Geometric and Topological Methods for Quantum Field Theory PDF
Preview Geometric and Topological Methods for Quantum Field Theory
Lecture Notes in Physics EditorialBoard R.Beig,Wien,Austria W.Beiglböck,Heidelberg,Germany W.Domcke,Garching,Germany B.-G.Englert,Singapore U.Frisch,Nice,France P.Hänggi,Augsburg,Germany G.Hasinger,Garching,Germany K.Hepp,Zürich,Switzerland W.Hillebrandt,Garching,Germany D.Imboden,Zürich,Switzerland R.L.Jaffe,Cambridge,MA,USA R.Lipowsky,Golm,Germany H.v.Löhneysen,Karlsruhe,Germany I.Ojima,Kyoto,Japan D.Sornette,Nice,France,andLosAngeles,CA,USA S.Theisen,Golm,Germany W.Weise,Garching,Germany J.Wess,München,Germany J.Zittartz,Köln,Germany TheEditorialPolicy forEditedVolumes TheseriesLectureNotesinPhysics(LNP),foundedin1969,reportsnewdevelopments inphysicsresearchandteaching-quickly,informallybutwithahighdegreeofquality. Manuscriptstobeconsideredforpublicationaretopicalvolumesconsistingofalimited number of contributions, carefully edited and closely related to each other. 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LNPHomepage(springerlink.com) OntheLNPhomepageyouwillfind: −TheLNPonlinearchive.Itcontainsthefulltexts(PDF)ofallvolumespublishedsince 2000.Abstracts,tableofcontentsandprefacesareaccessiblefreeofchargetoeveryone. Informationabouttheavailabilityofprintedvolumescanbeobtained. −Thesubscriptioninformation.Theonlinearchiveisfreeofchargetoallsubscribersof theprintedvolumes. −Theeditorialcontacts,withrespecttobothscientificandtechnicalmatters. −Theauthor’s/editor’sinstructions. H. Ocampo S. Paycha A. Vargas (Eds.) Geometric and Topological Methods for Quantum Field Theory ABC Editors ProfessorDr.HernanOcampo SylviePaycha UniversidaddelValle UniversitéBlaisePascal(ClermontII) ApartadoAereo25360 LaboratoiredeMathématiques Cali,Columbia ComplexedesCézeaux [email protected] 63177AubiereCedex,France [email protected] AndrésVargas UniversityofBonn InstituteofAppliedMathematics WegelerStr.6 53115Bonn,Germany [email protected] H.Ocampoetal.,GeometricandTopologicalMethodsforQuantumFieldTheory,Lect. NotesPhys.668(Springer,BerlinHeidelberg2005), DOI10.1007/b104936 LibraryofCongressControlNumber:2005923771 ISSN0075-8450 ISBN-10 3-540-24283-X SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-24283-3 SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springeronline.com (cid:1)c Springer-VerlagBerlinHeidelberg2005 PrintedinTheNetherlands Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorsandTechBooksusingaSpringerLATEXmacropackage Coverproduction:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:11374060 54/3141/jl 543210 A la memoria de Chepe Escobar Por su gran contribucio´n al desarrollo de las matema´ticas en Colombia y a su reconocimiento a nivel internacional. Preface This volume offers an introduction to some recent developments in several active topics at the interface between geometry, topology and quantum field theory: – String theory and Calabi-Yau manifolds – Knot invariants and configuration spaces – Euclidean quantum field theory and noncommutative geometry – Index theorems and noncommutative topology It is based on lectures delivered during a summer school “Geometric and Topological Methods for Quantum Field Theory” held at Villa de Leyva, Colombia,inJuly2003.Theshortcommunicationspresentedbyparticipants of the school appeared separately in the Annales Math´ematiques Blaise Pas- cal (Vol. 11 n.2, 2004) of the University Blaise Pascal in Clermont-Ferrand, France. These lecture notes, which are aimed at graduate students in physics or mathematics, start with introductory material before presenting more ad- vancedresults.Eachchapterisself-contained andcanbereadindependently of the others. The volume starts with an introductory course by Christine Lescop on knot invariants and configuration spaces, in which a universal finite type in- variant for knots is constructed as a series of integrals over configuration spaces. It is followed by an introduction by Raimar Wulkenhaar to euclidean quantum field