Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory PDF
Preview Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory
ESI Lectures in Mathematics and Physics Editors Joachim Schwermer (Institute for Mathematics, University of Vienna) Jakob Yngvason (Institute for Theoretical Physics, University of Vienna) Erwin Schrödinger International Institute for Mathematical Physics Boltzmanngasse 9 A-1090 Wien Austria The Erwin Schrödinger International Institute for Mathematical Phyiscs is a meeting place for leading experts in mathematical physics and mathematics, nurturing the development and exchange of ideas in the interna- tional community, particularly stimulating intellectual exchange between scientists from Eastern Europe and the rest of the world. The purpose of the series ESI Lectures in Mathematics and Physicsis to make selected texts arising from its research programme better known to a wider community and easily available to a worldwide audience. It publishes lecture notes on courses given by internationally renowned experts on highly active research topics. In order to make the series attractive to graduate students as well as researchers, special emphasis is given to concise and lively presentations with a level and focus appropriate to a student's background and at prices commensurate with a student's means. Previously published in this series: Arkady L. Onishchik, Lectures on Real Semisimple Lie Algebras and Their Representations Werner Ballmann, Lectures on Kähler Manifolds Christian Bär, Nicolas Ginoux, Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization Recent Developments in Pseudo-Riemannian Geometry, Dmitri V. Alekseevsky and Helga Baum (Eds.) Boltzmann's Legacy, Giovanni Gallavotti, Wolfgang L. Reiter and Jakob Yngvason (Eds.) Hans Ringström, The Cauchy Problem in General Relativity Emil J. Straube,Lectures on the (cid:1)2-Sobolev Theory of the ∂⎯ -Neumann Problem Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory Alan Carey Editor with the assistance of Harald Grosse and Steve Rosenberg Editor: Alan Carey Mathematical Sciences Institute Australian National University Canberra, ACT, 0200 Australia E-mail: [email protected] 2010 Mathematics Subject Classification (primary; secondary): 58B34, 11M55, 11G09 11M06, 11M32, 47G30, 81Q30, 81T15, 17A30; 16T30, 18D50, 46L80, 46L87, 19K33, 19K56, 58J42, 58J20, 81Q30 Key words: Arithmetic geometry, motives, noncommutative geometry, index theory, K-theory, pre-Lie algebras, Feynman integrals, zeta functions, Birkhoff–Hopf factorisation, renormalisation ISBN 978-3-03719-008-1 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. ©2011 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author's TEX files: I. Zimmermann, Freiburg Printed in Germany 9 8 7 6 5 4 3 2 1 Introduction This volume is a collection of expository articles on mathematical topics of current interest that have been partly stimulated or influenced by interactions with physical theory. Someofthesearticlesarebasedonlecturesgivenatvariousvenuesoverthe last few years, revised and written especially for this volume. They cover a diverse rangeofmathematicaltopicsstemmingfromdifferentpartsofphysics. Themajorunderlyingthemeofthevolumeinvolvestheinteractionsbetweenphysics andnumbertheory. Thisthemeismanifestedintwoways, firstthroughthestudyof Feynmanintegralsandrenormalisationtheory,andsecond,throughtheapplicationof methodsfromquantumstatisticalmechanics. Intheformer,theworkofBogoliubov– Parasuik–Hepp–Zimmermann (BPHZ) on renormalisation theory gave a method for stepbystepcontrolofdivergencesandoftheirregularisationinFeynman’sapproach toperturbativequantumfieldtheory. Already,intheevaluationofFeynmanintegrals, the occurrence of multizeta values hinted at deeper mathematical connections. This deeperunderlyingstructurewasfoundbyAlainConnesandDirkKreimerintheform ofassociatingtoeachrenormalisablequantumfieldtheoryaHopfalgebrathatprovided asystematicunderstandingoftheBPHZprocedureintermsofaBirkhofffactorisation inaLiegroupassociatedtotheHopfalgebra. Acontemporaryviewofthisfundamen- tal work is provided here by Christoph Bergbauer. The latter strand, the relationship betweenstatisticalmechanicsandnumbertheory,beganmuchearlierthroughthework of Jean-Benoît Bost andAlain Connes. Bost–Connes introduced a quantum statisti- calmechanicaldynamicalsystemthatcapturesinformationontheprimesandonthe Riemannzetafunction. Theydeterminedtheequilibriumstatesoftheirmodel(theso- called Kubo–Martin–Schwinger (KMS) states) and its phase