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Gauss-Manin Connection in Disguise PDF

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Hossein Movasati Gauss-Manin Connection in Disguise Calabi-Yau Modular Forms withappendicesbyKhosroM.ShokriandCarlosMatheus November17,2015 Publisher TomyparentsRogayehandAli,andmy familySaraandOmid Preface Theguidingprincipleinthismonographistodevelopatheoryof(Calabi-Yau)mod- ularformsparalleltotheclassicaltheoryof(elliptic)modularforms.Itisoriginated frommanyperiodmanipulationsoftheB-modelCalabi-Yauvarietyofmirrorsym- metryinTopologicalStringTheoryandtheearlierworksoftheauthorinwhichthe theoryof(quasi)modularformsisintroducedusingalargermodulispaceofellip- ticcurves,andtheRamanujandifferentialequationsbetweenEisensteinserieshave been derived from the corresponding Gauss-Manin connection. We have in mind an audience with a basic knowledge of Complex Analysis, Differential Equations, Algebraic Topology and Algebraic Geometry. Although the text is purely mathe- maticalandnobackgroundinStringTheoryisrequired,someofourcomputations areinspiredbymirrorsymmetry,andsothereaderwhowishestoexplorethemoti- vations,mustgototheoriginalPhysicsliterature.Thetextismainlywrittenfortwo primarytargetaudiences:expertsinclassicalmodularandautomorphicformswho wish to understand the q-expansions of Physicists derived from Calabi-Yau three- folds,andmathematiciansinenumerativeAlgebraicGeometrywhowanttounder- stand how mirror symmetry counts rational curves in compact Calabi-Yau three- folds.ExpertsinmodularformsarewarnedthattheywillnotfindsomuchNumber Theoryinthepresenttext,asthisnewtheoryofmodularformslivesitsinfancy,and yetmanyproblemsofcomplexanalysisnatureareopen.Wehavestillalongwayto dealwithmorearithmeticorientedquestions.Forourpurposewehavechosenapar- ticularclassofsuchq-expansionsarisingfromtheperiodsofaCalabi-Yauthreefold calledmirrorquintic,andingeneral,periodswhichsatisfyfourthorderdifferential equations.Theapplicationsofclassicalmodularformsarehugeandweareguided bythefactthatthisnewtypeofmodularformsmighthavesimilarapplicationsin thenearfuture,apartfromcountingrationalcurvesandGromov-Witteninvariants. Themaingoalistodescribeindetailmanyanalogiesanddifferencesbetweenclas- sicalmodularformsandthosetreatedhere.Thepresenttextisacomplementtothe available books on the mathematical aspects of mirror symmetry such as ”Mirror Symmetry and Algebraic Geometry” of D. A. Cox and S. Katz and and ”Mirror Symmetry” of C. Voisin. We hope that our text makes a part of mirror symmetry, whichisrelevanttonumbertheory,moreaccessibletomathematicians. vii viii Preface HosseinMovasati December2015 RiodeJaneiro,RJ,Brazil Acknowledgements The present text is written during the years between 2010 and 2014. First of all I wouldliketothankPierreDeligneforallhiscommentsontheorigin[Mov12b]of the present text. My sincere thanks go to E´tienne Ghys who drew my attention to the historical aspects of Ramanujan differential equations, that is, the contribution of Darboux and Halphen which is mainly neglected in number theory. My sincere thanksgotoCharlesDoran,StefanReiter,DucovanStraten,DonZagier,ConanLe- ung,BongLianandBabakHaghighatforusefulconversationsandtheirintereston thetopicofthepresenttext.IwouldalsoliketothankmathematicsinstitutesInsti- tutodeMatema´ticaPuraeAplicada(IMPA)inRiodeJaneiro(myhomeinstitute), Max-Planck Institute for Mathematics (MPIM) in Bonn, Institute for Physics and Mathematics(IPM)inTehran,InstituteforMathematicalSciences(IMS)inHong Kong, Center of Advanced Study in Theoretical Sciences (CASTS) in Taipei and Center of Mathematical Sciences and Applications (CMSA) at Harvard university forprovidingexcellentresearchambientduringthepreparationofthepresenttext. Final versionsofthe present text arewritten when the authorwas spending a sab- baticalyearatHarvarduniversity.Here,IwouldliketothankShing-TungYaufor theinvitationandforhisinterestandsupport.MysincerethanksgotoMuradAlim andEmmanuelScheideggerformanyusefulconversationsinmathematicalaspects ofTopologicalStringTheoryandmirrorsymmetry.IwouldliketothankCumrun Vafaforhisgenerousemailshelpingmetounderstandthemathematicalcontentof anomalyequations.DuringthepreparationofAppendixCweenjoyeddiscussions with many people. We would like to thank M. Belolipetsky for taking our atten- tiontotheworksonthingroups.ThanksalsogotoPeterSarnak,FritzBeukersand WadimZudilinforcommentsonthefirstdraftofthisappendix.Thesecondauthor of this appendix would like to thank CNPq-Brazil for financial support and IMPA foritslovelyresearchambient. ix Contents 1 Introduction................................................... 1 1.1 WhatisGauss-Maninconnectionindisguise?................... 