Modular Forms and String Duality 0821844849, 9780821844847

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Modular Form s and Strin g Dualit y

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http://dx.doi.org/10.1090/fic/054

FIELDS INSTITUT E COMMUNICATIONS T H E FIELD S INSTITUT E FO R RESEARC H I N MATHEMATICA L SCIENCE S

Modular Form s and Strin g Dualit y Noriko Yu i Helena Verril l Charles F . Dora n Editors

American Mathematica l Societ y Providence, Rhod e Islan d The Field s Institut e for Researc h i n Mathematica l Science s r Toronto, Ontari o r

i r r p\ r 1 C LD5

The Field s Institut e for Researc h i n Mathematica l Science s T h e Field s I n s t i t u t e i s a cente r fo r m a t h e m a t i c a l research , locate d i n Toronto , C a n a d a . O u r missio n i s t o provid e a supportiv e a n d s t i m u l a t i n g environmen t fo r m a t h e m a t i c s research, innovatio n a n d education . T h e I n s t i t u t e i s s u p p o r t e d b y t h e O n t a r i o Ministr y of Training , College s a n d Universities , t h e N a t u r a l Science s a n d Engineerin g Researc h Council o f C a n a d a , a n d seve n O n t a r i o universitie s (Carleton , M c M a s t e r , O t t a w a , Toronto , W a t e r l o o , W e s t e r n O n t a r i o , a n d York) . I n a d d i t i o n t h e r e ar e severa l affiliate d universitie s a n d c o r p o r a t e sponsor s i n b o t h C a n a d a a n d t h e U n i t e d S t a t e s . Fields I n s t i t u t e Editoria l Board : Car l R . R i e h m ( M a n a g i n g E d i t o r ) , B a r b a r a Le e Keyfitz (Directo r o f t h e I n s t i t u t e ) , Juri s S t e p r a n s ( D e p u t y Director) , J a m e s G . A r t h u r (Toronto), K e n n e t h R . Davidso n (Waterloo) , Lis a Jeffre y (Toronto) , T h o m a s G . Salis b u r y (York) , Norik o Yu i (Queen's) .

2000 Mathematics Subject

Classification.

P r i m a r y H F x x , 1 4Gxx , 1 4J32 , 1 4N35 , 33Cxx , 81T30.

Library o f Congres s Cataloging-in-Publicatio n Dat a Modular form s an d strin g dualit y / Norik o Yui , Helen a Verrill , Charle s F . Do r an, editors . p. cm . — (Field s Institut e Communications , ISS N 1 069-526 5 ; 54 ) Proceedings o f a worksho p hel d a t th e Banf f Internationa l Researc h Station , Jun e 3-8 , 2006 . Includes bibliographica l references . ISBN 978-0-821 8-4484- 7 (alk . paper ) 1. Forms , Modular—Congresses . 2 . Dualit y (Mathematics)—Congresses . 3 . Mirro r symmetry—Congresses. 4 . Number theory—Congresses . 5 . Strin g theory—Congresses . 6 . Parti cles (Nuclea r physics)—Congresses . I . Yui , Noriko . II . Verrill , Helena . III . Doran , Charle s F. , 1971QA243.M695 200 8 512.7'3—dc22 2008028 7

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C o p y i n g an d reprinting . Materia l i n thi s boo k ma y b e reproduce d b y an y mean s fo r edu cational an d scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y services tha t collec t fee s fo r deliver y o f document s an d provide d tha t th e customar y acknowledg ment o f th e sourc e i s given . Thi s consen t doe s no t exten d t o othe r kind s o f copyin g fo r genera l distribution, fo r advertisin g o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercial us e o f materia l shoul d b e addresse d t o th e Acquisition s Department , America n Math ematical Society , 20 1 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n also b e mad e b y e-mai l t o [email protected] . Excluded fro m thes e provision s i s materia l i n article s fo r whic h th e autho r hold s copyright . I n such cases , request s fo r permissio n t o us e o r reprin t shoul d b e addresse d directl y t o th e author(s) . (Copyright ownershi p i s indicate d i n th e notic e i n th e lowe r right-han d corne r o f th e first pag e o f each article. ) © 200 8 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Copyright o f individua l article s ma y rever t t o th e publi c domai n 2 8 year s after publication . Contac t th e AM S fo r copyrigh t statu s o f individua l articles . Printed i n th e Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . This publicatio n wa s prepare d b y th e Field s Institute . http://www.fields.utoronto.ca Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 21 1

3 1 2 1 1 1 0 09 0 8

Contents Acknowledgments vi

i

Introduction i

x

List o f Participant s xii

i

Schedule o f Workshop s x

v

Aspects o f Arithmeti c an d Modula r Form s Motives an d Mirro r Symmetr y fo r Calabi-Ya u Orbifold s 3 SHABNAM KADI R an d NORIK O Y U I

String Modula r Motive s o f Mirror s o f Rigi d Calabi-Ya u Varietie s 4

7

SAVAN KHAREL , MONIK A LYNKE R an d R O L F SCHIMMRIG K

Update o n Modula r Non-Rigi d Calabi-Ya u Threefold s 6

5

EDWARD L E E

Finite Inde x Subgroup s o f th e Modula r Grou p an d Thei r Modular Form s 8

3

LING LON G

Aspects o f Geometri c an d Differentia l Equation s Apery Limit s o f Differentia l Equation1 s o f Orde r 4 and 5 0

5

G E R T ALMKVIST , D U C O VA N STRATE N an d WADI M ZUDILI N

Hypergeometric System s i n Tw o Variables , Quivers , Dimer s and Dessin s d'Enfant s 2

5

J A N STIENSTR A

Some Propertie s o f Hypergeometri c Serie s Associate d wit h Mirror Symmetr y 6

3

D O N ZAGIE R an d ALEKSE Y ZINGE R

Ramanujan-Type Formula e fo r1 1 /n: A Secon d Wind ? 7 WADIM ZUDILI N

9

Contents

VI

A s p e c t s o f Physic s a n d Strin g T h e o r y 1 Meet Homologica l Mirro r Symmetr y 9 1 MATTHEW ROBER T BALLAR D

