Mirror Symmetry IV: Proceedings of the Conference on Strings, Duality, and Geometry, Centre De Recherches Mathematiques of the University De Montreal ... 2000 (Ams/Ip Studies in Advanced Mathematics) 0821833359, 9780821833353

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Mirror Symmetry IV

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AMS/IP

https://doi.org/10.1090/amsip/033

Studies in Advanced Mathematics Volume 33

Mirror Symmetry IV Proceedings o f th e Conferenc e o n Strings, Duality, an d Geometr y Centre d e Recherche s Mathematique s of th e Universite d e Montrea l (CRM) March 200 0

Eric D'Hoker, Duong Phong , and Shing-Tun g Yau, Editors

American Mathematical

Society

Centre de Recherches Mathematiques

International Press

Shing-Tung Yau , Genera l E d i t o r 2000 Mathematics Subject

Classification.

Primar

y 1 4-xx , 32-xx , 81 -xx .

Library o f Congres s Cataloging-in-Publicatio n D a t a Conference o n Strings , Duality , an d Geometr y (2000 : Montreal , Quebec ) Mirro r symmetr y IV : proceedings o f th e Conferenc e o n Strings , Duality , an d Geometry , Montreal , 2000/Eri c D'Hoker , Duong Phong , an d Shing-Tun g Yau , editors . p. cm . — (AMS/I P studie s i n advance d mathematics ; ISS N 1 089-3288 ; v . 33 ) Includes bibliographica l references . ISBN 0-821 8-3335- 9 (alk . paper ) 1. Mirro r symmetry—Congresses . 2 . Geometry , Differential—Congresses . 3 . Strin g models—Congresses. 4 . Dualit y (Nuclea r physics)—Congresses . I . D'Hoker , Eric , 1 956 II. Phong , Duon g H. , 1 953- . III . Yau , Shing-Tung , 1 949 - IV . Title . V . Series . QC174.17.S9C65 200 0 516.3 / 62-dc21 200203858

0

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 firs t pag e o f each article. ) © 200 2 b y th e America n Mathematica l Society , Internationa l Pres s an d th e Centre d e Recherche s Mathematiques . Al l right s reserved . The America n Mathematica l Society , Internationa l Pres s an d th e Centre d e Recherche s Mathematique s retai n al l right s except thos e grante d t o th e Unite d State s Government . 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 . Visit th e AM S hom e pag e a t http://www.ams.org / Visit th e Internationa l Pres s hom e pag e a t URL : h t t p : / / w w w . i n t l p r e s s . c o m / Visit th e Centr e d e Recherche s Mathematique s hom e pag e a t URL: http: //www. crm. umontreal. ca/ 10 9 8 7 6 5 4 3 2 1

07 06 05 04 03 02

Contents

Introduction I. Calabi-Ya u Manifolds , Mirro r Symmetry , an d Symplecti c Geometr y A Surve y o f Mirro r Principl e BONG H . LIAN , K E F E N G LIU , AN D SHING-TUN G YA U 3

Mirror Symmetry : Aspect s o f th e Firs t 1 0 Years B. R . G R E E N E

1

Lagrangian toru s fibration s o f Calabi-Ya u hypersurface s i n tori c varietie s and SY Z mirro r symmetr y conjectur e W E I - D O N G RUA N 3

3

Moduli Spac e o f Stabl e Map s GANG LI U 5

7

Cohomological propertie s o f rule d symplecti c structure s FRANQOIS LALOND E AN D DUS A M C D U F F 7

9

II. Supersymmetri c Gaug e Theorie s an d Integrabl e Model s Spectral La x pair s an d Calogero-Mose r system s J. C . HURTUBIS E 0

3

M-Theory teste d b y M = 2 Seiberg-Witten Theor y ISABEL P . ENNES , CARLO S LOZANO , STEPHE N G . NACULICH , HENRI C RHEDIN, 1 1 AN D HOWAR D J . SCHNITZE R

3

Seiberg-Witten curve s fo r ellipti c model s ISABEL P . ENNES , CARLO S LOZANO , STEPHE N G . NACULICH , 1 AND HOWAR D J . SCHNITZE R 2

7

The periodi c an d ope n Tod a lattic e I.1 KRICHEVE R AN D K . L . VANINSK Y 3

9

Exact Integratio n Method s fo r Supersymmetri c Yang-Mill s Theor y J E A N - L O U P GERVAI S 5

9

vi C O N T E N T

S

I I I . M-Theory , D-branes , an d Non-commutativ e Geometr y Nonabelian D-brane s an d Noncommutativ e Geometr y ROBERT C . MYER S 6

9

Evidence fo r Windin g State s i n Noncommutativ e Quantu m Fiel d Theor y P H I L I P P E POULIO T E T A L 8

7

The Discret e Boun d Stat e Spectru m o f th e Rotatin g DO-bran e System , an d its Deca y b y Emissio n o f Ramond-Ramon d Fiel d Radiatio n KONSTANTIN G . SAVVID Y 9

5

On th e correspondenc e betwee n D-brane s an d stationar y supergravit y solutions o f typ e I I Calabi-Ya u compactification s FREDERIK D E N E F 20

9

Phase-Transitions an d Tenso r Dynamic s i n M-theor y MICHAEL FAUX , D I E T E R LUST , AN D BUR T A . OVRU T 23 1

Duality, Eisenstei n Serie s an d Exac t Threshold s NIELS A . OBER S AN D BORI S PIOLIN E 25

5

IV. Strings , Gaug e Theories , an d A d S / C F T Correspondenc e Connectedness o f the Boundar y i n th e AdS/CF T Correspondenc e EDWARD W I T T E N AN D S.-T . YA

U 27

3

A not e o n th e topolog y o f th e Boundar y i n th e AdS/CF T Correspondenc e S.-T. YA U 28

9

Holographic dual s o f 4 D field theorie s M. PORRAT I AN D A . STARINET S 29 1

Black Hol e Thermodynamic s fro m Calculation s i n Strongly-Couple d Gaug e Theory DANIEL RABAT , GILA D LIFSCHYTZ , AN D DAVI D A . LOW E 29

Correlation function s fo r orbifold s o f th e typ e M

N

9

N

/S

O L E G1 LUNI N AN D SAMI R D . MATHU R 3 1

V. Ellipti c Gener a an d Automorphi c Form s Elliptic gener a o f singula r varieties , orbifol d ellipti c genu s an d chira l d e Rham comple x LEV A . BORISO V AN D ANATOL Y LIBGOBE R 32

5

c

On Famil y Rigidit y Theorem s fo r Spin Manifold s K E F E N G LI U AN D XIAONA N M A 34

Ample Divisors , Automorphi c Form s an d Shafarevich' s Conjectur e JAY JORGENSO N AN D ANDRE Y TODORO V 36 1

3

Introduction

Mirror Symmetr y wa s firs t discovere d i n th e earl y 1 990's . I t immediatel y sparked a vigorou s researc h progra m i n geometr y an d theoretica l physic s whic h continues unabate d t o thi s day . Th e searc h fo r a ful l geometri c understandin g o f mirror symmetr y ha s le d t o unexpecte d an d dee p connection s wit h othe r fields o f mathematics, i n particular algebraic , differential , an d no w symplectic geometry . A t the sam e time , th e notio n o f duality , o f whic h mirro r symmetr y i s a prim e exam ple, ha s experience d a tremendou s developmen t i n the las t 1 0 years. Th e discover y of M-Theory , Seiberg-Witte n theor y an d it s relatio n wit h integrabl e models , th e AdS/CFT correspondence , non-commutativ e quantu m fiel d theorie s ar e amon g th e most strikin g achievements . In Marc h 2000 , a week-lon g worksho p o n "Strings , Duality , an d Geometry " was held a t th e Centr e d e Recherches Mathematique s o f the Universit e o f Montrea l (CRM), unde r th e sponsorshi p o f th e 1 999-200 0 themati c yea r o n "Mathematica l Physics" a t th e Centre . Th e participant s wer e a n equa l mixtur e o f mathematician s and physicist s activ e i n th e field . Th e presen t volum e i s a recor d o f thei r contri butions. I t ca n b e viewe d a s a seque l t o th e thre e previou s volume s i n thi s series , "Mirror Symmetry " (ed . S.T . Yau) , "Mirro r Symmetr y II " (eds . B . Green e an d S.T. Yau) , an d "Mirro r Symmetr y III " (eds . D.H . Phong , L . Vinet, an d S.T . Yau) , published b y th e America n Mathematica l Societ y an d Internationa l Press . The volum e present s a broa d surve y o f man y particularl y noteworth y develop ments sinc e "Mirro r Symmetr y III" : In Sectio n I , a n overvie w o f progres s i n th e geometr y o f mirro r symmetry , including th e mathematica l proo f o f the mirro r principle , i s presented i n th e pape r by B . Lia n an d S.T . Yau . A n overvie w o f th e physica l aspect s i s i n th e pape r b y B. Greene . Progres s i n th e Strominger-Yau-Zaslo w progra m o f constructing mirro r manifolds a s Lagrangia n toru s fibration s i s explaine d i n th e pape r b y W.D . Ruan . Closely relate d an d importan t development s i n th e theorie s o f modul i o f stabl e maps an d fibration s i n symplecti c geometr y ar e covere d respectivel y i n th e pape r by G . Li u an d th e pape r b y F . Lalond e an d D . McDuff . Section I I i s devote d t o Seiberg-Witte n theory , integrabl e models , an d super symmetric gaug e theories . Th e pape r b y J . Hurtubis e provide s a ne w geometri c framework fo r th e recen t solutio n o f Calogero-Mose r system s b y D'Hoker-Phon g and A . Bordne r e t al. , whic h i s als o th e Seiberg-Witte n solutio n fo r certai n scal e invariant J\f — 2 supersymmetri c Yang-Mill s theories . I n th e tw o paper s b y I . Ennes e t al. , method s ar e describe d fo r computin g th e effectiv e prepotentia l an d vii

viii I N T R O D U C T I O

N

instanton contribution s fro m Seiberg-Witte n curves , thu s gettin g a n indirec t chec k of M-Theory . Th e pape r o f I . Kricheve r an d K . Vaninsk y extend s th e Tod a spec tral curve s fro m th e periodi c cas e t o th e ope n case , providin g a clos e analog y wit h brane constructions. Example s of exact solution s of 10-dimensional supersymmetri c Yang-Mills theor y ar e describe d i n th e pape r b y Gervais . Section II I cover s man y development s i n M-Theory , D-branes , an d Non-Com mutative Geometry . A descriptio n o f th e dynamic s o f non-abelia n D-brane s to gether wit h th e underlyin g non-commutativ e geometr y i s give n i n th e pape r b y R. Myers . Non-commutativ e geometr y als o play s a majo r rol e i n th e paper s b y P. Poulio t e t al . an d b y K . Savvidy , wher e thermodynamic s i n non-commutativ e quantum field theorie s an d bran e boun d state s o f the non-commutativ e spher e ar e explored respectively . Stationar y supergravit y solution s fro m BP S D-brane s i n Calabi-Yau manifold s ar e describe d i n th e pape r b y F . Denef . Orbifol d degener ations o f M-Theor y compactification s ar e describe d i n th e pape r b y M . Faux , D . Lust, an d B . Ovrut . Th e pape r b y N . Ober s an d B . Piolin e discusse s th e dualitie s in Typ e I I an d M-Theor y fro m th e poin t o f vie w o f duality-invarian t BP S mas s formulae an d generalize d Eisenstei n series . The paper s i n Sectio n I V ar e mos t closel y relate d wit h th e dualit y betwee n strings an d gaug e theories , an d th e AdS/CF T correspondence . Topologica l con straints fo r th e Penros e boundar y i n th e AdS/CF T correspondenc e ar e derive d i n the pape r b y E . Witte n an d S.T . Yau . I n th e subsequen t not e b y S.T . Yau , a n outline i s give n fo r ho w t o appl y th e method s o f Schoen-Ya u fo r positiv e scala r curvature an d construc t th e conforma l boundarie s o f Einstei n manifold s neede d for th e AdS/CF T correspondence . Whil e th e origina l AdS/CF T correspondenc e was conjecture d betwee n J\f = 4 supersymmetric Yang-Mill s theor y an d Typ e IIB , exciting result s hav e recentl y bee n obtaine d o n generalization s o f th e correspon dence t o case s wit h les s o r n o supersymmetry . Th e pape r b y M . Porrat i an d A . Starinets present s a general discussio n o f J\f = 0 and N = 1 deformations o f M — 4 super Yang-Mills and it s supergravity holographi c duals . Anothe r versio n of the cor respondence relate s problem s i n supergravity t o a n SU(N ) Yang-Mill s gauge theor y quantum mechanic s i n th e ' t Hoof t limit , an d i s use d i n th e pape r b y D . Kabat , G. Lifschyt z an d D . Low e to stud y th e entrop y o f certain non-extrema l blac k holes . Yet anothe r for m o f th e correspondenc e i s use d i n th e pape r b y O . Luni n an d S . Mathur t o relat e Typ e II B superstrin g theor y i n th e presenc e o f a D1 -D 5 bran e system t o th e two-dimensiona l conforma l field theor y give n b y a non-linea r sigm a model o n M n/Sn, whic h i s th e symmetri c orbifol d o f n copie s o f th e manifol d M. A method fo r th e computatio n o f correlation function s o n this orbifol d i s develope d in thi s paper . In Sectio n V ar e paper s relate d t o ellipti c gener a an d modula r forms . New , general version s o f th e famil y rigidit y theore m fo r ellipti c gener a ar e prove d i n the pape r b y K . Li u an d X . Ma . Two-variabl e ellipti c gener a fo r singula r varietie s together wit h generalization s o f the MacDonal d an d Zagie r generatin g function s fo r Euler characteristic s an d fo r signature s ar e reviewed i n the pape r b y L. Borisov an d A. Libgober. Th e paper b y J. Jorgenso n an d A. Todorov describes how automorphi c forms an d discriminan t divisor s ca n b e use d t o obtai n finiteness result s fo r familie s of algebrai c varietie s wit h a fixed type .

INTRODUCTION

IX

The editor s woul d lik e t o expres s thei r dee p appreciatio n t o Professo r Lu c Vinet, wh o organized th e 1 999-200 0 thematic year ; t o Professo r Jacque s Hurtubise , Director o f th e CRM ; t o Professo r Yva n Saint-Aubin , Associat e Directo r o f th e CRM; an d t o Mr . Loui s Peletier , fo r thei r invitatio n t o organiz e thi s worksho p an d for thei r war m hospitalit y i n Montreal . Eric D'Hoker , Duon g H . Phong , an d Shing-Tun g Yau , editors .

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I Calabi-Yau Manifolds , Mirror Symmetry , and Symplecti c Geometr y

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https://doi.org/10.1090/amsip/033/01 Mirror Symmetr y I V AMS/IP Studie s i n Advance d Mathematic s Volume 33 , 200 2

A Surve y o f Mirro r Principl e Bong H . Lian , Kefen g Liu , an d Shing-Tun g Ya u ABSTRACT. Thi s not e briefl y review s th e Mirror Principle a s develope d i n th e series o f paper s [1 8] , [1 9] , [20] , [21 ] , [22] . W e illustrat e thi s theor y wit h a few ne w examples . On e o f the m give s a n intriguin g connectio n t o a proble m of countin g holomorphi c disk s an d annuli . Thi s not e wil l appea r i n th e pro ceedings o f th e conferenc e o n Geometr y an d Strin g Theor y a t th e C.R.M . i n Montreal o f Sprin g 2000 .

1. Som e Backgroun d In th e aforementione d serie s o f paper s w e develo p th e mirror principle i n in creasing generalit y an d breadth . Give n a projective manifol d X , mirro r principl e i s a theor y tha t yield s relationship s fo r an d ofte n compute s th e intersectio n number s of cohomolog y classe s o f the for m 6(VD ) o n stabl e modul i space s M g^{d^X). Her e VD i s a certai n induce d vecto r bundle s o n M g^{d,X) an d b i s an y give n multi plicative cohomolog y class . I n th e firs t pape r [1 8] , we consider thi s proble m i n th e genus zer o g = 0 cas e whe n X — Pn an d VJJ i s a bundl e induce d b y an y conve x and/or concav e bundl e V o n P n . A s a consequence , w e hav e prove d a mirro r for mula whic h compute s th e intersectio n number s vi a a generatin g function . Whe n X = P n , V i s a direc t su m o f positiv e lin e bundle s o n P n , an d b is the Eule r class , a secon d proo f o f this specia l cas e ha s bee n give n i n [23] , [5 ] following a n approac h proposed i n [9] . Othe r proof s i n thi s cas e ha s als o bee n give n i n [3] , [8] , and whe n V include s negativ e lin e bundles , i n [6] . I n [1 9] , we develop mirro r principl e whe n X i s a projectiv e manifol d wit h TX convex . I n [20] , w e conside r th e g — 0 cas e when X i s an arbitrar y projectiv e manifold . Her e emphasis ha s bee n pu t o n a clas s of T-manifold s (whic h w e cal l balloo n manifolds ) becaus e i n thi s cas e mirro r prin ciple yields a (linear! ) reconstructio n algorith m whic h compute s i n principle al l th e intersection number s abov e for any convex/concave equivarian t bundl e V on X an d any equivariant multiplicativ e clas s b. Moreover , specializin g this theory t o the cas e of line bundles o n toric manifold s an d b to Eule r class , we give a proof o f the mirro r formula fo r tori c manifolds . I n bot h [20 ] an d [21 ] , we develo p mirro r principl e fo r higher genus . W e also extend th e theor y t o includ e the intersectio n number s fo r co homology classe s o f the for m ev*(0)6(Vb) . Her e ev : Mg^{d,X)— > X k i s the usua l evaluation ma p int o th e produc t X k o f k copies o f X , an d cj> i s an y cohomolog y class o n X k. ©2002 America n Mathematica l Societ y an d Internationa l Pres s 3

4

B.H. LIAN , K . LIU , AN D S.-T . YA U

For motivation s an d som e historica l backgroun d o f th e mirro r principle , w e refer th e reade r t o th e introductio n o f [1 8] , [19]. In sectio n 2 , we outline th e mai n idea s o f the mirro r principle , an d explai n on e of ou r mai n theorems . I n sectio n 3 , we discuss a fe w examples . Acknowledgment. W e thank C . Vafa for informing u s of his result o n the disk counting problem . W e als o than k th e organizer s fo r invitin g u s t o th e conferenc e on Geometr y an d Strin g Theor y a t th e C.R.M . i n Montrea l i n 2000 . B.H.L.' s research i s supporte d b y NS F gran t DMS-00721 58 . K.L.' s researc h i s supporte d by NS F gran t DMS-980323 4 an d th e Terma n fellowshi p an d th e Sloa n fellowship . S.T.Y.'s research i s supported b y DO E gran t DE-FG02-88ER2506 5 an d NS F gran t DMS-9803347. 2. Mirro r Principl e For simplicity, w e restrict ou r discussion s t o the genus zero theory, an d refe r th e interested reade r t o [21 ] fo r a theory o f higher genus . Le t X b e a projective n-fold , and d G H 2(X,Z). Le t Mo tk(d,X) denot e th e modul i stac k o f /c-pointed , genu s 0, degre e d, stabl e map s (C , /, x i , . .. ,£& ) on X [1 5] . (Not e tha t ou r notatio n i s without th e bar. ) B y [1 7 ] (cf . [4]) , each nonempt y Mo^(d,X) admit s a homolog y cycle LTo jfc(d, X) o f degree dim X-\-(ci(X), d)+k—3. Thi s cycle plays the role of the fundamental clas s i n topology , henc e LTo^(d, X) i s called th e virtua l fundamenta l class. Let V b e a conve x vecto r bundl e o n X (i.e. , i J 1 ( P 1 , f*V) = 0 for ever y holo morphic ma p / : P 1— • X). The n V induce s o n eac h Mo,k{d, X) a vecto r bundl e Vd, wit h fiber a t ( C , / , # ! , . . . ,Xk) give n b y th e sectio n spac e H°(C,f*V). Le t b be an y multiplicativ e characteristi c clas s [1 1 ] (i.e. , i f 0—> E' — > E — • E" — » 0 is a n exact sequenc e o f vector bundles , the n b{E) = b{E')b(E")). Th e proble m w e stud y here i s t o comput e th e characteristi c number s Kd : = / b(V

d)

JLTo,o(d,X)

and thei r generatin g function :

There i s a simila r an d equall y importan t proble m i f on e start s fro m a concav e vector bundl e V [1 8 ] (i.e. , H°(P 1 ,f*V) = 0 fo r ever y holomorphi c ma p / : P1—» X). Mor e generally , V ca n b e a direc t su m o f a conve x an d a concav e bundle . The roug h ide a o f th e Mirro r Principl e i s tha t th e classe s th e induce d bundle s Vd on th e stabl e modul i inheri t a numbe r o f universa l structure s (i.e. , exis t i n al l stable ma p modul i of any projective manifold) . Thes e structures combine d wit h th e multiplicative propertie s o f the classe s b(Vd) give rise to som e remarkable quadrati c identities. I t i s ofte n th e cas e (whe n sufficien t symmetr y i s presen t o n X) tha t these identitie s ar e stron g enoug h fo r a complet e reconstructio n o f the intersectio n numbers Kd- W e explai n thi s ide a furthe r belo w withou t proofs . Fo r details , se e [20]. Step 1 . Localization on the linear sigma model. Conside r th e modul i space s Md(X) : = M 0 ,o((l,^),P 1 x X). Th e projectio n P 1 x X - > X induce s a ma p 7T : Md{X)— » Mofi(d,X). Moreover , th e standar d actio n o f S 1 o n P 1 induce s a n S1 actio n o n Md(X). W e firs t stud y a slightl y differen t problem . Namel y conside r

A SURVE Y O F M I R R O R P R I N C I P L E

5

the classe s 7r*b(Vd) o n Md(X), instea d o f b(Vd) o n M 0,o(d, X). First , ther e i s a canonical wa y t o embe d fiber product s Fr = M

0>i(r,X)xxM0|i(d-r,X)

each a s a n S 1 fixed poin t componen t int o Md{X). Le t i r : F r— > M^(X ) b e th e inclusion map . Second , ther e i s a n evaluatio n ma p e : F r — • X fo r eac h r . Third , there i s a (produc t of ) projectiv e spac e W d equipped wit h a n 5 1 action , an d ther e is a n equivarian t ma p ip : Md(X)— > Wd, an d embedding s j r : X —* Wd, suc h tha t the diagra m

Fr -±> X -

M

^W

d{X)

d

commutes. Le t a denote s th e weight o f th e standar d S l actio n o n P 1 . Applyin g the localizatio n formul a [2] , [1 3] , this diagra m allow s u s t o recas t ou r proble m t o one o f studyin g th e S l-equivariant classe s Qd '•= Mo,i(d, X) induce d b y V , an d similarl y fo r U' d_r. Taking th e multiplicativ e characteristi c clas s 6 , we ge t th e identit y o n F r: e*b(V)b(i*rUd) = b(U' r)b(U'd-r)This i s wha t w e cal l th e gluing identity. Thi s ma y b e translate d t o a simila r qua dratic identity , vi a Ste p 1 , for Qd i n th e equivarian t Cho w group s o f Wd- Th e ne w identity i s called th e Eule r dat a identity . Step 3 . Linking theorem. Th e constructio n abov e i s functorial , s o tha t i f X comes equippe d wit h a toru s T action , the n th e entir e constructio n become s G — S 1 x T equivarian t an d no t jus t S 1 equivariant . I n particular , th e Eule r data identit y i s a n identit y o f G-equivarian t classe s o n W^ . Ou r proble m i s to first compute th e G-equivarian t classe s Qd o n Wd satisfyin g th e Eule r dat a identity . Note tha t th e restriction s Qd\ P t o th e T fixed point s p i n X o C Wd ar e polynomial s functions o n th e Li e algebr a o f G. Suppos e tha t X i s a balloo n manifold . Thi s i s a complex projectiv e T-manifol d satisfyin g th e followin g condition s [1 2] : (i) Th e T fixed point s ar e isolated . (ii) Le t p b e a T fixed point . The n th e T weight s A i , . . . , An o f th e isotropi c representation o n th e tangen t spac e T PX ar e pairwis e linearl y independent . We furthe r assum e tha t th e momen t ma p i s 1 - 1 o n th e fixed poin t set .

6

B.H. LIAN , K . LIU , AN D S.-T . YA U

In thi s case , th e classe s Qd ar e uniquel y determine d b y th e value s o f th e Qd\ P, when a i s som e scala r multipl e o f a weigh t A^ . Thes e value s o f Qd\ P, whic h w e call th e linkin g value s (se e [20 ] fo r precis e definition) , ca n b e compute d explicitl y by exploitin g th e momen t ma p [1 ] , [1 0 ] a s wel l a s certai n structur e o f a balloo n manifold. THEOREM 2. 1 ([20]) . The equivariant classes Qd — (f*7r*b(Vd), as a solution to the Euler data identity, can be completely recovered from the linking values.

Once th e linkin g value s ar e known , i t i s ofte n eas y t o manufactur e explicitl y the G-equivarian t classe s Qd usin g th e linkin g value s a s a guide . Man y explici t examples ar e discusse d i n [20] . Step 4 . Computing the K&. Onc e th e classe s Qd = 0 . wher e th e Li ar e respectivel y convex/concav e lin e bundle s o n X. Le t

n = Bo := c(v+)/c(v-) = JT^ + ci(4 + ))/ ![(* + Cl(L7)) i3

esi{Xo/Wd) x

.x ^

-(ci(L7),d)-l

X

II I

T(

« + ci(L7) + fca).

j fc=i

T H E O R E M 3.1 . There exist unique power series f(t),g(t) such ing formula holds:

that the follow-

where s : = r/ c F + — r/ c V~ — (n — 3), i := t + g. Moreover, f,g are determined by the condition that the integrand on the left hand side is of order 0(a~ 2).

7

A SURVE Y O F M I R R O R P R I N C I P L E

Note tha t whe n x —> 0 , th e formul a abov e reduce s t o th e cas e whe n b is th e Euler class . The tangent bundle on P n . Th e exampl e abov e deals , o f course , wit h direc t sum o f line bundle s only . W e no w giv e a n example startin g fro m th e tangen t bundle V = TX o n X = P n , whic h i s nonsplit . Conside r th e cas e wher e br the T-equivariant Cher n polynomial . Le t A ^ be th e weight s o f th e standar d T actio n on P n , an d pi b e th e it h fixed point . Recal l tha t Q := br{V) = \ r i i ( x + H — A*) , where H is the equivarian t clas s o n P n wit h H\ Pi = A^. Using th e T equivariant Eule r sequenc e 0 -> O - > ®? =0°(H - Ai ) - ^ T X - * 0 One ca n comput e th e linkin g value s i n thi s case . Ther e ar e give n b y

Ylflix + Xj-Xi-kx/d). k=0

Here p, q are th e j t h an d th e Zt h fixed point s i n P n , an d A = Xj — Aj . W e ca n us e this t o se t u p a system o f linea r equation s t o solv e fo r A(t) inductively . However , there i s an easie r wa y t o comput e A(t) i n thi s case . Usin g th e linkin g value s abov e as a guide, w e se t 1

Bd := ~l[Y[(x + H - \i - ka) fc=( i fc=0

and le t B,

B(t):^e-H-t^Y Then it can b e show n tha t th e serie s AU\

.

= e-H-t/a

y^ JoQd

e

d-t

^IliIlLi(H-\i-ka) is related t o B(t) b y A{t + g) = efl

a

B{t)

where / , g ar e explicitl y computabl e functions , simila r t o those i n the previou s example. Thi s relation, onc e again, allow s us to compute al l the K^ simultaneously . V : = O(-l) ® 0{—\) on P 1 . I n thi s case , we let b be th e Eule r clas s Ct op> an d we would lik e to comput e th e one-pointe d intersectio n number s / e*(H)bM). •/M0>i(d,Pi)

Here V' d is th e bundl e induce d o n Mo,i(d , P 1 ) b y V , H is th e hyperpian e clas s o n P 1 , an d e : Mo,i(d, P1)— * P 1 i s th e evaluatio n map . W e ca n easil y specializ e th e first exampl e abov e t o th e cas e o f X = P 1 an d V — 0{—1)0O(-1). I n thi s case , / = g = 0, an d w e get th e formul a

L

e-m/a__J2@d _ = a 2 P 1 Il (H-ma) 1 m=

a

_3(2 _

dt)Kd

8 B.H

. LIAN , K . LIU , A N D S.-T . YA U

Apply ^ t o bot h sides , an d combin e th e resul t wit h Theore m 3. 2 i n [1 8 ] (se e th e first eqn . o n p . 3 6 there). Wha t w e get i s = dK d = d- 2.

/ e*(H)b{V^) 1

./Mo.iCd.P )

The value s d~ 2 remin d u s o f a resul t tha t Vaf a obtain s vi a th e physic s o f local mirro r symmetry . H e consider s th e proble m o f countin g holomorphi c disk s in a Calabi-Ya u 3-fol d equippe d wit h a choic e o f Lagrangia n submanifold . Th e boundary o f the disks are required t o lie in the Lagrangian submanifold . A counting problem i s heuristically formulate d int o a proble m o f determining th e Eule r classe s of certai n yet-to-be-define d modul i spaces . I n thi s cas e th e physic s o f loca l mirro r symmetry indicate s tha t th e Eule r classe s shoul d b e a give n b y a "multiple-cover " contribution d - 2 , wher e d is the winding numbe r o f the disk' s boundary circl e alon g the Lagrangia n submanifold . Vafa's result suggest s the following interpretation. Th e Lagrangian submanifol d plays th e rol e o f a vanishin g cycl e i n a certai n limit . A holomorphi c dis k wit h boundary landin g o n th e Lagrangia n submanifol d woul d loo k lik e a P 1 wit h on e marked poin t i n thi s limit . Loca l mirro r symmetr y suggest s tha t w e shoul d us e the stabl e ma p modul i space s o f P 1 a s a mode l fo r thi s problem . Th e requiremen t that th e marke d poin t land s o n th e vanishin g cycl e ma y b e though t o f a s th e incidence conditio n o n th e ma p P 1— * P 1 wit h on e poin t mappe d t o th e cycl e H. The appropriat e modul i space s i n thi s mode l shoul d the n b e Mo,i(d , P1) , an d th e Euler classe s shoul d correspon d t o th e "multiple-cover " formul a fo r th e bundl e V'd induced hy V — G{—1)©0(-1). S o a good candidat e fo r th e intersectio n number s -2

Two Lagrangian S 3 in a CY 3-fold? Anothe r interestin g situatio n considere d by physicist s i s th e proble m o f countin g annul i i n a C Y 3-fol d equippe d wit h tw o Lagrangian 3-sphere s S' 3, subject t o the incidence condition tha t eac h of the bound ary circle s o f th e annulu s land s insid e on e o f th e 3-spheres . Agai n th e 3-sphere s plays th e rol e o f tw o vanishin g cycle , an d ar e allowe d t o contrac t t o point s x,y. The annulu s look s lik e a P 1 wit h tw o marke d point s anchore d t o x,y. B y analog y with th e previou s exampl e a s i n loca l mirro r symmetry , th e correspondin g stabl e map modul i i n thi s cas e shoul d b e th e two-pointe d modul i Mo,2(d , P1) , an d th e corresponding intersectio n number s shoul d b e

/ el(H)e*

2{H)bW).