3 1.2 WhymirrorquinticCalabi-Yauthreefold?...................... 4 1.3 Howtoreadthetext? ....................................... 5 1.4 WhydifferentialCalabi-Yaumodularform? .................... 5 2 Summaryofresultsandcomputations............................ 7 2.1 MirrorquinticCalabi-Yauthreefolds .......................... 7 2.2 Ramanujandifferentialequation .............................. 8 2.3 Modularvectorfields ....................................... 9 2.4 GeometricdifferentialCalabi-Yaumodularforms................ 11 2.5 Eisensteinseries ........................................... 12 2.6 Ellipticintegralsandmodularforms........................... 14 2.7 PeriodsanddifferentialCalabi-Yaumodularforms,I............. 15 2.8 IntegralityofFouriercoefficients ............................. 17 2.9 Quasi-ordifferentialmodularforms........................... 18 2.10 Functionalequations........................................ 19 2.11 Conifoldsingularity ........................................ 21 2.12 TheLiealgebrasl ......................................... 22 2 2.13 BCOVholomorphicanomalyequation,I ....................... 23 2.14 Gromov-Witteninvariants ................................... 24 2.15 PeriodsanddifferentialCalabi-Yaumodularforms,II ............ 25 2.16 BCOVholomorphicanomalyequation,II ...................... 28 2.17 Thepolynomialstructureofpartitionfunctions.................. 30 2.18 Futuredevelopments........................................ 30 3 Moduliofenhancedmirrorquintics.............................. 33 3.1 Whatismirrorquintic?...................................... 33 3.2 Modulispace,I ............................................ 34 3.3 Gauss-Maninconnection,I................................... 35 3.4 IntersectionformandHodgefiltration ......................... 36 xi xii Contents 3.5 AvectorfieldonS.......................................... 37 3.6 Modulispace,II............................................ 37 3.7 ThePicard-Fuchsequation................................... 38 3.8 Gauss-Maninconnection,II.................................. 39 3.9 ProofofTheorem2......................................... 41 3.10 Algebraicgroup............................................ 41 3.11 Anothervectorfield ........................................ 43 3.12 Weights................................................... 44 3.13 ALiealgebra.............................................. 45 4 TopologyandPeriods........................................... 47 4.1 Periodmap................................................ 47 4.2 τ-locus ................................................... 48 4.3 Positivityconditions ........................................ 50 4.4 GeneralizedPerioddomain .................................. 51 4.5 Thealgebraicgroupandτ-locus .............................. 52 4.6 Monodromycovering ....................................... 53 4.7 Aparticularsolution ........................................ 54 4.8 Actionofthemonodromy ................................... 54 4.9 Thesolutionintermsofperiods .............................. 56 4.10 Computingperiods ......................................... 57 4.11 Algebraicallyindependentperiods ............................ 59 4.12 θ-locus ................................................... 60 4.13 Thealgebraicgroupandtheθ-locus........................... 61 4.14 Comparingτ andθ-loci ..................................... 62 4.15 AllsolutionsofR , Rˇ ...................................... 63 0 0 4.16 Aroundtheellipticpoint..................................... 64 4.17 Halphenproperty........................................... 65 4.18 DifferentialCalabi-Yaumodularformsaroundtheconifold ....... 66 4.19 Logarithmicmirrormaparoundtheconifold.................... 67 4.20 Holomorphicmirrormap .................................... 69 5 Formalpowerseriessolutions ................................... 71 5.1 Singularitiesofmodulardifferentialequations .................. 71 5.2 q-expansionaroundmaximalunipotentcusp.................... 72 5.3 Anotherq-expansion........................................ 73 5.4 q-expansionaroundconifold ................................. 74 5.5 Newcoordinates ........................................... 75 5.6 Holomorphicfoliations...................................... 76 6 TopologicalStringPartitionFunctions............................ 77 6.1 Yamaguchi-Yau’selements .................................. 77 6.2 Proofoftheorem8 ......................................... 78 6.3 Genus1topologicalpartitionfunction ......................... 79 6.4 Holomorphicanomalyequation............................... 80

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