Calabi-Yau Varieties: Arithmetic, Geometry and Physics: Lecture Notes on Concentrated Graduate Courses PDF
Preview Calabi-Yau Varieties: Arithmetic, Geometry and Physics: Lecture Notes on Concentrated Graduate Courses
Fields Institute Monographs 34 The Fields Institute for Research in Mathematical Sciences Radu Laza Matthias Schütt Noriko Yui Editors Calabi–Yau Varieties: Arithmetic, Geometry and Physics Lecture Notes on Concentrated Graduate Courses Fields Institute Monographs VOLUME 34 The Fields Institute for Research in Mathematical Sciences FieldsInstituteEditorialBoard: CarlR.Riehm,ManagingEditor EdwardBierstone,IanHambleton MatheusGrasselli,DeputyDirectoroftheInstitute JamesG.Arthur,UniversityofToronto KennethR.Davidson,UniversityofWaterloo LisaJeffrey,UniversityofToronto BarbaraLeeKeyfitz,OhioStateUniversity ThomasS.Salisbury,YorkUniversity NorikoYui,Queen’sUniversity JurisSteprans,YorkUniversity TheFieldsInstituteisacentreforresearchinthemathematicalsciences,locatedin Toronto,Canada.TheInstitutesmissionistoadvanceglobalmathematicalactivity intheareasofresearch,educationandinnovation.TheFieldsInstituteissupported bytheOntarioMinistryofTraining,CollegesandUniversities,theNaturalSciences and Engineering Research Council of Canada, and seven Principal Sponsoring UniversitiesinOntario(Carleton,McMaster,Ottawa,Queen’s,Toronto,Waterloo, WesternandYork),aswellasbyagrowinglistofAffiliateUniversitiesinCanada, theU.S.andEurope,andseveralcommercialandindustrialpartners. Moreinformationaboutthisseriesathttp://www.springer.com/series/10502 Radu Laza • Matthias Schütt • Noriko Yui Editors Calabi-Yau Varieties: Arithmetic, Geometry and Physics Lecture Notes on Concentrated Graduate Courses 123 TheFieldsInstituteforResearch intheMathematicalSciences Editors RaduLaza MatthiasSchütt MathematicsDepartment InstitutfürAlgebraischeGeometrie StonyBrookUniversity LeibnizUniversitätHannover StonyBrook,NY,USA Hannover,Germany NorikoYui DepartmentofMathematicsandStatistics Queen’sUniversity Kingston,ON,Canada ISSN1069-5273 ISSN2194-3079 (electronic) FieldsInstituteMonographs ISBN978-1-4939-2829-3 ISBN978-1-4939-2830-9 (eBook) DOI10.1007/978-1-4939-2830-9 LibraryofCongressControlNumber:2015945196 MathematicsSubjectClassification(2010):14J32,11Gxx,24Dxx,14Cxx,32-xx,81-xx SpringerNewYorkHeidelbergDordrechtLondon ©SpringerScience+BusinessMediaNewYork 2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Coverillustration:DrawingofJ.C.FieldsbyKeithYeomans Printedonacid-freepaper SpringerScience+BusinessMediaLLCNewYorkispartofSpringerScience+BusinessMedia(www. springer.com) Preface The thematic program Calabi–Yau Varieties: Arithmetic, Geometry, and Physics washeldattheFieldsInstituteforMathematicalSciencesfromJuly1toDecember 31, 2013. It was organized by Mark Gross (UC San Diego/Cambridge), Sergei Gukov(Caltech),RaduLaza(StonyBrook),MatthiasSchütt(Hannover),Johannes Walcher(McGill),Shing-TungYau(Harvard),andNorikoYui(Kingston/Fields). ThismonographcontainsintroductorymaterialonCalabi–Yaumanifoldsandis basedonlectureswhichtookplaceduringtheintroductoryperiodfortheworkshops of the thematic program. These workshops (“Modular Forms Around String The- ory,” “Enumerative Geometry and Calabi–Yau Varieties,” “Physics Around Mirror Symmetry,”“HodgeTheoryinStringTheory”)andconsequentlythelectureshere explore various perspectives on Calabi–Yau varieties. Thus, the title “Calabi–Yau Varieties:Arithmetic,Geometry,andPhysics”isquiteappropriate. Thegoalofthisvolumeistogiveafriendlyintroductiontotherapidlydeveloping andvastresearchareasconcerningCalabi–Yauvarietiesandstringtheory.Ourhope isthatanyonewhowishestoworkonorisinterestedinsubjectsinthisareawillstart withthisbook.Moreprecisely,wewouldliketotellprospectivegraduatestudents that“ThisisabookyoushouldreadifyouareinterestedingettingintotheCalabi– Yauworlds:mathematicsandstringtheory.” The articles presented in this volume have been prepared by young researchers (mostly students and postdocs affiliated with the thematic program) with utmost enthusiasm, based on the concentrated graduate courses given by them during the thematicprogram.Theeditorswishtoexpresstheirgreatappreciationtoallofthem for preparing their