Geometry and Physics: A Festschrift in Honour of Nigel Hitchin, Volume I PDF
Preview Geometry and Physics: A Festschrift in Honour of Nigel Hitchin, Volume I
geometry and physics Geometry and Physics AFestschriftinHonourofNigelHitchin Editedby JØRGEN ELLEGAARD ANDERSEN ANDREW DANCER OSCAR GARCÍA-PRADA VOLUME I 1 3 GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©OxfordUniversityPress2018 Themoralrightsoftheauthorshavebeenasserted FirstEditionpublishedin2018 Impression:1 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressControlNumber:2018940609 TwovolumesetISBN 978–0–19–880200–6 VolumeoneISBN 978–0–19–880201–3 VolumetwoISBN 978–0–19–880202–0 DOI:10.1093/oso/9780198802006.001.0001 Printedandboundby CPIGroup(UK)Ltd,Croydon,CR04YY LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials containedinanythirdpartywebsitereferencedinthiswork. PREFACE InSeptember2016,aconferencewasheldinthreevenues(Aarhus,Oxford,Madrid)to markNigelHitchin’sseventiethbirthdayandtohonourhisfar-reachingcontributionsto geometryandmathematicalphysics. ThethreelocationschosenareallplaceswithwhichNigelhasbeencloselyassociated.He hasplayedakeyroleinthedevelopmentoftheCentreforQuantumGeometryofModuli Spaces(QGM)atAarhusandtheInstitutodeCienciasMatemáticasatMadrid.Hehasspent muchofhiscareeratOxford,firstasastudentworkingwithMichaelAtiyahandBrianSteer, thenasUniversityLecturerandFellowofStCatherine’sCollegeand,from1997,asSavilian ProfessorofGeometry. Inthesevolumes,wehaveincludedawideselectionofarticlesbasedontalksgivenin theconference,aswellasarticlesbymathematicians(SimonDonaldson,CarlosSimpson, Shing-TungYau)whoseworkhasbeencloselyassociatedwithNigel’sinterestsbutwhowere unabletoattendinperson. Whilethetalksspanawiderangeofgeometrictopics,reflectingthebreadthofNigel’s contributions,themeetinginAarhushadaparticularfocusonRiemanniangeometryand quantization,whilethatinMadridwasparticularlyconcernedwiththethemesofHiggs bundlemodulispacesandgeneralizedgeometry. Inmoredetail,thevolumesincludethefollowingtopics.ThearticlesbyDonaldson,by GauduchonandbyConti,MadsenandSalamondealwithissuesofspecialholonomy(the variationalapproachtoG -holonomy,Kählerandhyperkählergeometryandcohomogene- 2 ityoneconstructionsforquaternionicstructures,respectively).ChristianBär’sarticlewith SebastianHannesdealswiththeDiracoperator,thesubjectofmuchofNigel’searlywork, butthistimeinLorentziansignature. OneofNigel’smorerecentcontributionshasbeentheconceptofgeneralizedcomplex structures—thisrapidlygrowingareaisthesubjectofthearticlesbyBehrens,Cavalcanti, KlaasseandbyGualtieri.GeneralizedcomplexgeometryiscloselyintertwinedwithPoisson geometry,thesubjectofthepaperbyBrentPymandTravisSchedler.Poissongeometryalso linkstothetheoryofLiebialgebras,asubjectexploredinanotherarticlebyMerkulovand Willwacher.BothgeneralizedcomplexandPoissonstructurescan,ofcourse,beviewedas generalizationsofsymplecticstructures,anotherareawhereNigelhasworkedextensively. SymplecticgeometryandmomentmapsoccurinthearticlesbyJeffreyandMracekandby Hurtubise,Jeffrey,Rayan,SelickandWeitsman. OneofNigel’smostcelebratedpapersishis1987ProceedingsoftheLMSarticle‘The self-dualityequationsonaRiemannsurface’,whichintroducedtheconceptofHiggsbun- dlesandtheirmodulispaces.Thesecarryahyperkählerstructure,oneofwhosecomplex structures(theDolbeaultmodel)comesfromtheHiggsbundleviewpoint,whiletheothers (deRhamorBettimodel)arerelatedtoadescriptionintermsofflatbundles/localsystems. vi | preface Thesecomplexstructureshaveverydifferentproperties—thepaperbySimpsonexplores the idea of transferring the Hitchin fibration, which lives on the Dolbeault side, to the Betti/deRhampicture. Inrecentyears,thesemodulispaceshavebecomecentraltothestudyofthegeometric Langlandsprogramme.This(roughly)positsacorrespondencebetweenlocalsystemsand D-modulesonthemodulispaceofprincipalbundlesfortheLanglandsdualgroup.Work byGukov,KapustinandWitteninterpretsthisasamanifestationofaphysicsdualityon theHitchinspace,usingitsdoublenatureasaspaceoflocalsystemsandaspaceofHiggs bundles.ThepaperbyTeschnerlooksataconcretequantizationapproachtothegeometric Langlandscorrespondenceinthecasewherethelocalsystemhasanoperstructure.Quan- tizationissuesalsoarise(inparticular,theHitchinconnection)inthepaperbyEllegaard AndersenandRasmussen.Anotherkeylinkbetweengeometryandmathematicalphysics hasbeentheideaofmirrorsymmetry;inthecontextofHiggsmodulispaces,thisisexplored inthearticleofbyHausel,MellitandPei.Dedushenko,GukovandPutrov’sarticleinthe currentvolumeexploresaphysicalapproachto4-manifoldinvariants. An insight of Nigel’s