Mirror Symmetry III: Proceedings of the Conference on Complex Geometry and Mirror Symmetry, Montreal, 1995 0821811932, 9780821811931

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Selected Title s i n Thi s Serie s Volume 10 Duon g H . P h o n g , Lu c Vinet , an d S h i n g - T u n g Yau , Editor s Mirror Symmetr y II I 1999 9 Shing-Tun g Yau , Edito r Mirror Symmetr y I 1998 8 Jiirge n Jost , Wilfri d Kendall , U m b e r t o M o s c o , Michae l Rockner , and Karl-Theodo r Stur m New Direction s i n Dirichle t Form s 1998 7 D . A . Buel l an d J . T . Teitelbaum , E d i t o r s Computational Perspective s o n Numbe r Theor y 1998 6 Harol d Levin e Partial Differentia l Equation s 1997 5 Qi-ken g Lu , S t e p h e n S.-T . Yau , an d A n a t o l y Libgober , Editor s Singularities an d Comple x Geometr y 1997 4 Vyjayanth i Char i an d Iva n B , Penkov , Editor s Modular Interfaces : Modula r Li e Algebras , Quantu m Groups , an d Li e Superalgebra s 1997 3 Xia-X i Din g an d Tai-Pin g Liu , Editor s Nonlinear Evolutionar y Partia l Differentia l Equation s 1997 2.2 Willia m H . Kazez , Edito r Geometric Topolog y 1997 2.1 Willia m H . Kazez , Edito r Geometric Topolog y 1997 1 B . Green e an d S.-T . Yau , Editor s Mirror Symmetr y I I 1997

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Mirror Symmetry III

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AMS/IP

https://doi.org/10.1090/amsip/010

Studies in Advanced Mathematics V o l u m e 10

Mirror Symmetry III Proceedings of the Conferenc e on Complex Geometry and Mirror Symmetry, Montreal, 1 99 5 Duong H. Phong, Luc Vinet, and Shing-Tung Yau, Editors

America n Mathematica l

Society

Centr e d e Recherche s Mathematique s

Internationa l Pres s

Shing-Tung Yau , M a n a g i n g E d i t o r

1991 Mathematics Subject

Classification.

Primar

y 1 4-06 ; Secondar y 3 2 - 0 6 , 8 1 - 0 6 .

Library o f Congres s Cataloging-in-Publicatio n D a t a Conference o n Comple x Geometr y an d Mirro r Symmetr y (1 99 5 : Montreal , Quebec ) Mirror symmetr y II I : proceeding s o f th e Conferenc e o n Comple x Geometr y an d Mirro r Symmetry, Montreal , 1 99 5 / Duon g H . Phong , Lu c Vinet , an d Shing-Tun g Yau , editors , p. cm . — (AMS/I P studie s i n advance d mathematic s ; v. 1 0 ) Includes bibliographica l references . ISBN 0-821 8-1 1 93- 2 1. Mirro r symmetry—Congresses . 2 . Geometry , Differential—Congresses . 3 . Function s o f several comple x variables—Congresses . I . Phong , Duon g H. , 1 953 - . II . Vinet , Luc . III . Yau , Shing-Tung, 1 949 - . IV . Title . V . Title : Mirro r symmetr y three . VI . Series . QC174.17.S9C631 99 5 516.3 / 62—dc21 98-3764 3 CIP

C o p y i n g a n d reprinting . Materia l i n this boo k ma y b e reproduce d b y any mean s fo r educationa l and scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y service s that collec t fee s fo r deliver y o f documents an d provide d tha t th e customar y acknowledgmen t o f th e source i s given. Thi s consen t doe s no t exten d t o othe r kind s o f copying fo r genera l distribution , fo r advertising o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercia l us e o f material shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematica l Society , P. O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o b e mad e b y e-mai l t o reprint-permissionOams.org. Excluded fro m thes e provision s i s materia l i n article s fo r whic h th e autho r hold s copyright . I n such cases , request s fo r permissio n t o us e o r reprin t shoul d b e addresse d directl y t o th e author(s) . (Copyright ownershi p i s indicate d i n th e notic e i n th e lowe r right-han d corne r o f th e first pag e o f each article. ) © 1 99 9 b y th e America n Mathematica l Society , Internationa l Pres s an d th e Centr e d e Recherches Mathematiques . Al l right s reserved . The America n Mathematica l Society , Internationa l Pres s an d th e Centr e d e Recherches 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 URL : 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 U R L : h t t p : //www. crm. umontreal. c a / 10 9 8 7 6 5 4 3 2 1 0

