalgebra : explici t element s 51 8 7.2. Th e arithmeti c subalgebra : definitio n 52 1 7.3. Divisio n relation s i n the arithmeti c algebr a 52 7 7.4. KM S states 53 2 7.5. Actio n o f symmetries o n KM S states 54 1 7.6. Low-temperatur e KM S states an d Galoi s actio n 54 2 7.7. Th e hig h temperature räng e 54 3 7.8. Th e Shimur a variet y o f GL 2 54 5 7.9. Th e noncommutativ e boundar y o f modular curve s 54 6 7.10. Compatibilit y betwee n the System s 54 9 8. KM S states an d comple x multiplication 55 1 8.1. 1-dimensiona l K-lattice s 55 1 8.2. K-lattiee s an d Q-lattice s 55 3 8.3. Adeli c description o f K-lattices 55 4 8.4. Algebr a an d tim e evolutio n 55 6 8.5. K-lattice s an d ideal s 55 8 8.6. Arithmeti c subalgebr a 55 9 8.7. Symmetrie s 56 0 8.8. Low-temperatur e KM S states an d Galoi s actio n 56 3 8.9. Hig h temperature KM S states 56 9 8.10. Compariso n wit h other System s 57 2 9. Quantu m Statistica l mechanic s o f Shimura varietie s 57 4
Chapter 4 . Endomotives , thermodynamics , an d th e Wei l explici t formula 57 7 CONTENTS
1. Morphism s an d categorie s o f noncommutative Space s 58 2 1.1. Th e ÄTX-categor y 58 2 1.2. Th e cycli c category 58 5 1.3. Th e non-unita l cas e 58 8 1.4. Cycli c (co)homolog y 58 9 2. Endomotive s 59 1 2.1. Algebrai c endomotive s 59 4 2.2. Analyti c endomotive s 59 8 2.3. Galoi s actio n 60 0 2.4. Unifor m System s and measure d endomotive s 60 3 2.5. Compatibilit y o f endomotives categorie s 60 4 2.6. Self-map s o f algebraic varietie s 60 6 2.7. Th e Bost-Connes endomotiv e 60 7 3. Motive s an d noncommutativ e Spaces : highe r dimensiona l perspectives 61 0 3.1. Geometri e correspondence s 61 0 3.2. Algebrai c cycle s and if-theor y 61 2 4. A thermodynamic "Frobenius " i n characteristic zer o 61 5 4.1. Tomita' s theor y an d the modula r automorphis m grou p 61 6 4.2. Regulä r extrema l KM S states (cooling ) 61 8 4.3. Th e dua l Syste m 62 2 4.4. Fiel d extension s an d dualit y o f factors (a n analogy ) 62 3 4.5. Lo w temperature KM S states an d scalin g 62 6 4.6. Th e kerne l o f the dua l trace 63 1 4.7. Holomorphi c module s 63 4 4.8. Th e coolin g morphism (distillation ) 63 6 4.9. Distillatio n o f the Bost-Conne s endomotiv e 63 8 4.10. Spectra l realizatio n 64 8 5. A cohomologica l Lefschet z trac e formul a 65 0 5.1. Th e adel e dass spac e o f a global fiel d 65 1 5.2. Th e cycli c module o f the adel e dass spac e 65 2 5.3. Th e restriction ma p to the idel e dass grou p 65 3 5.4. Th e Morit a equivalenc e an d cokerne l fo r K = Q 65 4 5.5. Th e cokerne l o f p for genera l globa l fields 65 6 5.6. Trac e pairing an d vanishin g 67 0 5.7. Weil' s explicit formul a a s a trace formul a 67 1 5.8. Wei l positivity an d the Rieman n Hypothesi s 67 2 6. Th e Wei l proof fo r funetio n fields 67 4 6.1. Functio n fields and their zet a funetion s 67 5 6.2. Correspondence s an d divisor s inCxC 67 8 6.3. Frobeniu s correspondence s an d effectiv e divisor s 68 0 6.4. Positivit y i n the Wei l proof 68 2 7. A noncommutative geometr y perspectiv e 68 5 7.1. Distributiona l trac e o f a flow 68 6 7.2. Th e periodi c orbits o f the actio n o f Ck o n X^ 69 0 CONTENTS x i
7.3. Probeniu s (scaling ) correspondenc e an d the trace formul a 69 1 7.4. Th e Fubin i theorem an d trivia l correspondence s 69 3 7.5. Th e curv e insid e the adel e dass spac e 69 4 7.6. Vorte x configuration s (a n analogy ) 71 2 7.7. Buildin g a dictionary 72 1 8. Th e analog y betwee n Q G an d R H 72 3 8.1. KM S states an d the electrowea k phas e transition 72 3 8.2. Observable s i n Q G 72 7 8.3. Invertibilit y a t lo w temperature 72 9 8.4. Spectra l correspondence s 73 0 8.5. Spectra l cobordism s 73 0 8.6. Scalin g actio n 73 0 8.7. Modul i Space s fo r Q-lattice s an d spectra l correspondence s 73 1 Appendix 73 3 1. Operato r algebra s 73 3 1.1. C*-algebra s 73 3 1.2. Vo n Neumann algebra s 73 4 1.3. Th e passing o f time 73 8 2. Galoi s theory 74 1 Bibliography 74 9
Index 763 This page intentionally left blank Preface
The unifyin g theme , whic h the reade r wil l encounter i n differen t guise s throughout th e book , i s the interpla y betwee n noncommutativ e geometr y and numbe r theory , th e latte r especiall y i n it s manifestatio n throug h th e theory o f motives. Fo r us, this interwoven texture o f noncommutative Space s and motive s wil l become a tool i n the exploration o f two Spaces, whose rol e is central to man y development s o f modern mathematic s an d physics : • Space-tim e • Th e se t o f prime number s One ma y b e tempte d t o thin k that , lookin g fro m th e vantag e poin t o f those wh o si t ato p th e vas t edific e o f ou r accumulate d knowledg e o f suc h topics a s spac e an d numbers , w e ough t t o kno w a grea t dea l abou t thes e two Spaces . However , ther e ar e two fundamenta l problem s whos e difficult y is a clea r reminde r o f ou r limite d knowledge , an d whos e Solutio n woul d require a mor e sophisticate d understandin g tha n th e on e currentl y withi n our immediat e grasp : • Th e constructio n o f a theory o f quantum gravit y (QG ) • Th e Riemann hypothesi s (RH ) The purpos e o f thi s boo k i s t o explai n th e relevanc e o f noncommutativ e geometry (NCG ) i n dealin g wit h thes e tw o problems . Quit e surprisingly , in s o doin g w e shal l discove r tha t ther e ar e dee p analogie s betwee n thes e two problem s which , i f properl y exploited , ar e likel y to enhanc e ou r gras p of both o f them. Although th e boo k i s perhap s mor e aime d a t mathematician s tha n a t physicists, o r perhap s precisel y fo r tha t reason , w e choos e t o begi n ou r account i n Chapter 1 squarely on the physics side. Th e chapter i s dedicated to discussin g two main topics : • Renormalizatio n • Th e Standar d Mode l o f high energ y physic s We tr y t o introduc e th e materia l a s muc h a s possibl e i n a self-containe d way, taking int o consideratio n th e fac t tha t a significant numbe r o f mathe- maticians do not necessaril y hav e quantum field theory an d particle physic s as part o f their cultura l background . Thus , th e first hal f o f the chapte r i s dedicated to giving a detailed accoun t o f perturbative quantum field theory, presented i n a manne r that , w e hope, i s palatable t o th e mathematician' s taste. I n particular, w e discuss basic tools, suc h a s the effectiv e actio n an d
xiii XIV PREFACE the perturbativ e expansio n i n Feynma n graphs , a s wel l a s th e regulariza - tion procedure s use d t o evaluat e th e correspondin g Feynma n integrals . I n particular, w e concentrat e o n th e procedur e know n a s "dimensiona l regu - larization" , both becaus e o f it s bein g th e on e mos t commonl y use d i n th e actual calculations o f particle physics, and because o f the fact that i t admit s a very nice and conceptually simpl e Interpretation i n terms o f noncommuta- tive geometry, a s we will come to see towards the end o f the chapter. I n this first hal f o f Chapte r 1 we give a ne w perspectiv e o n perturbative quantu m field theory, whic h give s a clea r mathematica l Interpretatio n t o th e renor- malization procedur e use d b y physicist s t o extrac t finite value s fro m th e divergent expression s obtaine d fro m th e evaluation s o f the integral s assoei - ated to Feynman diagrams. Thi s viewpoint i s based on the Connes-Kreime r theory an d the n o n more recent result s b y the authors . Throughout thi s discussion , w e alway s assum e tha t w e wor k wit h th e procedure known in physics as "dimensiona l regularization and minimal sub- straction". Th e basi c resul t o f the Connes-Kreime r theor y i s then to sho w that th e renormalization procedur e correspond s exactl y to the Birkhof f fac - torization o f a loop 7(2: ) £ G associated to the unrenormalize d theor y eval - uated i n complex dimension D — z, wher e D i s the dimensio n o f space-tim e and z 7 ^ 0 i s th e comple x paramete r use d i n dimensiona l regularization . The group G is defined throug h its Hopf algebra o f coordinates, which is the Hopf algebr a o f Feynman graph s o f the theory . Th e Birkhof f factorizatio n of the loo p give s a canonica l wa y o f removing the singularit y a t z = 0 an d obtaining th e require d finite resul t fo r th e physica l observables . Thi s give s renormalization a clear an d wel l defined conceptua l meaning . The Birkhof f factorizatio n o f loop s i s a central too l i n the constructio n of Solutions to th e "Riemann-Hilber t problem" , whic h consist s o f finding a differential equatio n wit h prescribe d monodromy . Wit h time , ou t o f thi s original problem a whole area o f mathematics developed , under the name of "Riemann-Hilbert correspondence" . Broadl y speaking , thi s denote s a wa y of encoding objects o f differential geometri c nature, suc h as differential Sys - tems with specifie d type s o f singularities, i n terms o f group representations . In it s mos t genera l form , th e Riemann-Hilber t correspondenc e i s formu - lated a s an equivalenc e o f categories betwee n the tw o sides. I t relie s on th e "Tannakian formalism " t o reconstruc t th e grou p fro m it s categor y o f rep - resentations. W e give a general overvie w o f al l these notions, includin g th e formalism o f Tannakian categories and its application to differential System s and differentia l Galoi s theory . The main ne w result o f the first part o f Chapter 1 is the explici t identi - fication o f the Riemann-Hilber t correspondenc e secretl y presen t i n pertur - bative renormalization . At the geometri c level , the relevan t categor y i s that o f equisingular flat vector bundles. Thes e ar e vecto r bundle s ove r a bas e spac e B whic h i s a PREFACE xv principal G m(C) = C*-bundl e
Gm(C) - B -^ A over a n infinitesima l dis k A . Pro m the physica l poin t o f view, the comple x number z ^ 0 i n th e bas e spac e A i s th e paramete r o f dimensiona l reg - ularization, whil e th e paramete r i n th e fiber i s o f th e for m h\i z;, wher e K is the Planc k constan t an d /j, is a uni t o f mass . Thes e vecto r bundle s ar e endowed wit h a flat connectio n i n the complemen t o f the fiber ove r 0 G A. The fiberwise actio n o f G m(C) = C * i s give n b y h-^. Th e equisingularit y of the flat connectio n i s a mathematical translatio n o f the independence (i n the minima l subtractio n scheme ) o f th e counterterm s o n th e uni t o f mas s /i. I t mean s tha t th e singularit y o f th e connection , restricte d t o a sectio n z G Ai— > a(z) G B o f the bündl e ß, onl y depend s upo n th e valu e cr(0 ) o f the section . We show that th e categor y o f equisingular flat vecto r bundle s i s a Tan - nakian categor y an d w e identif y explicitl y th e correspondin g grou p (mor e precisely, affin e grou p scheme ) tha t encodes , throug h it s categor y o f finite dimensional linear representations, the Riemann-Hilbert correspondenc e un- derlying perturbativ e renormalization . Thi s i s a ver y specifi c proalgebrai c group o f the for m U * = U x\ Gm, whos e unipoten t par t U i s associate d t o the fre e grade d Li e algebr a ^(1,2,3, •••). with on e generator i n eac h degree . W e sho w that thi s grou p act s a s a uni - versal symmetry grou p o f all renormalizable theorie s and has the propertie s of th e "Cosmi c Galoi s group " conjecture d b y Cartier . I n man y way s thi s group shoul d b e considere d a s the prope r mathematica l incarnatio n o f th e renormalization grou p whose role, as a group encoding the ambiguity inher - ent to the renormalization proces s in quantum field theory, i s similar to that of the Galoi s group i n number theory . We conclud e th e first par t o f Chapte r 1 with a ver y brie f introductio n to the theory o f motives initiate d b y Grothendieck. W e draw some parallel s between th e Tannakia n formalis m use d i n differentia l Galoi s theor y an d i n particular i n our