Orbifold Gromov-Witte n Invariant s an d Topologica l String s 22

5

VINCENT BOUCHAR D

Conformal Fiel d Theor y an d Mappin g Clas s Group s 24

7

TERRY GANNO N

SL(2,C) Chern-Simon s Theor y an d th e Asymptoti c Behavio r o f the Colore d Jone s Polynomia l 26 1 SERGEI GUKO V an d HITOSH I MURAKAM I

Open String s an d Extende d Mirro r Symmetr y JOHANNES WALCHE R

279

Acknowledgments The editors wish to express their appreciatio n t o all the contributors for prepar ing thei r manuscript s fo r th e Field s Communicatio n Series , whic h require d extr a effort presentin g no t onl y curren t development s bu t als o the histor y o f the subject s treated i n thei r articles . All papers i n thi s volum e wer e refereed ver y rigorously . W e are deepl y gratefu l to al l ou r referee s fo r thei r time-consumin g effor t an d disciplin e i n evaluatin g th e articles. All paper s wer e copy-edite d b y Arthu r Greenspoo n o f Mathematica l Reviews . The editors an d th e Field s Institute ar e grateful fo r hi s help smoothing out , Englis h and mathematica l presentations . The worksho p wa s supporte d b y th e Banf f Internationa l Researc h Statio n (BIRS) throug h th e five-day workshop s program . W e than k BIR S fo r thei r financial support . W e enjoye d th e excellen t suppor t o f th e staf f a t BIRS , an d ar e grateful fo r thei r hospitality . I n addition , som e young participants fro m th e Unite d States were supported i n part b y Mathematical Science s Research Institute (MSRI ) Berkeley. Last bu t no t least , w e than k Debbi e Isco e o f th e Field s Institut e fo r he r hel p reformatting article s an d assemblin g thi s volum e fo r publication .

Noriko Yui , Helen a Verril l an d Charle s Do r an June 200 8

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Introduction Modular form s hav e lon g playe d a key rol e i n th e theor y o f numbers , includin g most famousl y th e proo f o f Fermat' s Las t Theorem . Throug h it s ques t t o unif y the spectacularl y successfu l theorie s o f quantu m mechanic s an d genera l relativity , string theory has long suggested dee p connections between branches of mathematic s such a s topology , geometry , representatio n theory , an d combinatorics . Les s well known are the emerging connections between strin g theory an d numbe r theor y - th e subject o f the worksho p Modular Forms and String Duality hel d a t th e Banf f Inter national Researc h Statio n (BIRS) , Jun e 3-8 , 2006 . Mathematician s an d physicist s alike converge d o n th e Banf f Statio n fo r a wee k o f introductor y lectures , designe d to educat e on e anothe r i n relevan t aspect s o f thei r subjects , an d o f researc h talk s at th e cuttin g edg e o f thi s rapidl y growin g field . The worksho p wa s organize d b y Charle s F . Doran , Helen a Verril l an d Norik o Yui. Th e worksho p wa s a hug e success . Altogethe r thirty-seve n mathematician s and physicist s converge d a t th e BIR S fo r th e fiv e da y workshop . Twenty-si x on e hour talk s wer e presented . Som e wer e introductor y lecture s b y mathematician s designed to prepare physicists in modular forms , quasimodula r forms , modularity of Galois representations, an d tori c geometry. A t th e sam e time, introductor y lecture s by physicist s wer e intende d t o educat e mathematician s o n som e aspect s o f mirro r symmetry an d strin g theor y i n connectio n wit h numbe r theory . Thes e introductor y lectures were scheduled i n the morning s of the earl y days of the workshop. Researc h talks were scheduled i n the afternoons an d late r days . The y covered recent advance s on variou s aspect s o f modula r forms , differentia l equations , conforma l fiel d theory , topological strings and Gromov-Witte n invariants , holomorphic anomaly equations, motives, mirro r symmetry , homologica l mirro r symmetry , constructio n o f Calabi Yau manifolds , amon g others . Summary o f scientifi c an d othe r objective s Physical dualit y symmetrie s relat e special limits of the various consistent strin g theories (Type s I , II , Heteroti c strin g an d thei r cousins , includin g M-theor y an d F theory) on e t o another . Th e compariso n o f th e mathematica l description s o f thes e theories ofte n reveal s quit e dee p an d unexpecte d mathematica l conjectures . Th e best know n strin g dualit y t o mathematicians , Typ e IIA/II B duality , als o calle d mirror symmetry , ha s inspire d man y ne w development s i n algebrai c an d arith metic geometry , numbe r theory , tori c geometry , Rieman n surfac e theory , an d in finite dimensiona l Li e algebras . Othe r strin g dualitie s suc h a s Heterotic/Typ e I I duality an d F-Theory/Heteroti c strin g dualit y hav e also , mor e recently , le d t o a series of mathematical conjectures , man y involvin g elliptic curves , K3 surfaces, an d modular forms . Modula r form s an d quasi-modula r form s pla y a central rol e in mir ror symmetry , i n particula r a s generatin g function s countin g th e numbe r o f curve s on Calabi-Ya u manifold s an d describin g Gromov-Witte n invariants . Thi s ha s le d ix