Here V£ i s th e induce d bundl e p\V' d o n M o ^ f ^ P 1 ) , wher e p 2 : M o ^ ^ P 1 ) - > Mo,i(d, P 1 ) i s th e ma p tha t forget s th e secon d marke d point , an d th e e ^ ar e th e usual evaluatio n map s o n Mo,2(d,P 1 )The intersectio n numbe r ca n b e easil y compute d i n a wa y analogou s t o th e previous example . B y writin g b(V£) = p 2K^d)^ w e & e^ f el(H)e*

2(H)b(V;')=

[

e*{H)b{V^)p

2^2{H).

9

A SURVE Y O F MIRRO R PRINCIPL E

By integratin g alon g a fibe r o f th e ma p p2, w e se e tha t th e las t facto r i n th e integrand contribute s a n overal l facto r d. Thu s w e ge t th e answe r

/

/M 0 , 2 (ci,P 1 ) JM

e*1(H)$(H)bW) = d- 1 .

The las t tw o exampl e suggest s th e ver y interestin g possibilit y tha t on e ma y be abl e t o us e stabl e ma p modul i space s a s model s fo r som e o f th e modul i space s in th e proble m o f countin g holomorphi c disk s an d annul i wit h suitabl e incidenc e conditions. Moreover , the appropriate intersectio n number s should come from Eule r classes o f induce d bundles , whic h i s exactl y wha t th e mirro r principl e i s designe d to study . Thi s possibilit y deserve s furthe r investigations . References M. Atiyah , Convexity and 1-15.

commuting Hamiltonians,

Bull

. Londo n Math . Soc , 1 4 (1 982) ,

M. Atiya h an d R . Bott , The moment map and equivariant cohomology, Topology , 2 3 (1 984) , 1-28. A. Bertram , Another way to enumerate rational curves with torus action, math . AG/9905159. K. Behren d an d B . Fentachi , The intrinsic normal cone, Invent . Math. , 1 2 8 (1 997) , 45-88 . G. Bini , C . D e Concini , M . Polito , an d C . Procesi , GiventaVs work relative to mirror symmetry, math.AG/9805097 . A. Elezi , Mirror symmetry for concavex bundles on projective spaces, math.AG/00041 57 . C. Faber , an d R . Pandharipande , Hodge Integrals and Gromov-Witten Theory, math.AG/9810173. A. Gathmann , Relative Gromov-Witten invariants and the mirror formula, math.AG/0009190. A. Givental , Equivariant Gromov-Witten invariants, alg-geom/9603021 . V. Guillemi n an d S . Sternberg , Convexity properties of the moment mapping, Invent . Math. , 67 (1 982) , 491 -51 3 . F. Hirzebruch , Topological methods in algebraic geometry, Springer-Verlag , Berlin , 1 995 , 3r d Ed. M. Goresky , R . Kottwitz , an d R . MacPherson , Equivariant cohomology,

Koszul duality and

the localization theorem, Invent . Math. , 1 3 1 (1 998) , 25-83 . T. Grabe r an d R . Pandharipande , Localization of virtual classes, alg-geom/9708001 . S. Katz , A . Klemm , an d C . Vafa , Geometric engineering of quantum field theories, Nucl . Phys., B 4 9 7 (1 997) , 1 73-1 95 . M. Kontsevich , Enumeration of rational curves via torus actions, i n 'Th e Modul i Spac e o f Curves,' ed . b y R . Dijkgraaf , C . Faber , G . va n de r Geer , Progres s i n Math. , 1 29 , Birkhauser , 1995, 335-368 . D. Morriso n an d R . Plesser , Summing the instantons: quantum symmetry in toric varieties, alg-geom/941 2236 .

cohomology

and mirror

J. L i an d G . Tian , Virtual moduli cycle and Gromov-Witten invariants of algebraic varieties, J. o f Amer . Math . S o c , 1 1 (1 ) (1 998) , 1 1 9-1 74 . B. Lian , K . Liu , an d S.T . Yau , Mirror Principle I , Asia n J . Math. , 1 (4 ) (1 997) , 729-763 . B. Lian , K . Liu , an d S.T . Yau , Mirror Principle II , Asia n J . Math. , 3(1 ) (1 999) . B. Lian , K . Liu , an d S.T . Yau , Mirror Principle III , math.AG/991 2038 . B. Lian , K . Liu , an d S.T . Yau , Mirror Principle IV , math.AG/00071 04 . B. Lian , C.H . Liu , an d S.T . Yau , A Reconstruction of Euler Data, math.AG/0003071 .

10

B.H. LIAN , K . LIU, AND S.-T . YA U

[23] R . Pandharipande, Rational curves on hypersurfaces (after givental), math.AG/98061 33 . [24] E . Witten, Phases of N = 2 theories in two dimension, hep-th/9301 042 . BRANDEIS UNIVERSITY , WALTHAM , M A 021 5 4

E-mail address: [email protected] u UNIVERSITY O F CALIFORNIA, LO S ANGELES , C A 90024-651 6

E-mail address: [email protected] u HARVARD UNIVERSITY , CAMBRIDGE , M A 021 3 8

E-mail address: [email protected] u

https://doi.org/10.1090/amsip/033/02 Mirror Symmetr y I V AMS/IP Studie s i n Advance d Mathematic s Volume 33 , 200 2

Mirror Symmetry : Aspect s o f t h e Firs t 1 0 Year s B.R. Green e

Through studie s i n superstrin g theor y i n th e lat e 1 980s , physicist s bega n t o suspect tha t ther e wa s an unexpecte d an d powerfu l lin k betwee n a priori unrelate d topologically distinc t Calabi-Ya u manifolds . Usin g technique s o f conforma l fiel d theory, thes e suspicion s wer e subsequentl y pu t o n firme r foundation , a s explici t examples o f mirror manifolds wer e found . Thes e ar e pair s o f topologicall y distinc t Calabi-Yau manifold s whic h giv e ris e t o isomorphi c strin g theorie s whe n eithe r is chose n fo r th e extr a spatia l dimension s require d b y th e theory . Throug h th e identical physics shared by these string compactifications, highl y nontrivial relation s between the members of a mirror pair ca n be found. Som e of these lead, for example , to calculationall y effectiv e method s fo r determinin g th e numbe r o f rationa l curve s of arbitrary degre e on various Calabi-Ya u spaces . Fro m a more physical standpoint , mirror symmetr y ha s als o playe d a pivota l rol e i n understandin g th e firs t know n examples o f topolog y changin g processes . Mathematically , ther e ha s bee n muc h recent progres s in understanding aspect s of mirror symmetr y usin g more traditiona l mathematical methods . 1. Introductio n Mirror symmetr y (fo r a n i n depth se t o f references, th e reade r ca n consul t [30] , [31]) i s a propert y o f Calabi-Ya u space s whic h relate s topologicall y distinc t man ifolds i n a surprisin g an d powerfu l way . I t wa s discovere d b y physicist s workin g on strin g theor y i n th e lat e 1 980 s an d earl y 1 990s . Althoug h muc h insigh t int o the mathematica l underpinning s o f mirro r symmetr y ha s bee n foun d ove r th e las t decade, it s mos t poten t expressio n i s stil l i n th e languag e o f strin g theor y wher e it ensure s tha t strin g propogatio n o n th e tw o manifold s o f a mirro r pai r lead s t o identical physica l theories . Amon g othe r things , thi s implies—fro m a purel y math ematical poin t o f view—surprisin g an d unexpecte d equalitie s betwee n geometrica l quantities o n th e tw o member s o f a mirro r pair . A s year s o f studie s hav e shown , these equalitie s ar e remarkabl y potent . Fo r example , i n som e instance s the y pro vide a dictionar y fo r translatin g difficul t question s o f mathematical interes t o n on e manifold int o fa r simple r question s o n th e other . In th e earl y day s o f mirro r symmetry , althoug h thes e mathematica l result s could b e state d withou t referenc e t o th e underlyin g physica l theory , ther e wa s n o means o f establishing thei r validit y withou t recours e t o physical methods . Bu t ove r ©2002 America n Mathematica l Societ y an d Internationa l Pres s 11

12

B.R. GREEN E

the years , som e o f the mos t famou s mathematica l implication s o f mirror symmetr y have bee n prove n wit h purel y mathematica l tools . A progra m whic h continue s t o be of central importance i s to find the full mathematica l framework whic h embodie s mirror symmetry an d hence ascertain its complete set of mathematical implications . The organizer s hav e aske d m e t o giv e a n introductio n t o an d overvie w o f som e of the highlight s o f the first te n year s of mirror symmetry , emphasizin g th e physica l intuition an d motivatio n whic h launche d thi s subject—intuitio n whic h continue s t o be an extremely fruitfu l guid e used by current research . A s the anticipated audienc e is one whos e interes t i n th e subjec t originate s i n it s mathematica l aspects , w e wil l try t o avoi d usin g detailed concept s fro m physic s whenever possible , an d emphasiz e those implication s o f th e result s fo r whic h th e geometri c interpretatio n i s mos t manifest. However , no w an d then—especiall y i n explainin g physicist' s motivatio n and th e concret e result s emergin g fro m strin g theory — w e wil l b e force d t o mak e use o f som e physica l methods . Thes e method s ar e standar d an d full y understoo d from th e physics viewpoint bu t ar e likely to be unfamiliar t o some readers. Happily , though, a s tim e goe s by , increasingl y larg e numbe r o f mathematician s ar e learnin g and usin g these physical methods, facilitatin g ye t mor e fruitful interaction s betwee n the tw o communities . Before enterin g i n upon som e of the detail s o f the discussion , w e want t o briefl y describe mirror symmetr y i n heuristic physical terms. Thi s informal descriptio n wil l facilitate ou r late r discussio n o f mirror symmetry , a s wel l a s introduc e th e physica l intuition whic h lead s t o th e mirro r symmetr y conjecture . I t i s no t require d fo r a n understanding o f th e mor e technica l descriptio n whic h follows . Studies i n string theory i n the mi d 1 980 s established a correspondence betwee n certain manifold s (and , i n fact , mor e genera l spaces , a s w e shal l discuss ) an d ob jects know n a s conforma l field theories. 1 Roughly , th e correspondenc e involve s considering th e manifol d a s the spacetim e i n which a string propagates , an d notin g that th e physic s of such strin g propagation i s described (i n a certain approximatio n scheme) b y a two-dimensional conforma l field theory . Man y o f the basi c mathemat ical propertie s o f a give n on e o f these manifold s (suc h a s it s cohomolog y ring ) find direct expressio n i n th e associate d conforma l field theory . I t turn s out , however , that thi s associatio n o f manifold s an d conforma l field theorie s i s no t one-to-one : two an d possibl y mor e distinc t manifold s may , i n fact , correspon d t o the sam e con formal theory . I t i s thu s worthwhil e t o partitio n th e spac e o f al l manifold s whic h have conformal field theoreti c realization s int o classe s o f those whic h have the same realization. Suc h distinc t manifolds , M an d M , whic h correspon d t o th e sam e con formal theor y ar e calle d a mirror pair i f the y hav e Hodg e diamond s satisfyin g th e relation h?«{M) = h d~p^{M) wher e h™ i s th e dimensio n o f th e (p,q) Dolbeal t cohomology grou p o f M , an d d i s th e comple x dimensio n o f th e manifold . Th e common conforma l theor y associate d t o a mirro r pai r provide s a previousl y un suspected lin k betwee n thes e topologicall y distinc t manifolds . B y passin g throug h this link , intrinsi c propertie s o f on e manifol d o f th e pai r ar e reflecte d i n th e other . Of muc h importanc e i s th e fac t tha t som e o f thes e intrinsi c propertie s ar e easie r to understan d an d calculat e whe n rephrase d o n th e mirror . Thi s i s th e origi n o f the famou s succes s o f [4 ] i n calculatin g th e numbe r o f rationa l curve s o f arbitrar y x In th e appendi x w e briefl y outlin e thi s Calabi-Ya u / conforma l field theor y corresondence . In th e earl y 1 990s , Witte n sharpene d certai n feature s o f this correspondenc e throug h th e us e o f socalled linea r sigm a models . Th e intereste d reade r ca n find a thoroug h revie w o f such development s in m y articl e i n [33] , fo r example .

MIRROR SYMMETR Y .. .

13

degree o n th e quinti c threefol d - a t th e time , a seemingl y impossibl e calculatio n which becam e thoroughl y tractabl e whe n rephrase d o n th e mirro r manifold . W e will discus s th e essentia l idea s underlyin g thes e calculation s an d illustrat e the m with explici t example s o f mirro r manifolds . In sectio n tw o we will give a mor e forma l discussio n o f the basi c point s covere d in thi s introductio n an d stat e th e origina l mirro r conjecture . I n sectio n thre e w e will review a somewhat les s ambitious versio n o f this mirro r conjectur e an d presen t the evidenc e supportin g i t whic h physicist s foun d i n th e earl y 1 990s . I n sectio n four w e wil l describ e th e mathematica l implication s o f mirro r symmetr y foun d b y physicists, an d w e wil l discus s som e o f th e explici t calculation s the y provide d t o verify thei r results . I n sectio n five w e wil l discus s som e o f th e highlight s o f recen t investigations int o the physic s an d mathematic s o f mirror symmetry , an d i n sectio n 6 w e will discus s a fe w o f th e majo r unresolve d issue s currentl y unde r study . Our purpose here is not t o give detailed analyse s and derivation s a s these can b e found, fo r example , i n th e extensiv e articl e [33] . Rather , thi s articl e i s a n informa l introduction an d overvie w o f mirro r symmetry . 2. Mirro r symmetry : Th e basi c framewor k The manifolds o f interest t o our discussion ar e of the so-called Calabi-Yau type . These wer e introduce d i n th e contex t o f strin g theor y i n [7] , [8] . Fo r ou r purpose s we take th e followin g definition : DEFINITION. A Calabi-Ya u manifol d i s a compact , comple x Kahle r manifol d with vanishin g first Cher n clas s an d zer o first Bett i number. 2

We not e a t th e outse t tha t i t i s insufficien t fo r ou r purpose s t o limi t attentio n to smoot h Calabi-Ya u manifolds . Rather , ou r discussio n wil l als o mak e us e o f orbifolds (sometime s als o calle d V-manifolds ) [8 ] of smoot h Calabi-Ya u manifold s by discret e grou p action s tha t genericall y hav e fixed points . W e shal l onl y nee d t o consider, though , orbifold s wit h a t mos t abelia n quotien t singularities . I t i s know n that thes e admi t resolution s t o smoot h manifold s [9 ] an d i t i s propertie s o f th e latter manifold s whic h wil l b e o f interes t t o us . We also note that whe n we speak of a Calabi-Yau manifold , w e generally assum e that a particula r choic e o f polarizatio n (i.e. , Kahle r structure ) an d comple x struc ture ha s been made . Differen t choice s of either th e comple x o r the Kahle r structur e on th e sam e underlyin g topologica l spac e ar e sai d t o constitut e differen t point s i n the modul i spac e o f th e Calabi-Ya u space . Infinitesima l deformation s o f th e com plex structure o f a Calabi-Yau manifol d ar e parametrized b y the cohomolog y grou p if2,1 whil e H 1 ,1 parametrize s Kahle r deformations . Pro m the point o f view of strin g theory, a s indicate d i n th e appendix , th e specificatio n o f a geometrica l solutio n o f the strin g consistenc y equation s require s no t onl y th e topolog y o f th e Calabi-Ya u manifold, bu t als o a choic e o f Ricci-fla t metri c (th e existenc e o f whic h i s guaran teed b y Yau' s theore m [1 0] ; it i s known tha t physica l processe s shif t th e Ricci-fla t metric t o on e tha t i s not Ricci-flat , bu t a perturbative treatmen t mus t star t wit h a metric tha t i s Ricci-flat). Th e spac e o f all Ricci-fla t metric s i s parametrized b y th e possible choices o f comple x an d Kahle r structures , thu s matchin g th e mathemati cal data specifie d above . Furthermore , althoug h th e cohomolog y grou p iJ 1 ,:L (M) i s 2 T h e requiremen t o f vanishin g firs t Bett i numbe r i s a convenienc e fo r ou r discussion , bu t essentially al l tha t w e describ e applie s t o tor i a s well .

14

B.R. GREEN E

naturally real , strin g theor y require s u s t o complexify th e Kahle r con e o f M. Con cretely, wha t w e mea n b y thi s i s a s follows . I f th e Kahle r metri c o n M i s denote d by gfj, the n i n local coordinates th e Kahle r for m ca n b e written J = ig^-dX 1 A dXJ'. In considerin g strin g propagatio n i t i s important t o includ e a n additiona l structur e on th e manifold , whic h lead s u s to conside r J — {ign + bq)dX % A dX^ wher e bq i s some chose n elemen t o f if 1 ' 1 (M). 3 I t turn s ou t tha t th e physica l mode l i s insensi tive t o shifts i n bq, th e so-calle d "antisymmetri c tenso r field," b y integra l element s of H 2(M). The mos t precis e definitio n o f a mirror pai r o f Calabi-Ya u manifolds , whic h w e give below , relie s upo n conforma l field theory . T o motivat e th e definitio n w e first spend a momen t o n a descriptiv e aspec t o f th e Calabi-Yau/conforma l field theor y connection. Immediatel y afte r givin g the definitio n o f mirror manifold s w e give two essential mathematica l implications . As mentioned , strin g theor y lead s u s t o associat e wit h a chosen Calabi-Ya u manifold a n objec t know n a s a conforma l field theory . Man y o f th e geometrica l properties o f the manifol d find a natural expressio n i n its associated conforma l field theory. I n particular, le t u s specialize to the case of complex dimension three for th e clarity o f the discussion , thoug h mos t o f the result s generaliz e t o othe r dimensions . The Hodg e diamon d fo r a Calabi-Ya u thre e fol d take s th e followin g form : 1 00 0 ft 1 (2.1)

ft

1 1

' 0

1 2

' ft

1 2

1 '

22

0 ft

' 0

00 1 We se e tha t th e onl y Hodg e number s whic h ar e no t uniquel y determine d ar e ft1'1 = ft 2'2 an d ft 2,1 = ft 1 '2. Recal l tha t deformation s o f M preservin g th e condi tion o f Ricci-flatnes s ar e parametrize d b y precisel y thes e cohomolog y groups : H 1 1' parametrizes infinitesima l deformation s o f the Kahle r clas s while H 2}1 parametrize s infinitesimal deformation s o f the comple x structure . Th e fac t tha t ft 3'0 = 1 reflect s the propert y tha t th e canonica l lin e bundl e o n M (mor e precisely , canonica l sheaf ) is trivial and henc e admits a global holomorphic sectio n ft. Th e three-form Q allows us t o establis h a canonica l isomorphis m betwee n H 2,1 (M) an d H l{M,T), wher e T i s th e holomorphi c tangen t bundle . Concretely , w e hav e u : Hl(T) — • H 2,l(M) given i n loca l coordinate s b y 1 (2.2) bUX 1 -— - J + VLinbidX^X^X . v % ' l dXJ As outline d i n th e appendix , thes e cohomolog y group s ar e relate d t o a specia l class o f fields i n th e conforma l field theory , whic h generat e deformation s o f th e theory preservin g it s superconforma l structure . Thes e fields ar e labele d b y thei r 3

hold.

Note tha t th e usua l positivit y condition s o n th e imaginar y par t o f J ar e stil l expecte d t o

M I R R O R S Y M M E T R Y .. .

15

transformation propertie s unde r th e symmetr y algebr a — th e N — 2 supersym metry algebra . Further , th e differenc e betwee n th e tw o kind s o f fields generatin g the tw o kind s o f deformation s lie s i n th e sign o f thei r charg e (eigenvalue ) unde r a U(l) subalgebra , th e choic e o f sig n bein g a matte r o f convention . I n othe r words , whereas element s o f H 1 *1 an d H 2'1 ar e vastl y differen t geometrica l objects , thei r conformal field theor y counterpart s hav e n o intrinsic differentiatin g characteristics . This i s strange : How , fo r instance , shoul d on e determin e intrinsically whethe r a given field i n a conforma l field theor y arise s fro m H 1 ,1 o r fro m H 2,1 ? I n th e lat e 1980s, thi s le d severa l physicist s [5] , [1 ] t o conjectur e th e existenc e o f a secon d Calabi-Yau manifold , whic h woul d correspon d t o th e same conforma l field theory , but suc h tha t th e identification s o f fields with geometrica l object s woul d follo w th e opposite conventio n fo r th e sig n o f th e U(l) eigenvalue . Tha t is , w e can't tel l intrinsically whethe r a given field in a conformal field theor y arise s fro m H 1 ,1 o r fro m H2'1 because , i n actuality , i t arises , say , fro m H 1 ,:L(M) and from H 2,1 (M), wit h M a differen t Calabi-Ya u space . We ca n no w defin e wha t i s meant b y a mirro r pair . DEFINITION. TW O Calabi-Ya u manifold s M an d M constitut e a mirro r pai r i f they correspon d t o th e sam e conforma l field theory , an d th e associatio n o f geomet rical object s o n th e tw o manifold s t o fields i n th e conforma l field theor y differ s b y a reversa l o f th e charge s unde r th e left-movin g 1 7(1 ) o f al l fields.

Technically, thi s i s really th e definitio n o f classical or perturbative mirro r sym metry becaus e i t turn s ou t tha t conforma l field theor y provide s th e framewor k fo r analyzing quantu m strin g theor y i n a perturbativ e expansion . Th e lowes t orde r term i n thi s expansion—classica l strin g theory—i s conforma l field theor y o n th e sphere; th e nex t ter m i s conforma l field theor y o n th e torus , an d s o forth . Quantum mirro r symmetr y doe s not onl y require a n isomorphis m betwee n th e conforma l field theorie s associate d t o tw o Calabi-Yaus , but , i n addition , a n isomorphis m be tween th e ful l quantu m strin g theorie s associate d t o each . A s w e briefl y discus s later, thi s ha s som e strikin g mathematica l implication s which , i n fact , ca n b e use d to motivat e th e first purel y mathematica l notio n o f mirro r symmetry . For now , stickin g t o classica l mirro r symmetr y (w e will dro p th e wor d classica l for mos t o f th e sequel) , let' s giv e two particular implication s whic h follo w fro m th e definition. Thes e embod y th e propertie s o f mirror manifold s whic h hav e bee n use d in mos t mathematica l application s t o date . A s a n aside , w e not e tha t som e math ematicians hav e loosel y refere d t o thes e implication s a s constitutin g th e definitio n of mirror manifolds . However , i t i s important t o emphasiz e tha t i n th e languag e o f physics, mirror manifolds , a s stated, correspon d t o th e sam e conforma l field theory . Thus, no t onl y ar e their respectiv e three poin t function s equa l (whic h is the conten t of implicatio n 2 below), bu t thei r n-poin t function s ar e als o equal , fo r arbitrar y n. The implications we give emphasize the three point function s a s these have the mos t direct geometrica l interpretation ; i t woul d b e o f interes t t o stud y th e geometrica l content o f th e othe r n-poin t functions . IMPLICATION. I

1) h™{M) = h

f M an d M ar e a mirro r pai r the n 3

~™(M)

B.R. GREEN E

16

2) H p,q(M) = H3 P ' 9 (M) unde r a n isomorphism preservin g the quantum tripl e products 4. The firs t o f these implie s tha t th e Hodg e diamon d o f M i s related t o tha t o f M by a reflection abou t a diagonal axis ; thi s i s th e origi n o f th e ter m "mirro r mani folds" [2] . I n particular , excep t i n ver y specia l cases , M and M are topologicall y distinct. Th e quantu m trilinea r product s ar e deformation s o f natura l mathemati cal triple product s base d o n th e ordinar y cu p produc t o f H*(M). Thei r motivatio n from th e poin t o f vie w o f strin g theor y i s describe d i n th e appendix ; the y ar e a n example o f wha t ha s com e t o b e know n mor e generall y a s quantu m cohomology . Here w e giv e th e explici t expression s fo r th e nontrivia l case s o f interes t t o us . T o do so , let u s firs t defin e th e natura l mathematica l tripl e product s refere d t o above . These ar e [1 1 ] , [1 2] : J 1 ' 1 : H1*1 x if 1 ' 1 x H 1 *1 - > C give n b y lll

(2.3) I

{B(i),B(j),B(k))= I

B(i)/\B(fi/\B(

k)

JM

and J 2 ' 1 : H2'1 x if 2'1 x H 2'1 - » C give n b y (2.4) I

1 2

flA^AA^A^,^

> (A{i),AUhAik))= I

JM

where fi/ mn i s th e holomorphi c three-for m o n M and A^ ar e element s o f H 21' (expressed a s element s o f H 1 (M , T) wit h thei r subscript s bein g tangen t spac e in dices). Notic e tha t 2. 3 i s nothin g bu t th e cu p produc t givin g ris e t o th e tripl e intersection matrix . Equatio n 2. 4 i s most easil y describe d a s the natura l ma p fro m Hl(T) x i fx ( r ) x H l(T) - > C arisin g fro m th e trivialit y o f th e canonica l bundle . As written , 2. 4 i s this ma p afte r th e isomorphis m 2. 2 ha s bee n employed . The quantu m tripl e product s ar e deformation s o f thes e natura l mathemtica l constructs i n a manner dictate d b y strin g theor y a s outline d i n th e appendix . In particular, thes e ar e I^ 1 : H1*1 x H 1 *1 x H 1 *1 - + C give n b y [1 3 ] (2.5) ig

' (£(*) , By), B( fc))

+ ^m-3e(-/'-^*(J)) / TJ 1 n,m

t{B®) In,m J

l L*{B&) In,m J

[

t*(B™ I C i s give n b y (2.6) I%\A

(i),A(j),A(k))=

f nA^A^AA^ftta . JM

In othe r words , th e relevan t tripl e produc t arisin g i n strin g theor y i s precisel y the natura l mathematica l construc t i n thi s cas e [151 . 4 Language become s a bit confusin g here . Althoug h w e ar e workin g a t th e leve l o f conforma l field theor y o n th e sphere — classica l strin g theory—thi s field theor y i s affecte d b y perturbativ e and nonperturbativ e physic s tha t i s ofte n refere d t o a s "quantu m corrections. " Fro m th e poin t of vie w o f space-tim e physics , thes e correction s ar e "stringy " correction s (tha t is , the y aris e fro m the extende d structur e o f th e string ) a s oppose d t o quantu m corrections , bu t th e latte r ter m is commonly used .

M I R R O R S Y M M E T R Y ...

17

To explai n thi s result , le t M an d M b e a mirro r pai r o f Calabi-Ya u manifold s which correspond t o the conformal fiel d theory K. Thi s implies that ever y construc t in K has two geometrical interpretations - one o n M an d on e o n M. I n particular , this is true of the correlation functions i n K. A s discussed i n the appendix, there ar e two special classes of correlation function s i n K. On e such clas s has the geometrica l interpretation o n M a s IQ 1 whil e the othe r i s given geometricall y b y IA . Now, th e crucial point o f mirror symmetr y i s that field s in K whic h correspond t o (1 ,1 ) form s ((2,1) forms ) o n M correspon d t o (2,1 ) form s ((1 ,1 ) forms ) o n M. Thus , correlatio n functions i n K which ar e geometricall y interpretabl e a s IA o n M necessaril y ar e geometrically interpretabl e a s IQ 1 o n M. Similarly , correlatio n function s i n K which ar e geometricall y interpretabl e a s IA o n M ar e geometricall y interpretabl e as IA o n M. Hence , we arrive a t th e equalit y state d i n implication 2 . I n summar y then, th e identitie s whic h aris e fro m mirro r symmetr y hav e thei r origi n i n th e fac t that a given object i n the conformal fiel d theor y ha s two vastly different geometrica l interpretations o n th e manifold s o f a mirro r pair . We now stat e th e mirror conjecture an d retur n shortl y t o a les s robus t versio n that ca n b e elevate d t o th e leve l o f a theorem . Fo r every 5 Calabi-Ya u manifol d M ther e exists a mirror Calabi Yau manifol d M. I n particular , fo r d = 3 CONJECTURE.

(2.7a) fffi

= h*7™

and (2.7b) I^(M)=I^(M) (2.7c) I%

1

(M) = I%

1

(M).

We note that b y the latter equalitie s of triple products we mean that ther e exist s a suitable choic e of bases of the relevan t cohomolog y group s suc h that th e equalitie s hold. I n the nex t sectio n w e review that fo r a certain subclas s o f Calabi-Yau space s we can establis h th e mirro r conjectur e an d explicitl y construc t th e mirro r manifol d M. 3. A constructio n o f mirro r manifold s In thi s sectio n w e focu s o n a subclass o f Calabi-Ya u space s fo r whic h w e ca n establish th e mirro r conjectur e describe d previously . Th e proo f involve s a n explici t construction o f the mirro r manifol d pair ; w e will see in sectio n fou r tha t thi s allow s the extractio n o f nontrivial informatio n abou t bot h member s o f the pair . W e agai n specialize th e discussio n t o th e cas e o f comple x dimensio n thre e fo r defmiteness , 5 T h e precis e meanin g o f 'every ' i n thi s contex t ha s neve r bee n mad e precise . Th e las t decad e has show n tha t th e conjectur e appear s t o b e tru e fo r Calabi-Ya u space s constructe d a s complet e intersections i n tori c varieties , bu t beyon d thi s clas s n o genera l statement s hav e bee n shown . In the las t sectio n w e wil l not e som e relativel y recen t insigh t o n thi s issue . Relatedly , i n th e earl y days o f mirror symmetry , a natura l questio n tha t wa s aske d was : wha t happen s whe n on e membe r of a supposed mirro r pai r i s rigid , i.e . h d~1 1, = 0? The n th e suppose d mirro r woul d appea r t o be non-Kahler . Thi s questio n ha s a natural solution , onc e on e realize s th e fact , discusse d later , that th e modul i spac e o f Calabi-Ya u manifold s ha s a cell structur e i n whic h certai n cell s hav e a natural physica l interpretatio n bu t n o geometrica l interpretation . Moreover , ther e ar e degenerat e moduli space s wher e onl y th e latte r cell s appear—thes e tur n ou t t o b e th e mirro r pair s o f rigi d Calabi-Yau manifolds .

18

B.R. G R E E N E

although mos t o f wha t w e sa y i s independen t o f thi s choice . W e begi n wit h th e main resul t o f [2] :

THEOREM. Let M be a Calabi-Yau hypersurface of degree n in weighted projective four space of Fermat type. Let G be the group of coordinate scaling symmetries which preserve the holomorphic three form on M. Then, the mirror M of M exists and is given by the quotient M/G.