formulatio n o f perturbative renormalizatio n an d the sam e formalism i n the context o f motivic Galois groups. I n particular w e signal the fact that the group U* also appears (albeit via a non-canonical identification ) as a motivic Galois group in the theory of mixed Täte motives. Thi s "motivi c nature" o f the renormalization grou p remain s to b e full y understood . While the discussio n i n the first par t o f Chapte r 1 applies t o arbitrar y renormalizable theories , th e secon d par t o f this chapte r i s concerne d wit h the theor y which , a s o f the writin g o f this book , represent s th e bes t o f ou r current knowledg e o f particl e physics : th e Standard Model. Thi s par t i s based o n Joint wor k o f the author s wit h Al i Chamseddine . Our mai n purpos e i n the secon d par t o f Chapte r 1 is to sho w that th e intricate Lagrangia n o f the Standar d Mode l minimall y couple d t o gravity , XVI PREFACE where w e incorporate the terms that accoun t fo r recen t findings i n neutrin o physics, can be completely derived from very simple mathematical data. Th e procedure involves a modification o f the usual notion o f space-time geometry using the formalis m o f noncommutative geometry . Again w e d o no t assum e tha t th e reade r ha s an y familiarit y wit h par - ticle physics , s o w e begi n thi s secon d par t o f Chapte r 1 by reviewin g th e fundamental fact s abou t th e physics o f the Standard mode l and its couplin g with gravity , i n a formulatio n whic h i s a s clos e a s possibl e t o tha t o f th e physics literature . A mai n poin t tha t i t i s important t o stres s her e i s th e fact tha t th e Standar d model , i n all its complexity, wa s built ove r the year s as a resul t o f a continuin g dialogu e betwee n theor y an d experiment . Th e result i s striking i n it s dept h an d complexity : eve n just th e typesettin g o f the Lagrangian i s in itself a time-consuming task . After thi s introductor y part , w e procee d t o giv e a brie f descriptio n o f the mai n tool s o f noncommutativ e geometr y tha t wil l b e relevan t t o ou r approach. The y includ e cycli c an d Hochschil d cohomologie s an d th e basi c paradigm o f spectral triples (A,H,D). A n importan t ne w featur e o f suc h geometries, which is absent i n the commutative case, is the existence of inner fluctuations o f the metric . A t th e leve l o f symmetries, thes e correspon d t o the subgrou p o f inne r automorphisms , a norma l subgrou p o f the grou p o f automorphisms whic h i s non-trivial precisel y i n the noncommutativ e case . We then begi n the discussio n o f our model. Thi s can be thought o f as a form o f unification, base d o n the symplectic unitary grou p i n Hubert space , rat her than o n finite dimensiona l Li e groups. Th e interna l symmetrie s ar e unified wit h th e gravitationa l ones . The y al l aris e a s automorphism s o f the noncommutativ e algebr a o f coordinate s o n a produc t o f a n ordinar y Riemannian spi n manifol d M b y a finite noncommutativ e spac e F. On e striking featur e tha t emerge s fro m th e computation s i s the fac t that , whil e the metri c dimensio n o f F i s zero , it s if-theoreti c dimensio n (i n rea l K- theory) i s equal to 6 modulo 8 . A lon g detaile d computatio n the n show s ho w th e Lagrangia n o f th e Standard Mode l minimall y couple d wit h gravit y i s obtaine d naturall y (i n Euclidean form ) fro m spectra l invariant s o f th e inne r fluctuations o f th e product metri c o n M x F. This mode l provide s specifi c value s o f som e o f th e parameter s o f th e Standard Mode l a t unificatio n scale , an d on e obtain s physica l prediction s by running them down to ordinary scales through the renormalization group, using the Wilsonia n approach . I n particular, w e find that th e arbitrary pa - rameters o f the Standar d Model , a s wel l as those o f gravity, acquir e a clea r geometric meanin g i n thi s model , i n term s o f modul i Space s o f Dira c Op - erators o n the noncommutativ e geometr y an d o f the asymptoti c expansio n of the corresponding spectral action functional. Amon g the physical predic - tions ar e relations betwee n som e o f the parameters o f the Standar d Model , such a s the mergin g o f the couplin g constant s an d a relatio n betwee n th e fermion an d boso n masse s a t unification . PREFACE xvii
Finally, i n th e las t sectio n o f Chapte r 1 , w e com e t o anothe r applica - tion o f noncommutative geometr y t o quantu m field theory , whic h bring s u s back t o th e initia l discussio n o f perturbativ e renormalizatio n an d dimen - sional regularization . W e construc t natura l noncommutativ e Space s X z o f dimension a comple x numbe r z, wher e th e dimensio n her e i s mean t i n th e sense o f th e dimension spectrum o f spectra l triples . I n thi s way , w e find a concrete geometri c meanin g fo r th e dimensiona l regularizatio n procedure . We sho w that th e algebrai c rule s du e t o ' t Hooft-Veltma n an d Breiten - lohner-Maison o n ho w t o handl e chira l anomalie s usin g th e dimensiona l regularization procedur e ar e obtained , a s fa r a s on e loo p fermioni c graph s are concerned , usin g th e inne r fluctuation s o f the metri c i n the produc t b y the Space s X z. Thi s fits wit h th e simila r procedur e use d t o produc e th e Standard Mode l Lagrangia n fro m a produc t o f a n ordinar y geometr y b y the finite geometr y F an d establishe s a relatio n betwee n chira l anomalies , computed usin g dimensiona l regularization , an d th e loca l inde x formul a i n NCG. Towards th e en d o f Chapte r 1 , on e i s als o offere d a first glanc e a t th e problem pose d b y a functional integra l formulatio n o f quantum gravity . W e return onl y a t th e ver y en d o f th e boo k t o th e proble m o f constructin g a meaningful theor y o f quantu m gravity , buildin g o n th e experienc e w e gai n along th e wa y throug h th e analysi s o f quantu m Statistica l mechanica l Sys - tems arisin g fro m numbe r theory , i n relatio n t o the statistic s o f primes an d the Rieman n zet a function . Thes e topic s for m th e secon d par t o f the book , to whic h w e no w turn .
The theme o f Chapter 2 is the Riemann zet a functio n an d it s zeros. Ou r main purpos e i n thi s par t o f th e boo k i s to describ e a spectra l realizatio n of th e zero s a s a n absorptio n spectru m an d t o giv e a n Interpretatio n a s a trace formul a o f the Riemann-Weil explici t formul a relatin g the statistic s o f primes t o th e zero s o f zeta . Th e rol e o f noncommutativ e geometr y i n thi s chapter i s twofold . In th e first place , th e spac e o n whic h th e trac e formul a take s plac e i s a noncommutativ e space . I t i s obtaine d a s th e quotien t o f th e adele s AQ by the actio n o f non-zer o rationa l number s b y multiplication . Eve n thoug h the resulting spac e X = AQ/Q* i s well defined set-theoretically , i t should b e thought o f a s a noncommutative space , becaus e the ergodicit y o f the actio n of Q * o n A Q prevents on e fro m constructin g measurabl e function s o n th e quotient X , a s w e sho w i n Chapte r 3 . I n particular , th e constructio n o f function Space s o n X i s don e b y homologica l method s usin g coinvariants . This wil l onl y acquir e a fül l conceptua l meanin g i n Chapte r 4 , usin g cycli c cohomology an d th e natura l noncommutativ e algebr a o f coordinates o n X. The spac e X ca n b e approximate d b y simple r Space s Xs obtaine d b y restriction t o finite set s S o f place s o f Q . W e us e thi s simplifie d setu p to obtai n th e relatio n wit h th e Riemann-Wei l explici t formula . Th e mai n point i s that , eve n thoug h th e spac e Xs i s i n essenc e a product o f term s XV111 PREFACE corresponding t o th e variou s places , th e trac e o f the actio n o f the grou p of idel e classe s become s a sum o f suc h contributions . I t i s in the proo f o f this ke y additivity propert y tha t w e use another too l o f noncommutativ e geometry: th e quantized calculus . In the simplest instance, the Interpretation o f the Riemann-Weil explici t formula a s a trace formul a give s a n Interpretation a s symplectic volum e i n phase spac e fo r th e mai n ter m o f the Rieman n countin g functio n fo r th e asymptotic expansion o f the number o f non-trivial zeros of zeta of imaginary part les s than E. W e show that a füll quantum mechanical computation then gives the complete formula . We en d Chapte r 2 by showin g ho w this genera l pictur e an d method s extend t o the zeta function s o f arithmetic varieties , leadin g t o a Lefschet z formula fo r the local L-factors associate d by Serre to the Archimedean places of a number field. Th e Serre formul a describe s the Archimedean factor s a s products o f shifted Gamm a function s wit h the shifts an d the exponents de- pending on Hodge numbers. W e derive this formula directly from a Lefschetz trace formul a fo r th e actio n o f the Wei l grou p o n a bündl e wit h bas e the complex lin e or the quaternions (fo r a real place ) an d with fiber the Hodge realization o f the variety.
The origi n o f the relatio n describe d abov e betwee n th e Rieman n zet a function an d noncommutative geometr y can be traced to the work o f Bost- Connes. Thi s consist s o f the construction, usin g Heck e algebras , o f a quan- tum Statistica l mechanica l Syste m whos e partition functio n i s the Rieman n zeta functio n an d whic h exhibit s a surprisin g relatio n wit h th e dass field theory o f the field Q . Namely , th e Syste m admit s a s a natural symmetr y group th e group o f idel e classe s o f Q modul o th e connecte d componen t o f the identity . Thi s symmetr y o f the Syste m i s spontaneously broke n a t th e critical temperature give n by the pole o f the partition function . Belo w thi s temperature, the various phases o f the System are parameterized b y embed- dings Q cycl— > C o f the cyclotomi c extensio n Q cyd o f Q . Thes e differen t phases are described i n terms o f extremal KMS ß states , where ß — ^ i s the inverse temperature. Moreover , anothe r importan t aspec t o f this construc - tion i s the existenc e o f a natural algebr a o f "rationa l observables " o f thi s quantum Statistica l mechanica l System . Thi s allow s one to defin e i n a con- ceptual manne r a n actio n o f the Galoi s grou p Gal(Q C2/d/Q) o n the phase s of th e Syste m a t zer o temperature , merel y b y actin g o n the value s o f the states o n the rational observables , values which turn ou t to provide a set of generators fo r Q cycl, th e maximal abelia n extensio n o f Q. Our mai n purpos e i n Chapter 3 is to present extension s o f this relatio n between number theory and quantum Statistica l mechanics to more involve d examples tha n th e cas e o f rationa l numbers . I n particula r w e focu s o n two cases . Th e first correspond s t o replacin g the role o f the group GL i i n the Bost-Conne s (BC ) Syste m wit h GL2 . Thi s yield s a n interestin g non - abelian case , whic h i s related t o the Galoi s theor y o f the field o f modula r PREFACE xix functions. Th e secon d i s a closel y related cas e o f abelian das s fiel d theory , where th e field Q i s replace d b y a n imaginar y quadrati c extension . Th e results concernin g thes e tw o quantu m Statistica l mechanica l System s ar e based, respectively , o n wor k o f th e author s an d o n a collaboratio n o f th e authors wit h Niranja n Ramachandran . We approach these topics by first providing a reinterpretation o f the B C System i n term s o f geometri c objects . Thes e ar e th e Q-lattices , i.e . pair s (A, (ß) o f a lattice A C Kn ( a cocompact fre e abelia n subgroup o f M n o f rank n) togethe r wit h a homomorphism o f abelian group s 0 : Qn/Zn — > QA/A. Two Q-lattices ar e commensurable i f and onl y i f
QAi = QA 2 an d (ßi = 4> 2 mo d A i + A 2.