4 03 02 01 00 9 9

Contents

Introduction Aspects o f Quantu m Geometr y Brian R. Greene Introduction t o Enumerativ e Invariant s Shing-Tung Yau Compactified Modul i Space s o f Pseudo-Holomorphi c Curve s Thomas H. Parker Mirror Symmetr y fo r Hyper-Kahle r Manifold s Misha Verbitsky Connecting th e Web : A Prognosi s Mark Gross Strong Couplin g Singularitie s an d Non-Abelia n Gaug e Symmetrie s i n String Theor y Albrecht Klemm and Peter Mayr Remarks o n (0,2 ) Calabi-Ya u Model s Shamit Kachru Relations Amon g Fixe d Poin t Kefeng Liu An Analyti c Discriminan t fo r Polarize d Algebrai c K3 Surface s Jay Jorgenson and Andrey Todorov Through th e Lookin g Glas s David R. Morrison An Updat e o n (Small ) Quantu m Cohomolog y Bernd Siebert

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Introduction This book ha s it s origin i n a workshop o n Comple x Geometr y an d Mirro r Sym metry hel d i n th e framewor k o f a them e yea r progra m i n Geometr y an d Topolog y at th e Centr e d e recherche s mathematique s (CRM) , Universit e d e Montreal , i n th e spring 1 995 . I t i s published thank s t o a n Agreemen t betwee n th e America n Math ematical Society , th e CRM , an d Internationa l Pres s an d i s i n som e sens e a seque l to "Mirro r Symmetr y I " an d "Mirro r Symmetr y II " publishe d b y th e America n Mathematical Societ y an d Internationa l Press . During th e workshop , physicist s an d mathematician s activel y discusse d ques tions relate d t o th e developmen t o f strin g theory . Bria n Green e gav e a beautifu l overview o f th e subject . Topic s o n enumerativ e geometry , vecto r bundle s i n supe r Calabi-Yau manifolds , localizatio n techniques , an d analyti c invariant s associate d to K3 surfac e ar e covere d i n thi s collectio n o f papers . We ar e gratefu l t o th e CR M fo r hostin g suc h a fruitfu l meeting . The Editor s

IX

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https://doi.org/10.1090/amsip/010/01 AMS/IP Studie s i n Advance d Mathematic s Volume 1 0 , 1 99 9

Aspects o f Q u a n t u m Geometr y Brian R. Green e ABSTRACT. Thes e lecture s ar e devote d t o introducin g som e o f th e basi c fea tures o f quantu m geometr y tha t hav e bee n emergin g fro m compactifie d strin g theory ove r th e las t coupl e o f years. I n particular , w e discus s th e wa y i n whic h N = 2 superconforma l field theory , i n th e contex t o f strin g theory , give s ris e to geometrica l model s o f spacetim e whos e quantitativ e an d qualitativ e proper ties diffe r significantl y fro m thei r classica l (Genera l Relativistic ) counterparts . After a n overvie w o f som e genera l consequence s o f N = 2 superconformal sym metry, w e specializ e t o concret e fiel d theoreti c representation s o f thi s algebra . We indicat e ho w a numbe r o f familia r construct s fro m classica l geometr y natu rally aris e i n thes e theories , bu t i n a modifie d for m givin g ris e t o wha t i s calle d "stringy" geometry . Ou r mai n focu s i s o n mirro r symmetr y an d a mil d for m of spacetim e topolog y change . W e the n g o o n t o discus s ho w inclusio n o f a particular clas s o f nonperturbativ e quantu m effect s give s ris e t o physica l pro cesses resultin g i n drasti c spacetim e topolog y chang e i n th e contex t o f typ e I I string theory .