X

Introduction

to a realization that th e time is ripe to asses s the role of number theory, in particula r that o f modula r forms , i n mirro r symmetr y an d strin g dualitie s i n general . One o f th e principa l goal s o f this worksho p wa s t o loo k a t modula r an d quasi modular forms , congruenc e zeta-functions , Galoi s representations , an d L-serie s fo r dual families o f Calabi-Yau varieties with the aim of interpreting dualit y symmetrie s in term s o f arithmeti c invariant s associate d t o th e varietie s i n question . Ove r th e last two decades, a great dea l of work has been done on these problems. I n particula r it appear s tha t w e nee d t o modif y th e classica l theorie s o f Galoi s representation s (in particular , th e questio n o f modularity ) an d modula r forms , amon g others , fo r families o f Calabi-Ya u varietie s (threefolds ) i n orde r t o accommodat e "quantu m corrections". As dictated b y the research interests o f the participating members , the researc h activities wer e focuse d o n th e followin g themes : (A) Arithmeti c o f Calabi-Ya u varietie s define d ove r numbe r fields: Arith metic o f elliptic curves , K3 surfaces, Calabi-Ya u threefolds , an d highe r dimensiona l Calabi-Yau varietie s define d ove r number fields i n connectio n wit h strin g dualities . These include d th e followin g topic s an d problems : Interpretatio n o f strin g dual ity phenomen a o f Calabi-Ya u varietie s i n term s o f zeta-function s an d L-serie s o f the varietie s i n question , th e modularit y conjecture s fo r Calabi-Ya u varieties , th e conjectures o f Birc h an d Swinnerton-Dye r fo r ellipti c curve s an d Abelia n varieties , the conjecture s o f Beilinson-Bloch o n specia l values of L-series an d algebrai c cycles, and intermediat e Jacobian s o f Calabi-Ya u threefolds . Calabi-Ya u varietie s o f CM (complex multiplication ) typ e an d thei r possibl e connection s t o rationa l conforma l field theorie s and , i n th e ellipti c fibered case , behavio r unde r F-Theory/Heteroti c string duality . (B) Mirro r symmetr y fo r familie s o f Calabi-Ya u varieties : Characterizatio n of mirro r map s i n connectio n wit h th e mirro r moonshin e phenomeno n and , vi a the Fourier-Laplac e transform , th e classificatio n o f Q-Fan o threefolds . I n particu lar, differentia l equation s associate d t o modula r an d quasi-modula r form s relate d to GKZ-hypergeometri c system s and , mor e generally , t o Picard-Fuch s differentia l systems wer e investigated . (C) Modula r an d quasi-modula r form s i n strin g duality : Modula r form s an d quasi-modular form s hav e appeare d frequentl y i n mirro r symmetr y contexts , e.g. , in the generatin g function s countin g th e numbe r o f simply ramified cover s of elliptic curves with marke d points , i n Gromov-Witte n invariants , an d als o as mirror maps . The appearanc e o f modular an d quasi-modula r form s i n string dualities , e.g. , i n th e Harvey-Moore conjecture s o f Heterotic-Typ e I I duality , wer e investigated . Under standing why modular an d quasi-modula r form s pla y central roles in string dualitie s was on e o f ou r goals . Talks presente d a t th e worksho p ma y b e classifie d int o no t clearl y disjoin t set s of th e followin g seve n subjects . The y are : (a) Modular , quasimodular , bimodula r forms , an d their application s (D . Zagier , M. Kaneko , J . Stienstra ) (b) Topologica l strin g theor y an d modula r form s (E . Scheidegger , A . Clingher , A. Klemm , V . Bouchard ) (c) Modularity , an d arithmeti c question s (R . Livne , S . Gukov , N . Yui , R . Schimmrigk)

Introduction

XI

(d) Mirro r symmetry : variou s version s (S . Hosono, J . Walcher , M . Aldi ) (e) Tori c geometr y (H . Verrill , C . Do r an) (f) Differentia l equation s (G . Almkvist , Masahik o Saito , A . Sebba r ) (g) Miscellaneou s topic s (N.-H . Lee , J . Stienstra , C . Herzog ) About thi s Volum e This volum e consist s o f the proceeding s o f the workshop , presentin g article s o n recent development s i n th e interfac e o f numbe r theor y an d hig h energ y physics , i n particular, modula r form s an d strin g duality . Som e o f th e article s ar e written-u p versions o f the talk s presente d a t th e workshop , whil e other s repor t o n subsequen t developments o n th e subjec t matte r o f th e workshop . I n addition , som e article s were solicite d b y th e editor s fro m non-participants , i.e. , invitee s wh o coul d no t make th e worksho p du e t o variou s constraints . All thirtee n contribution s include d her e hav e bee n referee d rigorously , an d a number o f article s wen t throug h extensiv e revision s t o reac h th e standar d impose d by th e Field s Communication s Series . I n particular , al l contributor s wer e aske d t o make extr a effort s t o includ e expositor y component s i n thei r article s t o reac h ou t to a wide r rang e o f readers . Here i s a brie f descriptio n o f th e contribution s assemble d i n thi s volume . Th e papers ar e divide d roughl y int o thre e categories , namely , Arithmeti c an d Modula r Forms, Geometri c an d Differentia l Equations , an d Physic s an d Strin g Theory . • Arithmeti c an d Modula r Forms : Th e pape r o f Kadi r an d Yu i consider s motives arisin g fro m certai n Calabi-Ya u orbifold s o f monomia l deformation s o f Fermat hype r surf aces, and establishe s the monomial-motiv e correspondence , a t th e Fermat poin t i n th e modul i space . The articl e o f Schimmrig k e t al . discusse s th e mirror s o f rigi d Calabi-Ya u varieties, i n particular , i t consider s th e cubi c sevenfol d an d th e quarti c fivefold, showing tha t thei r motivi c L-function s agre e wit h th e L-function s o f thei r rigi d mirror Calabi-Ya u varieties . Lee present s a n updat e o n th e modularit y o f Calabi-Ya u threefold s define d over th e field o f rationa l numbers , i n particular , h e construct s a non-rigi d Calabi Yau threefol d a s a quotien t o f th e Schoe n quinti c threefold , an d establishe s it s modularity. The articl e o f Lon g report s o n aspect s o f modula r form s fo r noncongruenc e arithmetic subgroups : th e unbounde d denominato r property , modularit y o f th e Galois representation s fo r noncongruenc e cuspforms , an d Atki n an d Swinnerton Dyer congruences .

