Geometry and Physics: A Festschrift in Honour of Nigel Hitchin, Volume I

382 Pages·2018·7.957 MB·English

by  Jørgen Ellegaard Andersen (Editor), Andrew Dancer (Editor), Oscar García-Prada (Editor)

geometry and physics

Geometry and Physics
AFestschriftinHonourofNigelHitchin
Editedby
JØRGEN ELLEGAARD ANDERSEN
ANDREW DANCER
OSCAR GARCÍA-PRADA
VOLUME I
1
3
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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
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• • • • • •
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