Let C n denot e th e se t o f commensurabilit y classe s o f n-dimensiona l Q - lattices. Eve n i n the simples t one-dimensiona l cas e ( n = 1 ) the spac e C n i s a noncommutativ e space . I n fac t i n th e one-dimensiona l cas e i t i s closel y related t o the adel e dass spac e X — AQ/Q* discusse d i n Chapter 2 . We first construc t a canonica l isomorphis m o f th e algebr a o f th e B C System with th e algebr a o f noncommutative coordinate s o n the quotien t o f C\ b y the scalin g actio n o f R+. Followin g Weil' s analog y betwee n trigono - metric an d ellipti c functions, w e then sho w that th e trigonometric analogu e of the Eisenstei n serie s generate , togethe r wit h th e commensurabilit y wit h division points , th e arithmeti c subalgebr a o f "rationa l observables " o f th e BC System. Thi s opens the wa y to the highe r dimensional cas e and muc h o f Chapter 3 is devoted to the extension o f these results to the two-dimensiona l case. The Syste m fo r th e GL 2 cas e i s mor e involved , bot h a t th e quantu m Statistical level , wher e ther e ar e tw o phas e transition s an d n o equilibriu m State above a certain temperature, an d a t th e number theoretic level , where the cyclotomi c field Q cycl i s replaced b y the modula r field. We en d Chapte r 3 wit h th e descriptio n o f ou r Join t result s wit h Ra - machandran o n th e extensio n o f th e B C Syste m t o imaginar y quadrati c fields. Thi s i s base d o n replacin g th e notio n o f Q-lattice s wit h a n anal - ogous notio n o f 1-dimensiona l K-lattices , wit h K th e imaginar y quadrati c extension o f Q . Th e relatio n betwee n commensurabilit y o f 1-dimensiona l K-lattices an d o f the underlyin g 2-dimensiona l Q-lattice s give s the relatio n between the quantum Statistica l mechanical Syste m fo r imaginary quadrati c fields an d th e GL2-system . Thi s yield s th e relatio n betwee n th e quantu m Statistical mechanics o f K-lattices and the explicit dass field theory o f imag- inary quadratic fields. Underlying ou r presentatio n o f th e mai n topic s o f Chapte r 3 ther e i s a unifyin g theme . Namely , th e thre e differen t case s o f quantum Statistica l mechanical System s tha t w e presen t i n detai l al l fit int o a simila r genera l picture, wher e an ordinar y modul i spac e i s recovered a s the se t o f classica l XX PREFACE points (zer o temperature states ) o f a noncommutative spac e with a natura l time evolution. I n the setting o f Chapter 3 the classica l Space s are Shimur a varieties, which can be thought o f as moduli Spaces of motives. Thi s general picture wil l provide a motivating analog y fo r ou r discussio n o f the quantu m gravity proble m a t th e en d o f the book . The spectra l realizatio n o f zero s o f zet a an d L-function s describe d i n Chapter 2 i s base d o n th e actio n o f th e idel e das s grou p o n th e noncom - mutative spac e X o f adele s classes . Nevertheless , th e construction , a s w e describe i t i n Chapte r 2 , makes little us e o f the formalis m o f noncommuta - tive geometry an d n o direct us e o f the crossed product algebr a A describin g the quotien t o f adeles b y the multiplicativ e grou p Q* . In Chapter 4 , the last chapter o f the book, w e return to this theme. Ou r main purpos e i s to sho w that th e spectra l realizatio n describe d i n Chapte r 2 acquire s cohomologica l meaning , provide d tha t on e reinterpret s th e con - struction i n terms o f the crosse d product algebr a A an d cycli c cohomology . This chapter i s based o n our Joint wor k with Caterin a Consani . We begi n th e chapte r b y explainin g ho w to reinterpre t th e entir e con - struction o f Chapter 2 in "motivic " term s usin g • A n extensio n o f the notio n o f Artin motive s to suitabl e projectiv e limits, whic h w e call endomotives. • Th e category o f cyclic modules a s a linearization o f the category o f noncommutative algebra s an d correspondences . • A n analogu e o f the actio n o f the Frobeniu s o n £-adi c cohomology , based o n a thermodynamical procedure , whic h w e call distillation. The construction o f an appropriate "motivi c cohomology" with a "Frobe- nius" actio n o f R!j _ fo r endomotive s i s obtained throug h a very general pro- cedure. I t consist s o f thre e basi c steps , startin g fro m th e dat a o f a non - commutative algebr a A an d a stat e (f. On e consider s th e tim e evolutio n at G Aut.4, fo r t G R , naturall y associate d t o the stat e cp. The firs t ste p i s what w e refer to a s cooling. One considers the space £ß of extremal KMS ß states, fo r ß greate r than critical . Assumin g these state s are o f type I , on e obtains a morphis m
1 TT : A X3 a R -• S{ßß x R;)
HC0(D(A, This procedur e i s quit e genera l an d i t applie s t o a larg e clas s o f dat a (»A, <£>) , producing spectral realizations o f zeros of L-functions. I t give s a rep- resentation o f the multiplicative grou p M.+ whic h combines with the natura l representation o f the Galoi s grou p G when applie d t o the noncommutativ e space (analyti c endomotive ) associate d t o a n (algebraic ) endomotive . In th e particula r cas e o f the endomotiv e associate d t o th e B C System , the resultin g representatio n o f G x R+ give s the spectra l realizatio n o f th e zeros o f the Riemann zet a function an d o f the Artin L-function s fo r abelia n characters o f G. On e sees in this example that thi s construction play s a role analogous to the action o f the Weil group on the £-adic cohomology. I t give s a functor fro m the category o f endomotives to the category o f representations of the group G x R^_ . Her e w e think o f the actio n o f R+ a s a "Frobeniu s i n characteristic zero" , hence o f G x R^ _ as the correspondin g Wei l group. We als o sho w that th e "dualization " step , i.e . the transitio n fro m A t o Ax^lR, i s a very good analog in the case of number fields of what happens fo r a function field K i n passing to the extensio n K ®^ q ¥q. I n fact , i n the cas e of positiv e characteristic , th e unramifie d extension s K ®F Q F^n , combine d with th e notio n o f places, yiel d th e point s C(¥ q) ove r F ^ o f the underlyin g curve. Thi s ha s a goo d paralle l i n the theor y o f factor s an d thi s analog y plays an important rol e in developing a setting in noncommutative geometr y that parallel s th e algebro-geometri c framewor k tha t Wei l use d i n hi s proo f of RH fo r functio n fields . We end the numbe r theoreti c par t o f the boo k b y a dictionary betwee n Weil's proo f an d th e framewor k o f noncommutative geometry , leavin g ope n the proble m o f completin g th e translatio n an d understandin g th e noncom - mutative geometr y o f the "arithmeti c site" . We end the book by Coming back to the construction o f a theory o f quan- tum gravity . Ou r approac h her e Start s b y developin g a n analog y betwee n the electroweak phas e transition i n the Standard Mode l and the phase tran- sitions i n the quantu m Statistica l mechanica l System s describe d i n Chapte r 3. Throug h this analog y a consistent pictur e emerge s which make s i t possi - ble to dehn e a natural candidat e fo r th e algebr a o f observables o f quantu m gravity an d t o conjectur e a n extensio n o f the electrowea k phas e transitio n to the fül l gravitational sector , i n which the geometry o f space-time emerge s through a symmetry breakin g mechanis m an d a cooling process . A s a wit - ness to the unit y o f the book , i t i s the constructio n o f the correc t categor y of correspondence s which , a s i n Grothendieck' s theor y o f motives , remain s the mai n challeng e fo r furthe r progres s o n both Q G an d RH . Acknowledgment. W e wis h t o than k ou r friend s an d collaborator s Ali Chamseddine , Kati a Consani , Miche l Dubois-Violette , Dir k Kreimer , Yuri Manin , Henr i Moscovici , an d Niranja n Ramachandran , whos e idea s and contribution s ar e reflecte d i n muc h o f th e conten t o f thi s book . W e thank Jorg e Plazas fo r acceptin g to be the officia l "tes t reader " o f the boo k and fo r th e man y usefu l comment s h e provided . W e als o than k al l th e XX11 PREFACE people who offered to send us comments and suggestions, among them Pierre Cartier, Masou d Khalkhali, Peter May, Laura Reina, Abhijnan Rej , Thomas Schücker, Walte r va n Suijlekom , Josep h Värill y an d Do n Zagier . Variou s institutions provide d hospitalit y an d suppor t a t variou s stage s durin g th e preparation o f the book: w e thank the IHES and the MPI for making several mutual visit s possible, a s wel l as the Kavl i Institute, th e Newto n Institute , and Vanderbil t Universit y fo r som e ver y productiv e extende d stays . W e thank Arthur Greenspoo n an d Patrick Io n fo r a very careful proofreadin g o f the manuscript an d the editors at both the AM S and HB A fo r their infinit e patience. Thi s wor k wa s partially supporte d b y NS F grant s DMS-065216 4 and DMS-0651925 . This page intentionally left blank Bibliography S.L. Adler, Axial-vector Vertex in spinor electrodynamics, Phys . Rev., Vol. 177 (1969) 2426-2438. Y. Andre , Une introduction aux motifs, Panorama s e t Syntheses , Vol . 17, Societ e mathematique d e France, 2005 . H. Araki, Expansional in Banach algebras, Ann. Sei . Ecole Norm. Sup . (4 ) 6 (1973), 67-84. H.Arason, D.J . Castano , B . 