We note tha t M/G i s a singula r spac e sinc e G generally act s wit h fixed points . However, du e t o [9 ] w e kno w tha t M/G admit s a desingularizatio n t o a smoot h Calabi-Yau manifold ; i t i s th e cohomolog y o f th e latte r whic h w e refe r to . (Th e role o f singularit y resolutio n i n mirro r symmetr y wil l b e discusse d shortly. ) A relate d poin t an d on e tha t mus t b e emphasize d i s that althoug h w e perfor m the explici t constructio n o f th e mirro r fo r a specifi c manifol d representin g a ver y special poin t i n th e paramete r spac e (th e Ferma t point ) th e theore m i n fac t allow s us to construc t th e mirro r manifol d i n (a t least ) a neighborhood o f this point . Thi s is becaus e mirro r symmetr y yield s a n isomorphis m o f th e tangen t space s t o th e paired modul i spaces , allowin g us to relat e deformation s o f one manifold awa y fro m the Ferma t poin t t o deformation s o f th e mirro r manifol d fro m th e Ferma t mirror . The isomorphis m relate s deformation s o f th e comple x structur e o f M t o deforma tions o f th e (complexified ) Kahle r structur e o f M an d vic e versa . Not e tha t i n general th e deforme d versio n o f M an d th e correspondingl y deforme d versio n o f M are no t relate d b y a n orbifoldin g operatio n (a s ar e M an d M) an d ar e no t o f Fer mat typ e - nonetheles s the y ar e a mirro r pair . Th e resul t tha t thes e correspondin g deformations preserv e th e mirro r symmetr y propert y i s bes t understoo d fro m th e conformal field theor y poin t o f view. I t reflect s th e fac t tha t w e have one conforma l theory wit h tw o geometri c interpretations . Deformin g th e Kahle r structur e o n M corresponds to a particular deformatio n o f the conformal field theory (b y turning o n a vacuu m expectatio n valu e fo r som e trul y margina l operator) . Thi s deformatio n of th e conforma l theor y (b y thi s particula r margina l operator ) i s interpretabl e o n M a s a comple x structur e deformatio n (b y th e correspondin g comple x structur e modulus). Afte r deformation , then , w e agai n hav e on e conforma l theor y ( a defor mation o f the original ) wit h tw o geometrical interpretation s (eac h a deformation o f one o f the orgina l spaces) . In th e earl y day s o f mirro r symmetry , physicist s emphasize d thei r belie f tha t there i s n o obstructio n t o extendin g thi s loca l deformatio n argumen t t o th e whol e of modul i spac e thu s yieldin g th e result : if a Calabi-Yau space lies in the same moduli space as a Fermat hypersurface then it has a mirror. Thi s statemen t mad e some mathematicians uncomfortable , especiall y i n light o f the fac t th e globa l struc ture o f the (complexified ) Kahle r modul i spac e o f one Calabi-Ya u an d th e comple x structure modul i spac e o f it s mirro r appea r t o b e quit e different : th e forme r i s a bounded domai n whil e th e latte r i s a quasi-projectiv e variety . Resolvin g thi s issu e took a numbe r o f year s an d uncovere d a substantia l amoun t o f interestin g physic s and mathematics , a s w e we briefly indicat e shortly . An immediat e corollar y o f th e theore m whic h underlie s it s mathematica l ap plications thu s fa r is :

M I R R O R S Y M M E T R Y .. .

19

COROLLARY.

(3.1a) / ^ ( M ) = / ^ ( M (3.1a) I%\M)

) =

I

1 1 cj

(M).

As note d i n [2] , there i s a n interestin g limi t o f thi s corollar y - namely , i f th e 'radius' o f M, say , should ten d t o "infinity " 6 the n IA (M ) tend s t o the intersectio n matrix o n M , i.e. , 7 1 ' 1 (M). Then , (3.1 ) relate s thi s topologica l quantit y o n M t o a quasi-topologica l (depend s o n th e comple x structur e a s well ) quantit y o n th e topologically distinc t spac e M. W e wil l retur n t o thes e idea s shortly . Before movin g o n t o a descriptio n o f th e proo f o f th e theorem , w e illustrat e the resul t wit h on e well known example ; w e will retur n t o thi s exampl e i n th e final section, a s th e objec t o f stud y i n th e first application s o f mirro r symmetr y Consider th e quinti c Ferma t hypersurfac e i n CP . Th e coordinat e scalin g sym metries constitut e a (Z5) 5 . On e o f thes e (th e diagonal ) i s trivia l becaus e w e wor k in projectiv e space . Anothe r doe s no t preserv e th e holomorphi c thre e form . Thus , G i s the grou p (Z5) 3 . Hence , (3.2a) Z\

+ Z% + Z\ + Z\ + Z f = 0

and (3.2b)

Z\ + Z\ + Z\ + Z\ + Z f = 0 (Z5)3

constitute a mirro r pair . W e not e tha t th e forme r ha s h 1 ,1 = 1 an d /i 2 ' 1 = 1 0 1 while th e latte r ha s fe 1 ,1 = 1 0 1 an d /i 2 ' 1 = 1 . I n particular , x ( ^ 0 = -20 0 an d x(M) — 200, wher e \ denote s th e Eule r characteristic . I n tabl e 1 w e lis t som e other explici t example s o f th e construction . We no w tur n t o a brie f summar y o f ho w w e establis h th e mirro r symmetr y theorem state d abov e [2] , [31 ] . I t i s a t thi s poin t i n th e discussio n tha t conforma l field theory idea s play a crucial role. Th e strateg y i s based o n Fermat hypersurface s and i s a s follows : W e see k a nontrivia l automorphism O o f th e conforma l field theory K whic h realizes an operation o n the corresponding geometrica l constructio n which i s not a n automorphis m o f th e correspondin g Calabi-Ya u hypersurfac e M. Rather, w e see k a n O suc h tha t O : M — • M wit h M bein g Calabi-Ya u an d no t isomorphic t o M . I n thi s way , M an d M woul d b e intimatel y linke d b y thei r common conforma l field theory . I n particular , a s mentione d above , i f O act s o n the conforma l field theor y b y reversing th e sig n o f all charges unde r th e left-movin g U{1) symmetry , w e have th e followin g situation : Bot h M an d O(M) correspon d t o the sam e conforma l field theor y sinc e K an d 0(K) ar e isomorphic . Furthermore , since th e explici t isomorphis m betwee n K an d O(K) i s th e reversa l o f th e sig n o f the lef t movin g U(l) charges , th e identificatio n o f conforma l fields wit h differentia l forms i s reversed o n O(M) relativ e t o M. Thu s M an d O(M) woul d constitut e a 6 T h e meanin g o f "infinity " i n thi s contex t prove s t o b e bot h subtl e an d crucial . A s w e shal l see, th e Kahle r modul i spac e o f a Calabi-Ya u shoul d b e enlarge d b y attachin g th e Kahle r cone s of particula r Calabi-Yau s tha t ar e birationa l t o th e original , amon g others , alon g commo n walls . Then, withi n eac h o f th e Kahle r cone s fo r th e birationa l Calabi-Yau s ther e i s a notio n o f goin g t o "infinity" — i.e. , approachin g th e dee p interio r poin t o f th e complexifie d Kahle r cell , a s w e wil l discuss.

20

B.R. GREEN E

mirror pair , i f suc h a n operatio n O coul d b e found . I n particular , w e would hav e (3.3a) h^j-

= h,Q( M)

(3.3b) hjM

= hp^

My

In [2 ] suc h a n operatio n O wa s constructed . It s existenc e relie s o n th e notio n of orbifolding . I f a manifol d M i s invarian t unde r a grou p o f symmetrie s G (i n our cas e thes e wil l generall y b e holomorphi c automorphisms ) w e ca n conside r th e quotient spac e M = M/G. W e restric t attentio n t o group s G whic h ensur e tha t M i s als o Calabi-Yau . Suc h G ar e calle d 'allowable'. 7 . Similarly , i f a conforma l theory K respect s a symmetr y grou p G , w e can conside r th e quotien t theor y K/G. The crucia l propert y o f thes e operation s fo r ou r argument , i s tha t th e quotien t conformal theor y describe s propagatio n o n th e quotien t o f th e underlyin g Calabi Yau manifol d b y th e sam e allowabl e grou p action s G 8 Th e relationshi p betwee n group action s i n th e tw o picture s i s furnished b y th e identificatio n o f homogeneou s coordinates o n th e manifol d wit h primar y field s i n the conforma l field theor y give n by th e argument s o f [1 6] , [1 7 ] an d mor e precisel y i n th e descriptio n o f [29] . The correspondence betwee n conformal field theories and Calabi-Ya u manifold s is mos t explicitl y understoo d fo r wha t ar e know n a s 'minima l model ' conforma l field theories an d represen t Calabi-Ya u hypersurface s i n weighte d projectiv e spac e [18], [1 6] , [1 7] . Th e proo f o f th e theore m give n i n [2 ] relie s o n th e algebrai c construction o f these 'minima l model ' conforma l field theorie s (derive d fro m simpl e representations o f th e supe r conformal algebra) . Usin g thi s formulation , on e ca n show that th e operation O defined b y taking the quotient b y the group of 'allowable' scaling symmetries G is an automorphism o f the conformal field theory satsfying th e conditions mentioned. Namely , the explicit isomorphis m involve s reversing the U(l) eignevalues o f al l fields. Le t u s not e her e th e point s i n th e argumen t a t whic h th e conformal field theor y enter s i n a crucia l way . Initially , argument s fro m conforma l field theory motivate d th e constructio n o f the particula r quotien t describe d above ; in a sens e thi s i s inessential, sinc e i n principl e th e constructio n itsel f ca n b e carrie d out wit h n o referenc e t o th e physica l interpretation . On e simpl y consider s a give n Fermat Calabi-Ya u hypersurfac e togethe r wit h th e appropriat e orbifold s thereof . One the n finds tw o manifold s wit h Hodg e diamond s relate d i n th e requisit e way . The crucia l contributio n o f th e conforma l field theor y argumen t i s i n showin g tha t the relatio n goe s quit e a bi t deeper , givin g th e isomorphis m betwee n th e relevan t cohomology classe s an d th e equalit y o f th e quantu m tripl e product s (an d mor e generally quantu m n-products ) a s well a s the extensio n o f these equalitie s throug h deformation arguments . Before discussin g th e implication s o f mirro r symmetry , w e paus e her e t o em phasize thre e importan t points . First , th e constructio n abov e work s equall y wel l on an y orbifol d o f th e theorie s unde r discussion . Tha t is , th e spac e o f al l orbifold s of a give n theor y ca n b e partitione d int o mirro r pairs . W e illustrat e thi s wit h th e quintic hypersurfac e i n tabl e 2 . W e not e tha t th e mirro r o f a theor y M/H wit h H C G i s give n b y M/H* wher e H* i s th e complemen t o f H i n G (tha t is , th e 7 We not e tha t G i s allowabl e i f i t preserve s th e holomorphi c d-for m presen t o n th e initia l Calabi-Yau d-fold . 8 To b e mor e precise , th e quotien t wil l hav e singularitie s a t th e fixed point s o f th e G action . The associate d conforma l fiel d theory , though , i s perfectl y wel l defined .

MIRROR SYMMETR Y .. .

21

smallest grou p containin g H an d H* i s G). Second , th e argument s ar e no t spe cific t o comple x dimensio n thre e an d immediatel y generaliz e t o othe r dimensions . Third, a s discusse d earlier , althoug h ou r discussio n t o thi s poin t ha s bee n tie d t o very specia l point s i n th e respectiv e Calabi-Ya u modul i space s b y deformatio n ar guments w e ca n immediatel y exten d ou r result s t o mor e genera l point s i n modul i space. Moreover, a s mentioned i n the introduction, man y believe that th e phenomeno n of mirro r symmetr y extend s wel l beyon d thi s clas s o f examples . Concret e evidenc e for thi s come s fro m a variet y o f sources . First , simultaneou s wit h th e abov e con struction o f a clas s o f mirro r manifolds , i n [3 ] Candelas , Lynke r an d Schimmrig k completed a thoroug h compute r searc h o f Calabi-Ya u manifold s i n weighte d pro jective fou r spac e (no t restricte d t o Ferma t type) . The y foun d tha t th e se t o f al l such constructions i s almost invarian t unde r th e interchang e o f ft 1 '1 an d ft 2'1 . Whil e finding tw o manifold s differin g b y th e interchang e o f thes e Hodg e number s b y n o means assure s tha t th e manifold s ar e a mirro r pai r (t o b e a mirro r pai r the y mus t correspond t o th e sam e conforma l field theory ) thei r result s wa s a n earl y indicato r that mirro r dualit y extend s beyon d th e particula r clas s o f model s treate d i n [2] . Second, a few years after th e papers of [2 ] and [3] , Batyrev [34 ] (inspire d i n par t by earlie r wor k o f Roan [6] ) use d method s o f toric geometr y t o exten d significantl y the clas s o f purporte d mirro r pair s o f Calabi-Ya u spaces . I n particular , Batyre v noticed tha t a connecte d famil y o f Calabi-Yau s realize d a s hypersurface s i n tori c varieties i s define d b y combinatoria l dat a involvin g a pai r o f reflexiv e polyhedra . This naturall y le d Batyre v t o conjectur e tha t mirro r symmetr y amount s t o th e interchange o f thes e tw o reflexiv e polyhedra— a conjectur e whic h h e supporte d b y much circumstantia l evidenc e suc h a s th e correc t interchang e o f Hodg e number s o f the purporte d Calabi-Ya u mirrors , a s well as the fac t tha t thi s conjecture reduce s t o the mirro r constructio n vi a orbifoldin g discusse d earlier , whe n applie d t o tha t case . Since Batyrev' s pape r (a s wel l a s a follow-u p generalizatio n t o complet e intersec tions i n tori c varietie s don e i n collaboratio n wit h Le v Boriso v [35] ) physicist s hav e subjected hi s conjectur e t o a grea t dea l o f detaile d test s an d i t ha s survive d al l i n tact. Mos t strin g theorist s believ e tha t Batyrev' s conjectur e doe s i n fac t construc t pair o f mirror Calabi-Ya u manifolds . I n fact , recen t wor k tha t w e breifly discus s i n section 5 , may wel l have proven mirro r symmetr y fo r Calabi-Ya u manifold s realize d as complet e intersection s i n product s o f tori c varieties . 4. Application s o f mirro r symmetr y In thi s sectio n w e wil l revie w tw o application s o f mirro r symmetry . Th e first centers o n th e consequence s o f (2.7b) , whic h w e reiterat e her e explicitly : (4.1) f f J A ^ A B U j A ^ n ,

™

JM

=/

B

(i) A J5(j) A S(fe)

JM

+ ]Tm- 3 e ( - / '«.™ t * (J)) J TJ

t

*(B«>) I i*(BW) f L'(B^)

In.m J

In.m ^

In.m

where w e remind th e reade r tha t th e A a ar e (2,1 ) form s (expresse d a s elements o f HX(M,T) wit h thei r subscript s bein g tangen t spac e indices ) O is th e holomorphi c

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B.R. GREEN E

three form , th e B l ar e (1 ,1 ) form s correspondingt o th e n£\ I n,m i s an m-fold cove r of a rational curve on M o f degree n, i : In,m < ~^ M th e inclusion, an d J i s the Kahle r form o f M. I n th e physic s literature , th e term s i n th e secon d lin e o f 4. 1 ar e calle d instanton correction s (an d constitut e a n exampl e o f th e "quantum " correction s referred t o earlier) . W e recal l th e crucia l rol e playe d b y th e underlyin g conforma l field theory i n derivin g this equation . I f we simply ha d tw o manifolds whos e Hodg e numbers wer e interchange d w e coul d no t mak e an y suc h statement . Thi s resul t [2] i s rathe r surprisin g an d clearl y ver y powerful . W e hav e relate d expression s o n a priori unrelate d manifold s whic h prob e rathe r intimatel y th e structur e o f each . Furthermore, th e lef t han d sid e o f 4. 1 i s directl y calculabl e whil e th e righ t han d side requires , amon g othe r things , knowledg e o f the rationa l curve s o f every degre e on th e space. 9 In perhap s th e mos t famou s o f the earl y paper s o n mirro r symmetry , Candelas , de La Ossa, Gree n and Parke s analyzed 4. 1 for the case of the quintic-mirror-quinti c pair (3.2) . I n particular, the y considered a one parameter famil y o f mirror manifold s given b y deformin g alon g th e singl e Kahle r modulu s (comple x structur e modulus ) on th e quinti c (mirro r quintic) , a s w e discusse d i n th e genera l contex t i n sectio n 2. Throug h a carefu l analysi s o f th e ma p betwee n Kahle r an d comple x structur e deformations, th e author s wer e abl e t o extrac t informatio n abou t th e righ t han d side of 4.1 from direc t calculatio n o f the lef t han d sid e usin g standard technique s of variation o f Hodg e structure . I n practice , thi s amounte d t o solvin g certai n Picard Fuchs differentia l equation s whic h gover n th e dependenc e o f th e period s o f th e holomorphic three-for m 0 o n th e comple x structur e o f M. A compariso n t o th e right han d sid e the n yielde d a relatio n betwee n th e number s o f rationa l curve s o f various degrees on M an d the coefficients i n an asymptoti c expansio n o f solutions t o these Picard-Fuch s equation s o f M. Thi s enable d thes e author s t o find th e number of rationa l curve s (holomorphi c instantons ) o f an y desire d degre e o n th e quinti c hypersurface simpl y b y carrying out thi s asymptotic expansio n to any desired order . This resul t too k mathematician s b y surprise , an d le d t o a goo d nature d show down at th e Mathematical Science s Research Institute meeting on Mirror Symmetr y in 1 991 . Fo r degre e 1 and 2 curves, the wel l known numbe r o f rational curve s (d=l : 2875, d=2 : 609250 ) wer e reproduce d b y thi s mirro r symmetr y calculation . Th e degree 3 case, though , prove d puzzling . Candela s an d hi s collaborator s found , us ing mirro r symmetry , tha t th e numbe r o f curve s o f degre e 3 wa s 31 7206375 . Bu t Ellingsrud an d Stromm e found , usin g mor e conventiona l mathematica l methods , that th e numbe r o f curves of degree 3 on th e quinti c wa s 2682549425 . Beyon d sort ing out thi s on e particular result , ther e wa s a great dea l o f motivation t o determin e who was right becaus e th e mirro r symmetr y calculatio n coul d easil y be extended t o curves o f arbitrar y degre e while , a t tha t time , i t seeme d nearl y impossibl e t o simi larly exten d th e mathematica l calculation . Althoug h th e questio n o f who was righ t was no t settle d a t th e MSR I meeting , shortl y thereafte r Ellingsru d an d Stromm e found a n erro r i n their compute r code , which whe n corrected , confirme d th e result s emerging fro m mirro r symmetry . Since then , a grea t dea l o f effor t ha s bee n expende d o n bot h verifyin g th e for mulae fo r rationa l curve s predicte d b y mirro r symmetr y an d o n finding a mean s of 9

We recal l a remar k mad e earlie r - 4. 1 i s bu t on e o f a n infinit e serie s o f equalite s implie d by havin g a pai r o f mirro r manifolds . A s yet , th e ful l geometrica l conten t o f an y o f th e othe r equalities ha s no t bee n studie d i n muc h detail .

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performing justifie d calculation s o f thes e number s usin g mor e rigorou s mathemat ical tools . Man y mathematician s hav e playe d a pivota l rol e i n thes e development s as w e will briefl y indicat e i n sectio n 5 . The second application of mirror symmetry has both mathematical an d physica l implications. A s mentione d earlier , fro m th e poin t o f vie w o f physics , i n th e earl y 1990s it wa s conjectured tha t onc e a mirror pai r o f Calabi-Yau space s i s found, thi s ensures tha t th e mirro r relationshi p ca n b e extende d t o al l point s i n th e connecte d component o f modul i spac e t o whic h eithe r Calabi-Ya u spac e belongs . Th e basi c reasoning behin d thi s conjectur e relie d o n deformatio n theory : ever y continuou s deformation t o on e membe r o f a pai r ha s a mirro r deformatio n define d o n th e mirror Calabi-Ya u whic h modifie s th e underlyin g physica l mode l i n precisel y th e same wa y (thereb y preservin g th e mirro r relationship) . Take n a t fac e value , thi s physics reasonin g lead s t o th e mathematica l statemen t tha t th e comple x structur e moduli spac e o f on e Calabi-Ya u manifol d M i s isormorphi c t o th e (complexified ) Kahler modul i spac e o f it s mirro r M. Bu t i s thi s true ? A s allude d t o earlier , the mathematica l structur e o f thes e modul i space s i s plai n differen t an d henc e th e answer i s clearl y no . Th e resolutio n t o thi s apparen t conflic t i s simple , bu t quit e interesting. The wor k o f [29 ] an d [28 ] showe d tha t strin g theor y require s tha t w e enlarg e the Kahle r modul i spac e of a Calabi-Ya u manifol d b y adjoining a number o f relate d moduli space s alon g th e variou s wall s o f a give n Kahle r cell . Som e o f thes e cell s correspond t o the complexified Kahle r modul i spaces of Calabi-Yau manifold s whic h are birationa l t o th e origina l Calabi-Ya u M bu t diffe r fro m i t b y flops o f rationa l curves. Physically , suc h cell s contain th e Kahle r modul i o f a conforma l field theor y built o n th e flopped Calabi-Ya u model . Othe r cell s onl y admi t a mor e abstrac t physical interpretatio n whic h w e will not discus s here . Th e essentia l point , though , is that th e totality o f these Kahler cell s — the so-called fully enlarge d Kahle r modul i space — is isomorphi c t o th e comple x structur e modul i spac e o f M , clearin g u p the th e previou s conflict . Mirro r symmetry , therefore , map s point s i n th e comple x structure modul i spac e o f M t o point s i n the full y enlarge d Kahle r modul i spac e of M—not simpl y t o th e Kahle r modul i spac e o f M a s wa s initiall y thought . From th e poin t o f view o f physics thi s ha s a n importan t implication : Singular ities i n the comple x structur e modul i spac e o f M ar e a t mos t comple x codimensio n one. Sinc e the Kahle r parameter s o f M ca n be held fixed at a "large " valu e ( a value ensuring the validity of perturbative methods) , we can conclude that physica l singu larties ar e also a t mos t comple x codimensio n one . (I n fact, subsequen t researc h ha s shown tha t eve n thi s statemen t ca n b e significantl y relaxed. ) B y mirro r symmetry , the sam e statemen t mus t hol d tru e fo r th e full y enlarge d Kahle r modul i spac e o f M. Bu t thi s implie s tha t a generi c pat h i n th e full y enlarge d Kahle r modul i spac e of M whic h penetrate s throug h cel l wall(s), passes throug h smooth physica l models . In othe r words , path s alon g whic h th e topolog y o f a Calabi-Ya u change s vi a a flop transition ar e perfectl y smoot h fro m th e poin t o f vie w o f physics , establishin g th e first concret e exampl e o f smoot h topolog y changin g processe s i n physics . 5. Importan t recen t development s The first ha s t o d o wit h attempt s t o establis h th e mirro r symmetr y result s i n enumerative geometr y b y usin g mor e traditiona l mathematica l methods . A lon g list o f impressive result s hav e bee n achieve d b y mathematicians includin g Givental ,

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B.R. GREEN E

Ellingsrud, Stromme , Li , Tia n Ruan , Manin , Kontsevich , an d b y Lian , Liu , an d Yau, whic h collectivel y hav e full y establishe d tha t th e curv e countin g argument s originating i n mirro r symmetr y ar e mathematicall y correct . The secon d development , foun d b y Strominger , Yau , an d Zaslo w [32 ] involve s a carefu l applicatio n o f notion s emergin g fro m th e secon d superstrin g revolution — techniques whic h giv e access to nonperturbativ e effect s i n string theory — t o mirro r symmetry. Thei r wor k focuse s o n th e stronge r notio n o f quantum mirro r symmetr y which require s that M an d M satisf y th e constraint s o f mirror symmetr y a s used t o this point , but , additionally , M an d M mus t giv e ris e t o isomorphi c compactifie d quantum strin g theories . Althoug h i t woul d tak e u s to o fa r afiel d t o revie w al l th e physics underlying their results, let's summarize the mathematical content . Accord ing t o Strominger , Yau , an d Zaslow , fo r M an d M t o constitut e a quantu m mirro r pair, M mus t b e th e modul i spac e (suitabl y compactified ) o f specia l Lagrangia n (real thre e dimensional ) submanifold s o f M. (Jus t b y wa y o f physic s motivation , these Lagrangia n submanifold s mak e a n appearanc e becaus e the y ar e associate d with so-calle d Dirichle t 3-brane s whic h wra p aroun d "supersymmetric " 3-cycle s i n a Calabi-Ya u space . Realizin g th e Calabi-Yau , locally , a s a supersymmetri c T 3 fibration, T-dualit y o n the fibers take s u s to a n isomorphi c theor y i n which 3-brane s turn int o 0-branes. A s the modul i spac e of O-branes o n a Calabi-Yau spac e M i s the Calabi-Yau itself , w e learn tha t M i s als o isomorphi c (b y T-duality ) t o th e modul i space o f supersymmetri c 3-cycle s o n M.) Puttin g th e physic s motivatio n aside , this characterizatio n o f mirro r symmetr y i s intrinsi c i n th e sens e tha t i t doe s no t require referenc e t o the underlyin g strin g theory. Therefore , i t is amenable t o direc t mathematical analysi s withou t th e mathematica l imprecisio n tha t accompanie s a quantum mechanica l theory . A numbe r o f mathematician s hav e an d continu e t o pursue thi s directio n whic h hold s th e hop e o f a complet e translatio n o f the physic s of mirro r symmetr y int o rigorou s mathematics . Third, a great dea l of progress has been made on establishing the connectivity of the full spac e of Calabi-Yau manifolds— a notio n contemplate d b y both mathemati cians an d physicist s ove r man y years . Specifically , throug h th e conifol d transitio n results o f [36 ] an d numerou s generalizations , w e no w hav e a mean s o f perform ing physicall y smoot h deformation s whic h tak e u s betwee n Calabi-Ya u manifold s with differen t Hodg e numbers . Mathematically , thes e transition s involv e degener ating rea l 3-cycle s i n a n origina l Calabi-Ya u spac e (satisfyin g nontrivia l homolog y relations) an d the n resolvin g th e resultin g singula r spac e b y performin g smal l res olutions. Sinc e w e no w kno w tha t suc h manipulation s ar e perfectl y smoot h fro m the poin t o f physics , w e ca n exten d th e deformatio n argument s give n earlie r an d claim tha t mirro r symmetr y i s establishe d fo r al l Calabi-Ya u space s tha t ar e par t of the hug e we b o f Calabi-Yau s joine d b y conifol d transition s an d ther e generaliza tions [37] , [38] . Som e believe that thi s we b may wel l include al l Calabi-Ya u spaces , thereby ensurin g tha t mirro r symmetr y hold s for all . Thi s has yet t o be established . Fourth, a n importan t questio n i s whethe r on e ca n prov e mirro r symmetr y for Calabi-Ya u manifold s realize d a s complet e intersection s i n product s o f tori c varieties—the clas s o f Calabi-Yau manifold s fo r whic h ther e i s a combinatorial pro cedure fo r producin g wha t appea r t o b e mirro r pairs , a s mentione d earlier . Th e first significan t ste p towar d suc h a proo f wa s take n b y Morriso n an d Plesse r [39 ] in whic h a singl e gauge d linea r sigm a mode l wa s propose d t o b e a commo n "par ent" theor y t o th e tw o Calabi-Ya u sigm a model s o f a mirro r pair . Roughly , the y

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attempted t o sho w tha t b y startin g wit h a n initia l gauge d linea r sigm a model , in tegrating ou t tw o differen t set s o f field s whil e carefull y accountin g fo r quantu m corrections, the y arriv e a t wha t appea r t o b e tw o distinc t theories , bu t whic h i n reality ar e th e effectiv e theorie s fo r strin g propagatio n o n th e tw o manifold s o f a mirror pair . A versio n o f thi s approac h ha s bee n take n u p an d vigorousl y purse d by Hor i an d Vaf a [40] , leading to , amon g othe r results , wha t ma y wel l b e a proo f of mirro r symmetr y fo r thi s clas s o f Calabi-Ya u spaces . 6. Conclusion s Mirror symmetr y i s now a decade old , an d thes e first te n year s have establishe d it a s a powerful, unexpected , an d a s yet incompletel y understoo d structur e i n strin g theory an d i n algebrai c geometry . Relativ e t o the earlies t day s o f the subject , ther e is no w a substantia l amoun t o f mathematica l insigh t int o mirro r symmetry , bu t there i s certainl y a wa y t o g o before complet e analyti c understandin g i s achieved . Moreover, i t i s almost a certainty tha t mirro r symmetry , a s currently formulated , i s but a smal l piec e o f a large r physica l an d mathematica l structure . Namely , i n het erotic string theory, th e data definin g a string model includes al l that wa s describe d above ( a Calabi-Ya u manifol d M , a n elemen t o f H 2(M), a s well a s additiona l dat a required t o defin e th e ful l quantu m strin g theory ) but , additionally , i t include s th e specification o f a stable , holomorphi c vecto r bundl e V whos e firs t Cher n clas s van ishes an d whos e secon d Cher n clas s equal s tha t o f th e tangen t bundl e o f M. T o this point , w e hav e implicitl y chose n V t o b e identicall y TM an d hav e therefor e not calle d attentio n t o thi s data . Thi s choic e give s ris e t o a so-calle d (2 , 2) model , where these number s refe r t o the numbe r o f left an d righ t movin g supersymmetries . More genera l choices fo r V giv e ris e t o (0 , 2) models . At th e leve l o f heteroti c strin g theory , mirro r symmetr y a s s o fa r describe d establishes a physica l isomorphis m betwee n strin g propagatio n o n (M,TM) an d string propagatio n o n (M,Tj^). But , althoug h les s wel l understood , on e ca n mak e different choice s fo r th e bundl e V. Then , on e expect s ther e t o b e a mor e genera l version of mirror symmetr y establishin g a n isomorphism betwee n string propagatio n on (M , V) an d (M , V). A little more precisely, the (2 , 2) mirror relationshi p betwee n (2,1) form s o n M an d (1 ,1 ) form s o n M an d vic e versa, arise s from a n isomorphis m between H^^M^TM) an d i J - ^ M , ! ^ ) (an d vic e versa ) a t th e leve l o f quantu m cohomology. Th e (0,2 ) versio n o f thi s statemen t i s likel y t o relat e ^(M.V) an d ^ ( M , V") , again, a t the level of an isomorphism betwee n the quantum cohomologie s of H*(M, A*V) i7*(M,A*y) . Recen t wor k headin g towar d thi s goa l ha s bee n carried ou t b y C . Vafa , E . Sharpe , R . Blumenhagen , an d other s bu t ther e i s muc h further t o go before (0 , 2) mirror symmetr y i s understood a t th e leve l of (2, 2) mirro r symmetry. Thi s i s a n importan t proble m fo r th e future .