1. Introductio n 1.1. Wha t i s quantu m geometry ? Simpl y put , quantu m geometr y i s th e appropriate modificatio n o f standar d classica l geometr y t o mak e i t suitabl e fo r de scribing th e physic s o f strin g theor y W e ar e al l familia r wit h th e succes s tha t many idea s fro m classica l geometr y hav e ha d i n providin g th e languag e an d tech nical framewor k fo r understandin g importan t structure s i n physic s suc h a s genera l relativity an d Yang-Mill s theor y I t i s rathe r remarkabl e tha t th e physica l prop erties o f thes e fundamenta l theorie s ca n b e directl y describe d i n th e mathematica l language o f differentia l geometr y an d topolog y Heuristically , on e ca n roughl y un derstand thi s b y notin g tha t th e basi c buildin g bloc k o f thes e mathematica l struc tures i s tha t o f a topologica l space—whic h itsel f i s a collectio n o f points groupe d together i n some particula r manner . Pre-strin g theorie s o f fundamental physic s ar e also base d o n a buildin g bloc k consistin g o f points—namely , poin t particles . Tha t classical mathematic s an d pre-strin g physic s hav e th e sam e elementar y constituen t is on e roug h wa y o f understandin g wh y the y ar e s o harmonious . Thinkin g abou t 1991 Mathematics Subject Classification. 81 T30 . This wor k i s supporte d b y a Nationa l Youn g Investigato r award , b y th e Alfre d P . Sloa n Foundation an d b y th e Nationa l Scienc e Foundation . Some o f th e materia l i n thi s articl e wa s previousl y publishe d i n anothe r for m b y Worl d Scientific i n String theory on Calabi-Yau manifolds, Field s Strings , an d Duality , pp . 543-726 . © 1 99 9 America n Mathematica l Society , Internationa l Press , an d Centr e d e recherche s mathematique s 1

2

BRIAN R . G R E E N E

things i n thi s manne r i s particularl y usefu l whe n w e com e t o strin g theory . A s th e fundamental constituen t i n th e latte r i s no t a poin t bu t rathe r a one-dimensiona l loop, i t i s natura l t o suspec t tha t classica l geometr y ma y no t b e th e correc t lan guage fo r describin g strin g physics . I n fact , thi s conclusio n turn s ou t t o b e correct . The powe r o f geometry , however , i s no t lost . Rather , strin g theor y appear s t o b e described b y a modified for m o f classica l geometry , know n a s quantu m geometry , with th e modification s disappearin g a s th e typica l siz e i n a give n syste m become s large relativ e t o th e Planc k scal e o f 1 0~ 33 cm. W e shoul d stres s a poin t o f termi nology a t th e outset . Th e ter m quantu m geometry , i n it s mos t precis e usage , refer s to th e geometrica l structur e relevan t fo r describin g a full y quantu m mechanica l theory o f strings . A larg e fractio n o f thes e lectures , though , focu s upo n tre e leve l string theory—tha t is , conformal field theor y o n th e sphere—whic h capture s nove l features associate d wit h th e extende d natur e o f the string , bu t doe s s o at th e clas sical level . A s such , th e ter m quantu m geometr y i n thi s contex t i s a bit misleadin g and a ter m suc h a s "stringy " geometr y woul d probabl y b e mor e appropriate . I n the las t lectur e w e shall trul y includ e quantu m effect s int o ou r discussion . I n fact , nonperturbative quantu m effect s associate d wit h solitoni c degree s o f freedom. Un derstanding th e geometrica l significanc e o f thes e quantu m effect s finally justifie s using th e ter m quantu m geometry . Havin g mad e thi s distinctio n clear , w e will no t be carefu l i n ou r us e o f th e ter m quantu m vs . string y geometry ; typicall y w e shal l use th e former . As w e shal l see , whe n th e typica l siz e i n a strin g compactificatio n doe s no t meet thi s conditio n o f bein g substantiall y large r tha n th e Planc k scale , quantu m geometry differ s bot h quantitativel y an d qualitativel y fro m classica l geometry . I n this sense , on e ca n thin k o f strin g theor y a s providin g u s wit h a generalizatio n of ordinar y classica l geometr y whic h differ s fro m i t o n shor t distanc e scale s an d reduces t o i t o n larg e distanc e scales . I t i s th e purpos e o f thes e lecture s t o discus s some o f the foundation s an d propertie s o f quantu m geometry .