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This page intentionally left blank Index C(7) (counterterms) , 6 4 Zc, 66 1 C*-algebra, 123 , 150 , 198 , 387, 439, Z C (T), 66 3 444, 577 , 582 , 584, 598, 600, 607, A£ = GLI(AK ) (grou p o f ideles o f th e 615, 617 , 626 , 630, 636 , 651, 733 global fiel d K) , 40 7 CK (idd e dass grou p o f the globa l fiel d AK = GLI(AK) (grou p o f ideles o f th e K), 343 , 397, 399, 400, 410, 414, global fiel d K) , 397 , 55 4 580, 651 , 669 AK (adele s o f the globa l fiel d K) , 344 , Dq (connecte d componen t o f 1 G CQ), 396, 406 , 409, 412 382 AQ, 64 1 DT(A, commensurable, 437 , 452, 454, 480, + 19 1 482, 483, 486, 72 9 e(r) (characte r o f AQ,//Z) , 45 7 divisibility, 454 , 457, 466 DK,D,DV (Wei l explici t distributions) , invertible, 441 , 452, 475, 641, 729 398 one-dimensional, 45 8 Fa (Fourie r transfor m fo r a bicharacte r scaling, 73 0 a), 35 7 Q-pseudolattice, 54 7 4>r, 63 3 R/3, 62 8 Z (profinit e completio n o f Z), 38 2 E (place s o f K), 34 3 K /' (principa l value) , 367 , 369, 395, 406, Eoo (Archimedea n places) , 41 3 426 Trdistr(T), 68 7 m (maxima l idea l o f ö) , 39 6 S , 69 5 K £, 377 , 390, 408, 638, 641, 651, 669, 67 4 SK/CK,I, 69 5 cohomological, 63 8 HK.V, 69 6 X (summatio n map) , 388 , 644, 656, U, 13 0 661, 66 9 U*, 13 0 ß(v,s) = v~ is (bicharacte r o f R+ x R), ß-decay, 16 4 373 Ai = C*(öi/R+ ) (B C algebra) , 45 7 04-theory, 6 4 A = C*(ö /C*) (C*-completio n o f 2 2 3-theory, 23 , 33, 42, 45, 53, 54, 57, 59, A {U )), 49 1 C 2 62, 65 , 66, 75, 87 A (U ) (convolutio n algebr a o f C 2 massless, 78 , 88 U = Ö2/C*) , 49 0 2 04-theory, 33 , 61, 62, 7 7 £>-module, 5 0 (Tf : = [F , /] (quantize d differential) , 37 2 VK (categor y o f differential modules) , Hu, 13 0 Q2 = {(g,p,a) e GL+(Q ) x M 2(Z) x C(H), 73 4 GL+(R); gpeM 2(Z)}, 48 8 Cn (commensurabilit y classe s o f Ü2 = {(g,p,z) e GL+(Q ) x M2(Z) x M : n-dimensional Q-lattices) , 45 2 gpeM2(Z)}, 49 0 £u, 13 0 W2 = GL+(Q ) x M 2(AQJ) x H , 49 7 ö (maxima l compac t subrin g o f a #a (additiv e scalin g action) , 342 , 357, non-Archimedean loca l field), 39 6 362, 378 , 391, 392, 407, 432, 658, S(Iß), 63 4 659, 691 , 709 S(Iß,Ac), 62 8 £j, 65 6 £K Dedekin d zet a function , 43 9 algebraic closure , 434 , 615, 623, 741, c(A) (normalize d covolum e 745 2iric(Z) = 1) , 462 algebraic curves , 145 , 580, 614, 671, 712 2lxi{z+i) ec(z) = e - (basi c character o f over finite fields, 145 , 615, 623, 651, C), 40 3 675, 68 0 2n lx em(x) = e ~ ' (basi c character o f R), zeta function , 67 5 354, 368 , 402 algebraic cycles , 4, 139-141 , 147 , 150 , ei,a(A, 0) (trigonometri c generators) , 592, 593 , 612, 72 2 462 algebraic endomotives , 60 4 q(z), 66 3 algebraic equations , 12 2 WA (Prolat e spheroida l wav e Operator) , algebraic extension, 12 6 358 algebraic hüll , 122 , 126 , 12 7 7^Q(r,r0) (Heck e algebra) , 46 0 algebraic points , 142 , 146 , 580, 591, MS(/) (minima l subtractio n principa l 598, 600 , 694, 71 2 value), 36 8 algebraic torus, 149 , 15 2 1PI graphs, 2 , 37-39, 43, 47, 63, 74, 76, algebraic varieties , 127 , 137 , 344, 78-80, 72 3 421-423 of graphs, 15 9 abelian category , 119 , 129 , 141 , 147, self-maps, 60 6 152, 153 , 580, 585, 590, 592, 616, zero-dimensional, 149 , 606 636, 64 9 ambiguity, 1 , 2, 12 , 19 , 21, 78, 113 , 242, abelian semigroup , 599 , 60 7 244, 315 , 70 7 abelian varieties , 151 , 152, 578, 610, 61 4 analytic continuation , 2 , 3, 62 8 abelianization, 602 , 65 7 analytic endomotives , 599 , 600 , 60 4 absolute cohomology , 154 , 590, 61 6 measured, 60 3 action functional , 8 , 10 , 21, 280, 310, Andre, 13 7 313 anomalies, 7 , 47, 165 , 318-33 9 bosonic part, 23 1 cancellation, 161 , 165 Euclidean, 3 0 anomalous graphs , 7 , 329, 33 5 adele dass space , 343 , 381, 383, 423, Anosov foliation , 73 8 437, 481, 482, 579 , 651, 685, 69 0 anti-matter, 162-164 , 175 , 238, 241, of a globa l field, 40 7 251, 27 4 semi-local, 38 4 anti-vortices, 71 6 additive category , 140 , 582, 59 7 antilinear, 192 , 205, 233, 237, 268 , 276, enveloping, 58 4 313-315, 61 6 additive group , 68 , 101 , 131 antipode, 67 , 75, 76, 83, 596 adjoint action , 115 , 164 , 205, 220, 221, inductive, 7 5 232, 235 , 243, 267, 276 , 28 0 approximate Solution , 71 9 adjunction o f exponentials, 12 6 approximate unit s adjunction o f primitives, 12 6 for C*-algebras , 45 2 Adler-Beil-Jackiw (ABJ ) anomaly , 171, in Sobole v Spaces , 344 , 41 4 319, 320 , 33 5 Arakelov geometry , 71 2 affine grou p scheme , 3 , 67, 82, 114 , 15 5 Araki expansional , 9 7 Lie algebra, 6 9 Archimedean pro-reductive, 12 2 component, 70 6 representations, 12 1 local factor , 144 , 156 , 344, 345, 421, universal, 12 8 423, 424 , 427, 430, 431, 434, 435, Albanese an d Picar d varieties , 146 , 15 3 650 algebra local field, 402 , 47 0 characters, 477 , 598-600, 606 , 608, non-Archimedean loca l field, 396 , 47 1 649 non-Archimedean place , 344 , 395, 56 9 finite dimensional , 23 4 place, 144 , 344, 382, 413, 421, 422, finite dimensiona l reduced , 59 3 424, 433 , 473, 54 8 766 INDEX places, 48 3 positive part, 8 6 Archimedes law , 2 8 recursive formula , 8 3 arithmetic infinity , 71 2 same negativ e part, 10 7 arithmetic structure , 542 , 599 , 60 9 bivariant dass , 72 2 arithmetic subalgebra , 462-470 , bivariant K-theory , 58 2 518-532, 559-560 , 581 , 608, 615, Bloch, 151 , 160, 61 4 619, 71 0 Bloch-Esnault-Kreimer, 15 9 Artin, 139 , 144 , 67 8 Bogoliubov-Parasiuk homomorphism, 471 , 472 method, 5 3 reciprocity, 47 3 preparation, 6 4 Artin motives , 4 , 146 , 149 , 150 , 153, Bogoliubov-Parasiuk-Hepp- 577, 591 , 604, 72 2 Zimmermann (BPHZ) , 3 , 59, 64, associated graded , 128 , 13 2 66, 85 , 87 asymptotic expansion , 5 , 6, 207 , 209, procedure, 5 9 273, 280 , 281, 300, 30 8 Bogomolny equations , 71 7 asymptotic formula , 48 3 Boltzmann constant , 44 3 Atiyah-Bott-Lefschetz, 431 , 686 Borel (A) , 15 9 Atiyah-Singer, 71 3 Borel (E ) Operator, 18 9 function, 73 6 Auerbach basis , 66 8 set, 73 5 augmentation, 83 , 84, 119 , 58 8 space, 73 8 automorphic forms , 14 6 Born, 8 automorphisms, 48 2 Born-Heisenberg-Jordan, 7 fiber functor , 12 2 bosonic field, 12 , 1 3 inner, 18 6 bosons, 6 , 161 , 162, 164 , 170 , 232, 269, axial current, 32 6 726 axial-vector current , 31 9 from inne r nuctuations , 231 , 246, 259, 265, 28 0 bare action , 72 4 Goldstone, 173 , 232 bare mass , 4 4 weak interaction , 27 5 bare parameters, 3 , 29, 43, 46, 5 3 Bost-Connes algebra , 44 0 base point, 100 , 110 , 11 1 Bost-Connes System , 438 , 440, 442, Becchi-Rouet-Stora (BRS) , 172 , 32 2 458-470, 473, 474, 476, 482, 518, Beilinson-Soule vanishing , 154 , 15 9 551, 560 , 561, 572-575, 578 , 579, Belkale-Brosnan, 15 9 607, 615 , 617, 626, 627, 632, 633, Bernoulli numbers , 463 , 504 636-638, 641 , 648, 651, 652, 657, Bernstein, 4 9 658, 699 , 710 , 731, 740, 74 1 Berry, 35 2 algebraic endomotive, 60 8 beta decay , 16 2 arithmetic subalgebra , 46 2 beta function , 3 , 89, 92, 95, 96, 102 , decay estimate, 64 2 135, 171 , 283 dual System , 63 9 Betti number , 34 , 38, 432 dual trace , 64 3 bicharacter, 354 , 356, 357, 361, 363, semigroup action , 60 8 373, 375 , 399, 640 symmetries, 461 , 474 bicomplex, 589 , 61 4 boundary condition , 378 , 38 0 big desert hypothesis , 6 , 28 3 Boyd, 37 8 bimodules, 582 , 59 3 Breitenlohner-Maison, 7 , 31 8 cyclic morphisms, 58 8 Broadhurst, 15 9 tensor product , 59 3 Brout-Englert-Higgs, 23 2 Virtual, 60 5 BRS cohomology , 73 1 Birkhoff factorization , 3 , 67, 81, 102, Bruhat-Schwartz space , 366 , 367, 104, 106 , 16 0 390-392, 396 , 407, 412, 64 6 negative piece , 86 , 102 , 108 , 11 1 Burnol, 372 , 373, 395 INDEX 767 Cabibbo angle , 16 6 charge conjugation , 16 3 Cabibbo-Kobayashi-Maskawa (CKM ) Chebyshev-von Mangold t function , 34 9 matrix, 166 , 167 , 174 , 254, 284 , 30 8 Chen iterate d integral , 9 7 Calabi-Yau, 23 4 Chern character , 317 , 318, 614 canonical dass, 67 1 Chern classes , 8 2 canonical commutatio n relations , 1 2 Chern-Simons, 5 , 209 , 22 4 canonical divisor , 676 , 68 3 chiral anomalies , 318-33 9 canonical quantization, 1 , 1 2 chiral fermions , 16 4 Cantor set , 704 , 70 5 chiral gaug e transformations, 7 cap product, 61 2 chiral theories , 4 7 Cartier, 89 , 9 5 chirality, 164 , 168 , 175 , 241, 257, 278, Castelnuovo positivity , 71 2 279, 285 , 286, 29 0 category Choquet simplex , 447 , 449, 540, 56 7 C*-monoidal, 12 3 Chow /c-linear, 12 0 group, 14 1 abelian, 119 , 129 , 14 1 motive, 14 1 additive, 140 , 14 9 Christoffel Symbols , 180 , 21 6 derived, 15 3 dass dimension, 120 , 121 , 123, 14 8 group, 473, 552, 56 1 equivalence, 146 , 15 6 number, 439 , 561, 564 Euler characteristic , 12 0 dass field theory, 437 , 440, 470-479, 71 1 filtered vecto r Spaces , 12 9 classical action , 1 1 monoidal, 14 0 classical fields, 11 , 38, 40, 72 3 neutral Tannakian , 12 1 Euclidean, 2 1 of differential modules , 12 6 classical points , 479 , 480, 546, 579, 580, of equisingular fla t vecto r bundles , 622, 695 , 699, 73 1 128 classifying space , 58 6 of mixed motives , 15 3 Clifford, 31 9 pseudo- abelian, 141 , 149 co-invariants, 39 2 rigid /c-linea r tensor , 13 4 co=rules, 6 7 rigid tensor , 120 , 126 , 14 7 coassociativity, 7 2 semi-simple, 122 , 140 , 141 , 148 cobordism, 727 , 73 0 Tannakian, 121 , 126, 148 , 435 weighted, 73 0 tensor, 12 0 coboundary, 70 9 triangulated, 15 3 cocycle, 70 9 causality, 1 , 13 , 1 6 coherent sheaves , 61 2 Cayley transform, 440 , 466, 46 8 cohomology ceiling function, 70 1 Betti, 138 , 148 , 15 7 central projections , 73 0 crystalline, 13 8 CERN, 16 1 de Rham, 138 , 15 7 Chamseddine, 4 , 161 , 206 etale, 138 , 14 4 Chamseddine-Connes, 4 , 5 , 223, 227, Singular, 13 8 231 Weil, 13 9 characteristic classes , 33 3 cokernel, 57 9 characteristic function , 230 , 323, 372, Coleman, 2 8 391, 394 , 457, 458, 466, 469, 499, color, 231 , 245, 275, 321 500, 536 , 545, 562, 592, 606, 645, charge, 16 2 709 index, 164 , 168 , 170 , 241, 242, 246, characteristic values , 191 , 252 251, 26 8 characteristic zero , 67, 69, 121 , 122, unbroken, 25 1 138, 139 , 148 , 149 , 151 , 158, 407, commensurable 412, 417, 471, 579, 580, 591, 592, Q-lattices, 437 , 452, 454, 480, 482, 610, 615, 624, 692 , 741, 745 483, 486 , 639, 641, 710 768 INDEX commutant, 17 , 314, 616, 620, 73 4 composition, 142 , 584 , 592 , 593, 597, compactification, 118 , 119 , 254, 308, 600, 611-613, 679, 72 2 384, 385 , 435, 444, 449, 545-547 , degree, 678 , 691, 693, 72 2 588, 73 2 effective, 680 , 72 2 Morita, 546 , 54 7 etale, 59 7 one-point, 444 , 445, 58 8 geometric, 61 1 comparison isomorphisms , 13 8 graph, 64 8 complex multiplication, 149 , 438, 473, involution, 67 8 492, 551 , 568, 610 perturbation, 72 2 complexified dimension , 3 , 48, 10 7 principal, 68 2 compressed algebra , 59 4 quantization, 72 2 concatenation product, 3 9 trace, 680 , 68 4 configuration space , 8 , 14 , 15 , 381, 483, trivial, 67 9 716 Virtual, 59 2 connected components , 32 , 34, 60, 72, cosmic Galois group, 95 , 128 , 136 , 155 , 87 158 connections, 97 , 100 , 106 , 108 , 110 , 111, cosmological terms, 180 , 209 , 223, 273, 114-116, 118 , 126 , 130 , 171 , 204, 308 216, 217 , 270, 273, 588, 71 6 cotangent space , 8 , 1 4 equivalence, 10 8 counit, 8 3 equivalence relation, 10 6 counterterms, 2 , 3, 43, 45, 53, 64, 86, flat, 9 9 87, 113 , 724 Hermitian, 204 , 21 7 coupling constants, 4 3 Levi-Civita, 216 , 31 3 field strength, 4 3 Riemannian, 21 7 mass, 4 3 spin, 212 , 219, 221, 270, 29 1 multiplicative, 6 4 Connes, 5 , 353, 439, 458, 481, 622 non-local, 5 4 Connes-Karoubi, 61 4 non-scalar, 5 8 Connes-Kreimer universal, 13 5 motivic, 16 0 coupling, 4 2 Connes-Kreimer theory , 3 , 7 , 6 6 effective, 4 3 Connes-Marcolli-Ramachandran, 55 1 gauge fields, 178 , 28 0 Connes-Moscovici, 5 gauge-fermion, 178 , 28 0 Connes-Skandalis, 61 0 Higgs, 178 , 179 , 28 0 Consani, 577 , 68 5 to gravity , 1 continuous field, 65 8 Yukawa, 