• Geometri c an d Differentia l Equations : Almkvist , va n Straten an d Zudili n consider Aper y limit s o f differential equation s o f order 4 and 5 of Calabi-Ya u type , identifying the m wit h som e know n transcendenta l numbers . Stienstra's pape r i s devote d t o surveyin g som e relation s betwee n plan e model s of statistical mechanic s an d hypergeometri c system s i n two variables . Th e stud y o f embeddings of quivers (i.e. , oriented graphs ) int o Riemann surface s turne d ou t t o b e crucial i n severa l areas . On e directio n i s Grothendieck's theor y o f dessins d'enfant s which relate s th e combinatoric s o f suc h embedding s wit h th e Galoi s grou p o f th e field o f rationa l numbers . Anothe r directio n i s dime r models , whic h ar e periodi c

Xll

Introduction

2-dimensional statistica l systems . Th e paper explain s several exciting interrelation s of thes e differen t directions . The articl e o f Zagie r an d Zinge r show s tha t certai n hypergeometri c serie s use d to formulat e mirro r symmetr y fo r Calabi-Ya u hypersurfaces , i n strin g theor y an d algebraic geometry, satisf y a number o f interesting properties . Thes e properties ar e used, fo r instance , t o verif y th e Bershadsky-Cecotti-Ooguri-Vaf a (BCOV ) predic tion fo r th e genu s on e Gromov-Witte n invariant s o f a quinti c threefold . Zudilin's pape r give s a survey o n method s o f proofs o f Ramanujan-type formu lae fo r 1 /TT. Th e method s includ e ellipti c curves , modula r forms , creativ e telescop ing, an d hypergeometri c series . • Physic s an d Strin g Theory : Ballard' s article gives a survey on homologica l mirror symmetry , focusin g o n th e simples t example s o f curve s o f genu s zer o an d one. Bouchar d present s a surve y o n th e computatio n o f orbifol d Gromov-Witte n invariants usin g mirro r symmetr y an d topologica l strin g theory , focusin g o n th e orbifold C 3 / Z 3 . Gannon treats rational conformal fiel d theor y an d mapping clas s groups. Ratio nal conformal fiel d theorie s produce a tower o f finite-dimensional representation s of surface mappin g clas s groups, acting on the conforma l block s of the theory. Ganno n gives review s o n thi s formalis m wit h explici t examples . Gukov an d Murakam i tak e th e reade r t o th e real m o f th e colore d Jone s poly nomial an d th e Chern-Simon s gaug e theory . I t i s a physic s predictio n tha t th e asymptotic behavio r o f th e colore d Jone s polynomia l i s equa l t o th e perturbativ e expansion o f th e Chern-Simon s gaug e theor y wit h comple x gaug e grou p SX(2 , C) on th e hyperboli c kno t complement . Th e pape r make s th e firs t ste p towar d verify ing thi s prediction . Walcher give s a n expositio n o n recen t progres s i n mirro r symmetry , i n partic ular, extendin g th e classica l close d strin g result s t o th e ope n strin g sector . Th e ideas ar e worked ou t fo r th e particula r exampl e o f the quinti c hypersurfac e an d th e mirror quintic . Th e pape r als o explain s th e relevanc e o f mirror symmetr y fo r ope n strings t o homologica l mirro r symmetry .

List o f Participant s We ha d i n tota l 3 7 participant s fo r th e workshop , o f whic h te n wer e eithe r graduate student s o r postdoctora l fellows , an d tw o wer e officia l observers . W e had five las t minut e cancellation s Terr y Ganno n (Alberta , Canada) , Ame r Iqba l (Washington), Bon g H . Lia n (Brandeis , USA) , Joh n McKa y (Concordia , Canada) , and Andre y Todoro v (Sant a Cruz , US A an d MPI M Bonn , Germany ) fo r variou s reasons. 1. Marc o Ald i (Northwester n University , USA) (Ph.D . student ) 2. Ger t Almkvis t (Lund s University , Sweden ) 3. Maii a Bakhov a (Louisian a Stat e University , USA ) (Ph.D . student ) 4. Matthe w Ballar d (Universit y o f Washington , USA ) (Ph.D . student ) 5. Vincen t Bouchar d (MSRI , USA ) (Postdoctora l fellow ) 6. Adria n Clinghe r (Stanfor d University , USA ) 7. Chuc k F . Dora n (Universit y o f Washington , USA ) 8. Georg e Elliot t (Universit y o f Toronto , Canada ) (Observer ) 9. Sharo n Frechett e (Colleg e o f th e Hol y Cross , USA ) 10. Serge i Guko v (Californi a Institut e o f Technology , USA ) 11. Christophe r Herzo g (Universit y o f Washington , USA ) 12. Sinob u Hoson o (Universit y o f Tokyo , Japan ) 13. Simo n Jude s (Columbi a University , USA ) (Ph.D . student ) 14. Shabna m Kadi r (Universit y o f Hannover , Germany ) (Postdoctora l fellow ) 15. Masanob u Kanek o (Kyush u University , Japan ) 16. Albrech t Klem m (Universit y o f Wisconsin , USA ) 17. Edwar d Le e (UCLA , USA ) (Postdoctora l fellow ) 18. Nam-Hoo n Le e (KIAS , Korea ) (Postdoctora l fellow ) 19. Ro n Livn e (Hebre w Universit y o f Jerusalem , Israel ) 20. Lin g Lon g (Iow a Stat e University , USA ) 21. Stephe n L u (Universit e d u Quebe c a Montreal , Canada ) 22. Richar d N g (Iow a Stat e University , USA ) (Observer ) 23. Matthe w Papanikola s (Texa s A & M University , USA ) 24. Mik e Rot h (Queen' s University , Canada ) 25. Masahik o Sait o (Kob e University , Japan ) 26. Emanue l Scheidegge r (Universit a de l Piemont e Orien t ale Amadeo Avogadro , Italy ) 27. Rol f Schimmrig k (Indian a Universit y Sout h Bend , USA ) 28. Abdella h Sebba r (Universit y o f Ottawa , Canada ) 29. Ahme d Sebba r (Universit y o f Bordeau x I , France ) 30. Ja n Stienstr a (Universit y o f Utrecht , th e Netherlands ) 31. Hiroyuk i Tsutsum i (Osak a Universit y o f Healt h & Spor t Sciences , Japan ) 32. Helen a Verril l (Louisian a Stat e University , USA ) 33. Johanne s Walche r (Institut e fo r Advance d Study , USA )