transitions. Subsequent extensions of this basic idea to number theoretic questions resulted in the theory of ‘endomotives’. Motives also arise in the study of Feynman integrals and hence we have provided here an introduction to these ideas in the Ramdorai–Plazas–Marcolli article. Wenowgiveabriefoverviewofthecontentsofthisvolume. Bergbauer’sarticlediscussesFeynmanintegrals,regularizationandrenormalization followingthealgebraicapproachtotheFeynmanrulesdevelopedbyBloch, Connes, Esnault,Kreimer,andothers. Itreviewsseveralrenormalizationmethodsfoundinthe literaturefromasinglepointofviewusingresolutionofsingularities,andincludesa discussionofthemotivicnatureofFeynmanintegrals. Motives are explained in much greater detail in the article of Sujatha Ramdorai andJorgePlazas. Theconstructionofthecategoryofpuremotivesisexplainedhere startingfromthecategoryofsmoothprojectivevarieties. Theyalsosurveythetheory ofendomotivesdevelopedbyD.C.CisinskiandG.Tabuada,whichlinksthetheoryof motivestothequantumstatisticalmechanicaltechniquesthatconnectnumbertheory and noncommutative geometry. The appendix to this article, contributed by Matilde Marcolli,elaboratestheselatterideasprovidingausefulintroductionto,andsummary vi Introduction of, the role of KMS states. Also described is the view of motives that arises from noncommutativegeometryalongwithadetailedaccountoftheinterweavingofnumber theorywithstatisticalmechanics,noncommutativegeometryandendomotives. Hopf algebras associated to rooted trees are known to systemetise the combina- toricsofFeynmangraphsthroughtheworkofConnes–Kreimer. DominiqueManchon describesanotheralgebraicobjectassociatedtorootedtrees,namelypre-Liealgebras. His article reviews the basic theory of pre-Lie algebras and also describes how they arisefromoperads. Theirrelationtootheralgebraicstructuresandtheirapplicationto numericalanalysisaredescribedaswell. Multiple zeta values arise from the evaluation of Feynman integrals. Sylvie Pay- cha’s contribution discusses generalisations of renormalised multiple zeta values at nonpositive integers. As with some of the other articles, the exposition is partly in- spiredbyrenormalisedFeynmanintegralsinphysicsusingpseudodifferentialsymbols. Zetaresiduesandpseudodifferentialanalysis, bothofwhichplayaroleinearlier articles, alsoariseinindextheory. Inturn, indextheoryiswellknowntoplayarole ingaugefieldtheoriesviathestudyofanomalies. Thesearedescribedmathematically bythefamiliesindextheorem. UndertheinfluenceofAlainConnesandothers,anon- commutativeapproachtoindextheory,partlyinspiredbyquantumtheory,hasemerged overthelasttwodecades. Thisnoncommutativeindextheoryisdescribedinthecontri- butionofAlanCarey,JohnPhillipsandAdamRennie,beginningwithadiscussionof classicalindextheoremsfromanoncommutativepointofview. Thisisfollowedbya reviewofK-theoryandcycliccohomologythatculminatesinadescriptionofthelocal index formula in noncommutative geometry. The article concludes with an example thatunderliesrecentapplicationsofnoncommutativegeometrytoMumfordcurves. All of the articles in this volume were partly inspired by a program in Number TheoryandPhysicsheldattheErwinSchrödingerInstitutefromMarch2toApril18, 2009. TheeditorwouldliketothankESIforitshospitalityandsupportfortheNumber Theory and Physics program,Arthur Greenspoon for his enthusiastic assistance with proof-reading and Steve Rosenberg and Irene Zimmermann for their very valuable contributionstothepreparationofthisvolume. Contents Introduction.................................................................v NotesonFeynmanintegralsandrenormalization byChristophBergbauer......................................................1 Introductiontomotives. WithanappendixbyMatildeMarcolli bySujathaRamdorai,JorgePlazas,andMatildeMarcolli.......................41 Ashortsurveyonpre-Liealgebras byDominiqueManchon .....................................................89 Divergentmultiplesumsandintegralswithconstraints: acomparativestudy bySylviePaycha...........................................................103 Spectraltriples: examplesandindextheory byAlanCarey,JohnPhillips,andAdamRennie...............................175 ListofContributors ........................................................267 Index.....................................................................269 Notes on Feynman integrals and renormalization ChristophBergbauer Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 FeynmangraphsandFeynmanintegrals . . . . . . . . . . . . . . . . . . . 