177 , 28 0 Banach algebras , 65 4 coupling constants, 3 , 23, 43, 88, 136, contracted graph , 62 , 63, 7 3 164, 16 8 external structure, 8 0 bare, 4 5 contragredient bimodule , 73 0 effective, 45 , 88 convolution, 72 9 covariant functor , 6 7 convolution product, 596 , 639, 67 0 covariant representation , 444 , 63 2 cooling, 579 , 699, 73 1 covolume, 440 , 458, 462, 464, 482, 493, warming and cooling , 451, 567 496, 557-55 9 cooling map, 636 , 638, 641, 650 CPT symmetries , 16 3 cokernel, 636 , 64 8 creation an d annihilatio n Operators , 1 5 cooling process, 62 2 critical temperature, 448 , 475 coproduct, 67 , 68, 71-73 , 75 , 80, 84, 86, critical zeros , 408, 64 1 89, 93 , 119, 12 2 cross sections, 1 , 1 7 examples, 7 6 crossed product, 623 , 624 correspondences, 140 , 578, 582, 69 1 time evolution, 62 3 codegree, 678 , 69 1 crossings, 3 1 INDEX 769 curvature, 184 , 217, 221, 222, 270, 273, degree o f divergence, 3 , 59, 8 6 274, 308 , 33 8 degrees o f freedom, 14 , 175 , 209, 231, Gaussian, 18 0 271, 277 , 309, 313, 321, 381, 383, Ricci, 18 0 443, 48 3 Riemann, 179 , 206, 27 3 Deligne, 122 , 123 , 127, 139 , 144 , 147 , scalar, 180 , 183 , 206, 210, 211, 213, 148, 150 , 152 , 153 , 158, 678 228 Deligne-Goncharov, 15 5 Weyl, 184 , 223, 271, 306 Deninger, 42 3 cusps, 505 , 506, 511, 545, 54 8 density matrix , 620 , 62 1 cutoff, 27 , 43, 44, 46, 341, 342, 354, 356, derivation, 15 , 99, 71 1 357, 360 , 377, 379, 394, 432, 72 4 derivative interactio n term , 3 2 function, 23 0 derived category , 127 , 15 3 infrared, 32 3 bounded, 15 3 ultraviolet, 2 , 27, 43, 44, 4 6 determinant lin e bündle, 71 4 cycle map, 139 , 140 , 57 8 diagonal cycles (Q,d,J), 19 4 self-intersection, 69 2 cobordism, 19 7 diffeographisms, 3 , 78, 88, 96, 135 , 13 6 cyclic diffeomorphism group , 18 4 3-cocycle, 202 , 22 4 differental idele , 398 , 406, 671, 672, 69 2 cyclic category, 58 5 differential algebra , 12 5 cyclic cohomology, 5 , 194 , 579, 589, 614 differential equations , 97 , 101 , 103, 10 5 fundamental exac t sequence , 19 7 existence o f Solutions, 101 , 106 periodic, 19 7 monodromy, 4 , 10 1 cyclic covering , 70 6 nonlinear elliptic , 715 , 718 cyclic homology , 616 , 638, 657 ordinary, 12 4 cyclic modules, 577 , 582, 585, 636, 65 3 differential field, 99 , 105 , 12 4 cyclic morphism, 638 , 65 5 differential form s cokernel, 648 , 65 7 algebraic, 15 7 cyclic object, 58 6 differential Galoi s grou p cyclic separat ing vector, 61 9 solvable, 12 6 cyclotomic differential Galoi s theory, 4 , 12 3 action, 441 , 514 differential ideal , 12 5 condition, 55 1 differential module , 12 4 extension, 513 , 524 differential Operato r field, 440 , 473, 475, 477, 510 , 517, polynomial, 4 9 524, 71 1 differential Operators , 71 2 polynormal, 47 7 ring, 5 0 theory, 47 6 differential Systems , 103 , 10 5 tower, 479 , 54 5 regular-singular, 12 7 dilaton field, 5 , 227, 22 8 D-modules, 12 7 dimension spectrum , 7 , 191 , 198, 199, Darboux, 44 2 201, 207 , 208, 210, 223, 228, 233, de Rham cohomology , 138 , 139 , 15 7 324, 327 , 329, 33 1 Dedekind simple, 191 , 198 7] function , 512 , 52 9 dimensional analysis , 10 , 52, 59, 87, 9 1 domain, 552 , 55 8 dimensional regularizatio n (DimReg) , 1 , ring, 56 7 3, 7 , 46, 48, 53, 64, 91, 370, 43 5 zeta function , 439 , 442, 478, 558, 559, geometry, 5 2 563, 57 5 dimensionless, 9 1 deficiency indices , 35 9 DimReg+MS, 3 , 46 deformation complex , 71 6 Dirac, 8 , 11 , 27 degree Dirac Hamiltonian, 72 7 of correspondences, 58 0 Dirac masses , 16 2 770 INDEX Dirac Operators , 6 , 190 , 71 4 Ecalle, 12 3 Classification, 6 , 246-25 2 Edwards, 34 5 moduli space , 252-25 5 effective action , 2 , 37, 39, 42, 43, 58, 723 direct limit , 594 , 622, 64 0 effective divisors , 676 , 679, 682, 714, 71 5 directed System , 59 8 effective fiel d theory , 5 disconnected graphs , 3 6 effective potential , 723 , 72 6 discrete symmetries , 16 3 renormalized, 72 4 discriminant, 671 , 692 effectiveness, 679 , 68 2 distillation, 616 , 636, 648, 658, 73 1 Einstein, 8 distilled A-module , 636 , 64 8 Einstein-Yang-Mills, 22 0 distinguished triangles , 58 5 Einstein equation , 18 0 distribution, 38 , 65, 7 9 Einstein's fluctuation formula , 8 external structure , 7 9 Einstein-Yang-Mills action , 5 homogeneous, 344 , 394 , 409, 410, 41 4 Einstein-Hilbert action , 18 0 of prime numbers , 34 5 Eisenstein Operator valued , 1 7 series, 438, 441, 462, 464, 506, 519, tempered, 366-368 , 417, 419 527, 529 , 727, 73 1 Weil, 34 3 summation, 46 3 distribution theoreti c trace , 580 , electromagnetic field , 8 , 2 9 electromagnetism, 2 , 29 , 162 , 16 4 686-688 electron, 8 , 22 , 27, 29, 162 , 163 , 174, divergences, 7 , 8 , 30 , 52, 107 , 113 , 12 8 287 geometric characterization , 10 3 electroweak, 163 , 164 , 170 , 171, 174, non-local, 5 5 319, 321, 322 non-local coefficients , 5 9 electroweak scale , 72 5 division algebra , 6 , 255 , 310, 423, 428, elementary C*-algebra s 434 continuous section , 63 6 division relations , 440 , 441, 466, 467, field of , 63 6 469, 470 , 52 7 elementary particl e physics , 4 , 7 , 4 6 divisors, 580 , 676, 678, 71 2 elliptic curve , 149 , 610, 680 , 68 5 complement, 12 7 elliptic curves , 61 4 effective, 67 6 elliptic functions , 438 , 462, 51 8 linear equivalence , 67 9 emergent phenomenon , 72 9 principal, 67 6 endomorphisms, 217 , 441, 450-452, 470 , Dixmier trace , 191 , 201, 225 482, 496-500, 527 , 541, 545, 561, Dixmier-Douady invariant , 636 , 63 7 562, 577 , 578, 593, 594, 598 , 607, Doplicher, 12 3 608, 610 , 709 , 711, 712 Doplicher-Haag-Roberts, 45 0 inner, 450 , 56 0 double commutant, 616 , 73 4 endomotives, 381 , 441, 560, 578 , 594, double complex , 58 9 637, 648 , 699, 710, 72 2 double dual , 45 2 algebraic, 578 , 594, 615 , 622 dual system , 481 , 482, 579, 580, 615, analytic, 578 , 615 622-624, 699 , 73 1 uniform, 578 , 615, 62 2 irreducible representations , 62 7 energy-momentum conservation , 18 1 regularity, 62 7 enveloping abelia n group , 59 4 dual trace, 632 , 633, 646, 65 2 equations characters, 63 4 regular-singular, 12 7 kernel, 633 , 636, 64 3 equations o f motion, 9 , 294 , 32 0 dual weight , 63 2 equilibrium duality, 120 , 147-149 , 15 3 unstable, 7 5 Dwork, 14 4 equisingular, 10 8 Dyson, 20 , 9 7 equisingular connections , 4 , 96, 127 , Dyson-Schwinger equation , 16 0 345, 43 5 INDEX 771 and Li e algebr a elements , 10 9 non-linear dependence , 7 9 Classification, 109 , 11 9 external structure , 3 , 56, 58, 66, 7 8 equivalence, 112 , 11 6 distributions, 6 5 flat, 109 , 11 0 external tenso r products , 58 4 geometric form , 11 8 equisingular flat vecto r bundles , 128 , factors, 481 , 622, 73 6 156 Classification, 622 , 62 5 equisingular vecto r bundles , 4 continuous decomposition , 63 2 equisingular W-connection , 12 9 Faddeev-Popov ghost , 17 2 equivalence relatio n family index , 71 4 homological, 14 2 Fermi energy , 29 5 numerical, 14 0 Fermilab, 16 6 rational, 14 1 fermion doubling , 6 ergodic, 343 , 651 fermion propagato r ergodic flows, 62 5 Euclidean, 72 7 etale fermionic action , 275-28 0 action, 597 , 59 9 fermions, 6 , 47 , 161 , 164, 724, 72 6 cohomology, 579 , 671, 677 Dirac, 175 , 17 6 correspondence, 597 , 60 0 Majorana, 175 , 176 , 27 9 covers, 12 2 Feynman, 1 1 groupoid, 454 , 488, 556, 598, 599, 63 9 ie trick, 1 , 2, 2 0 etale cohomology , 137 , 139 , 144 , 14 5 diagrams, 2 2 Euclidean field theory , 2 1 integral, 1 , 15 9 Euclidean signature , 21 , 37, 18 1 integral ove r histories , 1 7 Euler Feynman graphs , 1 , 2 , 22 , 31, 60, 62, Archimedean factor , 65 0 79, 159 , 72 6 characteristic, 120 , 145 , 184 , 223, extended, 6 0 271, 68 4 Hopf algebra , 7 1 constant, 345 , 368, 369, 39 5 discrete, 7 1 factor, 144 , 434 füll, 7 9 product, 144 , 345, 353, 422, 434, 664, subgraphs, 57 , 6 2 675 Feynman rules , 33 , 38, 48, 57, 65, 89 totient function , 466 , 474 fiber functor , 121 , 135, 146 , 158 , 592 evaluation, 12 , 86, 540 , 70 6 automorphisms, 12 1 evaluation map , 66 9 fiber product , 597 , 611, 613 evanescent, 32 6 over a groupoid, 597 , 60 0 even grading , 7 0 field o f coefficients , 62 4 exact sequence , 657 , 73 9 field o f constants, 99 , 105 , 12 4 excess intersection, 71 8 field with on e element, 69 9 expansional, 9 7 filtered vecto r bündle , 12 8 ODE, 9 7 filtered vecto r Spaces , 12 9 expectation value , 11 , 21, 165, 282, 294, filtration, 50 , 152 , 51 1 296, 44 2 fine structure constant , 168 , 282 Ext functor , 155 , 589, 59 0 finite extended Feynma n graphs , 6 0 adeles, 55 1 extension o f scalars, 13 7 ideles, 540 , 55 9 external legs , 38-40, 65 , 74, 8 9 place, 397 , 56 6 ordering, 7 8 finite field, 61 0 external lines , 31, 33, 34, 43, 60, 62, 78, finite noncommutativ e geometry , 6 , 232, 81 251, 255 , 293, 311, 314 external momenta , 38 , 39, 53, 55, 64, irreducible, 31 4 723 finiteness, 4 3 differentiation, 5 6 fixed points , 118 , 14 2 772 INDEX flatness, 99 , 110 , 73 0 Galois flavor action, 43 8 index, 16 8 cosmic Galoi s group , 9 5 flow, 580, 686 , 688, 690 Grothendieck-Galois correspondence , Fock space , 4 1 146, 14 9 formal diffeomorphisms , 8 8 Galois action , 437 , 442, 474, 475, 524, coordinates, 8 9 591, 600 , 601, 609, 615, 649, 675, formal primitive , 12 5 711, 71 2 formal Solution , 102 , 12 5 Galois group, 122 , 158 , 578, 57 9 Fourier components , 1 5 absolute, 13 8 Fourier Inversio n formula , 1 9 cosmic, 12 8 Fourier transform , 18 , 35, 38, 41, 48, 55, differential, 96 , 123 , 13 6 341, 343 , 357, 359, 363, 366-368, motivic, 4 , 70 , 136 , 146 , 149 , 155 , 372, 373 , 377, 379, 392, 394, 399, 158, 59 1 401, 404 , 409, 414, 417, 419-421, representations, 59 2 424, 432, 609, 629, 631, 635, 640, Tannakian, 12 1 643, 646 , 650 , 660, 670, 691, 694 Galois symmetries, 9 6 Fredholm, 715 , 716 Galois theory, 146 , 438, 741-74 7 free energy , 17 , 36, 72 6 Gamma function , 47 , 5 4 free field, 10 , 13 , 38, 72 6 gaps, 41, 705 free field theory , 2 0 gauge, 170 , 28 4 free Gaussia n bosons, 162 , 167 , 203, 232, 265, 280, normalization factor , 1 8 310, 72 6 BRS, 17 4 free grade d Li e algebra, 130 , 15 5 classes, 71 9 Fricke functions , 50 6 conjugate, 10 6 Frobenius, 142-145 , 422, 434, 471, 472, coupling constants , 16 2 579, 580 , 610 , 616, 671, 675, 677, equivalence, 108 , 112 , 115 , 116 , 12 3 680, 684, 695, 699, 712 , 72 2 Feynman, 168 , 172 , 173 , 298 eigenvalues, 145 , 678 Feynman-'t Hooft , 168 , 172 , 17 3 geometric, 14 4 Feynman-'t Hooft , 16 9 orbits, 68 0 fields, 164 , 171 , 209 Fubini theorem , 580 , 693, 694 fixing, 5 , 173 , 18 3 füll subcategory , 65 3 group, 123 , 172, 186 , 206, 232 , 48 3 function fields, 398 , 400, 422, 434, 439, harmonic, 18 3 471, 473, 580, 610, 624, 651, 