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 . For additiona l informatio n an d Update s o n thi s book , visi t www.ams.org/bookpages/coll-55 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 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , or multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e Hindusta n Boo k Agency . Request s fo r permissio n fo r commercial use of material should be addressed to the Hindustan Boo k Agency (India) , P 1 9 Green Park Extention , Ne w Delhi 11 0 016, India. Request s can als o be made by e-mail to hba@vsnl. com. © 200 8 by the Hindusta n Boo k Agency . Al l rights reserved . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 1 3 1 2 1 1 1 0 0 9 0 8 To our love d ones fo r thei r patienc e This page intentionally left blank Contents 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.