Appendix A . Conforma l Fiel d Theor y an d Algebrai c Geometr y This appendi x attempt s t o briefl y describ e th e correspondenc e betwee n confor mal fiel d theorie s an d Calabi-Ya u manifold s give n b y strin g theory . Th e inten t i s more to convey a glossary o f terms an d th e spiri t o f the relationshi p tha n t o presen t a ful l derivatio n o f th e result s mentioned .

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A . l . Fro m Manifold s t o Conforma l Fiel d Theories . Perturbativ e strin g theory describe s th e physic s o f one-dimensiona l extende d objects . A s the y propa gate, string s swee p ou t a tw o dimensiona l "worldsheet" , an d calculation s i n strin g theory lea d u s t o conside r a quantu m field theor y o n thi s two-dimensiona l surface . A quantu m field theor y o n a surfac e E i s compose d o f tw o essentia l ingrediants : fields an d a n action. Th e forme r ar e section s o f chose n bundle s o n E whil e th e latter i s a rea l value d functiona l o f th e sections . Fo r th e cas e a t hand , th e fields are simply coordinate s parametrizin g th e embedding o f E i n the chose n Calabi-Ya u manifold M , an d th e actio n i s basicall y th e are a o f th e imag e o f E i n th e induce d metric (mor e preceisly , thi s i s tru e afte r solvin g th e Euler-Lagrang e equation s fo r the metri c o n E) . 1 0 Explicitly , th e actio n S i s define d b y (A.l) S

= -L/

cPay/\h\(h a^Gij{X)daX%X^ +

e

afi

Bij(X)daX%X^).

In A. l th e 'fields ' X % compris e th e ma p X : E —> M embeddin g th e worldshee t in a (spacetime) manifol d M , h ap i s a metric on E, d 2ay/\h\ a n invariant integratio n measure on E, an d Gij a metric on M; th e antisymmetric tenso r B^ i s an additiona l term, o f crucia l importanc e i n strin g theor y bu t absen t i n discussion s o f particl e propagation; a' i s a dimensionfu l paramete r relate d t o th e 'tension ' o f th e string . The fundamenta l object s i n a quantu m field theor y ar e th e correlation functions of fields 1 Q- 1

2,l

where 7 n j m i s an m-fol d cove r o f a rationa l curv e o n M o f degre e n , X : E— > i" njm, and J i s th e Kahle r for m o f M. Th e facto r m - 3 i n (A.5a ) wa s discovere d phe nomenologically i n [4 ] by a n applicatio n o f mirror symmetr y an d wa s put o n a firm footing i n [1 4] . Th e fac t tha t J 2 ' 1 i s not correcte d wa s shown i n [1 5] . Heuristically , the nonrenormaiizatio n theorem s tel l u s tha t th e saddle-poin t approximatio n t o A.2 i s exact . Th e equation s (A.5 ) expres s th e contributio n (o r it s absence ) fro m nontrivial extrema . A.2. Fro m Conforma l Fiel d Theorie s t o Manifolds .1 6 We hav e describe d ho w strin g theor y associate s a conforma l field theor y t o a geometrica l spac e o f a n appropriat e type . Th e construction , however , i s quit e complicated an d doe s not see m lik e a very promisin g approac h t o studyin g th e geo metric situatio n itself . Additionally , variou s question s o f convergenc e an d rigou r are lef t unsatisfactoril y addressed . Fo r a particula r clas s o f Calabi-Ya u spaces , this situatio n wa s dramaticall y improve d i n [1 8] . I n thi s work , Gepne r constructe d 15 An exceptio n ar e th e condition s o n G. Thes e receiv e correction s a t ever y order ; th e con vergence o f thi s serie s ha s no t bee n proved . 16 To avoi d an y confusio n w e emphasiz e tha t onl y a limite d subclas s o f conforma l theorie s have a geometrica l interpretation .

M I R R O R S Y M M E T R Y .. .

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superconformal field theorie s whic h ar e exactl y solvabl e i n th e sens e tha t th e cor relation function s A . 2 can b e explicitl y evaluated . Thi s i s achieved b y constructin g a twiste d produc t o f simple , well-studie d representation s o f th e N = 2 supercon formal algebra . Wha t i s remarkable i s that thi s purel y algebrai c constructio n lead s to exactl y th e sam e fiel d theor y a s th e geometrica l constructio n describe d above . More precisely , on e ca n obtai n i n thi s wa y th e conforma l fiel d theor y correspond ing t o an y Calabi-Ya u manifol d give n b y th e vanishin g locu s o f a polynomia l o f Fermat typ e i n a weighte d projectiv e space , wit h a prescribe d Kahle r structur e corresponding t o a "radius " o f orde r 1 (in unit s o f ya!) . Thi s correspondenc e wa s conjectured b y Gepne r o n th e basi s o f th e representatio n conten t o f th e theorie s under larg e discret e symmetr y groups , an d furthe r strengthene d b y th e wot k o f [15] i n whic h th e value s o f J 2 , 1 wer e show n t o agre e exactl y betwee n th e tw o ap proaches (t o withi n a n undetermine d normalisation) . A forma l argumen t provin g the equivalenc e base d upo n pat h integral s wa s late r give n i n [1 6] , [1 7 ] an d a mor e rigorous derivatio n o f som e o f th e result s usin g mor e traditiona l method s i n [27] . For thes e specifie d point s i n certai n modul i space s w e thus hav e exac t formula s fo r the correlatio n function s o f interest . A s mentione d above , w e als o hav e a n explici t identification o f deformation s o f th e conforma l fiel d theor y wit h deformation s o f the geometrica l structure . Thi s allow s u s t o exten d th e solutio n t o a neighborhoo d of the exactl y solve d point . Globally, th e validit y o f th e identificatio n o f string y modul i spac e wit h th e geometric constructio n i s no t guaranteed , an d on e ma y expec t som e additiona l identifications betwee n inequivalent point s in the geometrical parameter spac e when considered a s strin g theories . Fo r example , on e readil y see s tha t A. 2 i s unchange d by addin g t o B^ a n elemen t o f i7 1 ' 1 (M, Z) s o thi s variabl e i s t o b e though t o f as periodic , an d ther e ar e othe r suc h identifications . Mirro r symmetr y ma y b e considered a highl y nontrivia l generalizatio n o f these, relatin g tw o manifold s whic h are i n fac t topologicall y distinct .

A.3. A Simpl e Example . A n exampl e ma y serv e t o clarif y som e o f the pre ceding discussion . Th e simples t exampl e o f a compact Ricci-fla t manifol d i s a toru s (this i s no t three-dimensiona l bu t w e ca n imagin e a tenso r produc t o f thi s wit h a "trivial " theory) . Sinc e M i s i n fac t flat , th e discussio n simplifie s considerabl y and th e conforma l fiel d theor y i s exactl y solvabl e throughou t th e modul i space . In th e discussio n above , ther e ar e n o higher-orde r correction s t o G i n thi s case . The infinitesima l deformation s o f th e conforma l fiel d theor y ar e generate d b y tw o (complex) field s - on e corresponding t o a deformation o f the comple x structure an d one t o a Kahle r deformation . Indeed , i n thi s exampl e th e globa l modul i spac e i s known: th e fia t metric s G ar e parametrize d b y th e comple x structur e labelle d b y a poin t r i n th e fundamenta l domai n o f SL(2,Z) , an d th e volum e \G\. Whe n w e incorporate th e B fiel d w e ar e le d t o th e comple x combinatio n p = B1 2 + fc\/[G| parametrizing th e uppe r hal f plane . I n fact , th e conforma l field theor y i s invarian t under B — > B + l a s mentioned above , but als o under p —» — 1/p s o that inequivalen t conformal field theories o n a torus ar e parametrized b y two copies of the fundamen tal domain . Th e "modula r group " [4 ] is thus SX(2,Z ) x 5L(2,Z) . Finally , ther e i s an additiona l identification , representin g mirro r symmetry , whic h interchange s th e roles o f comple x structur e an d Kahle r structur e deformations . I n thi s exampl e th e symmetry i s simply r ( 2 i , 2 2 , 2

3,a24,a

4

25)

5

with a = 1 . References W. Lerche , C . Vafa , an d N.P . Warner , Nucl . Phys. , B 3 2 4 (1 989) , 427 . B.R. Green e an d M.R . Plesser , Nucl . Phys. , B 3 3 8 (1 990) , 1 5 . P. Candelas , M . Lynker , an d R . Schimmrigk , Nucl . Phys. , B 3 4 1 (1 990) , 383 . P. Candelas , X.C . d e l a Ossa , P.S . Green , an d L . Parkes , Nucl . Phys. , B 3 5 9 (1 991 ) , 21 ; Phys. Lett. , 2 5 8 B (1 991 ) , 1 1 8 . L. Dixon , unpublished . S.-S. Roan , Int . Jour , o f Mathematics , 2 (1 991 ) 439 . P. Candelas , G . Horowitz , A . Strominger , an d E . Witten , Nucl.Phys , B 2 5 8 (1 985) , 46 . L. Dixon , J . Harvey , C . Vafa , an d E . Witten , Nucl . Phys. , B 2 6 1 (1 985) , 620 ; B 2 7 4 (1 986) , 285 . S.-S. Roa n an d S.-T . Yau , Act a Math . Sinica , Ne w Series , 3 (1 989) , 256 . S.-T. Yau , Proc . Nat . Acad . Sc i USA , 7 4 (1 977) , 1 798 . A. Strominger , Phys . Rev . Lett. , 5 5 (1 985) , 2547 . A. Strominge r an d E . Witten , Comm . Math . Phys. , 1 0 1 (1 985) , 341 . M. Dine , N . Seiberg , X . G . Wen , an d E . Witten , Nucl . Phys. , B 2 7 8 (1 987) , 769 ; B 2 8 9 (1 987) , 31 9 . P.S. Aspinwal l an d D.R . Morrison , Topological field theory and rational curves, Commun . Math. Phys. , 1 5 1 (1 993) , 245 . J. Distle r an d B.R . Greene , Nucl . Phys. , B 3 0 9 (1 988) , 295 . B.R. Greene , C . Vafa , an d N.P . Warner , Nucl . Phys. , B 3 2 4 (1 989) , 371 . E.Martinec, i n 'V.G . Knizhni k Memoria l Volume ' (L . Brin k et . al. , eds.) ; Phys . Lett , B 2 1 7 , 431. D. Gepner , Phys . Lett. , 1 99 B (1 987) , 380 ; Nucl. Phys. , B 2 9 6 (1 987) , 380 . B.R. Green e an d M.R . Plesser , i n preparation . P.S. Aspinwall , C.A . Liitken , an d G.G . Ross , Phys . Lett. , 2 4 1 B (1 990) , 373 . D.R. Morrison , Picard-Fuchs equations and mirror maps for hyper surf aces, Duk e Preprin t DUK-M-91-14. P.S. Aspinwal l an d C.A . Liitken , Nucl . Phys. , B 3 5 5 (1 991 ) 482 . D. Morrison , Mirror Symmetry and Rational Curves on Quintic Threefolds: Mathematicians, Duk e Preprin t DUK-M-91 -01 . S.-S. Roan , Int . Jour . Math. , 1 (1 990) , 21 1 .

A

Guide for

E. Witten , thes e proceedings , an d reference s therein . E. Witten , Nucl . Phys. , B 2 6 8 (1 986) , 79 . S. Cecotti , Nucl . Phys. , B 3 5 5 (1 991 ) , 755 . P. Aspinwall , B . Greene , an d D . Morrison , Nucl . Phys , B 4 1 6 (1 994) , 41 4 . E. Witten , Nucl . Phys. , B 4 0 3 (1 993) , 1 59 . See Essays in Mirror Manifolds, I , S.-T . Ya u (ed.) , Internationa l Pres s (1 991 ) . See Essays in Mirror Manifolds, II , B . Green e an d S.-T . Ya u (eds.) , Internationa l Pres s (1997). A. Strominger , S.-T . Yau , an d E . Zaslow , Nucl . Phys. , B 4 7 9 (1 996) , 243 . Fields, Strings, and Duality, Proceeding s o f th e Theoretica l Advance d Stud y Institute , 1 996 , C. Efthimio u an d B . Green e (eds.) , Worl d Scientific .

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[34] V . Batyrev , J . Algebrai c Geom. , 3 (1 994) . [35] V . Batyre v an d L . Borisov , i n Mirror Symmetry, I I (B . Green e an d S . T . Yau , eds.) , Inter national Pres s (1 997) , 71 . [36] B . Greene , D . Morrison , an d A . Strominger , Nucl . Phys. , B 4 5 1 (1 995) , 1 09 . [37] A.C . Avram , P . Candelas , D . Jancic , M . Mandelberg , Nucl.Phys. , B 4 6 5 (1 996) , 458-472 . [38] T . Chiang , B . Greene , M . Gross , Y . Kanter , Nucl . Phys . Proc . Suppl. , 4 6 (1 996) , 82-95 . [39] D . Morrison , R . Plesser , Nucl . Phys . Proc . Suppl. , 4 6 (1 996) , 1 77 . [40] K . Hori , C . Vafa , Mirror Symmetry, hep-th/0002222 . COLUMBIA UNIVERSITY , N E W YORK , N Y 1 002 7

https://doi.org/10.1090/amsip/033/03 Mirror Symmetr y I V AMS/IP Studie s i n Advance d Mathematic s Volume 33 , 2002

Lagrangian toru s fibrations o f Calabi-Ya u hypersurface s i n toric varietie s an d SYZ mirror s y m m e t r y conjectur e Wei-Dong Ruan

1. Introductio n In recen t years , we hav e witnessed quit e som e excitin g development s i n mathematics tha t ar e related t o physics, especiall y th e so-called "fina l theor y i n physics" — strin g theory . Amon g them , mirro r symmetr y conjectur e concernin g th e ge ometry o f Calabi-Ya u manifolds , althoug h relativel y old , is still on e of the mos t intriguing. Mirro r Symmetr y conjectur e originate d fro m physicists ' wor k in confor mal field theory an d strin g theory . I t proposes tha t fo r a Calabi-Yau 3-fol d X ther e exists a Calabi-Yau 3-fol d Y a s its mirror. Th e quantum geometr y o f X an d Y are closely related . I n particular on e can compute th e number o f rational curve s i n X by solvin g th e Picard-Fuchs equatio n comin g fro m variatio n o f Hodge structur e of Y. Th e first thing tha t stunne d mathematician s wa s the following tabl e (give n in [2]) predictin g number s o f rational curve s i n a quintic Calabi-Ya u 3-fold . k

nk

1 2875 2 609250 3 317206375 4 242467530000 5 229305888887625 6 248249742118022000 7 295091050570845659250 8 375632160937476603550000 9 503840510416985243645106250 10 704288164978454686113488249750 k — degree, nk — number o f degree k rationa l curve s i n quintic. Partially supporte d b y NS F Gran t DMS-9703870 . ©2002 America n Mathematica l Societ y an d Internationa l Pres s 33

34

W . - D . RUA N

Despite th e grea t impac t an d succes s mirro r symmetr y conjectur e bring s t o th e understanding o f th e geometr y o f Calabi-Ya u manifold s an d thei r modul i spaces , the fundamenta l questio n o f ho w t o construc t th e mirro r Calabi-Ya u manifol d fo r a give n Calabi-Ya u manifol d i n genera l wa s no t clea r a t al l fro m th e origina l mir ror conjecture . Althoug h mirro r manifold s wer e worke d ou t i n certai n cases , th e construction o f mirro r fo r genera l Calabi-Ya u stil l seeme d ver y elusiv e an d myste rious. Th e mos t genera l constructio n s o fa r wa s give n b y Batyre v fo r Calabi-Ya u hypersurfaces i n tori c varieties . Fro m th e tori c geometr y standin g point , Batyre v proposed tha t fo r Calabi-Ya u hypersurfac e X i n the tori c variety P ^ correspondin g to a reflexive polyhedro n A , th e mirror shoul d b e the Calabi-Ya u hypersurfac e Y i n the tori c variet y PA V correspondin g t o th e dua l reflexiv e polyhedro n A v . Batyre v computed amon g othe r thing s th e Hodg e number s o f Calabi-Ya u hypersurfac e X and th e mirro r Y an d showe d tha t the y behav e a s predicte d b y mirro r symmetry . Batyrev's mirro r constructio n include s man y specia l case s o f mirro r constructio n discussed previousl y b y man y physicist s an d mathematicians . Later , ther e ar e lot s of important wor k surroundin g Batyrev' s mirro r construction , includin g computin g numbers o f rationa l curves , etc . In 1 99 6 Strominger , Ya u an d Zaslo w ([1 6] ) propose d a geometri c constructio n of mirro r manifol d vi a specia l Lagrangia n toru s fibration. Accordin g t o thei r pro gram (w e will call it SY Z construction) , a Calabi-Ya u 3-fol d shoul d admi t a specia l Lagrangian toru s fibration . Th e mirro r manifol d ca n b e obtaine d b y dualizin g th e fibres. O r equivalently , th e mirro r manifol d o f X i s th e modul i spac e o f specia l Lagrangian 3-toru s i n X wit h a flat (7(1 ) connection . In a sense , SY Z constructio n appear s t o b e mor e fundamenta l t o mirro r sym metry phenomeno n an d mor e classica l tha n quantu m mirro r symmetry . Mor e im portantly, SY Z constructio n ha s th e potentia l t o explai n th e mathematica l reason s behind mirro r symmetry . Fo r example , SY Z constructio n give s u s a possibl e wa y to construc t th e mirro r manifol d i f w e understan d ho w t o construc t dua l singula r fibres. Remark. Th e origina l SY Z mirro r conjectur e i s rathe r sketch y i n nature . More detaile d knowledg e o f the fibration suc h a s singula r locus , singula r fibres an d duality o f singula r fibres ar e necessar y t o b e worke d ou t i n orde r t o formulat e th e precise SY Z mirro r conjecture . Withou t th e precis e formulation , on e woul d not reall y b e abl e t o construc t th e mirro r manifol d completely . According t o th e SY Z construction , specia l Lagrangia n submanifold s an d spe cial Lagrangia n fibrations fo r Calabi-Ya u manifold s see m t o pla y ver y importan t roles i n understandin g mirro r symmetry . However , despit e it s grea t potentia l i n solving th e mirro r symmetr y conjecture , ou r understandin g o n specia l Lagrangia n submanifolds i s very limited . Th e know n example s ar e mostl y explici t loca l exam ples o r example s comin g fro m 2-dimensiona l case . Ther e ar e ver y fe w example s of specia l Lagrangia n submanifold s o r specia l Lagrangia n fibrations fo r dimensio n higher tha n two . M . Gross , P.M.H . Wilso n an d N . Hitchi n ([3] , [5] , [7] ) di d som e important wor k i n thi s are a i n recent years . Thes e work s mainl y concer n loca l geometric structur e o f the specia l Lagrangia n fibrations an d case s tha t ca n b e reduce d to 2-dimensiona l situation . O n th e othe r extreme , i n [1 7] , Zharko v constructe d some non-Lagrangia n toru s fibration o f Calabi-Ya u hypersurfac e i n tori c variety . Despite al l thes e efforts , SY Z constructio n stil l remain s t o b e a beautifu l drea m t o us.

35

LAGRANGIAN TORU S FIBRATION S

For ou r discussion , w e will relax the specia l Lagrangia n condition , ye t stil l kee p the Lagrangia n condition , whic h w e think i s a goo d compromise . W e wil l conside r Lagrangian toru s fibrations o f Calabi-Ya u manifolds . Accordin g t o ou r discus sions i n [9] , ther e ar e majo r difference s betwee n C°° -Lagrangian fibrations an d general Lagrangia n fibrations, le t alon e non-Lagrangia n fibrations. I n a sense , C°° Lagrangian fibrations shoul d alread y captur e th e symplecti c topologica l structure s of th e correspondin g specia l Lagrangia n fibrations. W e wil l construc t Lagrangia n torus fibrations tha t exhibi t th e same topological structure a s C°°-Lagrangian toru s fibration. I n particular , singula r locu s o f th e fibrations wil l b e o f codimensio n 2 . In [1 4 ] w e smooth ou t ou r Lagrangia n fibrations t o a grea t extent , ye t w e stil l fal l short o f makin g i t int o a C°°-Lagrangia n fibration everywhere . Finally, i n [1 1 ] , we were able to construct Lagrangia n toru s fibrations o f generic Calabi-Yau hypersurface s i n tori c variet y correpondin g t o a reflexiv e polyhedro n in complet e generality . Wit h thes e detaile d understandin g o f Lagrangia n toru s fibrations o f generic Calabi-Yau hypersurface s i n toric variety, we were able to prov e the symplecti c topologica l SY Z mirro r conjectur e fo r Calabi-Ya u hypersurface s i n toric variety . Mor e precisel y T H E O R E M 1 .1 . For generic Calabi-Yau hypersurface X and its mirror CalabiYau hypersurface Y near their corresponding large complex limit and large radius limit, there exist corresponding Lagrangian torus fibrations

X^b) ^

X

Yb

^- > Y

II dAv dA

w

f

with singular locus T C dA v and T C dA w, where (/) : dA w— » dA v is a natural homeomorphism, 0(r' ) — V. For b G dAw\T', the corresponding fibres X^^ and Yb are naturally dual to each other. For detaile d notation s an d results , pleas e refe r t o late r section s o f thi s pape r and ou r paper s i n th e reference . Our wor k essentiall y indicate s tha t Batyre v mirro r construction , whic h wa s proposed purel y fro m tori c geometr y stan d point , ca n als o b e understoo d an d jus tified b y th e SY Z mirro r construction . Thi s shoul d giv e u s greate r confidenc e o n SYZ mirro r conjectur e fo r genera l Calabi-Ya u manifolds . 2. Ferma t typ e quinti c cas e Our ide a o f constructio n i s a ver y natura l one . W e star t fro m th e natura l Lagrangian toru s fibration a t the "Larg e Complex Limit" give n by moment ma p an d try t o use a gradient flow to get Lagrangia n toru s fibration o n nearby hypersurfaces . This metho d wil l i n principl e b e abl e t o produc e Lagrangia n toru s fibrations i n general Calabi-Ya u hypersurface s i n tori c variety . To illustrat e ou r idea , i t i s helpful t o explor e th e historicall y mos t famou s cas e of Ferma t typ e quinti c Calabi-Ya u threefol d famil y {X^} i n C P 4 define d b y 55

1 fc=l

36

W.-D. RUA

N

near th e larg e comple x limi t X^ define d b y

° II Zk = °'

Po =

fc=i

in som e detail . Mos t o f th e essentia l feature s o f th e genera l cas e alread y sho w u p there. Let u s star t wit h th e natura l Lagrangia n toru s fibration o f th e "Larg e Comple x Limit" XOQ. XQQ i s a unio n o f five C P 3 ' s . Ther e i s a natura l degenerat e T 3 fibration structure fo r X^. Le t {Pi,i = 1 , • • • , 5 } b e five point s i n R 4 t h a t ar e i n genera l position. Conside r th e natura l m a p F : C P 4 — > R 4 .

A = Image (F) i s a 4-simplex . Xo o i s naturall y fibered ove r dA vi a thi s m a p F with genera l fibre bein g T 3 . This i s precisel y th e T 3 specia l Lagrangia n fibration fo r XOQ as indicate d b y SYZ constructio n wit h respec t t o th e natura l flat Kahle r metri c dxi A dx

E. . ,-Ci 1n+ I T , 1 22

on Xoo . Therefor e thi s fibration i s a reasonabl e on e t o star t with . For ou r porpose , w e wil l conside r th e Fubini-Stud y metri c u;FS = ddlog(l +

\x\

2

).

F i s exactl y th e momen t m a p o f th e natura l rea l n-toru s T n actio n e*e(x) = (e^x 1 ,eld*x2,->- ,e

w

"xn)

on C P n wit h respec t t o UJFSConsider meromorphi c functio n

El 4 4

defined o n C P . Le t V / denot e th e gradien t vecto r field o f th e rea l functio n / = Re(s) wit h respec t t o th e Kahle r metri c g. Notice t h a t V/= H

h.

(Hh i s t h e Hamiltonia n vecto r field generate d b y h = Jra(s). ) Thi s implie s th e following L E M M A 2.1 . The gradient flow of f leaves the set {Im(s) = 0 } invariant and deforms Lagrangian submanifolds in XOQ to Lagrangian submanifolds in X^ . W i t h thi s lemm a i n mind , th e constructio n o f Lagrangia n toru s fibration o f X^ for ip larg e i s immediate . Deformin g th e canonica l Lagrangia n toru s fibration o f Xoo ove r dA alon g th e gradien t flow o f / wil l naturall y induc e a Lagrangia n toru s fibration o f X^ ove r dA fo r ip large an d real . T h e detaile d structur e o f th e resultin g Lagrangian toru s fibration o f X^ i s describe d i n th e followin g theorem . Fo r detail s of th e proof , pleas e refe r t o [8] .

LAGRANGIAN TORU S FIBRATION S

37

T H E O R E M 2.1 . The flow ofV will produce a Lagrangian fibration F : X^— > oo. We defin e a ne w sequenc e o f J-holomorphi c map s {gi} a s follows . Let D /(xij p) b e th e close d dis c o f radiu s p centere d a t Xi a t th e tangen t spac e TXiM an d D(x^p) b e it s imag e unde r th e exponentia l map . However , th e metri c we us e o n D(xi,p) i s th e on e induce d fro m th e metri c o n E whic h i s uniforml y equivalent t o th e Euclidea n metri c o n D'(xi,p). Le t ki — rrii • p an d defin e Ti : D(ki) —> D(xi,p) t o b e th e contractio n ma p wit h th e contrac t facto r ^ - . Not e that Ti is uniqu e u p t o th e rotations . Le t g^ : D(ki)— > M t o b e th e compositio n

fi°ri. For ou r late r use , w e denot e r^(l ) i n E b y ^ . Not e tha t yi i s gettin g infinitel y close t o Xi a s i tend s t o infinity . PROPOSITION 2.1 . Let fi : (E,j )— > (M,J), i = 1 ,2 , •• • , be a sequence of (j, J)-holomorphic maps of class A. (i) If rrii is bounded, there exists a subsequence of {fi}, still denoted by the same notation, and a (j , J)-holomorphic map f^ : E— > M such that {fi} is C°°-convergent to /oo . (ii) If rrii is unbounded. We may assume thatlimrrii = oo. Then {gi} constructed above is locally C°°-convergent to a J-holomorphic map g^ : C —> M, i.e., it is C°°-convergent to g^ on any compact part of C. Moreover, g^ can be extended to a non-trivial J-holomorphic map defined on S 2. PROOF, (i ) Sinc e \dfi\ < C fo r som e constan t C , th e Sobole v nor m ||/i||i, p i s uniformly bounded . Her e w e choos e p > 2 . I t follow s fro m th e standar d ellipti c estimate fo r d^j tha t al l H/iH^p' s ar e also uniformly bounded . Thi s implie s that fo r any /c , the C^-nor m ||/f||c fc o f fi i s als o uniforml y bounde d b y Sobole v embeddin g theorem. Th e conclusio n no w follow s fro m th e Ascol i theorem . (ii) Fo r an y compac t se t K C C , w e ma y assum e tha t K C D(ki) b y choosin g large i. The n sup | ^ 0 ) | < su xeK Wi

p \dfi(x)\ xeD(xi,p)

m

e 0 depending only on J such that the energy E(f) for any non-trivial J-holomorphic sphere f is greater than or equal to A . P R O O F . I f th e conclusio n i s no t true , ther e exis t a sequenc e o f J-holomorphi c spheres fi \ S2 -^ M suc h that liim . E(fi) = 0 . It follow s fro m th e abov e propositio n that fi i s C°°-convergent t o a J-holomorphi c spher e f^ wit h E(f (X>) = 0 . Thi s ca n happen onl y whe n f^ i s a constan t map . Thi s implie s tha t whe n i larg e enough , the imag e o f fi i s i n a smal l neighborhoo d o f som e constan t map . Hence , th e homology clas s [fi] of fi i s zero . W e conclud e tha t fi i s a constan t ma p whe n i i s large enough . •

Clearly, th e sam e proo f als o works fo r J-holomorphi c map s o f the highe r genu s as lon g a s th e comple x structure s o f th e domain s sta y i n a compac t regio n o f M g. Step II : deformatio n o f th e domai n To iterat e th e abov e bubblin g process , w e nee d t o defor m th e domai n o f fi. By usin g th e stereographi c projectio n p Xi : 5 2— » T XiM an d composin g wit h th e exponential map , w e ma y identif y D(xi,p) i n E wit h a bal l Di i n S 2 centere d a t oo. Not e tha t her e w e have place d o o o f S 2 a t Xi o f T XiT, to b e th e projectio n p x%. Let y[ be th e correspondin g poin t o f yi unde r thi s identification . W e ge t a rationa l curve (5 2 ; 0,1, y[, oo) o f fou r marke d points . A s th e cros s rati o (0,1 , y[, oo) = y[ is the onl y invarian t involved , w e als o hav e th e followin g tw o models : (i): Le t (S£ , 0L, 1 L > ° ° L) an d (S^ , 0#, 1#, OOR) b e tw o copie s o f the uni t spher e with standar d spherica l metric . Le t w an d w' b e th e comple x coordinate s centere d at 0 L an d 0 ^ respectively . W e define Si t o be (S1 \{ooL}^S 2l\{ooR})/{w^w, = y 2'}, which has the fou r marke d points . Hence , S 2 i s obtained fro m S\ an d S\ b y cuttin g off th e bal l \w\ > \y[\* an d gluin g bac k alon g th e boundarie s o f the remainin g part s by rotatin g arg(^) . Giv e Si th e induce d spherica l metri c /i^ s . (ii): Th e secon d mode l i s to stretch th e annulu s par t \w\ > 1 , \w'\ > 1 in Si int o the cylinde r (0,lo g \y[\) x S 1 wit h th e standar d cylindrica l metric . W e wil l denot e this cylindrical-lik e metri c o n Si b y /i^ c . We no w giv e Di, whic h i s centere d a t OR of Si now , thes e tw o metric s an d push-forward the m vi a p x. t o D(xi, p). Not e tha t th e ^-coordinat e o f OR i s w = oo . We wil l denot e E ^ equippe d wit h on e o f thi s metric s b y (E^,^ 5C ) o r (E^,/i^ s ). The distanc e o f th e tw o marke d point s Xi an d ^ no w become s TT i n thes e ne w metrics. To iterat e th e bubblin g i n ste p (I) , w e wil l us e th e cylindrica l metri c /i^ c fo r E^. Sinc e th e injectiv e radiu s o f E ^ i s bounde d below , i f sup x G S . \dfi{x)\^ ic i s stil l unbounded, w e ca n first repea t th e abov e bubblin g proces s t o produc e th e secon d bubble, an d the n repea t th e abov e proces s t o defor m th e domain . Not e tha t b y the bubblin g process , fo r an y fixed R, swp X£DR{xi) Wi\ i s alway s bounded . Her e Dii(xi) C D{xi,p) i s th e push-forwar d imag e o f th e hal f spher e o f Si centere d a t OR wit h a cylinde r o f lengt h R attache d i n th e secon d mode l above . Therefore , the secon d bubblin g ca n onl y tak e plac e awa y fro m Dn{xi). Le t Xi an d yi b e the tw o ne w marke d point s obtaine d fro m th e secon d bubblin g an d Dn{xi) b e th e corresponding regio n afte r th e deformatio n o f th e domai n th e secon d time . The n DR(xi) n D R(xi) = 0 whe n i i s larg e enough . Bu t E(f\ DR^x%){jDR^x%)) i s clos e t o

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2A, wher e A i s the universa l constan t i n Cor.2.1 . Sinc e the energ y o f fi i s bounde d above, afte r finite steps , thi s proces s wil l stop , an d w e arriv e a t th e following : (i): Th e domai n E ^ consist s o f fou r parts : There ar e thre e compac t parts : S(xi ik), k — 1, • * * , ^, whic h i s th e hal f spher e centered a t th e bubbl e poin t Xi :k1 Ci :i, I = 1 , • • • l[, which i s a spher e wit h a t leas t three discs removed, an d B x whic h is the genus g curve E with several discs removed. The fourt h par t Tij,j = 1 , • • • ,j[ consist s o f a collectio n o f cylinder s whic h connect th e first thre e compac t part s o f E; . A s i tend s t o infinity , th e lengt h o f In tends t o infinit y also . Note tha t eac h tim e whe n a ne w bubbl e i s forme d i n th e middl e par t o f thes e T^j, a ne w C^ j appears . Since k'^ l\ an d j[ ar e determine d b y the bubblin g an d th e deformatio n domai n process, the y ar e independen t o f i. W e will omit th e subscrip t i i n the notation s fo r them. I t i s eas y t o se e tha t j ' < 3k'. We giv e E ^ th e cylindrica l metri c /x^ c. I t i s th e metri c whic h i s spherical-lik e along S an d C par t an d i s th e induce d metri c fro m E alon g Bi. O f course , i t i s cylindrical alon g T^j's . There ar e 2k' marke d point s £i,fc,yi,fc ,A; — 1, • • • , A/, lyin g i n S(xi^) obtaine d from th e iteratio n o f th e bubblin g process . (ii): Le t Si^R be th e unio n o f al l Safe' s togethe r wit h al l attache d cylinder s o f length R. Similarl y defin e CI^R and BiR. Fo r a fixed R, whe n i i s larg e enough , these thre e part s d o no t intersec t eac h other . Sinc e \dfi\^ % c i s bounded , b y takin g a subsequence , fi\s iR i s C°° -convergent t o a J-holomorphi c ma p fs R : SR— > M , where SR i s th e unio n o f th e k' hal f sphere s eac h wit h a cylinde r o f lengt h R attached. Bu y lettin g R ten d t o infinit y an d takin g th e diagona l sequence , w e ge t k' J-holomorphi c spheres , denote d b y f$, whic h ar e al l th e to p bubble s o n th e bubble tree . Similarly, alon g C^# , w e get th e limi t fc R, an d fro m this , th e limi t fc define d again o n a disjoin t unio n o f V S 2's. Finally , th e sam e argumen t give s th e limi t fs defined o n E . Now w e stud y th e limi t behavio r o f fi alon g thos e connectin g cylinder s Tij. For tha t purpose , w e defin e th e non-negativ e numbe r 7 = UJ(A) — {E(fs) + E(fc) + E(IB)}- I f 7 = 0 , we move to ste p (IV) . Otherwise , w e go to th e nex t step . Step III : Convergenc e alon g cylinder s We wil l sho w tha t i n th e cas e tha t 7 > 0 , ther e exist s a t leas t on e mor e ne w non-trivial J holomorphi c spher e i n the limi t o f fi alon g these connectin g cylinders . Let Ti^R = T,i\{Si iRU Ci^RU ^ ? J R } , whic h i s a unio n o f cylinder s containe d i n the unio n o f T^j's . LEMMA 2.1 . There exists a subsequence of {%} and a corresponding sequence of Ti^Ri in T 0 0;

(ii) length of each cylinder in Ti R Z tends to infinity as i tends to infinity; (iii) EUi\ Ti,Ri)>\l. PROOF.