1.2. Th e ingredients . Recen t development s i n strin g theor y hav e take n u s much close r t o understandin g th e tru e natur e o f the full y quantize d theory . Rathe r surprisingly, i t appear s tha t perturbativ e tools—judiciousl y used—ca n tak e u s a long way . I n thes e lecture s w e shal l focu s primaril y o n th e tool s necessar y fo r analyzing strin g theory i n perturbation theory—i n fact , tre e leve l string theor y wil l occupy ou r attentio n unti l th e las t lecture . Ou r ai m i n this sectio n i s to giv e a brie f overview o f perturbativ e strin g theor y i n orde r t o hav e th e languag e t o describ e a number o f recen t developments . As is well known, it is most convenient t o formulate first quantize d strin g theor y in term s o f a 2-dimensiona l quantu m field theor y o n th e worl d shee t swep t ou t b y the string . Th e delicat e consistenc y o f quantizing a n extende d objec t place s sever e constraints o n thi s 2-dimensiona l field theory . I n particular , th e field theor y mus t be a conforma l field theor y wit h centra l charg e equa l t o fifteen. More precisely , w e wil l alway s discus s superstrin g theor y i n whic h cas e thi s field theor y mus t b e superconformall y invariant . (Withou t los s o f confusion , w e will ofte n dro p th e prefi x super.) I n fact , i n essentiall y al l o f our discussio n w e will limit th e scop e o f our attentio n eve n furthe r an d requir e thes e field theories t o hav e two independen t supersymmetrie s o n th e worl d shee t an d henc e w e discus s N — 2

ASPECTS O F QUANTU M GEOMETR Y

3

superconformal field theorie s wit h centra l charg e (bot h lef t an d right ) equa l t o fifteen 1 . Th e reaso n fo r thi s i s two-fold . First , i t i s onl y suc h N = 2 theories whic h can b e show n t o yiel d solution s t o th e strin g equation s o f motio n t o al l order s i n string perturbatio n theory . Hence , th e restrictio n t o N = 2 bolsters ou r confidenc e that ou r conclusion s trul y appl y t o perturbativ e strin g theory . Second , strin g the ories wit h N — 2 worldsheet supersymmetr y giv e ris e t o spacetim e physic s whic h is itself supersymmetric . Spacetim e supersymmetr y i s our bes t hop e fo r addressin g the hierarch y proble m an d henc e ha s stron g theoretica l motivation . Finally , th e constrained structur e of N = 2 theories ha s allowe d u s t o hon e a numbe r o f pow erful tool s whic h greatl y ai d i n thei r understanding . W e ca n thu s pus h bot h ou r physical an d ou r mathematica l analysi s o f thes e theorie s muc h further . Our stud y o f perturbativ e strin g theorie s wit h spacetim e supersymmetr y thu s boils dow n t o a stud y o f 2-dimensiona l N = 2 superconforma l field theorie s wit h central charg e fifteen. Ho w d o w e buil d suc h field theories ? W e wil l stud y thi s question in some detail in the ensuing sections; for no w let u s note the typical setup . In studyin g thes e strin g models , w e will generally assum e tha t th e underlyin g N = 2, c = 1 5 conformal theor y ca n b e decompose d a s th e produc t o f a nT V = 2 , c = 6 theory wit h a n N = 2 , c = 9 theory. Th e forme r ca n the n b e realize d mos t simpl y via a fre e theor y o f tw o comple x chira l superfields , (a s w e wil l discus s explicitl y shortly)—that is , a free theory of four rea l bosons and their fermioni c superpartners . We ca n interpre t thes e fre e boson s a s th e fou r Minkowsk i spacetim e coordinate s o f common experience . Th e c = 9 theor y i s the n a n additiona l "internal " theor y required b y consistenc y o f string theory . Wherea s w e were directl y le d t o a natura l choice fo r th e c = 6 theory , ther e i s n o guidin g principl e whic h lead s u s t o a preferred choic e fo r th e c = 9 theory fro m th e know n huge numbe r o f possibilities . The simples t choice , again , i s si x fre e chira l superfields—tha t is , si x fre e boson s and thei r fermioni c superpartners . Togethe r wit h th e c = 6 theory , thi s yield s ten dimensiona l flat spacetime—th e aren a o f th e initia l formulatio n o f superstrin g theory. I n this cas e the "interna l theory " i s of the sam e for m a s the usua l "externa l theory" an d hence , i n reality , ther e i s n o natura l wa y o f dividin g th e two . Thus , for man y obviou s reasons , thi s wa y o f constructin g a consisten t strin g mode l i s of limite d physica l interes t thereb y supplyin g stron g motivatio n fo r seekin g othe r methods. Thi s problem—constructin g (an d classifying ) N = 2 superconforma l theories wit h centra l charg e 9 t o pla y th e rol e o f th e interna l theory—i s on e tha t has bee n vigorousl y pursue d fo r a numbe r o f years . A s ye t ther e i s n o complet e classification bu t a wealt h o f construction s hav e bee n found . The most intuitive of these constructions ar e those in which six of the ten spatia l dimensions i n th e flat spac e approac h jus t discusse d ar e "compactified" . Tha t is , they ar e replace d b y a smal l compac t si x dimensiona l space , sa y M thu s yieldin g a spacetim e o f th e Kaluza-Klei n typ e M 4 x M wher e M 4 i s Minkowsk i fou r space . It i s crucia l t o realiz e tha t mos t choice s fo r M wil l no t yiel d a consisten t strin g theory becaus e th e associate d tw o dimensiona l field theory—whic h i s no w mos t appropriately describe d a s a nonlinea r sigm a mode l wit h targe t spac e M 4 x M — will not b e conformall y invariant . Explicitly , th e actio n fo r th e interna l par t o f thi s

