Table Of ContentFields 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
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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
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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