Fi x R, whe n i i s larg e enough , w e hav e \E(fsR) ~

E(f SRti)\
7-g7>27Fix suc h a n i an d se t Ri — R. B y takin g a n increasin g sequenc e o f R an d i , w e can inductivel y defin e Ri wit h th e desire d properties . • Since eac h T ^ ha s a t mos t 3/c ' components , withou t los s o f generality , w e may assum e tha t on e o f th e component s ha s th e propert y i n th e lemma . W e stil l use th e sam e notatio n T ^ ^ t o denot e th e component . Assum e tha t th e lengt h Ni of th e cylinde r T^# . i s a n intege r an d decompos e T ^ a s a unio n o f Ni adjacen t cylinders Dij o f uni t length , j — I-- - ,A^ . Sinc e \dfi\^ ic i s bounded , fo r an y sequence {ji}iZi wit h 1 < ji < Ni, {fi\D itj.}i^=i n a s a convergen t subsequence . LEMMA 2.2 . There exists a sequence {ji}^i such that (a subsequence of) {fi\Di • . }iZ\ converges to a non-trivial J-holomorphic map fry defined on D = S1 x [0,1 ] .

PROOF. Assum e th e conclusio n i n th e lemm a i s no t true . The n fi restricte d to th e tw o boundar y circle s o f T^ jRi converge s t o tw o constan t maps . Le t 6 i an d&2 be th e image s o f th e tw o constan t map s i n M. LEMMA 2.3 . There exists a constant S± such that for any 0 < S < Si and any i, there exists a Zi G T^#. such that both b\ and 6 2 are not in the ball B(fi(zi),5). P R O O F . Th e conclusio n i s obviou s i f 6 1 ^ 62 . We may assum e tha t b\ — 6 2 = b. Choos e Si smalle r tha n bot h o f the constant s appeared i n th e isoperimetri c inequalit y an d monotonicit y lemma . If image of {fi\r iyR.) C B(b, Si), b y our assumptio n an d isoperimetric inequality , we hav e

i 7 < £(/ita,« 4 )

0.

This i s a contradiction . Choos e Zi in suc h a wa y tha t fi(zi) £ B(b, Si). • It follow s fro m thi s lemm a tha t whe n i i s larg e enough , th e tw o boundar y components o f fi\T i)R. n e outsid e o f B{xi,S). Le t Li b e th e larges t sub-cylinde r i n T^;JR. containin g Xi suc h tha t fil^ onl y intersec t th e boundar y o f B(fi(xi),S) a t its tw o boundar y circles . B y ou r assumption , fi restricte d t o th e tw o boundar y components o f Li converge s t o tw o constan t map . W e conclude tha t fo r an y S' < 5, the imag e o f fi restricte d t o th e tw o boundar y component s o f Li li e i n th e shel l between th e boundarie s o f th e tw o ball s B(fi(zi),S) an d B(fi{zi),S f). I t follow s from isoperimetri c inequalit y an d th e monotonicit y lemm a tha t c • S' 2 < E(fi\ Li)
0 , w e ca n ad d m ' ne w marke d points, Zij^i • • • , Zij^m'. on eac h T i:j suc h tha t (i) for an y fixed i?, all of m'j cylinders Tij i7riiR(zijirn) o f length 2R wit h marke d point Zij iTn lyin g o n th e centra l circle s ar e mutuall y disjoint ; (ii) Le t fij j7n,R b e th e restrictio n o f fi o n th e correspondin g cylinder , the n {fi,j,m,R}i^i i s C°°-convergen t t o a J-holomorphic ma p fj,m,R define d o n the cylin der Tj^^R o f lengt h 2R. B y lettin g R ten d t o infinity , w e ge t a ne w non-trivia l J-holomorphic ma p fj^ m define d o n th e sphere , denote d b y Tj^ m(zj^rn). Her e Zj :Tn is the marke d poin t o n th e sphere . Let fr b e th e collectio n o f al l thes e limi t ma p / j , m . The n w e have , E(fs) + E(fc) + E(fs) + E(fr) = ^(A), i- e-> ther e i s n o energ y los s anymore . Step IV : convergenc e We defin e th e limi t o f {/i} , denote d b y f^, t o b e th e unio n o f al l thes e J holomorphic maps , fs, fd SB and fr wit h th e domai n togethe r wit h adde d marke d points describe d i n th e first thre e steps . Recall tha t s o fa r f^ consist s o f fou r parts : (i) ther e i s a union o f top bubble s fs, whos e domain (5 , x, y) i s a disjoint unio n of k' S 2ls wit h marke d point s Xk an d y^, k = 1 , • • • k'. Her e w e hav e use d x an d y t o denot e th e collectio n o f correspondin g marke d points , fs i s obtaine d fro m {fi\siiR}- Th e domai n S ^ togethe r wit h th e marke d point s i s locall y convergen t to (S , x, y) a s i an d R tend s t o infinity ; (ii) ther e i s a unio n o f V stable bubble s fc 1 whic h i s obtaine d fro m {fi\d R }Again, th e domai n CI,R i s locall y convergen t t o th e domai n C. Not e tha t eac h component o f CI^R ha s a t leas t thre e cylinder s o f lengt h R attached . Thi s implie s that th e component s i n C ar e stabl e assumin g th e lemm a below ; (iii) there is a union of connecting bubbles JT we just describe d a t th e end of the last step . Th e domai n Tj jrn(zjjTn) i s obtained a s th e limi t o f {Jij,ra,i?(^,j,m)}^:i ; (iv) away from th e places where those bubbling occur, {fi}\^Z 1 locall y converge s to fs define d o n E . Now th e limi t f^ alread y capture s th e ful l energ y UJ(A). Thi s implie s th e following lemma . LEMMA 2.4 . Let V %,R = £ * \ {S ijR U C ljR U Jjrn T ijJ^R(zlJjrn) U be any of its components. Then

lim li m diamiyl) =

S i j j R }, and V(

0.

P R O O F . B y th e definitio n o f V( R, eac h boundar y componen t o f V( R i s als o appear a s on e o f th e boundarie s o f cylindrica l end s o f th e fou r part s describe d above. I t follow s tha t whe n bot h R an d i larg e enough , th e diameter s o f thes e

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boundary component s ar e les s tha n \8, fo r a give n 5. No w i f th e diamete r o f V( R is greate r tha n o r equa l t o 3^ , we ca n find p i n Vi^R suc h tha t fi\dVi R ^ es outsid e of B(f(p),5). No w assum e tha t 5 her e satisfie s th e conditio n i n th e monotonicit y lemma, w e hav e E{fi\y ) > c • S 2 fo r som e constan t c. On th e othe r hand , sinc e lim.ji^ 00\im.i^00 E(f i\vi R ) = 0 , w e als o hav e E(fi\v.' ) < c • S 2 whe n R an d i ar e large . Thi s i s a contradiction . • Now fi map s eac h componen t o f thes e V ^ ' s int o an y prescribe d smal l neigh borhood o f a particula r poin t i n M . I t follow s tha t th e image s o f al l o f thos e limi t points along the ends of the four part s of f^ obtaine d from th e theorem of removable singularities ar e pairwisely identifie d wit h eac h other. Therefore , w e can identif y al l those limi t point s o f ends pairwisel y i n Eo o t o for m a connected domain . Fro m ou r description before , i t i s clea r tha t al l o f thes e joint point s appea r i n E ^ a s doubl e points, an d tha t E ^ become s a semi-stable curve . Le t E^ # b e the unio n o f the fou r parts, Si,R, Ci^n, Bi^R an d Tij }niiR(zijjrn). W e no w switc h t o th e spherica l metri c Hi^s. Then U#(E^# ; x, y,z) ca n b e identifie d wit h (Eoo,/ ? \ {double point s},x,y,z). Note tha t bot h (Ei]x,y,z) an d ( E ^ x , y,z) ar e stabl e curves , an d th e abov e iden tification simpl y mean s tha t i n th e modul i spac e M. g,m+m', E ^ i s convergen t t o Eoo. Her e m' i s the numbe r o f th e ne w adde d marke d point s durin g th e bubbling . Moreover, w e have tha t fi i s C°° -convergent t o f^ alon g an y £ # ( = £; ;J R = ^OO,R) for an y fixed R an d tha t firm, E(fi) = E(f OQ). This prove s tha t {fi} i s weekly convergen t t o f^ i n th e cas e that E ^ is a singl e point i n M g,m- Clearly , th e proo f fo r th e cas e tha t E ^ stay s i n a compac t par t o f M.g,m, i s almos t identical . W e leav e i t t o th e readers . Proof fo r th e cas e I I Assume tha t (E^,p. , ji) i s convergen t t o ( E ^ p , joo) i n M. g,m- W e wil l onl y deal wit h th e cas e tha t g > 2 or g = 0 and leav e th e cas e g — 1 to th e readers . As w e mentione d i n th e introduction , M g,m i s a n orbifold . Give n a poin t EQO G Ai g,mi th e uniformizer a t i t ca n be described b y an elementary gluin g proces s at eac h o f it s doubl e points . Mor e precisely , a t eac h doubl e poin t Eoo , as i — • o o in the sens e tha t ther e exist s a Eo o G M g,m+m'•> whic h is th e minima l stabilizatio n o f Eo o b y addin g ra'-marked point s an d E ^ i n £ Mg,m+m' obtaine d fro m E ^ b y als o addin g ml marke d points , suc h tha t Ei i s convergent t o E ^ i n G Mg^m+m'- Hence , there exis t loca l deformatio n parameters c ^ G C* with o^— > 0 such that E ^ can be identified wit h (Eoo)a zLet i : ( E Q O ) ^— > E f b e th e identificatio n maps . (ii) Fo r each compact se t K C E 0 0 \{double points} , le t Ki b e the correspondin g subset i n (Eoo) ^ whe n i i s larg e enough . The n p ^ . = (fi o E ^ such that f[ — faoipi. Note that xjji doe s not preserv e markings i n general. The n w e have two limi t stabl e map s /o o an d f^ wit h domain s Eoo an d E ^ i n M g,m+k an d M g,m>+k> respectively . We nee d t o prov e tha t k = k' an d ther e exist s a n identificatio n preservin g marked point s ^o o : E^ - > Eo o suc h tha t / ^ = /o o ° ^oo • To prove this , w e note tha t i f we contract al l those component s i n the domain s of Eoo , which correspon d t o trivia l bubbl e components , int o point s t o ge t s o calle d cuspidal maps , th e resultin g ma p i s unique u p t o identificatio n o f domains a s cusp idal curves . Th e ke y observation the n i s that thos e trivia l bu t stabl e bubbles,whic h are obtaine d fro m par t o f th e C^RS, ar e determine d b y th e othe r part s o f th e convergence o f th e sequence .

GANG LI U

68

More precisely , conside r th e imag e o f /oo . A s a se t o f V consistin g o f al l limi t points o f im({[fi]}), i t i s well-defined , no t dependin g o n an y particula r param eterization o f th e domain s o f {[/*]} . Le t /£ > b e th e cuspida l ma p obtaine d fro m /oo b y shrinkin g al l it s trivia l components . Le t E ^ b e th e domai n o f /£> . The n /oo = loo ° ^5 where f c i s a simpl e cuspida l ma p an d T r i E ^ - ^ E ^ i s a continuou s surjective ma p betwee n th e tw o cuspida l domains , whic h i s a holomorphi c branc h covering o n eac h componen t o f E ^ . /£ > give s ris e t o a holomorphi c parameteriza tion o f im(f QO)1 whic h i s on e t o on e awa y fro m finit e points . I t i s eas y t o se e tha t such a simpl e conforma l parameterizatio n i s uniqu e u p t o a conforma l identifica tion o f the cuspida l domain s o f /£, . I n particular , th e imag e D = f00(D) o f doubl e points o f EQ O i s a well-define d finite se t o f V. In th e res t o f the proof , w e will only conside r th e cas e that th e genu s g > 2 and leave the othe r case s to th e readers . T o simplify ou r presentation , w e will make on e more assumptio n tha t m an d m f ar e equa l t o zero . Let E ^ b e a subse t o f Eoo , whic h i s th e unio n o f domain s o f al l nontrivia l components o f /oo ? an d denot e foolyN b y / ^ . W e defin e D — (f^ ))~1 (D). The n D contain s al l doubl e point s o f E ^ a s a subset . Fo r eac h e > 0 , le t N e b e th e e-neighborhood o f D i n E ^ an d defin e K e — E ^ \ N e. Th e fo r i larg e enough , th e compact se t K e i s als o containe d i n E ^ throug h gluing . W e us e K\ t o denot e th e Ke i n Ei . W e wil l sho w tha t th e limi t o f th e non-trivia l par t o f K\ a s i tend s t o infinity an d e tends t o zer o wil l determin e /oo - Fo r tha t purpos e w e nee d t o kno w how K\ an d (K')\, ar e relate d eac h othe r unde r th e ma p ^ . To that end , w e consider a n e-neighborhoo d N e o f D i n V , an d defin e compac t subsets C e = Eo o \ /^(N^ an d C\ = E z \ f ' 1 ^ ) i n Eo o an d E f respectively . Th e following tw o facts concernin g K e an d C e ar e crucial fo r th e proo f o f Hausdorffness :

(I)4cC £ l , KlcCl; ( I I ) C e 2 c £ e i , Ci 2CKi; where e\ oo{q'k)) a s a stabl e curv e wit h marke d points . Sinc e ther e ar e n o extr a marke d points , ther e exists a n automorphis m A of E ^ suc h tha t A(^ ) = ^oo(# 0 fo r i = 1 , • • • , k. Thi s implies tha t ( / ^ ) = (f^). 4. Virtua l modul i cycle s an d quantu m homolog y Although A4g,m(A, J) i s th e mos t natura l compactificatio n o f A / l^, m (A, J ), i t can no t b e use d t o defin e GW-invariant s an d quantu m homology . Th e mai n diffi culty i s that th e dimensio n o f the boundar y componen t M. g^m(A, J) ma y b e greate r than tha t o f M g,m(A, J ) . Th e failur e o f transversality alon g th e boundar y compo nent wil l caus e seriou s problem s whe n ther e ar e J-holomorphi c curve s o f negativ e first Cher n class . T o overcom e thi s difficulty , w e embe d Mg^ m(A,J) int o a large r and infinit e dimensiona l spac e o f stabl e L^-maps . Fix (k,p) suc h that k— - > 1 . We define B%(A) to be the collectio n o f all equiv alent L^-map s o f clas s A fro m a Rieman n surfac e o f genus g an d ra-marked point s to M . Her e u G B^(A) i s sai d t o b e stabl e L^-ma p i f i t satisfie s al l condition s i n the definitio n o f stable J-holomorphi c map s excep t th e P.D.E . condition , conditio n (ii) i n Se c 1 , which i s replaced b y (II) u i s locall y L pk hence a t leas t C 1 . We wil l conside r subspac e B^ C{A) o f B^(A) consistin g o f al l u wit h energ y E(u) < C. Becaus e o f th e uniforml y estimat e o n th e energ y o f an y elemen t i n M.g,m(A,J), w e ca n choos e a C onc e fo r all . W e wil l dro p th e upper-scrip t c together wit h (p , k) fro m ou r notatio n an d simpl y us e B(A) t o denot e th e subspac e in th e res t o f thi s paper . Each o f thes e modul i space s i s decompose d int o it s variou s strat a accordin g to th e topologica l type s o f th e domain s o f it s element s an d th e homolog y classe s represented b y th e correspondin g elements . W e wil l cal l thes e tw o piece s o f infor mation togethe r a n intersectio n pattern . W e us e X t o denot e th e collectio n o f al l intersection patterns . W e hav e obviou s decompositio n

H,m(A,J)=U^ | m (>l,J), lex

B{A)= ( J B\A),etc. lex

• Orbifol d Structur e o f B(A):

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Given (u) G M.9im(A, J ) , le t u G (u) wit h domai n (E,p) . Conside r th e minima l stabilization (E,p , q) of (E,_p) . For simplicity, assum e that (E,_p , g) lies on the lowest strata i n th e correspondin g modul i space . The n fo r eac h marke d poin t p G p o r q G q, w e associat e a loca l hypersurfac e o f codimensio n two . W e ma y assum e tha t the marke d poin t i s generic an d henc e th e loca l hypersurfac e ca n b e chosen s o tha t it i s transversa l t o u a t th e marke d poin t (se e [LiuTl ] fo r detail s an d references) . Let S = U pep{Sp} U q(Eq{Sq} b e th e collectio n o f thos e loca l hypersurfaces . No w consider th e deformatio n f a o f / , wher e f a : E a— * M give n b y a n obviou s gluin g process b y usin g cut-of f function s (se e [LiuTl ] an d [FO ] fo r details) . Give n e > 0 , let Ue(u,S) = {va\va : Ea - > M , \\v a - u a\\ < e,v a(p) G SR,va(q) G

S J,

where th e (/c,p)-nor m i s measure d wit h respec t t o th e metri c o n E a induce d fro m the metri c o n E throug h th e gluin g an d th e fixed metri c gj = a;(J— , —). Not e tha t two v a an d v af hav e th e sam e intersectio n patter n i f (i ) eac h o f thei r component s represents th e sam e homolog y clas s an d (ii ) an y componen t o f a i s zero i f and onl y if th e correspondin g componen t i n a' i s zero. Let T u — {(f)\(j) : E — » E,-u o Wi with covering group IV Moreover, the Bj-operator lifts to a collections ofTi-equivariant sections of Ci—- > Wi.

We remark tha t al l th e abov e construction s ca n b e carrie d ou t whe n restricte d to a particula r intersectio n pattern . • Loca l Transversalit y As w e mentione d before , Bj : Wi— > Ci i s no t transversa l i n genera l eve n fo r a generic choic e o f J . Le t I b e th e intersectio n patter n o f m. Then , becaus e o f th e Fredholm propert y o f elliptic operators ,

(DBj)Ui : Tu%Wl - ( A ' k has a finite dimensiona l cokernel . Le t R(ui) b e th e cokernel . W e defin e Li = {Ddj) u% 0 Id : TuWl 0

Rim) - C TUi.

By th e definition , Li i s surjective . Le t (/i,-- - , / r ) b e a basi s o f R(ui) an d /3€ some cut-of f functio n supporte d outsid e o f th e 6-neighborhoo d o f doubl e point s o f E = £„. . Le t / i>e = p e • fi an d ii €(wi) = (/i, e , • • • , /r,e). I t wa s prove d i n [LiuTl ] that fo r e small enough , i f we replace R(ui) i n th e domai n o f Li b y R e(ui), the n 1 ^ is stil l surjective . No w fo r eac h elemen t v G Re(ui), sinc e i / = 0 in a neighborhoo d of it s doubl e points , i t ca n b e though t a s a sectio n o f -Z>^_ 1 (A0'1 {ui)1 ^TM)). B y using a loca l trivializatio n fo r th e loca l bundl e Ci —* Wi (se e [LT ] fo r details) , we ca n regar d suc h a sectio n v a s a sectio n o f th e bundl e Ci — » Wi. Wit h thi s interpretation i n mind , w e have a loca l sectio n

dj®Id:WixRe(ui)-+Ci given b y (Bj 0 Id)(v,v) — Bjv + v. It s derivativ e a t (ix^O ) i s jus t Li whic h i s surjective, whe n restricte d t o T UiW* 0 R[{ui)^ wher e R{{ui) i s th e restrictio n o f Re{ui) t o Wl. No w i t follow s fro m th e Implici t Functio n Theore m tha t Bj 0 Id i s 1 still surjectiv e i n a smal l neighborhoo d o f Ui in W* x R e(ui) . Given a gluing parameter a , le t Bj, a b e the restriction of Bj t o W-* = U?(ui, 5 ) , where U^{ui,S) i s the collectio n o f al l elements i n VF ^ with a fixed domai n E a . B y refining th e gluin g techniqu e develope d i n [L3] , we proved tha t PROPOSITION

4. 3 ([LiuTl]) . When e is small enough, 9j, a ®Id:W?x Reiui)

10

- > C?

a

is surjective, where W£*, R e(ui) and Cf are the subspace of the corresponding objects, the domain of whose elements is E a . It follow s tha t fo r a generi c choic e Vi G R€(ui), th e sectio n is transversal t o zer o section . Therefore , th e z/^-perturbe d modul i spac e

M^=M^{A,J) =

{8%)-\0)

72

GANG LI U

is a smooth manifol d o f finite dimension, whos e dimension i s the same as the virtua l dimension expecte d b y th e Inde x Theorem . Le t A ; = {a} b e th e collectio n o f al l these gluin g parameters . Le t M Ui = LLe A -M% > It wa s prove d i n [LiuTl] , 4.4 . M Vi is homomorphic to M 1 ^ x A$. Here we have used M 1 ^ to denote all elements in M Vi whose domain is E . PROPOSITION

It follow s fro m thi s tha t jV[ Ui has th e dimensio n give n b y th e Inde x Theore m on al l o f it s strat a an d i s a stratifie d smoot h manifold . It s stratu m structur e i s induced fro m th e correspondin g structure s o f A^. This establishe s th e loca l transversality . • Virtua l Modul i Cycle s We no w globaliz e th e locall y define d modul i spac e above . Fo r thi s purpose , le t M = {J\J = (j u • • • ,jfc ), W h n---W jk^ 0 } be th e nerv e o f the coverin g {Wj}™^. We wil l us e Wj t o denot e Wj 1 f l • • • Wj k. Le t / Kjl : Wj t— > Wjt b e th e natura l projection. Le t Wf t = TTJ^WJ) an d TT J = Y[ jt n jr The n TT J : H^ W? - > Wj wit h the coverin g grou p Tj = f ^ F jl. We no w defin e th e fiber produc t WjJ =

{{UJ^--- lUj^Ujt

G

W^TTJ^UJ,) =

7r

jl/{ujl/)}.

c

J

We hav e TTJ : Wj — > Wj wit h coverin g grou p Tj. I f J —> J, ther e i s a par tially define d projectio n TTJ from th e obviou s subse t o f W^ 1 t o Wj J, give n b y 7Tj(^15 • • • ,Uim) = (ui i: • • • ,Uik). Her e w e hav e assume d tha t J = (ii , • • • ,ik) ^ I = (ii , • • • , z m ). W e hav e th e obviou s relatio n 7T j O 7T J = 71 7 . We ca n repea t abov e construction fo r bundle s Ci t o for m C/ . I n suc h a way, we get a syste m o f bundle s

(Cr,Wr) = {(C

r

/,W^);JeM}

J

together wit h a syste m o f coverin g map s 7r , coverin g group s T J an d bundl e map s At this point w e should mentio n some pathological featur e o f our bundle syste m ( £ r , W r). Locall y ( £ r , W T) i s not a stratifie d Banac h manifol d bu t onl y a Banac h variety i n th e sens e tha t i t i s a finite unio n o f (stratified ) Banac h manifolds . I n fact, give n u = (ui, • • • ,Uk) G Wj, sinc e {Wj}JL 1 ar e uniformizer s o f th e orbifol d W — u y ^ W j, ther e exist s a neighborhoo d Uj t containin g Uj l suc h tha t ther e ar e homeomorphisms \j ujv ' Uj l — • Ujlf an d isomorphism s $j ujlf ' F Uj— > T u. / suc h that fo r an y G TUji, Xjldl, o 0 = $ j i j V (0 ) o \jldi,. Her e r Uj.f = { 0 | 0 G Tjz , 0 • ^jz = u ji}1 S ^ n e isotrop y grou p o f Uj r Fi x a n / , say , I = 1 , an d denot e Aij , b y A/, / = 2 , • • • , fc for short . Ther e ar e | r n . | homomorphism s fro m £7 ^ t o VK j given b y u - > (w , 02A2(w), • • • , (f) kXk(u)), where 0 ; G r U i .. W e denot e E/ ^ b y U, (1 , 0i, • • • , 0*) b y 0 an d (1 , Ai, • • • , X k) b y A 7-

for short . The n th e abov e ma p ca n b e denote d b y 0 o A, 0 G T ^ = JJ, -=i T Uj wit h 01 = Id . I t i s easy t o see that 0 o \{u) i s indeed containe d i n Wj an d U0 G r u 0° A ^ ) covers a neighborhoo d o f u i n VFj . We wil l cal l eac h 0 o A a loca l coordinat e char t of W/ j nea r i z an d 0 o A(C/) a loca l componen t nea r t/, . I t i s eas y t o se e tha t th e notion o f loca l componen t i s intrinsic . I n particular , i f i n abov e proces s w e star t

M O D U L I SPAC E O F STABL E M A P S

73

with differen t I an d ru n throug h everythin g above , w e wil l en d u p wit h th e sam e components a s above. Simila r thin g happen s fo r th e loca l bundl e Cj J —> Wj J . W e have th e followin g loca l pictur e nea r u G Wj fo r th e bundl e C r/ - > W T3J : Ther e are \^UJ | ^ ^ (stratified) component s (ft o \(U), (ft G Tu passin g through u. Ove r eac h of thos e components , ther e i s a vecto r bundle , denote d b y (ft o X(C U). Therefore , near u, {£>/) —> • W j J i s a unio n o f |T U. \ k~x hones t vecto r bundles . I n particular , if T Uj ^ {Id}, th e fiber (£>j J)u ove r u i s a finite unio n o f vecto r spaces . Thi s pathological phenomeno n bring s i n som e difficult y t o ou r constructio n o f virtua l moduli cycles . Fo r instance , i f w e onl y allo w single-value d section s o f th e bundl e Cj—> • Wj, the n a continuou s sectio n nea r u ma y hav e t o tak e zer o a s it s value . O n the othe r hand , i n orde r t o achiev e eve n loca l transversality , i t i s necessar y t o us e perturbation whic h i s a non-zer o sectio n ove r thos e orbifol d point s correspondin g to bubbles . I n particular , a t thes e bubbl e points , |r u ^ | ^ 1 in general . To overcom e thi s difficulty , w e follo w th e metho d i n [LiuT2 ] t o construc t a desingularization o f th e bundl e syste m ( £ r , W r). Th e ke y propert y abou t loca l components ar e summarize d i n nex t lemma . LEMMA 4.3 . The notion of local components are functorial with respect to restriction and projection. More precisely, (i) Given u — {u\, • • • ,Uk) G Wj J , let (ft o X(U), (ft G T^, be the local component near u. Let y_ = (v\, • • • , Vk) G Wj J be another point contained in \J^Y u(ft ° X{U). Then there exists a neighborhood of V in Wj J such that all local components near v are just V P i (ft o X(U) with (ft G IV (ii) Assume that I — (n, • • • , im) ^- ^ J = (ji , • • • ,jk), then irj maps each local component (ft o X(U) near u to some local component in W^ 1 near 7TJ(U).