XIV

List o f P a r t i c i p a n t s

34. Ursul a Whitche r (Universit y o f Washington, USA ) (Ph.D . student ) 35. Jeng-Da w Y u (Harvar d University , USA) (Postdoctora l fellow ) 36. Norik o Yu i (Queen' s University , Canada ) 37. Do n Zagie r (Max-Planc k Institut e fo r Mathematic s Bonn , Germany/ Colleg e d e France , France )

Schedule o f Workshop s J U N E 4 , 200 6 9:00am-10:00am Do

n Zagie r (MPI M Bon n an d Colleg e d e Prance )

Introduction: Modula r Form s an d Differentia l Equation s 10:15am-l 1:15am Ro

n Livn e (Hebre w an d IA S Princeton )

Modularity o f Galoi s Representations : Overvie w ll:30am-12:30pm Helen

a Verril l (Louisian a State )

An Introductio n t o th e Batyrev—Boriso v Constructio n o f Tori c Calabi—Yau Varietie s 3:00pm-4:00pm Shinob

u Hoson o (Tokyo )

Introduction t o Differentia l Equation s i n Mirro r Symmetr y 4:15pm-5:15pm Emanue

l Scheidegge r (T U Vienna )

Topological String s o n K3 Fibration s an d Modula r Form s 5:30pm-6:30pm Ja

n Stienstr a (Utrecht )

From Multi-gri d t o Multi-helix ; Remarkabl e Geometrie s fro m AdS/CFT J U N E 5 , 200 6 9:00am-10:00am Do

n Zagie r (MPI M Bon n an d Colleg e d e France )

Quasimodular Forms , Rankin—Cohe n Bracket s an d Relate d Algebraic Structure s 10:15am-l 1:15am Masanob

u Kanek o (Kyushu )

Modular an d Quasimodula r Form s an d thei r Application s ll:30am-12:30pm Johanne s Walche r (IA S Princeton ) Opening Mirro r Symmetr y o n th e Quinti c 2:30pm-3:30pm Ger

t Almkvis t (Lund )

Apery-like Limit s Connecte d wit h Calabi-Ya u Differentia l Equation s 4:00pm-5:00pm Christophe

r Herzo g (Washington )

How Exceptiona l Collection s Stac k U p

Schedule o f Workshop s

XVI

5:15pm-6:15pm Nam-Hoo

n Le e (KIAS )

Constructing Calabi-Ya u Manifold s 8:00pm-9:00pm Lin

g Lon g (Iow a State )

Modular Form s fo r Noncongruenc e Subgroup s J U N E 6 , 200 6 9:00am-10:00am Adria

n Clinghe r (Stanford )

Geometry Underlyin g th e F-Theory/Heteroti c Strin g Dualit y i n Eight Dimension s 10:15am-l 1:15am Serge

i Guko v (CalTech )

Strings, Fields , an d Arithmetic , Par t I ll:30am-12:30pm Norik

o Yu i (Queen's )

Motives, Mirro r Symmetr y an d Modularit y 8:00pm-9:00pm Serge

i Guko v (CalTech )

Strings, Fields , an d Arithmetic , Par t I I J U N E 7 , 200 6 9:00am-10:00am Albrech

t Klem m (Wisconsin )

Modular, Quasimodula r Form s an d Gromov—Witte n Invariant s 10:15am-l 1:15am Ro

n Livn e (Hebre w an d IA S Princeton )

Explicit Description s o f Universa l K3 Familie s ove r Shimur a Curve s ll:30am-12:30pn Chuc

k Dora n (Washington )

Algebraic Topolog y o f Calabi—Ya u Threefold s i n Tori c Varietie s 2:30pm-3:30pm Marc

o Ald i (Northwestern )

Twisted Homogeneou s Cring s o f Abelia n Surface s vi a Mirro r Symmetry 4:00pm-5:00pm Masahik

o Sait o (Kobe )

Painleve Propert y o f ODE s an d Deformatio n o f Logarithmi c Symplectic Varietie s 5:15pm-6:15pm Rol

f Schimmrig k (Indiana , Sout h Bend )

String Modula r fi-Motive s an d Aspec t o f Mirro r Symmetr y 8:00pm-9:00pm Ahme

d Sebba r (Bordeaux )

Differential Thet a Relation s an d Galoi s Theor y fo r Riccat i Equation s

Schedule o f Workshop s

xvn

J U N E 8 , 200 6 9:00am-l0:00am Ja

n Stienstr a (Utrecht )

Bimodular Form s an d Holomorphi c Anomal y Equatio n 10:15am-l 1:15am Vincen

t Bouchar d (MSRI/Perimeter )

Topological Strings , Holomorphi c Anomaly , an d (Almost ) Modula r Forms A numbe r o f invitee s ha d t o cance l thei r participatio n t o th e worksho p a t th e last minut e b y variou s reasons . The y wer e • Terr y Gannon (Alberta) : Th e Monster , Modula r Function s an d R C F T • Ame r Iqba l (Washington ) • Bon g H . Lia n (Brandeis) : Introductio n t o Mirro r Symmetr y • Joh n McKa y (Concordia ) • Andre y Todoro v (Sant a Cruz/MPI M Bonn) : Regularize d Determinant s of C Y Metric s an d Application s t o Mirro r Symmetr y

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Aspects o f Arithmeti c an d Modula r Form s

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http://dx.doi.org/10.1090/fic/054/01 Fields Institut e Communication s Volume 54 . 200 8

Motives an d Mirro r S y m m e t r y fo r Calabi-Ya u Orbifold s Shabnam Kadi r Institute fo r Mathematic s University o f Hannove r Welfengarten 1 , 301 67 , Hannover , German y kadirOmath.uni-hannover.de