2 3 Regularizationandrenormalization . . . . . . . . . . . . . . . . . . . . . . 14 4 MotivesandresiduesofFeynmangraphs . . . . . . . . . . . . . . . . . . . 28 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1 Introduction InrecentyearstherehasbeenagrowinginterestinFeynmangraphsandtheirintegrals. Physicists use Feynman graphs and the associated integrals in order to compute certainexperimentallymeasurablequantitiesoutofquantumfieldtheories. Theprob- lemisthatthereareconceptualdifficultiesinthedefinitionofinteractingquantumfield theories in four dimensions. The good thing is that nonetheless the Feynman graph formalismisverysuccessfulinthesensethatthequantitiesobtainedfromitmatchwith thequantitiesobtainedinexperimentextremelywell. Feynmangraphsareinterpreted aselementsofaperturbationtheory,i.e.,asanexpansionofan(interesting)interacting quantumfieldtheoryintheneighborhoodofa(simple)freequantumfieldtheory. One thereforehopesthatabetterunderstandingofFeynmangraphsandtheirintegralscould eventuallyleadtoabetterunderstandingofthetruenatureofquantumfieldtheories, andcontributetosomeofthelongstandingopenquestionsinthefield. AFeynmangraphissimplyafinitegraph,towhichoneassociatesacertainintegral: theintegranddependsonthequantumfieldtheoryinquestion,butinthesimplestcase itisjusttheinverseofadirectproductofrank4quadraticforms,oneforeachedgeof thegraph,restrictedtoareallinearsubspacedeterminedbythetopologyofthegraph. For a general graph, there is currently no canonical way of solving this integral analytically. However,inthissimplecasewheretheintegrandisalgebraic,onecanbe convincedtoregardtheintegralasaperiodofamixedmotive,anothernotionwhichis notrigorouslydefinedasoftoday. AlltheFeynmanperiodsthathavebeencomputed so far, are rational linear combinations of multiple zeta values, which are known to be periods of mixed Tate motives, a simpler and better understood kind of motives. A stunning theorem of Belkale and Brosnan however indicates that this is possibly a coincidenceduetotherelativelysmallnumberofFeynmanperiodsknowntoday: They showedthatinfactanyalgebraicvarietydefinedoverZisrelatedtoaFeynmangraph 2 C.Bergbauer hypersurface(theFeynmanperiodisoneperiodofthemotiveofthishypersurface)in aquiteobscureway. ThepurposeofthisarticleistoreviewselectedaspectsofFeynmangraphs,Feynman integrals and renormalization in order to discuss some of the recent work by Bloch, Esnault, Kreimer and others on the motivic nature of these integrals. It is based on public lectures given at the ESI in March 2009, at the DESY and IHES inApril and June2009,andseveralinformallecturesinalocalseminarinMainzinfallandwinter 2009. Iwouldliketothanktheotherparticipantsfortheirlecturesanddiscussions. Much of my approach is centered around the notion of renormalization, which seems crucial for a deeper understanding of Quantum Field Theory. No claim of originalityismadeexceptforSection3.2andpartsofthesurroundingsections,which isareviewofmyownresearchwithR.BrunettiandD.Kreimer[10],andSection3.6 whichcontainsnewresults. Thisarticleisnotmeanttobeacompleteanduptodatesurveybyanymeans. In particular,severalrecentdevelopmentsinthearea,forexampletheworkofBrown[24], [26],AluffiandMarcolli[3],[2],[1],DorynandSchnetz[35],[76],andthetheoryof ConnesandMarcolli[32],[67]arenotcoveredhere. Acknowledgements. I thank S. Müller-Stach, R. Brunetti, S. Bloch, M. Kontse- vich, P. Brosnan, E.Vogt, C. Lange,A. Usnich, T. Ledwig, F. Brown and especially D. Kreimer for discussions on the subject of this article. I would like to thank the ESIandtheorganizersofthespring2009programonnumbertheoryandphysicsfor hospitalityduringthemonthofMarch2009, andtheIHESforhospitalityinJanuary andFebruary2010. MyresearchisfundedbytheSFB45oftheDeutscheForschungs- gemeinschaft. 2 Feynman graphs and Feynman integrals Forthepurposeofthisarticle,aFeynmangraphissimplyafiniteconnectedmultigraph where“multi”meansthattheremaybeseveral,paralleledgesbetweenvertices. Loops, i.e., edgesconnectingtothesamevertexatbothends, arenotallowedinthisarticle. Roughly, physicists think of edges as virtual particles and of vertices as interactions betweenthevirtualparticlescorrespondingtotheadjacentedges. Ifonehastoconsiderseveraltypesofparticles,onehasseveraltypes(colors,shapes etc.) ofedges. HereisanexampleofaFeynmangraph: (1) ThisFeynmangraphdescribesatheoreticalprocesswithinascatteringexperiment: a pair of particles annihilates into a third, intermediate, particle, and this third particle thendecaysintothetwooutgoingparticlesontheright.
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