1

Such theorie s ca n alway s b e converte d int o mor e phenomenologicall y viabl e heteroti c strin g theories bu t tha t wil l no t b e require d fo r ou r purposes .

4

BRIAN R . G R E E N E

theory i s (1.1) S = — ! - r / dzdzlgmnidX^dX 71 + 47rar J

dX

n

BXrn)

+Brnn{dxrn8xn - dx nBxrn) + ...] ,

where g m n i s a metri c o n M , ,B m n i s a n antisymmetri c tenso r field, an d w e hav e omitted additiona l fermioni c term s require d b y supersymmetr y whos e precis e for m will b e give n shortly . To mee t th e criterio n o f conforma l invariance , M must , t o lowes t orde r i n sigma mode l perturbatio n theory 2 , admi t a metri c g^ u whos e Ricc i tenso r R^ v vanishes. I n orde r t o contribut e nin e t o th e centra l charge , th e dimensio n o f M must b e six , an d t o ensur e th e additiona l conditio n o f N = 2 supersymmetry , M mus t b e a comple x Kahle r manifold . Thes e condition s togethe r ar e referre d to a s th e 'Calabi-Yau ' condition s an d manifold s M meetin g the m ar e know n a s Calabi-Yau threefold s (thre e her e refers t o thre e comple x dimensions ; on e can mor e generally stud y Calabi-Ya u manifold s o f arbitrar y dimensio n know n a s Calabi-Ya u d-folds). A consisten t strin g model , therefore , wit h fou r flat Minkowsk i spacetim e directions ca n b e buil t usin g an y Calabi-Ya u threefol d a s th e interna l targe t spac e for a nonlinea r supersymmetri c sigm a model . I f w e take th e typica l radiu s o f suc h a Calabi-Ya u manifol d t o b e smal l (o n the Planc k scale , fo r instance ) the n th e ten dimensional spacetim e M4 x M wil l effectivel y loo k just lik e M 4 (wit h th e presen t level o f sensitivit y o f ou r bes t probes ) an d henc e i s consisten t wit h observatio n a la Kaluza-Klein . W e wil l hav e muc h t o sa y abou t thes e model s shortly ; fo r no w we not e tha t ther e ar e many Calabi-Ya u threefold s an d eac h give s ris e t o differen t physics i n M4 . Havin g n o mean s t o choos e whic h on e i s "right" , w e lose essentiall y all predictiv e power . Calabi-Yau sigm a model s provid e on e means o f building N = 2 superconforma l models tha t ca n b e take n a s th e interna l par t o f a strin g theory . Ther e ar e tw o other type s o f construction s tha t wil l pla y a rol e i n ou r subsequen t discussion , s o we mentio n the m her e a s well . A ke y featur e o f eac h o f thes e construction s i s tha t at firs t glanc e neithe r o f them ha s anythin g t o d o with th e geometrica l Calabi-Ya u approach just mentioned . Rather , eac h approach yield s a quantum field theory wit h the requisite properties bu t i n neither doe s one introduce a curled u p manifold. Th e best wa y t o thin k abou t thi s i s that th e centra l charg e i s a measur e o f th e numbe r of degree s o f freedo m i n a conforma l theory . Thes e degree s o f freedo m ca n b e associated wit h extr a spatia l dimensions , a s i n th e Calabi-Ya u case , bu t a s i n th e following tw o construction s the y d o no t hav e t o be . Landau-Ginsburg effectiv e field theorie s hav e playe d a ke y rol e i n a numbe r o f physical contexts. Fo r our purpose we shall focus on Landau-Ginsburg theorie s wit h N = 2 supersymmetry. Concretely , suc h a theory is a quantum field theory base d on chiral superfield s (a s we shall discuss ) tha t respect s N = 2 supersymmetry an d ha s a uniqu e vacuum state . Fro m ou r discussio n above , to be of use this theory mus t b e conformally invariant . A simpl e bu t non-constructiv e wa y o f doin g thi s i s t o allo w an initia l non-conforma l Landau-Ginsbur g theor y t o flow toward s th e infrare d vi a the renormalizatio n group . Assumin g th e theor y flows t o a nontrivia l fixed poin t (an assumptio n wit h muc h supportin g circumstantia l evidence ) th e endpoin t o f th e 2 That is , t o lowes t orde r i n a!/R? wher e R i s a typica l radiu s o f th e Calabi-Ya u abou t whic h we shal l b e mor e precis e later .