Similar things happen to the bundle C Tj— > W

Tj

.

P R O O F . W e ca n us e th e loca l coordinat e (ft o A. The n th e proo f i s a direc t verification an d ca n b e lef t t o ou r readers . •

We no w defin e th e desingularizatio n

(tT,WT) =

{(t Tj,WTj), J

eM}.

We defin e Wj J t o b e th e disjoin t unio n UU,^OA([/(M))

, modulo certai n equivalenc e relations , wher e u G W3 J an d U(u) i s a neighborhoo d of u\ i n Wj 1 . Her e B with fiber M and structural group G is c-split. In particular, H£(M) = H*(M) ® H*(BG).

F. L A L O N D E A N D D . M C D U F F

82

PROOF.

I t i s enough t o prov e th e secon d statemen t sinc e MG = EGx GM—> BG

is th e universa l bundle . Ever y compac t Li e grou p G i s th e imag e o f a homomor phism T x H — • G, wher e th e toru s T map s ont o th e identit y componen t o f th e center o f G an d H i s th e semi-simpl e Li e grou p correspondin g t o th e commuta tor subalgebr a [Lie(G),Lie(G) j i n th e Li e algebr a Lie(G) . I t i s eas y t o se e tha t this homomorphis m induce s a surjectio n o n rationa l homolog y BT x BH — > BG. Therefore, w e ma y suppos e tha t G — T x H. Le t T m a x = {S x)k b e th e max imal toru s o f th e semi-simpl e grou p H. The n th e induce d ma p o n cohomolog y H*(BH) - > # * ( £ T m a x ) = Q[ai,...,a/c ] take s H*(BH) bijectivel y ont o th e se t of polynomial s i n H* (BTma^) tha t ar e invarian t unde r th e actio n o f th e Wey l group, b y th e Borel-Hirzebruc h theorem . Henc e th e map s BT max — > BH an d BT x i?T m a x—> • BG induc e a surjectio n o n homology . Therefor e th e desire d resul t follows fro m par t (ii ) o f th e followin g lemma : LEMMA 3.2 . Consider a commutative diagram P' -

•P

ii B1 ->

B

where P' is the induced bundle. Then: (i) IfP-^Bis c-split so is P' -+ B'. (ii) (Surjectio n Lemma ) If P' — • B' is c-split and H*(B f)— > H*(B) is surjective, then P — > B is c-split. P R O O F , (i) : Us e th e fac t that , b y th e Leray-Hirsc h theorem , P — > B i s c-spli t if an d onl y i f th e ma p H*(M) —> • H*(P) i s injective .

(ii): Th e induce d ma p o n th e l^-ter m o f the cohomolog y spectra l sequence s i s injective. Therefor e th e existenc e o f a nonzer o differentia l i n th e spectra l sequenc e P— > B implie s that th e correspondin g differentia l fo r th e pullbac k bundl e P' — * B' does no t vanis h either . • Smooth projectiv e bundle s constitut e a n interestin g specia l cas e o f th e abov e proposition. Th e resul t i n thi s cas e ca n b e derive d fro m th e Delign e spectra l se quence, o r mor e generall y b y th e followin g argumen t du e t o Blanchar d [4] . Let' s call a smoot h fibe r bundl e M c -^ P — * B c-Hamiltonia n i f ther e i s a clas s a G H2(P) whos e restrictio n CLM t o th e fibe r M i s c-symplectic , i.e. , (CLM) 71 ^ 0 wher e 2n = dim(M) . Recal l that a closed manifold M i s said t o satisfy th e har d Lefschet z condition wit h respec t t o th e clas s a ^ G H2(M, R ) i f th e map s U{aM)k •

Hn~k{M,R) -

> # n + / c ( M , M )1,

< k < n,

n k

are isomorphisms . I n thi s case , element s i n H ~ (M) tha t vanis h whe n cuppe d with (aM) fc+1 ar e calle d primitive , an d th e cohomolog y o f M ha s a n additiv e basi s consisting o f element s o f th e for m b U (aj^Y wher e b is primitive an d I > 0. PROPOSITION 3. 3 (Blanchar d [4]) . Let M — • P — • B be a c-Hamiltonian bundle such that 7Ti(B) acts trivially on if*(M , R). / / in addition M satisfies the hard Lefschetz condition with respect to the c-symplectic class aM, then the bundle c-splits.

COHOMOLOGICAL PROPERTIE S .. .

83

P R O O F . T h e proof i s by contradiction. Conside r t h e Leray spectra l sequenc e in cohomolog y an d suppose t h a t d p i s t he first no n zero differential . Then , p > 2 and t h e Ep ter m i n t he spectral sequenc e i s isomorphic t o t he E2 t e rm a n d so c a n be identifie d wit h t h e tensor produc t H*(B) (g ) H*(M). Becaus e o f t he produc t structure o n t he spectral sequence , on e of the differential s d^ 1 mus t b e nonzero. S o there i s b G E^% = H l{M) suc h t h a t d^ l{b) ^ 0 . W e may assume t h a t b is primitive (since thes e element s togethe r wit h CLM generate H*(M)). The n 6 U a ^p ^ 0 b u t 6Ua^+1=0. We ca n write d p(b) = J2j e j ® fj wher e ej G H*(B) an d fj G Hl(M) wher e £ < i. Henc e fj U a ^ 1 1 1 ^ 0 for all j b y t he Lefschet z property . Moreover , becaus e the E p t e r m i s a tenso r produc t (dp(6)) U a^i+l =

£ e, - ® (/,- U a j p+ 1) ? 0 .

But thi s i s impossible sinc e thi s elemen t i s t he image vi a dp o f t he trivia l elemen t bl)anM-i+1. Here i s another , perhap s easier , argument . Suppos e d = d p i s t he first nonvanishing differential . I t vanishe s o n H l(M) fo r i < p fo r reason s o f dimension . Therefore, b y t he Lefschetz propert y i t mus t vanis h o n H2n~l(M) fo r these i. B ut then i t ha s to vanish o n H%(M) fo r p < i < 2p. Fo r if not, tak e b in suc h H l(M) such t h a t d(b) ^ 0 . B y Poincare dualit y ther e i s c G H2n~i(M) fo r 0 < j < p suc h t h a t d(b) U c ^ O. B u t b U c = 0 for reasons o f dimension, an d so b U d(c) ^ 0 , a contradiction. I t follow s t h a t d vanishe s o n H 2n~i fo r p < j < 2p. No w conside r the nex t bloc k o f i: 2p < i < 3p a nd so on. D Another fundamenta l questio n abou t Hamiltonia n bundle s i s t h at o f their sta bility unde r smal l perturbation s o f t he symplecti c for m o n t he fiber. I f t he bundl e M— > P— > B ha s structure grou p H a m ( M , u) a n d uo' is som e nearb y form , a n elementary argumen t (give n i n Lemma 4. 6 below) show s t h a t i t ca n be given t he structure grou p S y m p 0 ( M , u'). However , i t is not at all obvious whethe r t h e latter group ca n be reduced t o H a m ( M, UJ'). I f this reductio n i s possible fo r all u/ close to CLJ, t he original Hamiltonia n bundl e i s said t o be stable. W h e n t h e structural grou p is a compac t Li e group, thi s clearl y boil s dow n t o t he following statement . T H E O R E M 3. 4 (Hamiltonian stability) . Let (M, cu) be a closed symplectic manifold, and let i : G — > H a m ( M, UJ) be a continuous homomorphism defined on a compact Lie group. Then, for each perturbation UJ' in some sufficiently small neighbourhood U of u in the space of all symplectic forms on M, there is a continuous homomorphism L' : G - > H a m ( M , c j / ) that varies continuously as

the form UJ' varies in hi.

P R O O F . W e begi n wit h a well-know n averagin g argument . Defin e r' t o be t h e average o f t he form s i{g)*(uj f), i.e. , se t T\V,W)= [

L(

gy{uj')(v,w)diiG,

v,weT*(M),

JG

where d\iQ is Haar measure . Sinc e G i s compac t a n d t(g)*(cj) = UJ for all g G G , T' i s a symplecti c for m wheneve r UJ' is sufficiently clos e t o UJ. Moreover , i t i s eas y to se e t h at i{gY{r') = r' fo r all g G G. Thu s i map s G int o S y m p ( M , r / ) . B u t ,

84

F . L A L O N D E AN D D . M C D U F F

since L(G) is als o containe d i n th e connecte d grou p Ham(M , CJ), th e element s o f G must ac t triviall y o n H 2(M). Therefor e r' i s cohomologous t o u / an d henc e equal s /*(o/) wher e / i s th e tim e 1 n.ap o f som e isotop y f t (agai n assumin g tha t a / i s sufficiently clos e t o a;. ) Thus , th e homomorphis m

takes G t o Symp 0 (M,a/). It remain s t o sho w tha t thi s homomorphis m i' ha s imag e i n Ham(M,a/) , i n other word s tha t th e composit e homomorphis m G - ^ Symp 0 (M,a/) ^ ^ ( M . R ) / ^

,

is zero. Sinc e the target i s an abelian grou p an d G is compact, i t suffice s t o conside r the cas e whe n G i s th e circle . Observ e als o tha t i' = i T> an d t = L U are th e sam e when considered a s maps int o Diff(M). Therefor e th e same vector field X generate s both S x-actions an d w e just nee d t o sho w tha t th e close d 1 -for m a r> which i s r'dual t o X i s actually exact . Bu t eac h o f its Morse-Bot t singula r set s hav e the sam e topology an d inde x a s doe s th e correspondin g singula r se t fo r th e c 2 is t he close d dis c o f radius 1 of t he plane.) I n wha t follow s w e always assume t h a t t h e base S 2 i s oriented, a n d with orientatio n induce d fro m D^. Not e t h a t this correspondenc e ca n b e reversed : give n a symplecti c bundl e ove r t h e oriente d 2-sphere togethe r wit h a n identificatio n o f on e fiber wit h M , on e can reconstruc t the homotop y clas s o f > 0 ) an d exten d th e fiberwise symplecti c structure . I t i s alway s possibl e t o choose Q s o tha t i t i s a produc t wit h respec t t o th e give n produc t structur e nea r the fibers M 0 a t 0 G D% and M ^ a t 0 G D^. Besides th e couplin g clas s i^ , th e tota l spac e P ^ carrie s anothe r canonica l second cohomolog y clas s ^=

Cl

(TP;ert)eff2(P^K)

that i s define d t o b e th e first Cher n clas s o f th e vertica l tangen t bundle . Both classe s u^,c^ behav e wel l unde r composition s o f loops . Mor e precisely , consider tw o element s 0 , 0 G 7Ti(Ham(M, u)) an d thei r composit e 0 * 0 . Thi s ca n be represente d eithe r b y th e produc t ip t ° 4>t o r b y th e concatenatio n o f loops . I t is no t har d t o chec k tha t th e bundl e P^* can b e realise d a s th e fiber su m P^^P^ obtained a s follows. Le t M^ 500 denot e th e fiber a t 0 G D^ i n P$ an d M^ o th e fiber at 0 G D2 i n P^. Cu t ou t ope n produc t neighborhood s o f eac h o f thes e fibers an d then glu e th e complement s b y a n orientatio n reversin g symplectomorphis m o f th e boundary. Th e resultin g spac e ma y b e realise d a s D+ x M U ^ ^ S

1

x [-1 ,1 ] u

aiM

D~ x M ,

where a0 5 _i(27rt,x) = (27r£ , - l , 0t ( x ) ) , a^

5 i(27rt,

1 , 0t (x)) = (2nt,x),

and thi s ma y clearl y b e identifie d wit h P^*^ . Se t V^ = D+ x M U S 1 x [-1 ,1 /2) , V

0= S

1

x (-1 /2,1 ] U D^ x M.

The nex t lemm a follow s imediatel y fro m th e constructio n o f the couplin g for m via symplecti c connections . LEMMA 4.2 . The classes u^^^ and c^* ^ are compatible with the decomposition P^f = V^UVcf) in the sense that their restrictions to V^PiV^ = ( — 1/2,1/2) xS 1 xM equal the pullbacks of [co] and c\ (TM).

We no w explai n th e proo f o f Theore m 4. 1 (se e [9 , 1 1 ] fo r mor e details) .

87

COHOMOLOGICAL PROPERTIE S .. .

SeidePs m a p s ^ , 0 We star t wit h th e definitio n o f th e quantu m homolog y rin g o f M. T o simplif y our formula s w e wil l denot e th e firs t Cher n clas s c\ (TM) o f M b y c . Let A be the usua l (rational ) Noviko v ring of the grou p H = H^M, Z ) / ~ wit h valuation v(.) wher e B ~ B' if ou(B - B') = c(B - B') = 0 , an d le t A R b e th e analogous (real ) Noviko v rin g base d o n th e grou p HR = # f (M , M)/ ~ . Thu s th e elements o f A have th e for m Ben where fo r eac h K there ar e onl y finitel y man y nonzer o A # G Q wit h UJ(B) < ft, an d the element s o f AR ar e BenR where Aj g G R an d ther e i s a similar finitenes s condition. 1 Se t QH*(M) = H*(M) A and QH*(M, A R) = H*(M)®A R. The n QH*(M) i s Z-graded wit h deg(a®e B) = deg(a) - 2c(B). I t i s best t o thin k o f QH*(M, A R) a s Z/2Z-grade d wit h QHa = H^M) ®

Afl , QH

odd

- H

odd(M) A R.

With respec t t o the quantum intersectio n produc t bot h version s of quantum homol ogy are graded-commutative ring s with unit [M]. Further , th e units in QH eu(M, A R) form a group QH eu(M1 A R)X tha t act s on QH*(M, A R) b y quantum multiplication . Now we describe ho w Seide l arrive s a t a n actio n o f the loo p o n th e quantu m homology o f M . Denot e b y C th e spac e o f contractibl e loop s i n th e manifol d M . Fix a constan t loo p x * G £, an d defin e a covering C of C with th e bas e poin t x* a s follows. Not e firs t tha t a pat h betwee n x * an d a give n loo p x ca n b e considere d a s a 2-dis c umM bounde d b y x. The n th e coverin g £ i s define d b y sayin g tha t tw o paths ar e equivalen t i f th e 2-spher e S obtaine d b y gluin g th e correspondin g disc s has CJ(S) = c(S) = 0 . Thu s th e coverin g grou p o f C coincide s wit h th e abelia n group H. Let n — 1 . I n th e genera l case , on e use s a version o f th e virtua l modul i cycl e fo r Gromov-Witte n invariant s tha t i s adapte d to th e fibered structure : se e [1 1 ] . Note finally that , b y Gromo v compactness , ther e ar e for each given energy leve l K only finitely man y homolog y classe s D wit h LU(D — a) < K that ar e represente d by J-holomorphi c curve s i n P^. Thu s ^ ^ ( a ) satisfie s th e finiteness conditio n fo r elements o f QH*(M, A) . 2

T h e minima l spherica l Cher n numbe r N i s th e smalles t nonnegativ e intege r suc h tha t th e image o f c = c±(TM) o n H^(M) i s containe d i n 7VZ . Th e weakl y exac t cas e N = 0 i s als o tractable b y thes e standar d methods .

COHOMOLOGICAL PROPERTIE S .. .

89

Since n(i(a) 1 i(b); D) = 0 unless 2c (j){D) + di m (a) + di m (6) = 2n , w e hav e

where dim(a 5 ) = dim(a ) + 2c (j)(D) = dim(a ) + 2c^(cr ) + 2c(J5) . Observe als o tha t When M i s sphericall y monoton e o r ha s minima l spherica l Cher n numbe r a t least n — 1 the followin g tw o result s ar e prove d b y Seide l [1 7] . Th e genera l cas e i s established i n [1 1 ] . 4.3 . If is the constant loop * and P* = M x S then ^ r*,Cr0 ^ s the identity map. LEMMA

CTQ

is the flat section pt x S 2 of

2

PROPOSITION 4.4 . Given sections a of P^ and a' of P^ let o'^a be of these sections in the fiber sum P^^P^ = P^^. Then

the union

The mai n ste p i n th e proo f o f thes e statement s i s t o sho w tha t whe n calculat ing th e Gromov-Witte n invarian t n(i(a),i(b); D) vi a th e intersectio n betwee n th e virtual modul i cycl e an d th e clas s i(a) 0 i(b) w e ca n assum e th e following : — th e representativ e o f i(a) C* ) i(b) ha s th e for m a x (5 where a,[3 ar e cycle s lying i n th e fibers o f P^ ; — th e intersectio n occur s wit h element s i n th e to p stratu m o f ^ 0 , 2 ^ ( ^5 J-> D) consisting o f section s o f P^. In th e semi-monoton e case , Lemma 4. 3 is then almos t immediate 3 , an d Propo sition 4. 4 ca n b e prove d b y th e well-know n gluin g technique s o f [1 6 ] o r [1 2] . COROLLARY 4.5 . ^ ^ is

an isomorphism for all loops (f) and sections a.

With thi s i n hand , w e ca n establis h th e proo f o f Theore m 4. 1 i n th e followin g way. Th e Gromov-Witte n invariant s ar e linea r i n eac h variable . Thu s i f i(a) — 0 for som e a ^ 0 , the n ^^^(a) = 0 , a contradictio n wit h th e fac t tha t ^ > a i s a n isomorphism. • 4.2. Genera l stability . I t turn s ou t tha t th e fac t tha t al l Hamiltonia n bun dles ove r S 2 ar e c-spli t i s enoug h t o establis h th e stabilit y o f genera l Hamiltonia n bundles ove r an y base . Thi s i s what w e now explain . Let 7 r : P— > B b e a symplecti c bundl e wit h close d fiber (M , a;) an d compac t base B. Moser' s homotop y argumen t implie s tha t thi s bundl e ha s th e followin g stability property . LEMMA 4.6 . There is an open neighborhood U of u: in the space S(M) of all symplectic forms such that TT : P— » B may be naturally considered as a symplectic bundle with fiber (M,UJ') for all uo' G U. 3

The proo f o f th e firs t lemm a i s surprisingl y har d i n th e genera l case . Th e difficult y lie s i n showing tha t invariant s i n classe s A + B wit h J B / O G H2(M) d o no t contribute . Th e reaso n i s that suc h curve s ca n neve r b e isolated : the y ar e graphs , an d reparametrization s o f th e ma p t o M giv e ris e t o familie s o f graphs . However , t o se e thi s i n th e genera l cas e involve s constructin g a virtual modul i cycl e tha t i s invarian t unde r a n S 1 -action . Se e [1 1 ] .

F. L A L O N D E A N D D . M C D U F F

90

PROOF. Firs t recal l tha t fo r ever y symplecti c structur e u / o n M ther e i s a Serre fibration Symp(M,o/) — > Diff(M) — > T? , whatever th e symplectic structur e UJ' may be. Therefore, w e are given a sectio n a o f W(UJ) an d our task i s to sho w ho w t o construct fro m i t a smoot h famil y o f sections a^' o f the bundle s W{UJ') fo r all UJ' near UJ. Le t Oi be the restriction o f a ove r Vi. The n (T^)* ^ i s a smoot h map V^—> 5a ; (constan t an d equal t o UJ i f the faj ar e chosen i n Symp(M, UJ)). Fo r each a/ nea r UJ an d b e Vi consider th e symplectic for m (X x {1 } ) x (M,u) > (x,l,t, x(y))

F. L A L O N D E A N D D . M C D U F F

92

Since i i s Hamiltonian , T^ an d i n [9 ] t h a t th e ran k o f T^ i s finite an d locally constant . Thu s becaus e th e m a p 7Ti(X t)— » T^ i s zer o fo r UJ' — UJ an d sinc e it depend s continuousl y o n UJ' ', we conclud e t h a t i t mus t b e zer o fo r al l UJ' and t , i.e . the induce d Hamiltonia n structur e o n Pt(u/ ) i s trivia l fo r eac h t an d i n particula r for t = 0 , 1 . Thi s mean s t h a t P ( u / ) i s defined a s th e quotien t o f (X x [0,1 ] ) x (M , UJ') by a n automorphis m o f th e trivia l Hamiltonia n bundl e fa :

(X x {0} ) x ( M , o / ) - + ( X x {1 } ) x ( M , u / ) ,

which i s homotopi c t o a m a p L' \ X -* H a m ( M , UJ'). Here i s a secon d mor e direc t proof . Firs t observ e t h a t i f K i s an y compac t se t of S y m p ( M , CJ) , the n fo r al l UJ' G Wjtl/ o *, homotop s int o H a m ( M , UJ') i f an d onl y i f th e element s i n ( ^ ^ o L)*(TTI(B)) li e i n th e kerne l o f th e homomorphis m F l u x ^ : 7ri(Symp 0 (M,(j / )) - » P ^ O ^ ) In fac t th e flux homomorphis m Flux^ / i s define d o n 7Ti(Diff(M)) , an d so , sinc e ^UJ,UJ'°L i s homotopic t o i a s maps t o Dif f (M) , i t suffice s t o sho w t h a t Flux^ / vanishe s on th e element s oii*.(yt\{B)). Bu t Flux ^ vanishe s o n L*(TTI(B)) b y construction , an d so Flux^ / als o vanishe s o n thes e element s b y th e "stabilit y o f Hamiltonia n loops " in [9] . Thi s i s just anothe r wa y o f expressin g th e stabilit y o f Hamiltonia n structure s over S 2. T o se e this , le t (\> = £*(T ) b e th e imag e o f a loo p 7 i n P , an d conside r th e associated bundl e P^ — > S2 constructe d usin g (j) as clutchin g m a p . Then , fo r an y closed for m r o n M , symplecti c o r not , Flux r() i s nothin g othe r t h a n th e valu e o f the Wan g differentia l d^ o f thi s bundl e o n th e clas s [r] . T h e stabilit y o f P^ — » S 2 implies t h a t d $([&'})— 0, an d therefor e Flux w /(0) = ^ ( [ o ; , ] ) = O , as required . 4.3. F r o m S

2

t o mor e genera l bases , usin g analyti c arguments .

P R O P O S I T I O N 4.1 1 . Let (M,UJ) be a closed symplectic manifold, and M c -> P— > B a Hamiltonian bundle over a CW-complex B. Then the rational cohomology of P splits if the base has the homotopy type of a symplectic manifold W for which

C O H O M O L O G I C A L P R O P E R T I E S .. .

93

some spherical Gromov-Witten invariant nw(pt,pt, c\, ... , c^; A) does not vanish, where k > 0, A G H2(W;Z) and the c^s are any cycles in W. Note that space s satisfying th e above condition includ e all products of complex projective space s an d their blow-ups . A specia l cas e o f this propositio n wa s proved i n [1 0] , and the general cas e will appear i n [7]. Th e proof i s a generalization of the argument s i n [9, 11]. Th e idea is to sho w tha t modul i space s o f J-holomorphic curve s i n ruled symplecti c manifold s P behav e lik e fibered modul i spaces , which implie s that appropriat e GW-invariant s in P ar e equal t o the product o f a GW-invarian t o f the base wit h a GW-invarian t of th e fiber. Indeed , suppos e tha t M C = S 2 an d np c count s th e number o f J-holomorphi c curves i n clas s a passin g throug h L(Q) an d t(b).) Thi s implie s tha t t(a) canno t vanish, an d therefore Pc c-split s b y the Leray-Hirsch theorem . No w take a n almost comple x structur e J 7 o n P suc h tha t th e projection TT : P — •» B i s (J\ J)holomorphic an d consider th e invariant i n P nP(i'(a), */(6) , 7r- 1(ci),... ,

T T - 1 ^ ) ; LP

C,P((T))

where n is the projecton P — > B, i' denotes the inclusion of the fiber in P, and Lp c,p is the inclusion o f Pc i n P. I t i s not hard t o see, a t leas t whe n th e moduli space s are well-behaved , tha t thi s las t invarian t mus t b e equa l t o th e sum , taken ove r the rationa l curve s C appearin g i n riB(pt,pt, c i , . . . ,c^;A) , o f the correspondin g numbers np c(t(a), t(b); a), with sign s according to orientations i n the moduli space . But becaus e th e Hamiitonian bundle s Pc an d Pc ar e isomorphic whe n C and Cf are homologou s i n B, thi s su m is actually th e product nB{pt,pt,cu... ,c

fc ;

A) x n

Pc{t{a),i{b)\a)

which doe s no t vanish. Therefore i'{a) cannot vanis h eithe r an d the bundle P c-splits . 4.4. Iteratin g bundles : geometri c arguments . Le t M ° ^ P — > B b e a Hamiitonian bundl e ove r a simply connecte d bas e B an d assume tha t al l Hamiitonian bundle s ove r M a s well a s over B c-split . W e explain i n this sectio n tha t an y Hamiitonian bundl e ove r P mus t als o be c-split. Thi s provide s a powerful recursiv e argument tha t extend s c-splittin g result s t o much mor e genera l bases . We begin wit h som e trivial observation s an d then discus s composite s o f Hamiitonian bundles . Th e first lemm a i s tru e fo r an y clas s o f bundle s wit h specifie d structural group .

94 F

. L A L O N D E AN D D . M C D U F F

L E M M A 4.1 2 . Suppose that TT : P— > B is Hamiltonian and that g : B'— > B is a continuous map. Then the induced bundle TT' : g*(P) — • B' is Hamiltonian. Recall t h a t an y extensio n r o f th e form s o n th e fiber s i s calle d a connectio n form. L E M M A 4.1 3 . If P — + B is a smooth Hamiltonian fiber bundle over a symplectic base (B, a) and if P is compact then there is a connection form Q K on P that is symplectic. P R O O F . Th e bundl e P carrie s a close d connectio n for m r . Sinc e P i s compact , the for m Q K — r + ft7r*(cr) i s symplecti c fo r larg e K. Observe t h a t th e deformatio n typ e o f th e for m fl K i s uniqu e fo r sufficientl y large K, since give n an y tw o close d connectio n form s r , r' th e linea r isotop y tr +

( 1 - t)r +

KIT*(a), 0

< t < 1

,

consists o f symplecti c form s fo r sufficientl y larg e K. However , i t ca n happe n t h a t there i s a symplecti c connectio n for m r suc h t h a t r + /^7r*(a ) i s no t symplecti c for smal l K > 0 , eve n thoug h i t i s symplecti c fo r larg e K. (Fo r example , suppos e P = M x B an d t h a t r i s t h e su m u + TT*(U;B) wher e ujp + o~ i s no t symplectic. ) Let u s no w conside r th e behavio r o f Hamiltonia n bundle s unde r composition . If ( i W » ^ P ^ X , an

d (F,G)-^X^B

are Hamiltonia n fibe r bundles , the n th e restrictio n TTP : W

= 7Tp 1 (F) — > F

is a Hamiltonia n fibe r bundle . Sinc e F i s a manifold , w e ca n assum e withou t los s o f generality t h a t W — > F i s smooth . Moreover , th e manifol d W carrie s a symplecti c connection for m f2£^ , an d i t i s natura l t o as k whe n th e composit e m a p TT : P — > B with fibe r (W , f2^) i s itsel f Hamiltonian . L E M M A 4.1 4 . Suppose that B is a simply connected CW-complex and that P is compact. Then TT — TTX o TTP : P —* B is a Hamiltonian fiber bundle with fiber (W, Sly/)' W ^ere ^w ~ T w + KTT P(O~), Tw is any symplectic connection form on W, and K is sufficiently large. PROOF. W e ma y assum e t h a t th e bas e B a s wel l a s th e bundle s ar e smooth . Let rp (resp . TX) b e a close d connectio n for m wit h respec t t o t h e bundl e TTP , (resp . 7Tx), an d le t TW b e it s restrictio n t o W. The n f i ^ i s th e restrictio n t o W o f th e closed for m tip =

TP + KTT

P{TX)-

By increasin g K i f necessar y w e ca n ensur e t h a t ft p restrict s t o a symplecti c for m on ever y fibe r o f TT not jus t o n th e th e chose n fibe r W. Thi s show s firstl y t h a t TT : P— > B i s symplectic , becaus e ther e i s a wel l define d symplecti c for m o n eac h of it s fibers , an d secondl y t h a t i t i s Hamiltonia n wit h respec t t o thi s for m Qyy o n the fibe r W. Henc e Lemm a 4. 7 implie s t h a t H 2(P) surject s ont o H 2(W). Now suppos e t h a t TW i s any close d connectio n for m o n TTP : W— » F. Becaus e the restrictio n m a p H 2(P)— > H 2(W) i s surjective , th e cohomolog y clas s [TW] i s th e restriction o f a clas s o n P an d so , b y Thurston' s construction , th e for m TW ca n b e extended t o a close d connectio n for m rp fo r th e bundl e TTp. Therefor e th e previou s argument applie s i n thi s cas e too . •

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95

Now le t u s conside r th e genera l situation , whe n TTI(B) ^ 0 . Th e proo f o f th e lemma abov e applie s t o sho w tha t th e composit e bundl e ?r : P— > B i s symplecti c with respect t o suitable Cl^ an d tha t i t ha s a symplectic connectio n form . However , even thoug h TTX • X— > B i s symplectically trivia l ove r th e 1 -skeleto n B\ th e sam e may no t b e tru e o f th e composit e ma p TT : P — * B. Moreover , i n genera l i t i s not clea r whethe r trivialit y wit h respec t t o on e for m flfy- implie s tha t fo r another . Therefore, w e ma y conclud e th e following : PROPOSITION 4.1 5 . If (M, u) - • P ^ X , and (F,a)-+X*-$ B are Hamiltonian fiber bundles and P is compact, then the composite TT — nx ° ^p • P —> B is a symplectic fiber bundle with respect to any form Q, 1 ^ on its fiber W — 7r~ 1 (pt), provided that K is sufficiently large. Moreover if TT is symplectically trivial over the 1-skeleton of B with respect to Q,yy then TT is Hamiltonian. In practice , w e will appl y thes e result s i n case s wher e iri(B) — 0. W e wil l no t specify th e precis e for m o n W , assumin g tha t i t i s Q,^- for a suitabl e K. LEMMA 4.1 6 . If (M,u) A p - » B is a compact Hamiltonian bundle over a simply connected CW-complex B and if every Hamiltonian fiber bundle over M and B is c-split, then every Hamiltonian bundle over P is c-split. PROOF.

Le t TTE • E— • P b e a Hamiltonia n bundl e wit h fiber F an d le t F - > W -> M

be it s restrictio n ove r M. The n b y assumptio n th e latte r bundl e c-split s s o tha t H*(F) inject s int o H*{W). Lemm a 4.1 4 implies that th e composit e bundl e E — • B is Hamiltonia n wit h fiber W an d therefor e als o c-splits . Henc e H*(W) inject s int o H*(E). Thu s H*(F) inject s int o H*{E), a s required . • 4.5. Topologica l arguments . W e now put togethe r th e result s an d method s of th e las t subsection s abou t c-splitting . Fo r mor e detail s se e [1 0] . LEMMA 4.1 7 . J / E is a closed orientable surface then any Hamiltonian bundle over S 2 x . . . x S 2 x E is c-split.