Noriko Yu i Department o f Mathematic s Queen's Universit y Kingston, O N K7 L 3N 6 Canad a [email protected]; y u i O f i e l d s . u t o r o n t o . c a

Abstract. W e conside r certai n familie s o f Calabi-Ya u orbifold s an d their mirro r partner s constructe d fro m Ferma t hypersurface s i n weighte d projective 4-spaces . Ou r focu s i s the topological mirro r symmetry . Ther e are a t leas t thre e know n ingredient s t o describ e th e topologica l mirro r symmetry, namely , integra l vertice s i n reflexiv e polytopes , monomial s i n graded polynomia l ring s (wit h som e grou p actions) , an d period s (an d Picard-Fuchs differentia l equations) . I n thi s paper , w e wil l introduc e Fermat motive s associate d t o thes e Calabi-Ya u orbifold s an d the n us e them t o giv e a motivi c interpretatio n o f th e topologica l mirro r sym metry phenomeno n betwee n mirro r pair s o f Calabi-Ya u orbifolds . W e establish, a t th e Ferma t (th e Landau-Ginzburg ) poin t i n th e modul i space, th e one-to-on e correspondenc e betwee n th e monomia l classe s an d Fermat motives . Thi s i s don e b y computin g th e numbe r o f Fq-rationa l points o n ou r Calabi-Ya u orbifold s ove r F g i n tw o differen t ways : Weil' s algebraic numbe r theoreti c metho d involvin g Jacob i (Gauss ) sums , an d Dwork's p-adi c analyti c metho d involvin g Dwor k character s an d Gaus s sums. W e wil l discus s specifi c example s i n detail .

2000 Mathematics Subject Classification. Primar y 1 4J32 , 1 4G1 0 . 1 1 G40 . 1 1 F80 . Key words and phrases. Calabi-Ya u threefolds , orbifoldings , Ferma t motives , Mirro r sym m e t r y monomials . Gaus s (Jacobi ) sums , motive-monomia l correspondence , Ferma t point . This wor k i s partiall y supporte d b y N . Yui' s Discover y Gran t fro m th e Natura l Scienc e Research Counci l o f Canad a (NSERC) . ©2008 America n Mathematica l Societ y 3

4

Shabnam Kadi r an d Norik o Yu i CONTENTS

1. Introductio n 4 2. Ferma t motive s 7 3. Constructio n o f Calabi-Ya u orbifold s 9 4. Constructio n o f mirro 1 r Calabi-Ya u orbifold s 2 1 5. Ferma t motive s an d mirro r map s 5 6. Batyrev' s mirro r symmetr y 2 1 7. Monomial s an d period s 2 5 8. Th e monomial-motiv e correspondence : Example s 2 7 9. Proo f o f th e monomial-motiv e correspondenc e 3 0 10. Conclusion s an d furthe r problem s 4 2 References 4 5

1 Introductio n This is a sequel to the articl e of Yui [Y05 ] where Calabi-Yau orbifold s o f Ferma t hypersurface threefold s i n weighte d projectiv e 4-space s wer e constructed , an d thei r L-series (associate d t o th e £-adi c Galoi s representations ) wer e determined . I t wa s often th e cas e tha t th e Galoi s representation s ha d ver y hig h rank , whic h mad e it rathe r impossibl e t o carr y ou t th e require d calculations . T o remed y this , w e introduced Ferma t motives , an d the n decompose d th e Calabi-Ya u threefold s int o Fermat motives . Vi a cohomologica l realization s o f thes e motives , w e wer e abl e to calculat e th e motivi c L-serie s fo r eac h motive . Th e globa l L-serie s wa s the n obtained b y gluin g th e motivi c result s together . These Calabi-Ya u orbifold s ar e al l non-rigi d (ti 2d > 0) , an d thei r mirro r Calabi-Yau threefold s exis t satisfyin g th e followin g conjecture . Conjecture 1 . 1 (Topologica l Mirro r Symmetr y Conjecture ) Give n a famil y of Calabi-Ya u threefold s X, ther e i s a mirro r famil y o f Calabi-Ya u threefold s X such tha t h2A{X) =

h 1 -1 (X) an

dh

lA

(X) =

h

2A

{X)

so tha t th e Eule r characteristic s ar e subjec t t o th e relatio n X(X)

= -

X(X).

The topologica l mirro r symmetr y describe d abov e ca n b e reformulate d i n th e toric geometric settin g a la Batyrev [Ba94] . I n Batyrev's theory , mirro r symmetr y i s described i n term s o f pairs o f reflexive pol y topes. Integra l vertice s o f reflexive poly topes ar e the mai n ingredient s i n Batyrev's theory , an d the y correspon d t o monomi als in graded polynomia l rings . B y Aspinwall, Green e and Morriso n [AGM93] , there is th e monomial-diviso r mirro r ma p (fo r th e correspondin g cohomolog y groups , Htoric an d H \ ) , whic h yield s a one-to-on e correspondenc e betwee n tori c divisor s of a Calabi-Ya u famil y an d monomial s i n th e mirro r Calabi-Ya u family . Based o n th e theor y o f Dwork , Kat z an d Griffith s o n period s (se e Co x an d Katz [CK99]) , Candela s e t al . [CORV00 , CORV03 ] (resp . Kadi r [Ka05] ) estab lished explicitl y a one-to-on e correspondenc e betwee n monomial s an d period s vi a Picard-Fuchs differentia l equations , fo r th e quin t ic one-paramete r (resp . th e octi c