manuscripts for the Fields Monograph Series, which required extra effort presenting not only current developments but also some background material on the topics discussed. All articles in this volume were peer-reviewed. Wearedeeplygratefultoalltherefereesfortheireffortsevaluatingthearticles,in particularinthelimitedtimeframe.ThisvolumewaseditedbyR.Laza,M.Schütt, andN.Yui. The thematic program was financially supported by various organizations. In addition to the Fields Institute, the program received substantial support from the NSF (DMS-1247441, DMS-125481), the PIMS CRG Program Geometry and v vi Preface Physics, and the Perimeter Institute. Additionally, several participants used their individual grants (e.g., NSF, ERC, or NSERC) to cover their travel expenses. We wishtothankalltheseinstitutionsfortheirsupport. Last but not least, our thanks go to everyone at the Fields Institute for making thisthematicprogramsosuccessfulandenjoyable. StonyBrook,USA RaduLaza Hannover,Germany MatthiasSchütt Kingston,Canada NorikoYui October2014 Contents PartI K3Surfaces:Arithmetic,GeometryandModuli TheGeometryandModuliofK3Surfaces.................................... 3 AndrewHarderandAlanThompson PicardRanksofK3SurfacesofBHKType................................... 45 TylerL.Kelly ReflexivePolytopesandLattice-PolarizedK3Surfaces ..................... 65 UrsulaWhitcher PartII HodgeTheoryandTranscendentalTheory AnIntroductiontoHodgeStructures.......................................... 83 SaraAngelaFilippini,HelgeRuddat,andAlanThompson IntroductiontoNonabelianHodgeTheory.................................... 131 AlbertoGarcía-RabosoandStevenRayan AlgebraicandArithmeticPropertiesofPeriodMaps ....................... 173 MattKerr PartIII PhysicsofMirrorSymmetry MirrorSymmetryinPhysics:TheBasics ..................................... 211 CallumQuigley PartIV EnumerativeGeometry:Gromov–Witten andRelatedInvariants IntroductiontoGromov–WittenTheory ...................................... 281 SimonC.F.Rose IntroductiontoDonaldson–ThomasandStablePairInvariants ........... 303 MichelvanGarrel vii viii Contents Donaldson–ThomasInvariantsandWall-CrossingFormulas .............. 315 YuechengZhu PartV Gross–SiebertProgram EnumerativeAspectsoftheGross–SiebertProgram ........................ 337 MichelvanGarrel,D.PeterOverholser,andHelgeRuddat PartVI ModularFormsinStringTheory IntroductiontoModularForms................................................ 423 SimonC.F.Rose LecturesonBCOVHolomorphicAnomalyEquations....................... 445 AtsushiKanazawaandJieZhou PolynomialStructureofTopologicalStringPartitionFunctions........... 475 JieZhou PartVII ArithmeticAspectsofCalabi–YauManifolds IntroductiontoArithmeticMirrorSymmetry................................ 503 AndrijaPerunicˇic´ Index............................................................................... 541 Contributors Sara Angela Filippini Institut für Mathematik, Universität Zürich, Zürich, Switzerland Alberto García-Raboso Department of Mathematics, University of Toronto, 40 St.GeorgeStreet,Toronto,ONM5S2E4,Canada AndrewHarder DepartmentofMathematicalandStatisticalSciences,632CAB, UniversityofAlberta,Edmonton,AB,Canada Atsushi Kanazawa Department of Mathematics, Center for Mathematical Sci- encesandApplications,HarvardUniversity,Cambridge,MA,USA Tyler L. Kelly Department of Pure Mathematics and Mathematical Statistics, UniversityofCambrigde,WilberforceRoad,CambridgeCB30WB,UK Matt Kerr Department of Mathematics, Washington University in St. Louis, St.Louis,MO,USA D.PeterOverholser DepartmentofMathematics,KULeuven,Leuven(Heverlee), Belgium AndrijaPerunicˇic´ DepartmentofMathematicsandStatistics,Queen’sUniversity, Kingston,ON,Canada CallumQuigley DepartmentofMathematicalandStatisticalSciences,University ofAlberta,Edmonton,ABT6G2G1,Canada Steven Rayan Department of Mathematics, University of Toronto, 40 St. George Street,Toronto,ONM5S2E4,Canada SimonC.F.Rose MaxPlanckInstituteforMathematics,Bonn,Germany HelgeRuddat MathematischesInstitut,UniversitätMainz,Mainz,Germany Alan Thompson Department of Pure Mathematics, University of Waterloo, Waterloo,ON,Canada ix
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