Conside r an y degre e on e ma p / fro m E — > S' 2. Becaus e Ham(M , UJ) is connected, B Ham(M , UJ) i s simply connected , an d therefor e an y homotop y clas s of maps fro m E -- > B Ham(M, uo) factor s throug h / . Thu s an y Hamiltonia n bundl e over E i s the pullbac k b y / o f a Hamiltonian bundl e ove r S* 2. Becaus e suc h bundle s c-split ove r 5 2 , th e sam e i s true ove r E b y Lemm a 3.2(i) . The statemen t fo r S 2 x . . . x S 2 x E i s no w a direc t consequenc e o f iterativ e applications of Lemma 4.1 6 applied to the trivial bundles S 2x.. .xS 2xY, ^ S 2. • PROOF.

COROLLARY

4.1 8 . Any Hamiltonian bundle over S 2 x . . . x S 2 x S 1 is c-split.

Conside r th e maps S 1 — * T2— > 5 1 give n by inclusion on the first facto r and projectio n ont o the first factor . Thei r compositio n i s the identity. Exten d the m to map s PROOF.

S2x...xS2xS1^S2x...xS2xT2^S2x...xS2xSl by multiplyin g wit h th e identit y o n th e S' 2 factors . The n a Hamiltonia n bundl e P on S 2 x . . . x S 2 x S 1 pulls-bac k t o a c-spli t bundl e P' o n S 2 x . . . x S 2 x T 2 b y Lemma 4.1 7 . B y naturality , it s pull-bac k P" t o S 2 x . . . x S 2 x S 1 i s c-split . Bu t P" = P. •

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F. LALOND E AN D D . MCDUF F

PROPOSITION

4.1 9 . For each k>l, every

Hamiltonian bundle over Sk c-splits.

PROOF. B y Lemma 4.1 7 and Corollary 4.1 8 there is for each k a /c-dimensiona l closed manifol d X suc h tha t ever y Hamiltonia n bundl e ove r X c-splits . Give n any Hamiltonian bundl e P — » S k conside r it s pullback t o X b y a ma p / : X— » S k o f degree 1 . Sinc e the pullback c-splits , the original bundl e does too by Lemma 3.2(iz) .

•

By th e Wan g exac t sequence , thi s implie s tha t th e actio n o f th e homolog y groups o f Ham(M) o n H*(M) i s always trivial . Here ar e some othe r example s o f situations i n which Hamiltonia n bundle s ar e c-split. LEMMA

4.20 . Every Hamiltonian bundle over C P n i x . . . x CP nfc c-splits.

PROOF.

Thi s i s an obvious applicatio n o f Lemma 4.1 6 . •

LEMMA 4.21 . Every Hamiltonian bundle over a compact CW-complex of dimension < 3 c-splits. P R O O F . Thi s i s because on e can first assum e tha t B i s simply-connected an d then construc t a homolog y surjectio n B' — > B wher e B' i s a wedg e o f 2 an d 3spheres. • PROPOSITION 4.22 . Every Hamiltonian bundle over a product of spheres csplits, provided that there are no more than 3 copies of S 1 .

B y hypothesis B = H ieI S 2mi x H jeJ S 2n^1 x T fc, where m > 0 an d 0 < k < 3. Set B' = J | C P m * x Y[ CP nz x T | J | x T £, PROOF.

iEl jGJ

where £ = k i f k -f | J\ i s even an d = k + 1 otherwise. Sinc e CP n* x S 1 map s ont o g2m-\-i k y a m a p o f degre e 1 , ther e i s a homolog y surjectio n B' — > B tha t map s the facto r T £ t o T k. B y the surjection lemma , i t suffice s t o show tha t th e pullback bundle P' — • B' i s c-split . Consider th e flbratio n T\J\

X T ^ 5 ' - >

JJCP

mi

x Y[ C P n i .

Since | J\ + £ is even, w e can think o f this a s a Hamiltonia n bundle . Moreover , b y construction, th e restriction o f the bundle P' — > B 1 t o T' J I X T £ i s the pullback of a bundl e ove r T fc, sinc e th e map T' J I— > B i s nullhomotopic. (Not e tha t eac h S 1 factor i n T ' J ' goe s int o a differen t spher e i n B.) Becaus e k < 3 , the bundle ove r Tk c-splits . Henc e w e can appl y th e argumen t i n Lemm a 4.1 6 to conclud e tha t P' - > B' c-splits . D COROLLARY 4.23 . Let B be a simply connected Lie group, or more generally any H-space whose rational fundamental group has rank less than 4 and whose homotopy groups are finitely generated in each dimension. Then c-splitting holds for all Hamiltonian bundles over B. P R O O F . Le t B b e suc h a i7-space . B y the theory o f minima l model s (se e [3] for instance ) whic h applie s i n thi s cas e becaus e th e fundamenta l grou p o f B act s trivially o n all higher homotop y groups , the rational cohomolog y o f B i s generated

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97

as a Q-vector spac e by cup-products o f elements that pai r non-trivially wit h spheres, ie each a E H*(B, Q ) ca n b e written a s a cup produc t U ^ ' s wher e there i s for eac h i a spherica l clas s c ^ i n rationa l homolog y wit h ai(ai) ^ 0 . I f w e denot e b y th e same symbo l oti : S ni— > B a ma p tha t realise s a non-zer o multipl e o f th e clas s Qi, the n th e obviou s ma p V ^ : ViS ni— > B extend s t o a ma p 0 a define d o n th e product o f thes e sphere s tha t pull s bac k th e elemen t a t o a generato r o f th e to p rational cohomolog y group . I f ther e wer e a Hamiltonia n bundl e P ove r B tha t did no t c-split , ther e woul d b e a n elemen t o f lowes t degre e a G iJ*(£;Q ) wit h non-zero differentia l i n th e spectra l sequenc e o f P an d therefor e th e differentia l o f the correspondin g to p elemen t o f H*(HiS ni) i n th e spectra l sequenc e o f th e pul l back bundl e M^-i —> • • • —> M i = S 2 o f Kahle r manifold s wher e eac h ma p M^+ i— > Mi i s a bundl e wit h fiber S 2. The y sho w tha t an y coadjoin t orbi t X ca n b e blow n u p t o a manifol d that i s diffeomorphi c t o a Bot t towe r M& . Moreove r th e blowdow n ma p M^ — • X induces a surjectio n o n rationa l homology . Ever y Hamiltonia n bundl e ove r M & csplits b y repeate d application s o f Lemm a 4.1 6 . Henc e th e resul t follow s fro m th e surjection lemma . • 5. Application s t o rule d symplecti c manifold s THEOREM 5. 1 (Obstruction s t o th e existenc e o f rule d symplecti c structures) . Let M be a closed manifold and P a smooth fiber bundle with fiber M over a simply connected manifold B and assume that B is either a compact CW-complex of dimension less than 4 or is a product of complex projective spaces, of spheres and of coadjoint orbits of arbitrary dimensions. Denote by i the inclusion of the fiber in P. Then the non-vanishing of the kernel of

U : #*(M) - • H*{P) is an obstruction to the existence of a ruled symplectic structure on P. By the Leray-Hirsc h theorem , th e vanishin g o f the kerne l i n the theore m abov e amounts t o th e cohomologica l splittin g H*(P) = H*(B) (g ) H*(M). Thu s thi s las t result ma y b e state d a s follows : unde r th e give n condition s o n B, a rule d structur e exists o n P onl y i f P split s cohomologically . Thi s impose s stron g topologica l con straints o n th e constructio n o f rule d symplecti c manifold s b y twiste d product s o f two give n ones . Theorem 5. 1 is an immediate corollary o f our results about c-splittin g an d of the characterization o f Hamiltonian fiber bundle s ove r simpl y connecte d base s i n term s of the existenc e o f a close d extensio n t o th e tota l spac e o f th e symplecti c form s o n the fibers. Thi s characterizatio n als o implie s th e followin g versio n o f Hamiltonia n stability. 5

We ar e gratefu l t o Jarosla w Kedr a wh o pointe d ou t a varian t o f thi s argumen t t o us .

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F. LALOND E AND D . MCDUF F

THEOREM 5. 2 (Stabilit y o f rule d symplecti c structures) . Let M ^ P -^ > B be a smooth compact fiber bundle over a simply connected manifold B. Suppose that P admits a ruled symplectic structure ft, that restricts to UJ on the M-fiber. Then the ruled symplectic structure on P persists under small deformations of UJ, i.e. there is a neighborhood U of UJ in the space of all symplectic forms on M such that each UJ' G IA extends to a ruled symplectic structure Q! on P, which varies continuously as UJ' varies in U. Observe t h a t th e abov e theore m remain s tru e fo r arbitrar y base s B provide d t h a t P — > B i s symplecticall y trivia l ove r th e 1 -skeleto n o f B. 6. C o n c l u d i n g r e m a r k s It i s stil l unclea r whethe r ever y Hamiltonia n fiber bundl e ove r an y compac t CW-complex c-splits . On e o f th e simples t unknow n case s i s a Hamiltonia n bundl e (M, Q) — > P— > B wit h bas e th e 4-toru s an d wit h fiber a symplecti c 4-manifol d t h a t doe s no t satisf y specifi c propertie s lik e th e har d Lefschet z property . T h e problem her e i s t h a t , i f on e trie s t o appl y Lemm a 4.1 6 t o ( M , UJ) —> P — > B wit h B = T 4 give n itsel f a s a bundl e T 2— * T 4 -^ T 2 , the n nothin g guarantee s t h a t th e composite fibration W — > P *•¥ T 2 i s trivia l ove r th e 1 -skeleto n o f th e base . I n fact, th e structura l grou p o f th e composit e fibration ma y wel l b e a disconnecte d subgroup o f th e symplectomorphis m grou p o f th e fiber W = (n o p)~ 1 (pt). Note , however, t h a t becaus e al l Hamiltonia n fibrations ove r T 3 c-split , w e d o kno w t h a t the element s o f thi s subgrou p ac t triviall y o n th e cohomolog y o f W. Thi s raise s the interestin g questio n o f whethe r on e ca n exten d ou r result s o n c-splittin g fo r Hamiltonian bundle s t o certai n disconnecte d extension s o f th e Hamiltonia n group . We hav e n o technique s a t presen t t o dea l wit h thi s question , sinc e bundle s ove r T 2 need no t admi t an y J-holomorphi c sections .

References M. Abre u an d D . McDuff , Topology of symplectomorphism groups

of rational ruled surfaces,

SG/9910057, t o appea r i n Journa l o f th e Amer . Math . Soc . (2000) . M.F. Atiya h an d R . Bott , The moment map and equivariant cohomology, Topology , 2 3 (1984), 1 -28 . M. Aubry , Homotopy theory and models, base d o n lecture s b y H.J . Baues , S . Halperin , an d J.-M. Lemaire , DM V Seminar , 24 , Birkhauser , 1 995 . A. Blanchard , Sur les varietes analytiques complexes, Ann . Sci . Ec. Norm. Sup. , 73(3 ) (1 956) . M. Grossber g an d Y . Karshon , Bott towers, complete integrability and the extended character of representations, Duk e Math . Journ. , 7 6 (1 994) , 23-58 . F. Kirwan , Cohomology of quotients in symplectic and algebraic geometry, Mathematic s Notes, 3 1 (1 984) , Princeto n Universit y Press . F. Lalonde , i n preparation . F. Lalonde , D . McDuff, an d L . Polterovich , On the Flux conjectures, CR M Proceeding s an d Lecture Notes , 1 5 (1 998) , 69-85 . F. Lalonde , D . McDuff , an d L . Polterovich , Topological rigidity of Hamiltonian loops and quantum homology, Invent . Math. , 1 3 5 (1 999) , 369-385 . F. Lalond e an d D . McDuff , Symplectic structures on fiber bundles, Topology , t o appear . [io; D. McDuff, Quantum homology of Fibrations over S 2, Internationa l Journa l o f Mathematics , [ii. 11 (2000) , 665-721 .

C O H O M O L O G I C A L P R O P E R T I E S .. .

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[12] D . McDuf F an d D.A . Salamon , J -holomorphic curves and quantum cohomology, Universit y Lecture Series , 1 994 , American Mathematica l Society , Providence , RI . [13] D . McDuf f an d D . Salamon , Introduction to Symplectic Topology, 2n d edition , 1 998 , OUP , Oxford, U K [14] Piunikhin , D . Salamon , an d M . Schwarz , Symplectic Floer-Donaldson theory and Quantum Cohomology, Contact and Symplectic Geometry, e d C . Thomas , Proceeding s o f th e 1 99 4 Newton Institut e Conference , CUP , Cambridge, 1996. [15] A.G . Reznikov, Characteristic classes in symplectic topology, Select a Math , 3 (1 997) , 601 -642 . [16] Y . Ruan an d G. Tian , A mathematical theory of quantum cohomology, J . Differentia l Geom. , 42 (1 995) , 259-367 . [17] P . Seidel , ir\ of symplectic automorphism groups and invertibles in rings, Geometri c an d Functiona l Analysis , 7 (1 997) , 1 046-95 . UNIVERSITE D U QUEBE C A MONTREA L

E-mail address: [email protected] a STATE UNIVERSIT Y O F N E W YOR K A T STON Y BROO K

E-mail address: [email protected] u

quantum cohomology

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II Supersymmetric Gaug e Theories and Integrabl e Model s

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https://doi.org/10.1090/amsip/033/06 Mirror Symmetr y I V AMS/IP Studie s i n Advance d Mathematic s Volume 33 , 2002

Spectral La x pairs a n d Calogero-Mose r system s J.C. Hurtubis e

1. Introductio n The Calogero-Mose r syste m i s deceptively simple . I n its most basi c manifesta tion, it describes the motion of n particles alon g the line, interacting with an inverse square potential . I n canonical coordinate s p = (pi),x — (a^), this ha s hamiltonian: (1.1) H

= p -p + Y^ m(xt ~

x

o)~2

One ca n generalise thi s somewhat , whil e preservin g integrability . On e firs t ste p is to replac e the functions (xi — Xj), whic h ar e roots i n the root syste m A n _i, b y the roots o f another roo t system . I t turn s ou t that on e can impose differen t "masses " for eac h roo t length , giving : (1.2) H

= p.p+ ] T m

H(a(£))-

2

,

Another modificatio n i s to change the form o f the potential . Ther e ar e two ways in which on e can do this: on e ca n replac e the function x~ 2 (rationa l case ) b y sin(x) - 2 (trigonometric case) , or by the Weierstrass p-functio n p(x) for a given elliptic curv e (elliptic case) . Th e rational case an d the trigonometri c cas e ca n be though t o f as degeneration s o f the elliptic case , give n b y degenerating th e elliptic curv e t o a cuspidal curv e an d a noda l curv e respectively . W e will therefor e concentrat e o n the ellipti c case : wit h som e reservations , man y o f the thing s w e say will go over to the rationa l an d trigonometric cases . W e therefore hav e the most genera l famil y of (elliptic) Calogero-Mose r Hamiltonians : (1.3) H

= p.p+ ^ ra

wp(a(a;)),

where p(x ) i s the Weierstrass p-function . The differen t Caloger o Mose r system s ten d t o pop up in a disconcerting arra y of places . On e of the earlies t surprise s wa s when i t occurre d i n describin g th e motion o f th e pole s o f rationa l solution s t o th e K P equation ; ther e hav e bee n others since . Fo r several surveys , we refe r th e reader t o the conference proceeding s [DV]. T o quote th e editors o f [DV]: "B y now i t is well establishe d tha t th e CM S systems pla y a rol e i n investigation s i n researc h area s rangin g fro m theoretica l The autho r o f thi s articl e woul d lik e t o than k NSER C an d FCA R fo r thei r support . ©2002 America n Mathematica l Societ y an d Internationa l Pres s 103

104

J.C. H U R T U B I S E

physics (such as , e.g., soliton theory , quantu m field theory , strin g theory , solvabl e models o f statistica l mechanics , condense d matte r physics , quantu m chaos , etc. ) to pur e mathematic s (suc h a s representation theory , harmoni c analysis , theor y of special functions , dynamica l systems , rando m matri x theory , comple x geometry , etc.)." One o f th e mor e recen t an d intriguin g place s wher e th e syste m occur s i s in the solutio n o f the N = 2 SUS Y field theorie s b y Seiber g an d Witten, wher e th e key geometrical piec e of information i s an algebraically integrabl e system . I n many cases, th e appropriate syste m turn s ou t be a Calogero-Mose r system . Thi s follow s from suc h consideration s a s degeneration , an d monodrom y arguments ; I d o not know o f an intrinsic reason . If on e wants a n algebraicall y integrabl e system , however , on e needs t o realis e the famil y o f Abelia n varietie s t o whic h th e syste m corresponds . Thi s involves, implicitly, finding th e compactification o f the Lagrangian leaves . On e easy wa y o f doing thi s i s to realise th e system a s a Lax pair wit h spectra l parameter : (1.4) 4>(z)

= [A(z),ct>(z)}

The relevan t Abelia n varietie s wil l be sub-Abelian varietie s o f the Jacobians o f the spectral curve s give n by det(0(z) — XT) = 0 . Th e aim of this not e i s to discuss som e aspects t o this problem . 2. Hitchi n system s an d Lax pair s One o f th e mos t genera l integrabl e systems , whic h encompasse s mos t o f the more frequentl y studie d integrabl e cases , i s the generalise d Hitchi n syste m [Ma , Bo]. T o define it , on e fixes a Rieman n surfac e E an d a positiv e diviso r D o n E . Choosing a complex Li e group G , one considers th e moduli spac e MG o f pairs: (G-bundle P G on E, trivialisation tr o f PQ at D) and it s cotangen t bundl e T* MG- Fo r PQ G MG, le t P g b e th e adjoin t bundl e associated t o PG, and P g* the associated coadjoin t bundle . Fo r any vector bundl e V, se t V(D) = V 0{D), V(-D) = V® 0{-D). Th e fibre of T*M G - > M G a t PG i s canonically identifie d wit h th e vector spac e -fiT°(E , PQ* ^ ^ ( D ) ), (wher e K^ is the canonical bundl e o f E ) so that T*MG i s a space o f triples (G-bundle PG on E, trivialisation tr a t D , section (j) o f P g* ® K^(D)) The grou p GD of maps fro m th e divisor D (though t n o longer a s a formal sum of point s bu t a s a scheme o f points wit h multiplicity ) int o G acts naturall y o n the trivialisations, an d so on MG an d on T*MG- W e take the quotient, whic h amount s to forgettin g th e trivialisations. Th e resulting spac e (T* MG)red o f pair s (G-bundle P G o n E, section 0 of P s * Kv(D))) is the n Poisson , wit h symplecti c leave s obtaine d b y fixing th e coadjoin t orbit s o f the pola r part s o f 0 ove r D. Th e Hamiltonians o f th e Hitchi n integrabl e syste m are obtaine d fro m pair s (F,u;) , wher e F i s a homogeneou s invarian t functio n o n g* o f som e degre e n , an d uo an elemen t o f i^ x (E, K^ n+1 (—nD)). Applyin gF to , one obtains a n elemen t F((j)) of H°(T>,K® n(nD)), an d s o on e can defin e th e Hamiltonian F u o n T*MG b y FU(PG•>) = (•^(0)» a;) wher e (, ) denote s th e Serr e duality pairing . Thes e functions descen d to the reduced spac e (T*MG) red, an d give us a n integrable system .

S P E C T R A L LA X PAIR S AN D C A L O G E R O - M O S E R S Y S T E M S

105

In term s o f trivialisation s fo r PQ ove r ope n set s U{, with transitio n function s Tij fo r PQ and expression s fa for th e section s , th e flo w ca n b e give n b y l J tij,i)

(2.1) (T-

= (dF(fr)-cv,0).

Of course, dealin g with vecto r bundles , w e are allowe d t o choos e trivialisations, an d indeed chang e the m alon g th e flow. Innnitesimally , i f gi G H°(Ui, P 0 ), w e have th e equivalent versio n o f the flow: (2.2) (T^tij,^)

=

(dF^-u +

AdiTr1)^)-^, ad*( 9i)4>i).

In particular , i f we ar e o n a bundle whic h i s rigid, w e ca n spli t th e th e cocycl e (T^~1Tij) int o —Ad(T^ 1 ){gj) + g^ s o that th e flows becom e (2.3) (Tr

1

tij,fa) =

(0, ad*(

9l)fa).

This i s the standar d La x pai r for m fo r th e flows. Returning t o the Calogero-Mose r system , Kricheve r [K ] showed ho w the ellipti c system for sl(r, C ) could be considered a s a Hitchin system. Th e curve E one chooses is elliptic, an d w e write i t a s C/(Z(2o;i ) + Z(2o;2)) - Th e diviso r D i s just th e origi n Po of th e curve . Ou r modul i spac e i s the n on e o f pair s (E, fa),where - E i s a rank n degree 0 bundle o n E wit h trivia l determinant , an d - (p is a section o f End(E) 0 K^ wit h a simple pole a t th e origi n whose residu e is a conjugate o f m • diag (1,1,1, ..1 , — n + 1 )) . Semi-stable bundle s o f degre e 0 on ellipti c curve s ar e genericall y sum s o f lin e bundles o f degre e zero . Thes e lin e bundle s ar e classifie d b y a point p x in E (corresponding t o x G C under th e projectio n fro m C t o E) , s o tha t th e lin e bundl e Lx correspond s t o th e diviso r p x — po- Mor e explicitly , w e hav e o n C th e standar d elliptic function s a{z),C s{z) wit h expansion s a t z = 0 a(z) = z + 0(z 5)

(2.4)

CW,=4 + 0 ( A

z and periodicit y relation s a(z + 2uJi) = -a(z) exp(2f]i(z (2.5) C(

+ 2 ^ ) ),

^ + 2 ^) = C(^) + 27 ?2,

with r\i = CO^O- W e hav e

(2.6) ^logw*)

) = c(*), ±a*) = -p(*),

where p(z) i s th e standar d Weierstras s p-function . Fro m th e periodicit y relations , one ha s tha t th e functio n (2.7) P

\x,z) =

^ ^ \ e ^ j)(x),z) (3.4) Y

=

y

£/cijEij^{(u,i-u>j)(x),z).

Here th e Cij are constants , wit h c^ — Cji. I t is shown b y d'Hoke r an d Phon g tha t the La x flo w

(3.5) 0=[A,0

]

is equivalent t o th e Calogero-Mose r flo w if: 1. Fo r al l a G 1Z, the constant s c^ ar e relate d t o the constant s m^ o f the Calogero-Moser hamiltonian s b y 2

(3.6) s

mfal= ] T

c2.

u>i —ujj=a

2. Fo r al l / ? € 1 1 , (3 = UJI - to k, I ^ k,

= Y,

(3.7) 0

c

Uvi'vi)

3. Fo r al l pair s i ^ j , (3.8) scij(di

-dj)= ^2

CikCkj[p(ui(x) - u k(x)) - p{u)k(x) ~ Uj{x))\

Conditions 1 . an d 2 . impl y tha t c^ - = 0 whenever uji — ujj is no t a root.

J.C. HURTUBIS E

108

4. Th e Bordner-Corrigan-Sasak i approac h Another versio n o f the La x pai r fo r Calogero-Mose r system s i s due t o Bordner , Corrigan an d Sasak i [BCS2] ; see also [BCSl] . T o obtain thei r La x pair,they defin e an abstrac t algebr a i n term s o f generator s an d relations , i n whic h th e Calogero Moser flow s occu r a s La x flows . Th e variou s matricia l La x pair s the y obtai n ar e then define d i n term s o f representation s o f th e algebra . Their algebr a i s define d a s follows : th e roo t syste m 1 Z is define d o n a vecto r space C r . Le t s a denot e th e reflectio n i n th e roo t plan e i n C r associate d t o th e root a. Le t a v denot e th e dua l roo t t o a . I n a n orthonorma l basis , w e writ e a = (ai , ...a r ), a v = (a^,... ; a^). On e defines a n algebr a with generators iJi , ...if r, and s a, a G 71. Th e s a satisf y th e sam e relation s amongs t themselve s a s d o th e standard generator s s a o f th e Wey l group , th e Hi commut e amongs t themselves , and on e ha s (4.1)

[HjiSo] = (Xj l^a^Hi I

s a.

Bordner, Corriga n an d Sasak i sho w tha t i f on e define s La x operator s ^piHi +

i

(4.2) A then th e associate d La x equatio n = [A , 0] i s equivalen t t o th e Calogero-Mose r flow. Her e z i s the linea r paramete r o n th e ellipti c curve . To ge t a G/(n,C ) La x pair , on e ha s t o represen t th e algebra . On e clas s o f representations ca n b e define d b y takin g C n t o b e a su m ©™ =1 C • v^^ wit h on e generator v^ fo r eac h weigh t UJ^ o f a set . Th e se t o f weight s i s take n t o b e Wey l invariant. On e ca n defin e th e actio n o f th e generator s o f th e algebr a o n th e basi s v^ b y

Here u ; ^ i s th e i-th componen t o f th e weigh t vecto r UJ^. Th e actio n o f th e Hi i s diagonal i n th e basi s v u . On e check s tha t thi s representatio n satisfie s th e relatio n (4.1). We not e tha t th e spac e C n doe s no t necessaril y correspon d t o a representatio n of th e grou p correspondin g t o th e roo t system , whe n suc h a grou p exists . 5. Hitchi n system s fo r a non-standar d group . One ca n wonde r wha t i s the geometri c conten t o f the preceedin g methods . On e explanation, take n fro m [HM] , i s tha t th e Calogero-Mose r syste m i s a Hitchi n system fo r a grou p G whic h i s no t semi-simple , bu t whic h ca n constructe d fo r an y root system . T o an y roo t syste m 7£ , on e ha s a n associate d toru s H an d a Wey l group W. Le t u s se t N t o b e th e semi-direc t produc t o f th e toru s an d th e Wey l

S P E C T R A L LA X PAIR S A N D C A L O G E R O - M O S E R S Y S T E M S

109

group: (5.1) 0-^H-^N-^W-^O. We defin e G to be the semi direc t produc t (5.2) &aenC a -+ G -TV , where C a i s the root spac e correspondin g t o the root a. Th e connected componen t of th e identit y i s the semi-direct produc t (5.3) ©cG^C

a

->G

0-*H.

On th e level of Lie algebras , w e have (5.4) e

aenCa

-

> g -> fj.

There ar e ^-invariant pairing s o n g, which exten d th e natural Weyl-invarian t pair ings o n f) by pairing th e root space s C a an d C _a . Th e space o f such pairing s i s 1+ th e number o f Weyl orbit s i n 1Z. One chooses a non-degenerate pairin g i n this family t o identif y g an d g*. W e note tha t th e identificatio n i s no t G-invariant , though i t is TV-invariant. On e can show: PROPOSITION 5.5 . The G — Ad*-invariant functions on g* only depend on the root space components, and correspond to the N-invariant functions on Yla^-aThe generic coadjoint orbit is 2r-dimensional, where r = dim(TV) , and is of the form

(5.6) f

N- orbi t i n ] T C

a

J x [>* .

Moreover, g * has a 2r-dimensional connected (W-invariant) coadjoint

orbit V.

We choose th e orbit V of the vector (1 ,1 ,1 ,... , 1 ) in © a ^ C Q . We tak e th e Hitchin phas e spac e M associate d t o the group G , the divisor p 0 and th e orbit V , that i s the space o f pairs (G-bundle PQ on D, section (j) o f P Q* (g) K^(po) wit h pola r par t i n the orbit V) This spac e i s indeed 2 r dimensional . Ove r a generic set , one finds that th e G bundles reduc e t o H, s o that i n fact on e is dealing wit h sum s o f line bundles . Th e function on e takes as Hamiltonian i s defined b y applying on e of the quadrati c form s of our famil y t o 0, and then pairin g i t via Serre dualit y t o a cocycle £ with a simple pole a t the origin. O n a section a)aen-, ( t)\))^ the - Hamiltonian i s then give n explicitly by

(5.7) re

s U I h ' h + Yl

Explicitly, a s for the s/(r)-case, on e can show tha t thi s give s th e Calogero-Mose r Hamiltonian. One ca n ask how this tie s i n with th e ansatze give n above . T o see this, one notes tha t give n an y sum of weight spaces , th e group H act s naturall y o n it. I f in addition, th e set is Weyl-invariant, on e has an actio n o f the group TV.On e woul d like t o think o f extending thi s t o an action o f the group G , so that th e G-Hitchi n system woul d ma p naturally t o a Gl(n) system . On ou r generic ope n set , there i s no problem i n gettin g fro m a G-bundle t o a G/(n)-system , a s over thi s se t the bundles reduc e t o H, an d the map from H

J.C. HURTUBIS E

110

into th e diagona l matrice s i n G/(n, C) give s u s G/(n)-bundles . Ther e remain s th e question o f ho w t o exten d thi s t o (j). One mus t exten d th e actio n t o the roo t space s C a. Ther e ar e two natural way s to thin k o f th e actio n o f a root spac e C a : on e i s t o thin k o f a n actio n o f th e typ e that occur s i n a representation, actin g a s "shif t operators" , fro m th e weigh t spac e C^ t o C w + Q ; th e othe r i s i n term s o f th e associate d reflectio n i n th e roo t plan e o f a, mappin g C ^ t o C a , _ ^ a v ^ a . I t turns ou t tha t th e first interpretatio n lead s u s i n a fairl y natura l wa y t o th e d'Hoker-Phon g ansatz , an d th e secon d t o th e Bordner Corrigan-Sasaki prescription . W e not e tha t neithe r actio n i s really a group action , but bot h wil l b e iV-equivariant . A s ou r bundle s ar e reduce d t o H, this i s al l tha t matters. More precisely , w e choos e fo r eac h pai r {w^Wj) o f weight s a constant Cij in a way tha t i t is invarian t unde r th e Wey l grou p an d s o tha t (o.o) Cij

= Cji.

We the n defin e a "shift" operato r fo r eac h roo t a V^-^V \&

iLoijij — Owi—Wj,aCij->

where w e index th e entrie s o f the matri x b y the weight s themselves. Th e coefficien t 8w-w',a i s the Kronecke r S. W e the n se t

(5.10) ct>

Gl{N)

= £(0 b ) + J2 (t> 0 for positiv e kineti c energies .

M - T H E O R Y .. .