660, invariance, 184 , 201, 202, 206, 73 1 671, 67 4 local gauge transformations, 16 4 functional equation , 144 , 677 non-abelian theories , 46 , 47, 16 3 functional integral , 7 , 309-318, 726 , 73 0 Potential, 205 , 270, 32 5 Euclidean, 45 , 309-31 8 evanescent, 32 6 functor, 67 , 69, 12 0 Potentials, 219 , 22 3 additive, 12 0 symmetries, 164 , 31 8 exact, 120 , 15 9 theories, 53 , 172, 23 1 Ext, 152 , 15 5 theory, 71 5 faithful, 120 , 15 9 transformations, 127 , 164 , 172 , 190 , fiber, 121 , 135 , 14 6 201, 202 , 280, 71 5 tensor, 12 1 chiral, 231 , 325 functors Gaussian integral , 47 , 48 of sheaves, 15 3 Gaussian Integration , 3 fundamental domain , 385 , 390 Gaussian measure , 16 , 21, 22 fundamental group , 122 , 73 0 imaginary, 1 8 profinite, 12 2 Gelfand wild, 123 , 12 6 transform, 73 4 INDEX 773 Gelfand triple , 31 7 gravitational sector , 72 7 Gelfand-Naimark, 598 , 73 4 graviton, 184 , 20 3 Gell-Mann matrices , 168 , 169 , 267 , 274, gravity, 162 , 180 , 231, 232, 31 0 285 observables, 20 6 Gell-Mann-Low formula , 2 , 1 8 unimodular, 18 3 general position , 612 , 61 3 Graßmann variables , 27 6 generalized section , 68 6 Green's functions , 2 , 18 , 35, 39, 41, 725 generating function , 35 , 36, 14 2 2-point, 1 9 generations, 6 , 161 , 245, 275 Euclidean, 22 , 31, 36 genus, 71 3 momentum space , 4 2 geometric correspondence , 597 , 611, 612 temperature, 72 5 geometric series , 3 9 time-ordered, 17 , 2 0 germs, 8 3 Gross-'t Hoof t relations , 96 , 10 2 ghost, 5 , 168-170 , 172-174 , 30 0 Grothendieck, 122 , 137 , 139 , 144 , 146 , Gibbs 147, 15 0 ensemble, 44 2 Grothendieck decomposition , 8 2 measure, 439 , 443, 739 Grothendieck group , 151 , 583, 734 states, 443 , 619, 72 5 Grothendieck ring , 15 9 Gilkey, 21 0 Grothendieck Standar d conjectures , 4 Gilkey's theorem , 5 , 216-21 7 Grothendieck-Galois, 146 , 149 , 591, 592 GL(2) System , 438 , 439, 441, 448, Grothendieck-Teichmüller, 9 5 483-548 ground states , 44 7 global character, 67 2 group global field, 343 , 396, 398, 421, 579, modulated, 364-36 5 624, 69 0 group ring , 60 8 definition, 47 1 group-like elements , 6 9 gluons, 163 , 164 , 170 , 274, 28 6 groupoid, 440 , 442, 455-458, 481, 488, GNS construction, 44 5 489, 577 , 581, 596, 651, 654, 72 9 GNS representation, 445 , 451, 494, 616, C*-algebra, 456 , 457, 482, 55 6 618-620, 624 , 62 6 algebra, 49 0 Goldstone bosons , 173 , 232 law, 48 9 Goncharov, 155 , 15 9 units, 73 0 Gordon, 37 8 Grössencharakter, 343 , 344, 397, 398, Ha, 442 , 57 5 407 Haar measure , 624 , 634 , 640 , 643, 652, graded-commutativity, 14 8 688, 689 , 693, 697 grading additive, 363 , 366, 392, 398, 412 internal lines , 7 7 dual, 354 , 356, 37 7 line number, 7 7 multiplicative, 357 , 390, 409, 41 8 loop number, 7 7 normalized, 342 , 364-366, 370 , 398, vertices, 7 7 399, 41 5 grading Operator , 90 , 102 , 107 , 110 , 115, selfdual, 366 , 368, 401, 403 132, 135 , 71 1 Hadamard, 34 7 grand unifie d theories , 6 , 281-283 , 309 half-lines, 32 , 7 0 graph insertion , 7 8 Hamiltonian, 1 , 8 , 9 , 12 , 14 , 15 , 21, graphs 341-343, 352-354 , 356 , 357, 361, automorphisms, 3 5 442, 444 , 451, 475, 532, 539, 559, connected, 2 , 37, 39, 41 563, 619 , 620, 629, 632 , 699, 701, geometric realization, 31 , 34, 6 3 725, 72 7 one-particle irreducible , 2 , 3 7 Dirac, 20 7 pairings, 3 2 shift, 70 6 planar, 3 1 spectrum, 1 6 symmetry factor , 3 4 Hamiltonian formulation , 1 774 INDEX hard Lefschet z condition , 13 9 homology, 31 1 harmonic oscillator , 14 , 1 5 Hodge, 71 2 Hasse, 67 8 Hodge-type conjecture , 15 0 Hasse-Weil, 143 , 156 mixed Hodg e structure, 152 , 153 , 15 6 heart, 59 0 mixed Hodg e structures, 345 , 435 heart o f a t-structure, 15 3 numbers, 344 , 42 3 Hecke structure, 423 , 479, 57 4 algebra, 440 , 459-462, 482, 521, 527, Hodge conjecture , 14 0 618 Hodge decomposition , 15 6 correspondence, 438 , 52 1 Hodge filtration, 156 , 15 8 modular algebra , 43 9 Hodge structures, 138 , 152 , 15 6 Heisenberg, 8 , 186 , 73 8 mixed, 4 time evolution , 2 mixed Täte , 15 7 Hermite-Weber approximation , 378-38 1 pure, 15 7 Hermite-Weber functions , 343 , 360, Täte, 15 7 378, 379 , 38 1 Hodge theory, 15 6 Higgs, 173 , 232, 252, 292 , 295, 310 holomorphic bundle s o n the sphere , 8 1 abelian theory , 71 5 holomorphic function , 44 6 coupling, 273 , 280 holomorphic function s field, 6 , 165 , 167 , 30 8 boundary values , 8 1 from inne r fluctuations, 26 1 germs, 8 3 heavy, 7 holomorphic modules , 63 4 mass, 167 , 302 , 304, 305, 30 9 holonomy, 38 6 mass terms, 273 , 274 homogeneous components , 132 , 13 5 mechanism, 232 , 32 1 homogeneous distributions , 64 5 minimal coupling , 179 , 274, 280 , 30 0 homogeneous elements , 9 3 Potential, 165 , 273, 274 Hopf algebra , 6 7 rescaling, 28 2 commutative, 3 , 67, 68, 70, 130 , 156 , scattering parameter , 168 , 304 158 vacuum, 282 , 294-29 6 connected, 7 7 Higgs field Connes-Kreimer, 3 , 7 1 effective potential , 72 3 dual, 69 , 81, 90 high temperature, 442 , 474, 543, 569, graded, 7 7 572, 581, 618, 619, 723, 725, 726, graded connected , 6 8 729 graded-commutative, 7 0 higher Cho w groups , 61 4 homomorphism, 8 8 Hubert of Feynman graphs , 6 7 12th problem, 472 , 47 6 over Q , 7 6 symbol, 57 4 Huygens principle , 37 9 transform, 372 , 39 4 hydrodynamics, 2 , 27, 2 9 Hubert 21s t problem , 12 7 hypercharges, 6 , 165 , 170 , 240, 243, 245, Hubert das s field, 61 0 265, 266 , 268, 275, 285, 321, 322 Hubert modules , 58 2 hypergeometric functions , 4 7 Hilbert's Nullstellensatz , 4 9 hypersurface, 4 9 Hilbert-Schmidt norm, 37 7 idele dass group , 343 , 384, 398, 462, Operator, 376 , 62 1 483, 56 1 Hironaka, 15 1 action, 69 0 Hochschild, 183 , 193 , 195 , 197 , 202, idempotent, 594 , 60 6 214, 225 , 227, 33 5 character o f Z*, 64 8 cocycle, 20 1 characters o f Ck,i, 65 8 cohomology, 5 , 194 , 201, 614 idempotent U(N), 660 , 66 6 dimension, 31 1 incompressible fluid, 2 7 INDEX 775 index, 71 8 jets, 661 , 669 index theorem, 712 , 72 2 Jordan, 8 index theory, 71 2 inertia, 14 4 K-orientation, 61 1 inertial mass, 2 9 K-theory, 612 , 614, 714 , 73 4 infinitesimal deformations , 71 5 algebraic, 154 , 155 , 614 infinitesimal disk , 82 , 91, 106, 107 , 110 , topological, 61 4 127 Kahler manifold , 43 1 infinitesimal loop , 8 2 Kümmer extension , 15 4 infinitesimals, 18 8 Künneth infrared cutoff , 35 7 Künneth-type conjecture , 15 1 infrared divergence , 5 4 Künneth formula , 138 , 13 9 inner Källen-Lehmann, 41, 42 automorphisms, 186 , 708, 73 9 Kaluza-Klein, 18 5 fluctuations, 5 , 6 , 203 , 209, 223, 232, Kapranov, 15 2 310 Karoubian envelope, 141 , 584 discrete part, 6 Kasparov, 58 2 Integration b y parts, 19 , 22, 31, 49, 50, cup product, 611 , 613, 72 2 173, 332 , 33 3 Kasparov modules , 582 , 712 , 72 2 pairings, 3 2 cup product, 58 4 interaction direct sum , 58 3 electroweak, 160 , 16 1 even, 58 3 strong, 160 , 161 , 164 homotopy equivalence, 58 3 interaction vertices, 3 9 unbounded, 58 4 intermediate bosons, 17 0 kinetic energy internal edges , 3 8 density, 2 8 internal lines , 31, 33, 60, 77 , 7 8 kinetic term, 11 , 58 grading, 7 0 KK-category, 58 2 intersection form , 72 9 KK-groups, 58 3 intersection number, 141 , 142, 678, 68 4 KK-theory, 577 , 582 , 72 2 intersection pairing, 6 equivariant, 60 4 intersection product, 142 , 592, 612, 613, Klein-Gordon equation, 1 6 678 KMS condition, 2 , 12 , 17 , 21, 446, 448, intertwining, 437 , 440, 442, 474-476, 450, 452 , 495, 534, 535, 539, 565, 479, 482 , 524 , 542 , 563, 601, 609, 566, 628 , 701, 723, 739, 74 0 711, 71 2 KMS states, 437 , 447, 474, 581, 617, invertibility, 72 9 619, 626 , 632, 652, 700 , 706 , 72 7 invertible extremal, 447 , 474, 541 , 546, 72 9 K-lattice, 55 8 low temperature, 43 8 Q-lattice, 45 2 zero temperature, 437 , 44 7 irregulär singular, 12 3 knot, 73 2 isogeny, 153 , 486 KO-dimension, 6 , 191 , 233, 234, 251, isometries, 56 2 257, 275 , 280, 310, 31 4 isotropy, 35 , 688, 69 0 Kontsevich, 15 2 iterated integrals , 97 , 10 3 KP equations , 61 5 Krajewski, 8 9 Jacobi Kramers, 2 9 j-invariant, 441 , 505 Kreimer, 159 , 16 0 modular function , 50 2 Kronecker-Weber, 440 , 470, 47 3 Jacobians, 145 , 152 , 614, 67 8 Kubo-Martin-Schwinger (KMS) , 12 , Jacobson topology , 629 , 63 0 446 pullback, 63 7 Kuga-Sato varieties, 14 6 Jannsen, 141 , 148 Kuranishi model , 71 5 776 INDEX 1-adic cohomology, 137 , 616, 68 1 Lie algebra, 69 , 80 L-functions, 58 1 free graded , 4 complete, 65 0 linear basis , 7 8 Hasse-Weil, 14 3 structure constants , 16 8 Hasse-Weil, 42 1 Lie bracket, 69 , 78, 81 of algebraic varieties , 142 , 422, 423, limit measure , 60 3 434 linear algebrai c group , 68 , 78, 100 , 12 2 of motives, 14 4 linear equivalence , 580 , 676, 679, 682, with Grössencharakter , 343 , 344, 393, 685 396-397, 407-409, 419-421, 483, linear forms , 80 , 590, 65 9 580, 645 , 648, 651, 658, 673, 692 linearity o n the right , 7 6 Laca, 442 , 54 4 Liouville Lagrange equations , 8 measure, 44 3 Lagrangian, 1 , 5 , 8 , 10 , 13, 21, 23, 31, theorem, 44 6 42-44, 53 , 57, 59, 75, 164 , 172 , 180 , local Euler factors , 14 4 232 local field, 361 , 366, 370, 396, 401, 423, Euclidean, 4 6 470, 471, 624, 69 3 interaction, 5 9 local index , 72 2 interaction part , 1 0 local inde x cocycle , 7 , 131 , 198-200, interaction terra , 31 , 32 335-339 monomials, 32 , 59, 65 local inde x formula , 5 , 131 , 198-20 0 Standard Model , 161 , 166 even case , 19 8 Landau, 35 8 local monodromy , 71 2 Langlands program , 14 6 local region , 1 7 Laplace equation , 2 8 local terms, 4 8 Laplace transform, 20 8 local trace formula , 362 , 36 6 Large Hadron Collider , 16 1 locality, 43 , 56 Larsen, 442 , 54 4 logarithmic derivative , 99 , 10 5 Laurent series , 51, 64, 9 4 loop number, 36 , 38, 39, 77, 78, 86, 90, convergent, 83 , 100 , 12 4 723 formal, 100 , 103 , 124 , 13 4 loops, 4 non- formal, 10 3 classL(G(C),/x), 10 3 Lawson-Michelsohn, 21 0 Classification, 11 1 Lefschetz regulär, 104 , 11 1 fixed poin t formula , 142 , 143 , 677 Lorentz, 2 9 formula, 138 , 13 9 Lorentz group , 4 1 Lefschetz motive , 143 , 147 , 151 , 152 Lorentz invariance , 4 6 Lefschetz trac e formula , 345 , Lorentz metric , 2 , 10 , 1 3 421-424, 427 , 428, 431, 434, 436, Lorentzian signature , 9 , 18 1 437, 483, 580, 69 5 low temperature, 437 , 438, 441, 475, Lefschetz-type conjecture , 15 0 480-482, 532 , 533, 538, 541, 545, Operator, 13 9 546, 563 , 575, 579, 581, 619, 621, Legendre transform, 2 , 8, 9 , 37, 40 700, 729 , 73 1 Lehmann-Symanzik-Zimmermann, 42 , lower semicontinuous , 70 4 44 lepton sector , 73 2 Möbius leptons, 6 , 161-164 , 167 , 168 , 170 , 177 , function, 466 , 47 4 233, 241 , 255, 256, 262, 267, 268, inversion formula , 46 6 275, 279 , 285 , 286, 