Gauss-Manin Connection in Disguise

198 Pages·2015·1.15 MB·english

by  Hossein Movasati

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