Motives an d Mirro r S y m m e t r y fo r Calabi—Ya u Orbifold s

5

two-parameter) famil y o f Calabi-Ya u threefolds . I n thei r calculations , period s de composed int o th e produc t o f subperiods . Thi s seem s t o sugges t tha t ther e shoul d be a motivi c interpretatio n fo r suc h factorizations . The concep t o f motive s ha s bee n emergin g i n th e physic s literature , an d th e purpose o f thi s pape r i s t o giv e a mathematicall y rigorou s discussio n o n motives , restricting ou r attentio n t o specifi c example s o f Calabi-Ya u threefolds . W e follo w the notio n o f motive s du e t o Grothendieck . Startin g wit h Ferma t hypersurface s i n weighted projectiv e 4-spaces , we define an d construc t explicitl y the so-called Ferma t motives fro m algebrai c correspondences , a s describe d i n Shiod a [Sh87] . Ou r goa l i s to interpre t th e topologica l mirro r symmetr y phenomeno n fo r th e mirro r pair s o f specific Calabi-Ya u orbifold s i n term s o f Ferma t motive s an d thei r cohomologica l realizations. A s our mai n result , w e establish a one-to-one correspondenc e betwee n monomials an d Ferma t motives . Thi s correspondenc e determine s Ferma t motive s which are invariant unde r th e mirror map . Sinc e Fermat motive s ar e defined onl y a t the Ferma t (th e Landau-Ginzburg ) poin t i n th e modul i space , thi s correspondenc e is established onl y a t th e Ferma t point . There ar e mor e monomial s tha n motives , an d w e observ e tha t monomial s as sociated t o conifol d point s see m t o b e associate d t o (mixed ) Tat e motives . Incidentally, a t th e Ferma t point , ou r Calabi-Ya u threefold s captur e th e struc ture o f CM typ e varietie s (se e Yui [Y05]) , an d henc e ou r motive s ar e als o C M typ e motives. (G . Moor e [Mo07 ] define d "attractive " Calabi-Ya u threefolds . Amon g our Calabi-Ya u orbifolds , ther e i s onl y on e suc h threefold , namely , m = 6 , Q = (1,1,1,1,2). I n fact , thi s i s th e onl y Calabi-Ya u orbifol d whos e weigh t motiv e i s rigid, i.e. , h 2d(MQ) = 0 and B 3{MQ) = 2. ) Now w e will describ e th e content s o f thi s paper . Section 2 is devoted t o the definition o f Fermat motive s an d thei r cohomologica l realizations. W e us e th e definitio n o f motive s du e t o Grothendieck , an d Mani n [Ma70], whic h i s based o n algebrai c correspondence s an d projectors . W e follow th e exposition o f Shiod a [Sh87 ] an d Gouve a an d Yu i [GY95] . In Section 3 , we construct Calabi-Ya u orbifold s i n weighted projectiv e 4-spaces . The startin g poin t i s th e Ferma t hypersurfac e V o f degre e m > 5 an d dimensio n 3, an d a finite abelia n grou p (whic h i s a subgrou p o f th e automorphis m grou p o f V). Thi s grou p wil l determin e a weight . W e tak e th e quotien t o f V b y suc h a group. Thi s give s ris e t o a quotien t threefol d wit h singularities . W e the n resolv e singularities b y taking th e crepan t resolutio n (whic h i s guaranteed t o exis t uniquel y (up to flops) for dimensio n < 3) . Th e smoot h threefol d thu s obtaine d i s our Calabi Yau threefold . Ther e ar e altogethe r 1 4 7 such Calabi-Ya u orbifolds . In Sectio n 4 , w e describ e th e Greene-Plesse r orbifoldin g constructio n o f mir ror partner s o f th e Calabi-Ya u orbifold s constructe d i n th e previou s section . W e review th e mirro r constructio n fro m th e pape r o f Green e an d Plesse r [GP90] . Th e mirror symmetr y i s interpreted a s the dualit y betwee n th e tw o finite abelia n group s associated t o th e mirro r pai r o f Calabi-Ya u threefolds . In Sectio n 5 , w e wil l construc t Ferma t motive s fo r th e Calabi-Ya u orbifold s in Sectio n 3 , an d comput e thei r invariant s (e.g. , motivi c Hodg e numbers , motivi c Betti numbers ) vi a their cohomologica l realizations . Fo r each mirror pai r o f Calabi Yau threefolds , w e als o determin e Ferma t motive s whic h ar e invarian t unde r th e mirror map . I n particular, w e observe that fo r eac h Calabi-Ya u orbifold , th e motiv e associated t o th e weigh t i s alway s invarian t unde r th e mirro r map . I f h ld = 1 , th e