289, 290, 293, Mackey, 62 5 308, 32 1 Majorana mas s term, 6 , 162 , 176-178 , level, 463, 502, 52 2 251, 252 , 254, 257 , 261, 271, 280, Levi-Civita connection , 216 , 31 3 284, 290 , 293 , 294, 298, 305, 30 8 Lichnerowicz formula , 21 7 Majorana spinors , 17 5 INDEX 777 Malgrange, 12 3 field, 438 , 441, 442, 501-518, 522, Manin, 147 , 159 , 479 524, 56 8 mapping cone , 58 5 automorphisms, 56 8 Martinet, 12 3 specialization, 54 2 Maslov index , 35 3 forms, 441 , 506, 511, 520, 529 , 53 1 mass eigenstates, 16 6 functions, 438 , 441, 454, 501, 502, mass parameter, 3 , 10 7 504, 510 , 511, 513 independence o f counterterms, 9 1 level one, 50 4 mass relation, 6 , 29 2 Hecke algebras, 43 9 mass scale , 52 , 10 7 tower, 545 , 54 8 mass-shell, 41, 46, 5 3 modular automorphis m group , 494 , 617, massless, 5 4 697, 708 , 74 0 matrix mechanics , 18 6 modular field, 74 1 maximal abelia n extension , 383 , 437, modular forms , 14 6 442, 471-473, 56 8 motives, 14 6 maximal compac t subring , 382 , 39 6 modulated group , 69 0 maximal subalgebra , 6 module Maxwell, 23 0 differential, 12 4 measure, 480 , 61 5 of a globa l field, 398 , 416 measured endomotive , 603 , 607 of a loca l field, 366 , 37 0 meromorphic continuation , 49 , 5 1 of a modulated group , 364 , 388, 390, 407, 415, 429, 430, 432 Taylor coefficient s a t p = 0 , 3 of factors, 62 5 meromorphic function s with connection , 124 , 58 8 germs, 8 3 moduli space , 575 , 715, 73 1 metric dimension , 191 , 210, 224 , 233, dimension, 71 8 251, 312 , 31 3 of curves, 146 , 15 9 metric noncommutativ e geometry , 5 of Dirac Operators , 6 , 73 1 Meyer, 642 , 67 2 Mohapatra-Pal, 29 5 Milne, 12 2 momentum conservation , 33 , 38 Milnor-Moore theorem , 69 , 70 , 7 8 momentum space , 4 1 minimal subtractio n (MS) , 1 , 3, 46, 51, momentum variables , 32-3 4 53, 91 , 342, 427 monodromy, 4 , 110 , 123 , 581 Minkowski signature , 3 7 representation, 100 , 123 , 12 7 Minkowski space , 35 , 41, 42, 16 3 trivial, 100 , 106 , 110 , 111 , 128 mixed Hodg e structure, 152 , 156 , 157 , monoidal structure , 141 , 147 159 monomials, 31 , 32, 34, 60, 65, 75, 78, mixed Hodg e structures, 15 8 79, 87 , 9 1 rational, 15 8 interaction, 8 7 real, 157 , 15 8 Morita compactification , 546 , 54 7 variations, 15 7 Morita equivalence , 203 , 255, 457, 458, mixed Hodge-Tat e structure , 15 7 480, 545-547 , 549 , 550 , 556, 572, mixed motives , 151 , 590 573, 579 , 581, 582, 636, 641, 697, mixed Tät e motives , 154 , 15 9 707 periods, 15 9 seif, 5 , 20 4 unramified, 15 5 Morita invariance , 63 8 mixing angles , 164 , 167 , 25 4 Morita-type morphisms , 58 2 Mod invarian t morphisms, 577 , 58 2 factors, 62 5 Moscovici, 43 9 global fields, 62 4 motives, 4 , 120 , 137 , 341, 421-423, 67 8 local fields, 62 4 1-motives, 151 , 152 modular Artin, 150 , 57 7 curves, 442 , 545 , 547, 57 4 Artin motives , 146 , 38 1 778 INDEX effective, 14 7 non-local terms , 6 6 extensions, 15 1 non-perturbative, 72 7 homological equivalence , 14 2 noncommutative Kümmer motives , 154 , 15 7 zero-dimensional space , 63 7 Lefschetz, 14 7 boundary, 54 6 Lefschetz motive , 15 1 space, 341 , 343, 383, 387, 421, 423, mixed, 15 3 434, 438 , 453, 454, 458, 479, 480, mixed motives , 15 1 545, 546 , 572 , 575, 577, 580, 582, mixed Täte , 4 , 70 , 154 , 156 , 15 9 590, 610 , 614 , 657 , 696, 72 9 modular forms , 14 6 noncommutative geometry , 47 , 144 , motivic sheaves , 15 3 156, 161 , 169 numerical effective , 14 1 noncommutative polynomials , 13 0 numerical equivalence , 142 , 14 7 noncommutative tori , 578 , 615 pure, 4 , 141 , 148, 423 Nori, 15 3 rational equivalence , 14 1 normal abelia n extension , 60 2 realizations, 59 3 normal bündle , 68 7 Seattle Conference , 13 7 normalization condition , 8 3 Täte, 138 , 14 7 normalization factor , 21 , 36 Virtual pure, 15 2 nuclear norm , 66 8 weight, 14 1 number field, 154 , 575, 591, 600, 610, with coefHcients , 149 , 15 0 624, 65 1 motivic cohomology , 154 , 614, 61 6 numerical equivalence , 147 , 59 2 motivic decomposition , 143 , 15 4 motivic Galoi s group, 13 6 observables, 2 , 13 , 444 motivic Integration , 15 2 classical, 1 1 moving frame , 2 8 local, 1 7 multiple zet a values , 159 , 16 0 spectral, 31 7 multiplicative group , 68 , 114 , 135 , 148, obstruction, 101 , 715 608, 63 9 odd spin , 6 multiplicative relations , 46 6 ODE, 12 4 multiplier, 445 , 497, 500, 518 , 634 on-shell condition , 2 2 algebra, 444 , 44 8 one-loop correction , 72 5 multivalued maps , 679 , 683, 713 one-parameter group , 12 , 67, 90, 96, composition, 67 9 102, 688, 70 8 Mumford-Tate group , 14 9 one-parameter semigroup , 71 1 muon, 163 , 168 , 17 4 Oppenheimer, 2 7 orbits, 694 , 696, 70 7 naturalness problem , 30 8 order on e condition , 7 , 192 , 193 , 237, negative energy , 2 2 239, 246 , 249 , 250, 258 , 313, 31 6 Neshveyev, 442 , 54 4 order parameter , 16 5 neutral Tannakia n category , 59 2 orientability, 7 , 311, 729 neutrino real, 31 7 masses, 169 , 23 3 orientation, 34 , 3 5 mixing, 169 , 23 3 real, 31 1 neutrino masses , 7 orthogonal case , 31 6 neutrino mixing , 1 , 5 , 161 , 162, 17 4 oscillatory term , 1 9 neutrino scattering , 16 6 outer action , 65 8 neutrinos, 16 3 left-handed, 16 8 Palatini action , 31 3 right-handed, 16 5 parallel transport, 9 7 new physics , 16 1 partial action , 480 , 481, 486, 487, 492, Newton's law , 2 8 496, 547 , 599, 64 0 Noether, 17 1 partial isomorphisms , 596 , 597 , 599 , 60 6 INDEX 779 partition function , 2 , 437-439, 441, 476, poles, 2 , 41, 44, 45, 47, 55, 144 , 184 , 478, 538 , 543, 558, 559, 563, 575, 511, 683 , 705, 71 6 701-703, 70 6 Pollak, 358 , 378 path integral , 1 polylogarithm, 157 , 160 , 475 paths Pontecorvo-Maki-Nakagawa-Sakata homotopy classes , 9 9 (PMNS) matrix , 174 , 176 , 177 , Paugam, 442 , 57 5 254, 30 8 Pauli matrices , 235 , 266, 267, 275, 28 6 Pontrjagin dual , 153 , 357, 392, 399, 412, perfect field , 13 8 458, 477 , 60 9 period isomorphism , 13 8 positive characteristic , 138 , 399, 409, period map , 15 7 471, 624 , 675, 699, 71 2 period matrix , 138 , 15 7 positive energy , 13 , 16 , 699, 701, 702, Kümmer motives , 15 7 730 periodic classica l points , 69 5 positive energ y representation , 44 4 periodic orbits , 580 , 685, 688-690, 695, positivity, 1 , 13 , 21, 110, 118 , 135 , 181, 699, 707 , 73 1 183, 201, 202, 227 , 343, 391, 445, periodicity ma p 5, 193 , 19 7 452, 476 , 572 , 580, 651, 671, 673, periods, 15 7 682, 72 2 perturbation, 61 1 Potential, 8 , 12 , 21, 75, 165, 273, perturbation theory , 20 , 3 8 723-725 perturbative expansion , 1 , 10 , 11, 36, opposite sign , 1 0 726 power counting , 5 8 perturbative renormalization , 128 , 15 9 Powers, 73 7 perverse sheaves , 12 7 prepared graph , 6 6 Pfaffian, 6 , 275-28 0 primitive elements , 69 , 81 phase space , 352 , 354, 356, 381, primitive ideals , 62 9 442-444, 48 2 principal bündle , 107 , 11 6 phase transition, 437 , 443, 448, 476, equivariant, 108 , 114 , 11 6 543, 569 , 572 , 619, 723, 729 principal divisors , 676 , 679, 680, 685, electroweak, 581 , 723 714, 715 , 722 photon, 15 , 22, 27 , 162 , 164 , 17 0 principal values , 366-372 , 672 , 69 2 massless, 246 , 30 1 pro-reductive group , 14 8 physical dimensions , 87 , 9 1 pro-unipotent group , 66 , 68, 78, 83, 90, physical units , 1 0 Picard group , 67 6 91, 128 , 131, 134, 155 , 15 8 Picard-Vessiot, 123 , 12 5 pro-varieties, 578 , 594, 597, 60 0 extension, 12 4 probability amplitude , 1 1 ring, 12 5 probability measure , 480 , 60 3 Planck constant , 10-12 , 18 , 36, 39, 107 , product geometry , 5 119 projective limit , 68 , 78, 99, 100 , 110 , unit o f action, 1 1 122, 577 , 594, 60 3 Poincare duality , 7 , 138 , 139 , 255, 31 6 projective modules , 203 , 313 Poincare group, 4 1 projective System , 102 , 593, 594, 598, Poincare return map , 68 8 600, 60 6 pointwise convergence , 9 3 uniform, 60 3 pointwise index , 714 , 72 2 Prokhorov extension , 60 3 Poisson bracket , 9 prolate spheroida l wav e functions, 342 , Poisson summatio n formula , 643 , 646 359, 377 , 378, 381 polar decomposition , 61 6 propagator, 21 , 24, 31, 33, 36, 44, 8 7 polar part , 5 1 Euclidean, 23 1 projection Operator , 8 5 external, 38 , 42, 47 pole part, 56 , 64 , 66, 9 1 fermion, 19 0 subtraction, 64 , 8 6 pseudo-abelian category , 582 , 584 , 64 8 780 INDEX pseudo-abelian envelope , 141 , 592, 597, rectangular contour , 666 , 66 7 600 recursive relations , 94 , 10 2 pseudo-manifolds, 31 4 reduced algebra , 594 , 596 , 59 8 pure motives , 591 , 592, 61 4 reduction ma p pure states , 601 , 609 cokernel, 70 7 pushforward ränge, 670 , 674, 69 3 of KMS states , 451-452 , 534 , 56 5 Reeb foliation , 73 8 regular-singular, 12 3 q-expansion, 441, 502, 503, 506, 510, regularity condition , 198 , 210, 228 , 31 3 511, 515 , 52 5 regulärization, 43 , 44, 46 quadratic divergence , 72 4 regulator map , 61 4 quantized calculus , 372-375 , 57 7 relativistic wav e equation, 8 quantized differential , 372 , 374, 395, 396 Rennie, 31 4 quantum corrections , 39 , 72 3 renormalizable quantu m field theor y quantum electrodynamic s (QED) , 8 , finite, 13 7 164 super-renormalizable, 13 7 quantum field theory , 1 , 7 , 13 , 29, 38, renormalizable theory , 8 , 59 , 78, 107 , 46, 481, 483, 727 113 algebraic, 2 , 17 , 123 , 450 renormalization, 1 , 2, 27, 31 8 quantum gravity , 581 , 723, 729, 73 1 renormalization group , 3 , 67, 92, 95, observables, 72 3 113, 135 , 158 , 310, 73 1 quantum mechanics , 7 equation, 170 , 283, 293, 297, 306, 30 8 quantum Statistica l mechanics , 437 , universal lift , 13 1 442-452, 481, 615, 700, 723 , 727 renormalization scale , 72 4 quark mixing , 16 6 renormalized value , 3 , 64, 8 6 quark section , 73 2 representation quarks, 6 , 161-164 , 166-168 , 170 , 174 , covariant, 44 4 177, 233 , 242, 245, 251, 254-256, positive energy , 44 4 262, 267 , 268, 274, 275 , 278, 279, regulär, 440 , 460, 482, 49 3 284-286, 289 , 293, 308, 32 1 rescaling, 42 , 8 9 masses, 16 2 residue, 44 , 4 5 top an d bottom , 16 3 residue field, 144 , 624, 67 5 quasi-central approximat e unit , 45 2 resolution, 58 9 quasi-character, 397 , 409, 41 0 restriction map , 579 , 580, 651, 659 quasi-invertible K-lattice , 56 6 ränge, 72 2 quasi-normal pairs , 459 , 48 2 to periodi c orbits , 70 8 quasi-states, 44 9 resummation, 12 6 quaternions, 6 , 232 , 234-236, 240 , 261, Ricci curvature, 180 , 21 0 266, 310 , 422-424, 428 , 429, 431 Riemann, 345 , 346, 349, 35 1 Quillen, 15 4 f function , 346 , 379, 650, 69 4 counting function , 34 2 Radon-Nikodym derivative , 603 , 604 counting function , 34 1 Ramachandran, 153 , 438 explicit formula , 343 , 348 ramification index , 12 6 flow, 352, 35 3 Ramis, 12 3 zeta function , 143 , 145 , 341, 