original paper was that restricting to flat bundles corresponding torepresentationsvaluedinarealformofthecomplexgroupgaveageneralizationofthe classicalTeichmüllerspace.ThisthemeispursuedinthearticlesbyCollier,byBradlow, García-Prada,GothenandHeinloth,byGarcía-PradaandRamananandbyMundetiRiera. ThepaperbyBaraglia,BiswasandSchaposnikinvestigatestheBrauergroupofHiggs bundlemodulispaces.InNigel’soriginalpaper,theHiggsfieldswereholomorphic,butit hasbecomeapparentthatmanyinterestingspacescanbeobtainedbyallowingpoles—this isdealtwithinthepapersbyBoalchandbyChekhov,MazzoccoandRubtsov.Thepaperby ForniandGoldmanlinksthetheoryofmodulispaces(inthedeRhammodel)todynamical systemsviatheactionofthemappingclassgroup.Biquard’sarticledealswithHiggsbundles foraninfinite-dimensionalgroupandalsoaddressesquantizationquestions.Thewiderange ofsubjectsdealtwithhereisatestamenttotheincrediblyrichandmultifariousstructureof theHitchinmodulispace. Although Nigel is usually viewed as a differential geometer, of course much of his work,fromtheADHMconstructioninthe1970storecentworkonHiggsbundles,hasa strongalgebraicflavour.Threepapersintheseproceedingshaveapurealgebraicgeometric theme—thosebyBogomolov,FuandTschinkel(onellipticcurvesinpositivecharacteris- tic),byMoriandProkhorov(onextremalcurvegermsinthreefolds)andbyGrushevsky, Hulek and Tommasi, with an appendix by Dutour Sikiric´ (on the topology of partial compactificationsofthemodulispaceofAbelianvarieties). Twopapersfocusedmoreonmathematicalphysics.ThesearethepapersbydelaOssa, LarforsandSvanes(onheteroticstringsandmanifoldswithG structure),andbyCollins, 2 XieandYau(ondeformedHermitian–Yang–Millsequationsandmirrorsymmetry). Theorganizingcommitteeoftheconferencecomprised:LuisÁlvarez-Cónsul(Madrid), JørgenEllegaardAndersen(Aahus),SteveBradlow(Urbana),AndrewDancer(Oxford), OscarGarcía-Prada(Madrid),FrancesKirwan(Oxford),HenrikPedersen(Odense),Yat SunPoon(UCRiverside)andAndrewSwann(Aarhus). Theconferencewassupportedbyawiderangeoffundingagencies,includingtheClay MathematicsInstitute,theLondonMathematicalSociety,EPSRC(throughtheSymmetries preface | vii andCorrespondencesgrant),QGMAarhus,theCarlsbergFoundation,theMadridInsti- tuto de Ciencias Matemáticas (via the Excellence Grant Severo Ochoa), the EU-IRSES project‘Moduli’andtheNSF-fundednetworkGeometricStructuresandRepresentation Varieties(GEAR).Wethankallthesebodiesfortheirgeneroussupport. We extend our thanks to the other members of the organizing committee, to all the administrativestaffwhomadesuretheconferencesransosmoothly,toGilCavalcantiand LauraSchaposnikforprovidingthephotographs,tothespeakersandcontributorstothis volume,andfinally,ofcourse,toNigelhimself. JørgenEllegaardAndersen AndrewDancer OscarGarcía-Prada 1 • • • • • • • Boundary Value Problems for the Lorentzian Dirac Operator christian bär sebastian hannes InstitutfürMathematik,UniversitätPotsdam,Karl-Liebknecht-Str.24-25,14476Potsdam, Germany,Email:[email protected],[email protected] Dedicated to Nigel Hitchin on the occasion of his seventiethbirthday Abstract. OnacompactgloballyhyperbolicLorentzianspinmanifoldwithsmooth space-likeCauchyboundary,the(hyperbolic)DiracoperatorisknowntobeFredholm whenAtiyah–Patodi–Singerboundaryconditionsareimposed.Inthispaper,weinvestigate towhatextenttheseboundaryconditionscanbereplacedbymoregeneralonesandhow theindexthenchanges.TherearesomedifferencestotheclassicalcaseoftheellipticDirac operatoronaRiemannianmanifoldwithboundary. 1. Introduction TheAtiyah–Singerindextheorem[1]forellipticoperatorsonclosedmanifoldsisoneof thecentralmathematicaldiscoveriesofthetwentiethcentury.Itcontainsfamousclassical resultssuchastheGauss–Bonnettheorem,theRiemann–RochtheoremandHirzebruch’s signaturetheoremas special casesand has numerousapplications in analysis, geometry, topology and mathematical physics. For instance, it has been used in [14] to obtain a topological obstruction to the existence of metrics with positive scalar curvature, and a refinementoftheindextheoremwasemployedin[12]toshowthat,onmanymanifolds, Date:14July,2017. 2010MathematicsSubjectClassification.58J20,58J45. Keywordsandphrases. Diracoperator,globallyhyperbolicLorentzianmanifold,Fredholmpair,Dirac–Fredholm pair,indextheorem. Bär,C.,Hannes,S.,BoundaryValueProblemsfortheLorentzianDiracOperator.In:GeometryandPhysics: AFestschriftinHonourofNigelHitchin,JørgenEllegaardAndersen,AndrewDancer,OscarGarcía-Prada(Eds): OxfordUniversityPress(2018).©OxfordUniversityPress.DOI:10.1093/oso/9780198802006.003.0001
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