115

The holomorphi c prepotentia l ca n b e expressed i n term s o f a perturbativ e par t and infinit e serie s o f instanto n contribution s a s oo ( 2 . 7 ) T{A)

=

^classica

l d=l

where w e have specialize d th e instanto n term s t o SU(iV ) a s a n illustration . Du e t o a non-renormalizatio n theorem , th e perturbativ e expansio n fo r (2.7 ) terminate s a t 1-loop, though there is an infinite serie s of non-perturbative instanto n contributions . In (2.7) , A i s th e quantu m scal e (Wilso n cutoff ) an d I(R) i s th e Dynki n inde x o f matter hypermultiplet(s ) o f representatio n R. The Seiberg-Witte n dat a whic h (i n principle ) allo w one t o reconstruc t th e pre potential are : 1) A suitabl e Rieman n surfac e o r algebrai c curve , dependen t o n modul i u^ o r equivalently o n th e orde r parameter s a^ . 2) A preferre d meromorphi c 1 -for m A = S W differential . 3) A canonica l basi s o f homolog y cycle s o n th e surfac e (Afc , Bk). 4) Computatio n o f perio d integral s (2.8) 2niak

= f A JAk '

, 2maD,k

= f A JB

,

k

where recal l arj,k = ^ a i s the dua l orde r parameter . Th e progra m is : i) find th e Rieman n surfac e o r algebrai c curv e appropriat e t o th e give n matter content , ii) comput e th e perio d integrals , iii) integrat e thes e t o find J 7(a), an d iv) tes t agains t result s fro m C m{CTO whe n possible . What classe s o f S W curve s ar e encountere d fo r simple , classica l group s (SU , SO, Sp) wit h matte r hypermultiplet s consisten t wit h asymptoti c freedom ? a) hyperellipti c curve s 2

(2.9) y

+ 2A(x)y + B{x) = 0 ,

for pur e gaug e theor y + matte r hypermultiplet s i n th e fundamenta l repre sentation. b) cubi c non-hyperellipti c curve s (2.10) y

3

+ 2A{x)y 2 + B(x)y + e(x) = 0 ,

which occur s fo r i) SU(iV ) + 1 antisymmetric +(Nf < N + 2 ) fundamenta l hypermulti plets. ii) SU(A Q + 1 symmetric +(Nf < N — 2 ) fundamenta l hypermultiplets . c) curve s o f infinit e orde r fro m i) ellipti c models , o r ii) decompactification s o f ellipti c models . Example. SU(A^ ) + 2 antisymmetric an d Nf < 4 fundamental hypermultiplets . The mai n tas k i n extractin g instanto n prediction s fro m curve s suc h a s (2.1 0 ) is the computatio n o f th e perio d integral s (2.8) , an d th e integratio n o f dJ r{a)/dak to obtai n J 7{a). Ther e ar e tw o principa l (complementary ) method s t o evaluat e the perio d integral s fo r hyperelliptic curves . Thes e ar e Picard-Fuch s differentia l

116

ENNES/LOZANO/NACULICH/RHEDIN/SCHNITZER

equations fo r th e perio d integral s [1 3] , and direc t evaluatio n o f the perio d integral s by asymptoti c expansio n [1 4 , 1 5 , 1 6] . The proble m w e face i s ho w t o evaluat e perio d integral s (2.11)

A

xdy y

i

for non-hyperellipti c curve s such as (2.1 0) . Fo r the cubic curve, the exact solutio n is too complicate d t o be useable, while for curve s of higher order , eve n exac t solution s are no t possible . Numerica l solution s ar e o f n o interest , a s w e wan t t o stud y th e analytic behavio r o f J 7(a) o n th e orde r parameters .

3. M-theor y an d th e Rieman n Surfac e The semina l wor k o n thi s subjec t i s b y Wi t ten [3] , wh o consider s II A strin g theory lifte d t o M-theory . I t i s convenien t t o us e th e languag e o f II A theor y i n describing th e bran e structure . Conside r SU(A ) gaug e theor y wit h eithe r a n anti symmetric o r symmetri c matte r hypermultiple t [4] . Th e M-theor y pictur e i s

:$

)06 [

XQ

Mirror imag e Figure 1 .

There ar e 3 paralle l N S 5-brane s wit h N D4-brane s suspende d betwee n each , and a n 06-plan e o n th e centra l N S 5-brane . I n th e absenc e o f th e orientifold , on e would hav e SU(iV ) x SU(7V ) wit h matte r i n th e (N,N) 0 (N,N) representation . The orientifol d "identifies " th e tw o SU(iV ) factors , projectin g t o th e diagona l sub group, givin g a singl e SU(A' ) facto r wit h on e hypermultiple t i n th e antisymmetri c representation fo r 0 6 _ , o r on e hypermultiple t i n th e symmetri c representatio n fo r 0 6 + [4] . I t i s important t o not e tha t th e orientifol d induce s a 7L 2 involution i n th e curve. Th e N S 5-branes ar e a t xj = x% = xg = 0, with classically fixed values of XQ. The D4-brane s hav e world-volume XO,XI,X2,XS,XG, wit h end s a t fixed value s o f XQ, which give s a macroscopic world-volum e o n th e D4-bran e a s d = 4 . Th e M-theor y picture give s ris e t o Rieman n surface , whic h i s the S W curv e fo r thi s situation .

117

M - T H E O R Y .. .

4. Hyperellipti c Perturbatio n Theor y We hav e develope d a systemati c schem e fo r th e instanto n expansio n fo r pre potentials associate d t o non-hyperellipti c curve s [5]-[ll] , whic h wil l b e illustrate d for th e case o f SU(iV ) gaug e theor y wit h on e hype r mult iplet i n th e antisymmetri c representation [5] . Th e curv e i s give n b y (2.1 0 ) wher e L 2 = A ^ 2 , wit h A th e quantum scal e o f the theory , an d e = L 6 ; 2A(x)

=

[f{x)+3L

=

L

2

],

TV

f(x)=x2l[(x-el); B(x)

2

[f(-x) +

3L 2 ] ,

L4 (4.1) involutio

n: y

—> • —, x

— » —x.

It i s fruitful t o regard th e last ter m e = L 6 i n (2.1 0 ) a s a perturbation. Th e intuitio n is tha t thi s involve s a muc h highe r powe r o f th e quantu m scal e i n (2.1 0 ) tha n th e other terms , an d geometricall y i t mean s separatin g th e right-mos t 5-bran e fa r fro m the remainin g tw o N S 5-branes . To zerot h approximatio n w e conside r (2.1 0 ) wit h e = 0 , whic h i s the n a hy perelliptic curve , an d ca n b e analyze d b y previousl y availabl e method s [1 4]-[1 6] . This approximatio n give s ^i-ioo p correctly , bu t i t i s no t adequat e fo r -Fi-mst , s o one need s t o g o beyon d th e hyperellipti c approximation . W e presen t a systemati c expansion i n e , which i s not th e sam e a s a n expansio n i n L , a s the coefficien t func tions A(x) an d B(x) depen d o n L. Th e perturbativ e expansio n i n e for th e solutio n is (4.2) yi

= Vi + Syi = yi + a {e + &e 2 H ,

i

= 1 , 2, 3,

which fo r ou r exampl e give s t o firs t order ,

2/2 (re) = (4-3) y

3(x)

=

L6(A + r) - A + r+ ^ +-.

.

~ | r +'• •

with subscript s denotin g th e appropriat e sheet , an d wit h r — \/A 2 — B. I t i s straightforward t o g o to highe r order s i n e . We not e tha t shee t 3 i s disconnecte d i n an y finit e orde r o f ou r perturbatio n expansion, s o w e nee d onl y conside r y\ an d y^. W e nee d th e S W differentia l fo r these tw o sheets , wit h th e S W differentia l fo r shee t 1 (4.4) A

1

=

x^, 2/1

and A 2 obtained fro m (4.4 ) b y r— > — r. Informatio n o n sheet 3 is obtained fro m th e involution symmetr y i n (4.1 ) . Th e expansio n (4.2 ) induce s a comparable expansio n

ENNES/LOZANO/NACULICH/RHEDIN/SCHNITZER

118

for A , whereb y (4.5)

Ai = = (Ai) / + (Ai)/ / + - .

(4L-BL\ X A A/ =

U 2B —

=—

1

(\rr.

V ^

(4.6)

Xn =

- - L^zMdx

up t o term s t h a t d o not contribut e t o perio d integrals . Notic e t h a t Xj i s t he SW differential obtaine d fro m t h e hyperelhpti c approximatio n ( e = 0 ) t o (2.1 0) , an d completely determine s J-i-\ 0oP an d a par t o f t h e 1 -instanto n term , whil e Xu ~ (9(L 2 ), s o i s o f 1 -instanto n order . Furthe r term s contribut e onl y t o 2-instanto n order an d higher, s o (4.6 ) is all t h at i s needed t o 1 -instanto n order . In orde r t o expres s t h e solutions t o our problem wit h economica l notation , w e define "residue functions", Rk(x), S(x), SQ(X), a n d Sfc(x) , wher e (4.7)

**

(x-ek) f(x)'

3

-

and f(-x) _ f2(x) x

(4.8) S(x)

S 0(x) _ S 2 (x-e

k(x) k)

2

The function s S(x) an d Sk(x) pla y a crucia l rol e fo r understandin g t h e genera l features o f t he instanton expansio n o f SW problems . T h e branch-cut s ar e centered o n t he e$ and connect sheet s y \ an d 2/2 a s show n in Fig . 2

x^ e\

X]_

Xfc

e

k

x^

e^

x^

v %N

Figure 2 . The orde r parameter s ar e compute d i n a canonica l homolog y basis . Fo r t he order parameter s a k w e have t h e basis A k , a s show n i n Fig. 3 ,

e

i

Figure 3 . and t h e basis B k fo r t he dual orde r parameter s CLD k

as

show n i n Fig. 4 ,

M - T H E O R Y .. .

119

ie\z

Figure 4 . The cycl e B k connect s sheet s y\ an d 7/2 , with th e soli d lin e on shee t yi an d th e dashed lin e o n y 2, wit h Bk passin g throug h th e branch-cu t a s shown . T o comput e (2.8), on e only need s (A i — A2), so that w e only nee d term s od d unde r r — > —v. Th e order paramete r i s

I \ ~ l {Xi

2ni ak

J Ah J

(4.9)

/

+ Xn +•••)

Ah

dx

(i-is)

r6

-L°

A2

(A-&)

&f

B_ A2

The secon d ter m doe s no t contribut e t o (9(L 2 ), a s ther e ar e n o pole s a t x = e& . Thus t o thi s orde r (4.10)

dSk (ek) - Rk{ek) dx

^h — ^h

+•

The computation o f the dual order parameter i s considerably mor e complicated , with th e resul t 2iriaD,k =

27ri(aD,k)i

+ 27ri(aD yk)u N

(4.11) -

27TZ^

- [Classica l + ^ 1 - l o o p ] + L

2

^ -[

- 25

0(0)

+ ^ S

k(ak)],

so tha t th e one-instanto n prepotentia l fo r SU(7V ) gaug e theor y wit h on e massles s antisymmetric hypermultiple t [5 ] turns ou t t o b e (4.12)

J~ 1 — inst — ^ .

-2S0(0)+^2Sk(ak)

2m which i s a predictio n o f M-theor y tha t ma y b e teste d agains t microscopi c calcula tions. Thi s i s presently possibl e fo r SU(iV ) wit h N < 4, sinc e SU(2 ) + g = SU(2 ) (pure gaug e theory) ; SU(3 ) + 0 = SU(3 ) + D ; an d SU(4 ) + g = SO(6 ) + • . I n each o f thes e thre e cases , (4.1 2 ) agree s wit h 1 -instanto n calculation s fro m £ miCro [17]. Fo rT V > 5 , (4.1 2 ) shoul d b e regarde d a s prediction s o f M-theory , awaitin g testing. Th e fac t tha t (4.1 2 ) agree s wit h microscopi c calculations , whe n available , after a lon g derivation , wit h distinc t methods , i s alread y impressive . There ar e furthe r application s o f hyperellipti c perturbatio n theor y [5]-[ll] , where th e analysi s i s very simila r t o tha t sketche d above . One ma y ad d hypermultiplet s i n th e fundamenta l representation , an d hyper multiplets wit h non-zer o masses . Fo r SU(7V ) gaug e theor y wit h a n antisymmetri c representation an d Nf < N + 2 , whic h i s describe d b y a cubi c S W curv e [4] , on e

120 E N N E S / L O Z A N O / N A C U L I C H / R H E D I N / S C H N I T Z E

R

finds [7] : N r

(4.13) 27nL7

i_inst = ^2s k(ak) -

2S

m(-±i

k=l

where (4.14) S

k(ak)

= /

1

x

K+

(4.15) S

|m)

m{-\m)

N 2

n^/cK-a02

nLiK + i^)

where Mj (ra ) i s th e mas s o f th e hype r mult iplet i n th e fundamenta l (resp . anti symmetric) representation . Eqs . (4.1 4 ) an d (4.1 5 ) agre e wit h scalin g limit s takin g Mj and/o r m — » oo. Eqs . (4.1 3)-(4.1 5 ) provid e additiona l test s o f M-theory , sinc e SU(2) + 0 + iV / D = SU(2 ) +N f n , SU(3 ) + 0 + W , D = SU(3 ) + (N f + 1) D , both o f whic h agre e wit h microscopi c instanto n calculation s [1 7] . An additiona l SU(7V ) theor y wit h a cubi c S W curv e i s SU(7V ) - f m 4 - N 1f ~ L with N f < N + 2 . Her e th e resul t i s [7] , [8]: TV

(4.16) 27riFi_

inst

= ^ S

(4.17) 5 fc(afc) = - ^

fc(afe),

—.

This agree s wit h th e microscopi c calculatio n o f Slate r [1 2] . Whenever th e prediction s o f the cubi c S W curves obtained fro m M-theor y hav e been tested, agreemen t ha s been found wit h those of microscopic calculations . How ever, ther e remai n numerou s furthe r opportunitie s t o subjec t M-theor y prediction s to testing . 5. Universalit y If one examines the case s discussed i n the previous section , on e observes certai n universal features : (i) Th e natura l variable s fo r thi s clas s o f problem s ar e th e orde r parameter s {ak} an d no t th e gaug e invarian t moduli . (ii) Th e 1 -instanto n contribution t o the prepotentia l ca n be written a s [6 , 7, 1 4 ] N 7

(5.1) 27riJ

i_inst = Y2 S

k(ak),

k=l

for SU(iV ) + N f • o r SU(N) + m + TV / • , an d TV

(5.2) 27ri7

r

i_ inst = ^ S

k(ak)

for SU(JV ) + 0 + N f • [5 , 7].

-

2S

m(-|ra),

M - T H E O R Y .. .

121

Define S(x) whic h generalize s (4.8 ) a s

(5.3) S(x)

Sk(x) _ Sm(x) 2 (x - a k) ( x + | m

)

We tabulat e th e know n S(x) fo r SU(7V ) i n th e firs t thre e entrie s o f Tabl e 1 , wher e we include al l generic case s o f asymptoticall y fre e M = 2 SU(iV) gaug e theories . A careful examinatio n o f the firs t thre e rows of Table 1 leads to the following empirica l rules. S(x) i s give n a s th e produc t o f th e followin g factors , eac h correspondin g t o a differen t Af = 2 multiplet i n a give n representatio n o f SU(7V): (1) Pur e gaug e multiple t facto r

nili(^- fl i) 2 (2) Fundamenta l representatio n • . A facto r

(5.5) (x

+ Mj)

for eac h hypermultiple t o f mas s Mj i n th e fundamenta l representation . (3) Symmetri c representatio n ED . A facto r

N

1 (5.6) ( - )

* (x + \mf Y[(x

+ ) =

( v + | m i ) - 6 ( v + m i - | m 2 ) 2 JJ(t > - a * + rai - m 2 ),

AT

2 =1

This i s equivalen t t o th e curv e give n i n ref . [1 0] , u p t o overal l multiplicatio n b y F(v) an d redefinitio n to = G(v)t, wher e F(v) an d G{v) ar e rationa l function s o f v an d th e hypermultiple t masses . Th e prepotentia l derive d fro m thes e curve s i s independent o f F(v) an d G(v). In Refs . [7 , 1 0] , SW curve s ar e give n fo r al l the othe r ellipti c model s discusse d in th e previou s section . In th e limi t m i— > m2 (i.e. , vanishin g globa l mass) , ther e ar e n o subleadin g terms, an d th e curv e (3.1 0 ) fo r SU(7V ) + m 4 - g reduces , upo n chang e o f variabl e t = t 0(v - f \rn) 2, t o

o = Y, i

n2/4 tn

NN

jl( u -«*) + £ «

neven i—\

f\ (3.17) -

0

3

n

N

n2/4

*" W- v -*i-

m

)

1 odd i—

f

\

N

( - J | 2 rj J[(v - a z) + 0 2 f ^ | 2 rJ J[(-v -

a * - m) ,

where g = exp(27rfr) , £ = exp(— iixzjuj\)^ an d # 2 (z/|r), #3(Z/|T ) ar e Jacob i thet a functions. I n ref . [1 0] , we hav e show n tha t eq . (3.1 7 ) i s equivalent t o th e curv e fo r this theor y give n b y Urang a [1 4] . 4. O n e - i n s t a n t o n p r e p o t e n t i a l Although w e hav e obtaine d a n infinit e orde r curv e fo r th e SU(7V ) - f C D - f [ ] theory, the one-instanton (0(q)) prepotentia l fo r this theory may be extracted [6 , 7] from th e quarti c truncatio n o f thi s curv e consistin g o f just thos e fiv e term s show n explicitly i n eq . (3.1 7) . Defin e UU QUA (4-1) S(v)=

Pi(v)P-i(v) . po{v)2

136

LP. E N N E S , C . LOZANO , S.G . NACULICH , H.J . S C H N I T Z E R

For thi s theory , S(v) ha s quadrati c pole s a t v = a k an d v = —\mi. A t thes e poles , we define th e residu e function s S k(v) an d S rni(v) b y Sk{v) S (v-ak)2 {v+\mi)

(4.2) S(v)

mi(v) 2

'

It ma y b e show n tha t th e one-instanto n prepotentia l i s give n b y N r

(4.3) 27nL7

i_inst = ^S k(ak) -2S

rni(-\mi).

fe=i

Although S(v) an d therefor e eq . (4.3 ) depen d explicitl y onl y o n thre e o f th e five coefficients i n eq. (3.1 7) , the entire quartic truncation (includin g the first subleadin g term) i s necessar y fo r th e consistenc y o f th e calculatio n t o 0(q) [3] . In Tabl e 1 below , w e lis t th e expression s fo r S(v) fo r al l th e ellipti c model s described i n sect . 2 [1 0] . Th e one-instanto n prepotentia l fo r eac h o f these theorie s is the n give n i n term s o f th e residu e function s define d i n eq . (4.2) . Fo r SU(7V ) + adjoint [1 8 ] N

(4.4) 27riFi_

[nst

= J2s

k(ak).

fc=i

For SU(A 0 + [ ] + m , SO(27V) + adjoint , an d SO(2N + 1 ) + adjoint , N r

(4.5) 27ri7

i_inst = ^2s k=i k(ak) -

2S

mi(-|rai),

where mi i s the mas s of the antisymmetri c o r adjoin t hypermultiplet . Fo r SU(7V ) +

20 + 4D,

N

(4.6) 27ri7

r 1

_ inst = ^S k(ak) -2S

rni(-\mi)

-

2S

rn2(-^m2),

k=l

where mi an d m^ ar e th e masse s o f th e antisymmetri c hypermultiplets . Fo r Sp(2iV)+ adjoint , an d Sp(2iV ) + 0 + 4 • , (4-7) 2 ^ ^ _

inst

= -2[S 0 (0)] 1 / 2

where (4.8) S(v)

= ^

defines th e residu e functio n a t th e quarti c pol e a t v = 0 . Al l o f thes e result s hav e been subjecte d t o a wid e variet y o f consistenc y checks , a s describe d i n ref . [1 0] .

SEIBERG-WITTEN CURVE S ...

jM

— 2 gaug e theor y

SU(iV) + 2 g ( m i , m 2 ) +4D(M,)

1 W)

137

\

nf=iKal+m1)nf=1Ka,+m2)nJ=1HMJ) (^+|m1)2(^+^m2)2 Vu = x (v-a%Y {v+\m2)2 nfLi(^+^+mi)n

; z

li(v+a2+m2)

SU(JV) + g ( m i) + m ( m 2 ) (^+^m 1 ) 2 nil 1 ^-a,) 2 nf=1[(^-^)2-m2]

SU(7V) + adjoint(m ) niLi(v-*i)2 v4 IlSLiKv-rnf-a*] nf=1[(^+m)2-a2]

SO(27V) + adjoint(m ) ( V +|m) 2 (^-|m) 2 n t i Ii(^ 2 -o 2 ) 2 , 2 (,+m)(,-m)nf = 1 [(^m) 2 -a?]nf = 1 [Km) 2 -aJ]

SO(27V + 1 ) + adjoint(ra ) (v+|m) 2 (v-|m) 2 n£i(* 2 -ai) 2 K|m)2(,-|m)2nf=1[(^-m)2-a2]nf=1[Km)2-a2]

Sp(2N) + adjoint(m )

^ 4 nf =1 (^ 2 -« 2 ) 2

nlLi^-mj^ajiniLi^+mj^a^n^!^2-^) Sp(2iV) + 0(m ) + 4D(Mj ) v±(v+\my{v-\m)* Tl?=i(v2-a2)2

Table 1 . Acknowledgement. W e ar e gratefu l t o th e organizer s o f thi s Workshop , E . D'Hoker, D.H . Phong , an d S.T . Yau , fo r th e opportunit y t o presen t thi s wor k i n such a pleasan t an d stimulatin g environment . References [1] N . Seiber g an d E . Witten , Nucl . Phys. , B 4 2 6 (1 994) , 1 9 ; erratum, ibid, B 4 3 0 (1 994) , 485 , hep-th/9407087 ; Nucl. Phys. , B 4 3 1 (1 994) , 484 , hep-th/9408099 . [2] E . D'Hoker , I.M . Krichever , an d D.H . Phong , Nucl . Phys. , B 4 8 9 (1 997) , 1 79 , hep-th/9609041; Nucl. Phys. , B 4 8 9 (1 997) , 21 1 , hep-th/96091 45 . [3] S . Naculich , H . Rhedin , an d H . Schnitzer , Nucl . Phys. , B 5 3 3 (1 998) , 275 , hep-th/98041 05 . [4] I . Ennes, S . Naculich , H . Rhedin , an d H . Schnitzer , Int . J . Mod. Phys. , A 1 4 (1 999) , 301 , hep-th/9804151. [5] I . Ennes , S . Naculich, H . Rhedin, an d H . Schnitzer , Nucl . Phys. , B 5 3 6 (1 998) , 245 , hep-th/9806144.

138

LP. ENNES , C . LOZANO , S.G . NACULICH , H.J . SCHNITZE R I. Ennes , S . Naculich , H . Rhedin , an d H . Schnitzer , Phys . Lett. , B 4 5 2 (1 999) , 260 , hep-th/9901124. I. Ennes , S . Naculich , H . Rhedin , an d H . Schnitzer , Nucl . Phys. , B 5 5 8 (1 999) , 41 , hep-th/9904078. I. Ennes , C . Lozano , S . Naculich , H . Rhedin , H . Schnitzer , hep-th/00061 41 . E. Witten , Nucl . Phys. , B 5 0 0 (1 997) , 3 , hep-th/97031 66 . I. Ennes , C . Lozano , S . Naculich , H . Schnitzer , Nucl . Phys. , B 5 7 6 (2000) , 31 3 , hep-th/9912133. A. Giveo n an d D . Kutasov , Rev . Mod . Phys. , 7 1 (1 999) , 983 , hep-th/9802067 . K. Landsteine r an d E . Lopez , Nucl . Phys. , B 5 1 6 (1 998) , 273 , hep-th/97081 1 8 ; K. Landsteiner , E . Lopez , an d D . Low e JHEP , 980 7 (1 998) , 01 1 , hep-th/98051 58 . R. Donag i an d E . Witten , Nucl . Phys. , B 4 6 0 (1 996) , 299 , hep-th/951 01 01 . A. Uranga , Nucl . Phys. , B 5 2 6 (1 998) , 241 , hep-th/9803054 . T. Yokono , Nucl . Phys. , B 5 3 2 (1 998) , 21 0 , hep-th/98031 23 . J. Erlich , A . Naqvi , an d L . Randall , Phys . Rev. , D 5 8 (1 998) , 046002 , hep-th/9801 1 08 . N. Seiber g an d E . Witten , hep-th/96071 63 . E. D'Hoke r an d D.H . Phong , Nucl . Phys. , B 5 1 3 (1 998) , 405 , hep-th/9709053 . MARTIN FISHE R SCHOO L O F PHYSICS , BRANDEI S UNIVERSITY , WALTHAM , M A 0245 4

E-mail address: [email protected] s E-mail address: [email protected] s E-mail address: schnitzerObrandeis.ed u DEPARTMENT O F PHYSICS , BOWDOI N COLLEGE , BRUNSWICK , M E 040 1

E-mail address: [email protected] u

1

https://doi.org/10.1090/amsip/033/09 Mirror Symmetr y I V AMS/IP Studie s i n Advance d Mathematic s Volume 33 , 200 2

The periodi c an d ope n Tod a lattic e I. Kricheve r an d K.L . Vaninsk y ABSTRACT. W e develo p algebro-geometrica l approac h fo r th e ope n Tod a lat tice. Fo r a finite Jacob i matri x w e introduc e a singula r reducibl e Rieman n surface an d associate d Baker-Akhieze r functions . W e provid e ne w explici t so lution o f invers e spectra l proble m fo r a finit e Jacob y matrix . Fo r th e Tod a lattice equation s w e obtai n th e explici t for m o f th e equation s o f motion , th e symplectic structur e an d Darbou x coordinates . W e develo p simila r approac h for 2 D ope n Toda . Explainin g som e th e machiner y w e als o mak e contac t wit h the periodi c case .

1. Introductio n Until no w th e method s o f integratio n fo r periodi c an d ope n Tod a lattic e wer e absolutely unrelate d t o eac h other . Th e periodi c Toda , i s a Hamiltonia n syste m o f iV-particles wit h th e Hamiltonia n N2

#= ^ y

N

+ Yl eqk~qk+1 ' 4;v

+i

= qi + Jo , Io

= const

fc=i fc=i

The solutio n o f equation s o f motio n wa s obtaine d b y Kricheve r usin g genera l alge bro-geometrical approac h base d on the concept o f Baker-Akhiezer functio n [1 ] . Th e solution wa s expresse d i n term s o f theta function s o f the spectra l curve , associate d with th e auxiliar y spectra l proble m fo r a n infinit e periodi c Jacob i matrix . Fo r th e open Tod a wit h th e Hamiltonia n N2

N

1

~

fe=l fc=l

the solutio n o f equation s o f motio n wa s obtaine d b y Mose r [2] . Mose r reduce d th e original proble m o f integratio n t o th e invers e spectra l proble m fo r a finite Jacob y matrix. Th e latte r wa s solved b y Stieltje s [3 ] more tha n a hundre d year s ag o usin g continuous fractions . Though the Hamiltonia n o f open Toda lattice can be obtained fro m th e periodi c one a s th e limi t IQ —> oo , th e method s o f solutio n ar e different . Thi s i s du e t o th e fact tha t auxiliar y spectra l proble m associate d wit h a finite Jacob y matrix , unlik e The firs t autho r i s supporte d i n par t b y Nationa l Scienc e Foundatio n unde r th e gran t DMS 98-02577 an d b y CDR F Awar d RP1 -21 02 . The secon d autho r i s partiall y supporte d b y NS F grant s DMS-9501 00 2 an d DMS-9971 834 . ©2002 America n Mathematica l Societ y an d Internationa l Pres s 139

140

I. K R I C H E V E R A N D K.L . VANINSK Y

the spectra l proble m fo r th e periodi c infinit e Jacob y matrix , doe s no t determin e a natural spectra l curve . Th e firs t attemp t t o singulariz e smoot h spectra l curv e o f the periodi c Tod a t o obtai n a spectra l curv e fo r th e ope n Tod a chai n goe s bac k to seventie s [4] . Recently , th e interes t i n thi s proble m wa s revive d [5 , 6 ] du e to connection s o f th e Tod a lattic e wit h Seiberg-Witte n theor y o f supersymmetri c SU(N) gaug e theor y [7] . ( A relativel y complet e lis t o f reference s o n ne w insigh t upon th e rol e of integrable system s i n Seiberg-Witte n theorie s [8]-[1 2 ] ca n b e foun d in [1 3]-[1 7 ] an d book s [5 , 1 8]. ) The spectra l curv e propose d i n [5 , 6 ] fo r th e ope n Tod a lattic e i s determine d by th e equatio n N

(1.1) w

Q

= P(E) = Y[(E - E k) fc=i

and ca n b e considere d a s th e limi t whe n J 0 — > o c o f th e hyper-ellipti c spectra l curve (1.2) w

o + — =P(E), A

2

= e~

/o

Wo

for th e periodi c case . From algebro-geometrica l viewpoin t th e limi t A — > 0 o f hyper-ellipti c curv e leads t o a singula r curve , whic h i s tw o copie s o f rationa l curve , correspondin g t o two sheet s o f hyper-ellipti c curve , glue d togethe r a t N point s E^. I t i s well know n that Baker-Akhieze r function s introduce d originall y fo r smoot h algebrai c curves , are als o a n importan t too l o f integratio n i n th e limitin g case o f singula r curves . Multi-soliton an d rationa l solution s o f integrabl e equation s ca n b e obtaine d withi n this approac h (se e i n [1 9]) . Th e mai n goa l o f th e presen t pape r i s t o demonstrat e that algebro-geometrica l approac h base d on the concept o f Baker-Akhiezer functio n can b e use d i n th e cas e o f reducible singular algebrai c curves . A s a n outcome , w e provide a solutio n t o th e invers e spectra l proble m fo r a finit e Jacob y matri x whic h is differen t fro m th e classica l Stieltjes ' solution . In th e recen t paper s b y Kricheve r an d Phong , [20 , 21 ] , th e ne w approac h using Baker-Akhieze r function s fo r constructio n o f Hamiltonia n theor y o f solito n type equation s wa s developed . I t provide s a universa l schem e fo r constructio n o f angle-act ion typ e variable s fo r thes e equations . I n Sectio n 4 w e illustrat e thes e ideas, an d sho w that , th e cas e o f th e ope n Tod a lattic e wit h singula r curv e ca n b e treated similarly . Finally, i n sectio n 5 we indicate ho w th e idea s introduce d i n thi s pape r ca n b e applied t o integratio n o f ope n two-dimensiona l Tod a lattice . 2. Periodi c Tod a lattice . To begin with, le t us first briefl y recal l algebro-geometric solutio n of the periodi c Toda lattice . Tha t wil l hel p u s late r t o clarif y algebro-geometri c origi n o f ou r approach t o th e ope n Tod a lattice . Most o f th e materia l i s standar d an d detail s ca n b e foun d i n [22 , 23] . I t i s convenient t o consider th e periodic Tod a lattic e system a s a subsystem o f an infinit e lattice. A s i t wa s foun d i n [24 , 25] , the equation s o f motio n (2.1)