theory from a statistical physics point of view, after which theauthordiscussespossiblerenormalizationtechniquesonnoncommutative spaces. The third lecture by Anamar´ıa Font and Stefan Theisen presents an elementary introduction to (string) compactification with unbroken super- symmetry. The authors show how this requirement leads to internal spaces ofspecialholonomyanddescribeCalabi-Yaumanifoldsindetail.Stringcom- pactification on orbifolds are presented as particular examples. The last lec- ture by Thierry Fack is devoted to a K-theory proof of the Atiyah-Singer index theorem and discusses some applications of K-theory to noncommuta- tive geometry. We hope that these lectures will give – as much as the school itself seems tohavegiven–youngstudentsthedesiretopursuewhatmightbetheirfirst acquaintance with some of the problems on the edge of mathematics and VIII Preface physics presented in this volume. On the other hand, we hope that the more advanced reader will find some pleasure reading about different outlooks on related topics and seeing how well-known geometric tools prove to be very useful in some areas of quantum field theory. We are indebted to various organizations for their financial support for this school. Let us thank first of all the French organizations “C.I.M.P.A.” andECOS-Nord,withoutwhichthisschoolwouldnothavetakenplace.This schoolispartofalong-termscientificprogramsupportedbyECOS-Nordbe- tween the Universit´e Blaise Pascal in Clermont-Ferrand and the Universidad de los Andes in Bogota´ in the areas of mathematics and physics. We are also deeply grateful to the I.C.T.P. in Trieste, for its constant financial support overtheyearsandespeciallyforthisschool.Wealsoaddressourthankstothe French Embassy in Bogot´a, and particularly to the former cultural attach´e, Jean-YvesDeler,forshowinginterestinthisprojectandsupportingusatdif- ficulttimes.WearealsoverymuchindebtedtotheUniversidaddelosAndes, which was our main source of financial support in Colombia. Other organi- zations such as C.L.A.F. in Brasil, Colciencias, I.C.E.T.E.X., and I.C.F.E.S. in Colombia, also contributed in a substancial way to the financial support needed for this school. Special thanks to Sergio Adarve (Universidad de los Andes), coorganizer of the school, who dedicated much time and energy to make this school possible in a country like Colombia where many difficulties are bound to arise along the way due to social, political and economic problems. This school would not have taken place without him. Our special thanks to Jos´e Rafael Toro, vice-rector of the Universidad de los Andes, for his constant encouragementsandefficientsupport.FromtheUniversidaddelosAndeswe also want to express our sincere gratitude to Carlos Montenegro, head of the DepartmentofMathematics,RolandoRolda´n,deanoftheFacultyofScience, and Bernardo Go´mez, head of the Department of Physics. We are extremely grateful to Juana Vall-Serra who did an extraordinary jobforthepracticalorganizationoftheschool,thequalityofwhichwasvery much appreciated by participants and lecturers. We are also very indebted to Marta Kovacsics, Alexandra Parra, Germa´n Barraga´n, Mar´ıa Fernanda Go´mez,Andr´esGarc´ıa,JuanEstebanMart´ınandAnderssonGarc´ıafortheir helpwithvariousessentialtasksneededforthesuccessfuldevelopmentofthe school and to the academic coordinators, Alexander Cardona and Andr´es Reyes, who jointly with Bernardo Uribe and Mario Fernando H. Arias par- ticipatedactivelywiththeirpermanentassistanceandcollaboration.Without thepeoplenamedhere,allofwhomhelpedwiththeorganizationinsomeway oranother,before,duringandaftertheschool,thisscientificeventwouldnot Preface IX have left such vivid memories in the lecturers’ and the participants’ minds. Last but not least, thanks to all the participants who gave us all, lecturers andeditors,theimpulsetopreparethisvolumethroughtheenthusiasmthey showed us during the school. February 2005 Hern´an Ocampo Sylvie Paycha Andr´es Vargas
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