6

S h a b n a m Kadi r an d Norik o Yu i

weight motiv e i s th e onl y motiv e invarian t unde r th e mirro r map . However , whe n /i 1 ' 1 > 1 , there ar e other motive s apart fro m th e weight motiv e that remai n invarian t under th e mirro r map . In Sectio n 6 , w e revie w th e constructio n o f mirro r pair s o f Calabi-Ya u hyper surfaces i n tori c geometr y du e t o Batyre v [Ba94] . W e wil l confin e ourselve s t o Calabi-Yau threefolds . Reflexiv e polytope s an d thei r dua l polytope s ar e th e mai n players i n Batyrev' s tori c mirro r symmetry . A pai r o f reflexiv e polytope s (A , A*) gives ris e t o a mirro r pai r o f Calabi-Ya u hype r surf aces. I t i s note d tha t th e origi n is the onl y integra l poin t containe d bot h i n th e reflexiv e polytop e an d it s dua l poly tope. (Thi s fac t play s a pivota l rol e i n provin g ou r mai n result. ) Integra l point s correspond t o monomial s i n grade d polynomia l rings . W e wil l discuss , i n particu lar, th e monomial-diviso r mirro r ma p o f Aspinwall, Green e and Morriso n [AGM93] , which give s the isomorphis m betwee n th e tw o space s H toric(X) an d H \ (X) fo r a mirror pai r (X,X). Thi s establishe s a one-to-on e correspondenc e betwee n integra l points i n th e reflexiv e polytop e o f X an d monomial s i n th e polynomia l rin g o f X. In Sectio n 7 , w e wil l discus s a one-to-on e correspondenc e betwee n monomi als an d period s vi a Picard-Fuch s differentia l equations . Th e metho d o f Dwork Katz-Griffiths determine s th e Picard-Fuch s differentia l equation s fo r Calabi-Ya u hypersurfaces i n weighte d projectiv e space s (o f an y dimension ) (se e Co x an d Kat z [CK99]). I n thi s section , w e wil l illustrat e thi s correspondence , focusin g o n th e concrete calculation s o f periods o f Candela s e t al . [CORV00 , CORV03 ] fo r th e one parameter deformatio n o f th e quinti c Calabi-Ya u threefol d i n th e ordinar y projec tive 4-space P 4 , an d o f Kadir [Ka05 ] fo r th e two-paramete r deformatio n o f the octi c Calabi-Yau threefol d i n the weighted projectiv e spac e P 4 (Q) wit h Q — (1,1, 2, 2, 2). Here w e observ e tha t th e Picard-Fuch s differentia l equatio n decompose s int o th e product o f lower orde r Picard-Fuch s differentia l equations . Thi s suggest s tha t suc h a decompositio n ough t t o hav e it s origi n i n th e motivi c decompositio n o f th e man ifold. Section 8 contain s ou r mai n example s an d th e mai n resul t o n th e monomial motive correspondenc e (Theore m 8.1 ) . W e establis h a one-to-on e correspondenc e between th e clas s o f monomial s an d Ferma t motive s a t th e Ferma t poin t fo r th e Calabi-Yau threefold s o f Sectio n 3 . W e prove tha t th e motive s whic h ar e invarian t under th e mirro r ma p correspon d t o th e clas s o f the constan t monomia l (an d henc e to th e origi n i n th e polytopes) . W e illustrate th e monomial-motiv e correspondenc e for th e quinti c an d th e octi c Calabi-Ya u orbifolds . Section 9 contain s ou r proo f fo r th e monomial-motiv e correspondence . W e compute th e numbe r o f F^-rationa l point s (an d henc e congruenc e zeta-functions ) for ou r Calabi-Ya u threefold s ove r finit e field s ¥ q i n two different ways . O n th e on e hand, w e comput e the m wit h Weil' s metho d usin g Jacob i (Gauss ) sums . O n th e other hand , w e compute them with Dwork's p-adic method usin g Dwork's character s and Gaus s sums . W e the n sho w tha t th e tw o approache s yiel d th e sam e resul t a t the Ferma t point . Away fro m th e Ferma t point , Calabi-Ya u orbifold s wit h deformatio n param eters yiel d mor e monomial s tha n motives . A conifol d poin t o n thes e Calabi-Ya u orbifolds wit h deformatio n parameter s i s a singularit y locall y isomorphi c t o th e projective quadri c surfac e X 2 + Y 2 + Z 2 - f T 2 = 0 , an d w e ma y associat e a motiv e (e.g., a mixe d Tat e motive ) employin g th e sam e lin e o f argument s a s i n Bloch Esnault-Kreimer [BEK05] .

Motives an d Mirro r S y m m e t r y fo r Calabi—Ya u Orbifold s 7

Finally, Sectio n 1 0 present s conclusion s o f thi s work , an d furthe r problem s and futur e projects . Th e mai n conclusio n i s t o brin g motive s int o th e real m o f topological mirro r symmetr y fo r ou r Calabi-Ya u orbifolds . In a recen t pape r [K07] , Kloosterman ha s extende d th e motive-monomia l cor respondence (establishe d a t th e Ferma t poin t i n thi s article ) t o an y quasismoot h member o f a one-paramete r monomia l deformatio n o f a Ferma t hypersurfac e i n a weighted projectiv e space , usin g a p-adi c (e.g. , Monsky-Washnitzer ) cohomolog y theory. Als o h e ha s give n explici t solution s t o th e p-adi c Picard-Fuch s differentia l equation associate d wit h monomia l deformation s o f Ferma t hypersurfaces . Acknowledgments Shabnam Kadi r hel d a Field s Postdoctora l Fellowshi p i n 2004-200 5 supporte d by th e Field s Institut e Themati c Progra m "Geometr y o f Strin g Theory" , an d N . Yui's NSER C Discover y Grant . During th e cours e o f thi s work , Norik o Yu i hel d visitin g position s a t a numbe r of institutions . Thes e includ e Universit y o f Leide n (Winte r 2004) , Max-Planck Institut-fur-Mathematik Bon n (Sprin g 2004 , Sprin g 2006) , Universit y o f Hannove r (Spring 2004 , 2006) , Universit y o f Main z (Sprin g 2004) , Tsud a Colleg e (Sprin g 2005) an d Kavl i Institut e fo r Theoretica l Physic s (KITP ) (Decembe r 2005 ) a t Uni versity o f Californi a Sant a Barbara . Th e researc h a t KIT P wa s supporte d i n par t by th e Nationa l Scienc e Foundatio n unde r th e Gran t No . PHY99-07949 . Sh e i s grateful fo r th e hospitalit y o f thes e institutions . Finally, i t i s ou r pleasur e t o than k th e refere e fo r usefu l comments , remark s and suggestion s fo r th e improvemen t o f th e paper . 2 Ferma t motive s We wil l emplo y th e definitio n o f motive s du e t o Grothendiec k an d Mani n [Ma70]. Th e mos t recommende d reference s fo r th e generalit y o f motive s migh t be Deligne-Miln e [DM82] , and Soul e [So84] . I n thi s pape r w e will confin e ourselve s to th e so-calle d "Fermat " motive s arisin g fro m Ferma t hypersurfaces . W e wil l re call th e constructio n o f Ferma t motive s fro m Shiod a [Sh87 ] an d Gouve a an d Yu i [GY95]. Th e constructio n work s fo r an y dimension , no t onl y fo r dimensio n 3 . We star t wit h th e Ferma t hypersurfac e o f degre e m an d o f dimensio n n i n th e projective spac e P n + 1 : V : Z ™ + Z™ + • • • + Z™ +2 - 0 C P n + 1 . Let fjL m be th e grou p o f 772-t h roots o f unit y an d le t