343-347, Ramis exponentia l torus , 126 , 12 8 378, 379 , 383, 424, 433, 475, 476, rapid decay , 50 , 666, 66 8 481-483, 607 , 638, 641, 648, 651, rational algebra , 71 0 731 rational function , 68 4 Riemann curvatur e terms , 27 3 rational homotop y groups , 15 6 Riemann Hypothesis , 145 , 343, 347, real structure, 191 , 192, 205, 73 0 348, 651 , 671, 678, 692 reality condition , 15 , 1 9 function fields, 684 , 685, 71 2 reconstruction o f spin manifolds , 31 3 Weil proof, 67 4 INDEX 781 Weil reformulation, 67 3 space, 358 , 366, 367, 391, 392, 396, Riemann surfaces , 61 5 407, 412, 419, 622, 623, 648 Riemann-Hilbert correspondence , 4 , 94, convolution, 62 8 96, 119 , 123 , 12 6 Schwinger for flat equisingula r bundles , 13 1 functions, 2 0 irregulär, 12 8 functions (2-point) , 2 1 regular-singular, 12 7 Parameters, 3 , 47, 48 Riemann-Hilbert problem , 4 scissor-congruence, 15 1 Riemann-Roch, 580 , 581, 676, 677, 682, secondary invariants , 61 4 698, 713 , 722 see-saw mechanism , 1 , 7 , 161 , 169, 233, Grothendieck, 611 , 612 251, 29 3 Riemann-Siegel, 350 , 37 3 Seeley-DeWitt coefncients , 5 , 209, Riemann-Weil explici t formula , 341, 216-217 349, 396 , 421, 483, 672 self-energy, 8 , 26 , 29, 30, 33, 42, 43, 45, Riemannian geometry , 5 47, 6 3 rigged Huber t space , 31 7 self-energy graph , 2 , 3 Roberts, 12 3 self-maps, 606 , 63 9 roots o f unity, 68 , 473, 475, 477, 478, semi-abelian varieties , 15 2 508, 52 5 semi-classical, 5 , 483 semi-local adele dass space , 383 , 384, 38 8 S-matrix, 40 , 42, 46 trace formula , 343 , 344, 381, 422, Saavendra, 12 2 433, 434 , 436, 577, 638, 671 scalar curvature , 180 , 183 , 206, semidirect product , 7 1 210-216, 228 , 270, 29 4 affine grou p schemes , 114 , 13 0 scalar field, 12 , 13 , 31, 37, 41, 724, 72 6 diffeographisms, 8 0 scalar theory , 7 5 Lie algebra, 8 1 scaling, 90 , 91, 104, 482, 562, 580, 609, semigroup, 49 6 697, 710 , 72 9 unital abelian , 59 4 equivariance, 627 , 635, 637, 641, 648 semigroup crosse d product , 59 8 flow, 356 , 361, 362, 37 0 separable C*-algebras , 58 2 group, 412 , 414, 433 separable closure , 13 7 Hamiltonian, 341 , 342, 35 4 separating vector , 31 4 onHC0(DT(Ac, coupling constants , 16 2 Standard conjectures , 141 , 142, 147 , gravitational parameters , 16 2 148, 150 , 15 4 Higgs mass, 16 2 Hodge-type conjecture , 15 0 Higgs quartic coupling , 16 2 Künneth-type conjecture , 15 1 lepton masses , 16 2 Lefschetz-type conjecture , 15 0 mixing angles , 16 2 Standard Model , 1 , 4, 53 , 160 , 723, 724 phase, 16 2 Lagrangian, 5 , 16 6 QCD vacuu m angle , 16 2 coupled t o gravity , 180 , 280, 72 9 quark masses , 16 2 mathematical input , 16 1 relations, 16 2 minimal, 5 , 16 1 smoothness, 71 5 minimally couple d t o gravity , 5 , 16 1 Sobolev Spaces , 317 , 344, 381, 407, 414, Parameters, 7 , 162 , 233, 254, 281, 308 420 with neutrin o mixing , 16 1 solvability o f equations, 12 5 State, 615 , 617, 61 9 source, 35 , 36, 40 faithful normal , 62 1 space o f dimension z , 7 , 5 2 ground, 44 7 arithmetic model , 7 , 48 1 periodic, 62 5 space-like Separation , 13 , 16, 1 7 regulär, 616 , 63 7 space-time, 11 , 13, 727 thermodynamical equilibrium , 62 2 special fiber, 71 2 state space , 63 0 special relativity, 7 states spectral action , 4 , 5 , 310 , 313, 724, 72 9 equilibrium, 44 6 asymptotic expansion , 6 , 20 8 stationary phase , 4 0 fermionic term , 6 stationary points , 9 spectral correspondences , 581 , 727, 729, stationary value , 8 730, 73 2 Statistical mechanics , 21 , 36, 44 3 spectral lines , 2 7 Stiefel-Whitney dass , 61 1 spectral observables , 31 7 Stokes phenomenon, 12 6 spectral realization , 341 , 343-345, 383, Stone-Cech, 44 4 407-421, 431 , 482, 577, 580 , 648, string theory , 23 4 650, 651, 658 strong continuit y spectral sequence , 15 4 pointwise, 63 8 spectral triple , 5 , 7 , 181 , 190, 72 7 strong convergenc e cup product, 6 , 7 pointwise, 63 7 even, 19 0 strong force , 16 2 finitely summable , 19 1 strong Schwart z space , 580 , 655, 670, manifolds wit h boundary , 73 0 673, 691 , 692, 72 2 metric dimension , 19 1 structure constants , 16 8 real, 191 , 205, 257, 311, 730 structure sheaf , 61 3 real part, 19 2 subdivergences, 3 , 52, 54-56, 59 , 60, 87, regulär, 198 , 210, 228 , 313 91 spectrum subgraph, 6 2 absorption, 341 , 342, 353, 433, 483 subgraphs, 5 7 emission, 341 , 353, 356 external structure , 8 0 spherical harmonic , 2 8 super-renormalizable, 75 , 13 7 spin, 16 3 Suspension, 58 4 spin connection , 219 , 270 , 29 1 symbol, 687 , 68 9 spin manifold , 6 Symmetrie algebra , 7 9 spin structur e Symmetrie produets , 152 , 71 5 antiperiodic, 72 5 symmetries, 310 , 442, 478, 482, 501, spontaneous symmetr y breaking , 16 5 524, 541 , 575, 61 9 stable equilibrium , 7 5 automorphisms, 449 , 475, 50 1 INDEX 783 endomorphisms, 438 , 439, 450, 501, periods, 62 5 709 time-ordered exponential , 4 , 96, 97, 102 , inner, 5 , 449, 45 0 103 internal, 18 6 Todd genus , 61 1 spontaneous symmetr y breaking , 23 2 Tomita, 460 , 579, 617, 626, 73 8 symmetry breaking , 581 , 723, 727 top quark , 29 8 electroweak, 72 9 topological algebras , 59 1 spontaneous, 439 , 443, 482, 569, 72 3 Tor functor , 589 , 613, 657 vacuum, 72 7 torsion, 72 2 symmetry factor , 31 , 34, 36, 52, 8 8 torsion points , 71 0 symplectic case , 31 6 totally disconnected , 578 , 598, 599, 70 5 symplectic unitar y group , 72 9 trace binuclear Operator , 66 8 t Hooft , 4 6 canonical dual , 63 2 t-structure, 153 , 590 dass, 362 , 376, 395, 409, 420, 73 5 tadpole, 25 , 33, 59, 32 9 continuity, 63 6 term o f order a , 20 9 distributional, 342 , 362, 68 6 constant, 16 8 of a correspondence, 68 0 vanishing, 26 , 209 , 224 , 225, 229, 335, on factors , 73 6 338 scaling, 63 3 Takesaki, 617 , 622, 73 9 trace dass , 589 , 619, 628, 654, 66 0 tangent space , 8 trace formula , 341-343 , 345, 349, 362, Tannakian category , 4 , 121 , 136, 148, 366, 381 , 388, 396, 421-424, 427, 158 428, 431, 432, 434, 483, 651, 660, characteristic zer o field, 14 8 691 neutral, 119 , 14 8 cohomological, 67 1 Tannakian formalism , 126 , 146 , 15 6 geometric side , 671 , 691 Täte semi-local, 343 , 381, 422, 433, 434, modules, 440 , 486, 61 0 577, 638 , 671 motives, 138 , 147-14 9 spectral side , 66 0 objects, 15 4 trace pairing , 670 , 67 3 twist, 148 , 15 7 radical, 67 4 tau, 16 3 traces Taylor coefficients , 5 1 cyclic morphisms, 58 7 Taylor expansion , 51 , 54, 94, 11 3 transversality, 611 , 687-689, 692 , 715, Taylor series , 9 3 718 temperature, 57 8 temperature states , 72 3 transverse space , 69 1 tempered growth , 634 , 645 , 666, 66 7 tree-level, 39 , 40, 42 tensor category , 120 , 597, 73 0 triangulated category , 153 , 154, 584, 59 0 tensor product , 16 , 41 trigonometric, 438 , 440, 462, 464, 470, homogeneous elements , 8 4 518 tensor produc t structure , 14 8 trivial correspondences , 679 , 682, 693, tensor structure , 13 4 713 test function , 688 , 68 9 type I , 579 , 61 9 thermodynamics, 57 8 type Io o factor, 61 9 theta functions , 61 5 type II , 481, 622, 699, 72 2 Thomson, 2 9 type III , 481, 618, 619, 622, 69 9 three lin e Interpolation, 62 8 continuous decomposition , 63 2 time evolution , 12 , 444, 458, 480, 482, 493, 532 , 549, 556, 572 , 622, 699, ultraviolet cutoff , 2 , 357, 72 4 727 unbroken phase , 72 9 canonical, 459 , 625 , 73 9 unconditional, 15 0 784 INDEX unification scale , 6 , 171 , 183, 184 , interaction, 6 3 281-283, 293 , 294, 296, 297, 300, vierbein, 219 , 314, 72 9 304, 309 , 724 , 72 9 Virtual dimension , 715 , 71 8 uniform endomotive , 60 3 Virtual particles, 22 , 2 7 uniform System , 60 3 Virtual states, 8 uniformizer, 62 4 Voevodsky, 15 9 unimodularity, 183 , 220, 267-269, 277 , volume form , 31 1 280, 46 0 von Mangold t function , 34 8 unitarity, 1 , 13 , 309, 32 1 von Neumann algebras , 2 , 17 , 616, 618, unitary case , 31 6 619, 622 , 73 4 unitary equivalence , 62 7 vortex configurations , 71 5 universal derivation , 19 4 vortex equations, 58 1 universal dga , 19 4 moduli space , 71 2 universal envelopin g algebra , 70 , 13 0 universal singula r frame , 96 , 130 , 131, W-connection 135 equisingular, 12 9 coefficients, 13 1 W-connections, 128 , 13 2 unramified, 60 6 equivalence, 12 8 unramified extension , 471 , 579, 580, W-equi valence, 13 4 624, 625 , 699 Wald, 18 0 unrenormalized value , 31, 34, 38, 47, 52, Ward identities , 172 , 319, 32 0 724 weak upper half-plane , 12 , 447, 483, 485, 490, currents, 16 4 501, 545-547 , 551 , 553, 574 isodoublet, 16 4 upper triangulä r matrices , 13 2 isosinglet, 16 4 isospin, 16 4 vacuum weak closure , 616 , 62 1 QCD angle , 16 2 weak continuity , 70 3 State, 13 , 26 weak convergenc e vacuum bubble , 24 , 36 pointwise, 63 7 vacuum representation , 16 , 1 7 weak Lefschet z condition , 13 9 vacuum state , 70 0 weak topology , 63 0 vacuum states , 2 , 12 , 13 , 16, 17 , 706, Weierstrass p function , 441 , 473, 519 727 weight, 506 , 511, 520, 697, 73 1 degeneracy, 70 6 weight filtration, 70 , 132 , 153 , 154 , 15 6 normalization, 1 8 Weil, 143 , 145, 438, 462, 645, 67 8 vacuum vector , 1 5 distribution, 409 , 411, 414 valence, 31, 60, 6 3 group, 421, 422, 428-431, 65 7 valuation, 382 , 470, 513, 536, 570, 58 1 principal value , 343 , 396, 401, 426 valuation Systems , 707 , 70 8 explicit formula , 341 , 344, 349, 396, van Suijlekom , 17 1 398, 399 , 406, 421, 428, 579, 651, vanishing, 652 , 69 3 682, 69 2 vanishing cycles , 15 6 Weil cohomology , 139 , 14 1 variational derivative , 3 8 Weil conjectures, 144 , 147 , 67 8 varieties Weil proof , 580 , 581, 674, 678, 683, 685, projective, 139 , 140 , 143-145 , 147 , 692, 712 , 72 2 151, 15 4 Weil-Deligne group , 581 , 712 zero-dimensional, 143 , 591, 594 Weinberg angle , 30 8 Varilly, 31 4 Weyl Veltman, 46 , 169 , 29 5 action, 22 7 vertices, 33 , 57, 60, 74 , 87, 9 1 curvature, 271, 306 2-leg, 5 7 curvature tensor, 184, 223 3-leg, 5 7 mode, 181 spinors, 17 5 Wick rotation, 2 , 20, 21, 181 Witt vectors , 13 8 Wodzicki residue , 19 1 Yang-Mills, 5 , 201, 209, 224, 232 , 273, 274 Einstein-Yang-Mills, 22 0 Yang-Mills functional , 715 , 71 6 York-Gibbons-Hawking, 73 0 Yukawa coupling, 177 , 233, 280, 29 6 Parameters, 6 , 252-25 5 Yukawa coupling , 724 , 72 6 Zariski closed , 6 8 Zariski closure , 123 , 12 6 Zariski topology, 12 6 zero temperature, 437 , 439-442, 446, 447, 451, 474-476, 543 , 567, 569, 706, 72 5 zero-point energy , 72 5 zeros critical, 40 8 nontrivial, 42 4 of £ function, 38 1 of L-functions, 407 , 421, 423, 433 of prolate spheroida l wav e functions , 359 of zeta, 356 , 377-379, 381, 383, 481 zeta functio n Dedekind, 439 , 442, 478, 558, 559, 563, 57 5 function field, 67 5 of a curve , 67 5 Riemann, 341 , 343-347, 378 , 379, 383, 424 , 43 3