ASPECTS O F QUANTU M GEOMETR Y

5

flow i s a conformall y invarian t N = 2 theory . Mor e explicitly , th e actio n fo r a n N = 2 Landau-Ginsburg theor y ca n b e writte n

(1.2) Jd

2

zd4OK(*u*u...,*n,*n)+nd2zd2OW(yu...,*n)+h.c\

where th e kineti c term s ar e chose n s o a s t o yiel d conforma l invariance 3 an d wher e the superpotentia l W, whic h i s a holomorphi c functio n o f th e chira l superfield s ^ i , i s a t leas t cubi c s o th e ^i ar e massless . (An y quadrati c term s i n W represen t massive fields tha t ar e froze n ou t i n th e infrare d limit. ) By suitably adjustin g th e (polynomial ) superpotentia l W governin g these fields we can achiev e central charg e nine . Mor e importantly , alon g renormalization grou p flows, the superpotentia l receive s nothing mor e than wavefunctio n renormalization . Its form , therefore , remain s fixed an d ca n thu s b e use d a s a label fo r thos e theorie s which al l belon g t o th e sam e universalit y class . Th e kineti c ter m K , o n th e con trary, doe s receive corrections alon g renormalization grou p flows and thu s achievin g conformal invarianc e amount s t o choosin g th e kineti c ter m correctly . W e d o no t know ho w t o d o thi s explicitly , bu t thankfull y muc h o f wha t w e shal l d o doe s no t require thi s ability . The final approac h t o buildin g suitabl e interna l N = 2 theories tha t w e shal l consider i s based upo n th e s o called "minima l models" . A s we shall discuss i n mor e detail i n th e nex t section , a conforma l theor y i s characterize d b y a certai n subse t of its quantum field algebr a know n a s primary fields. Mos t conforma l theorie s hav e infinitely man y primar y fields bu t certai n specia l examples—know n a s minima l models—have a finite number . Havin g a finite numbe r o f primar y fields greatl y simplifies th e analysi s o f a conforma l theor y an d lead s t o th e abilit y t o explicitl y calculate essentiall y anythin g o f physica l interest . Fo r thi s reaso n th e minima l models ar e ofte n referre d t o a s bein g "exactl y soluble" . Th e precis e definitio n o f primary field depends , a s w e shal l see , o n th e particula r chira l algebr a whic h a theory respects . Fo r non-supersymmetri c conforma l theories , th e chira l algebr a is tha t o f th e conforma l symmetr y only . I n thi s case , i t ha s bee n show n tha t only theorie s wit h c < 1 can b e minimal . On e ca n tak e tenso r product s o f suc h c < 1 theorie s t o yiel d ne w theorie s wit h centra l charg e greate r tha n on e (sinc e central charge s ad d whe n theorie s ar e combine d i n thi s manner) . I f ou r theor y has a large r chira l algebra , sa y th e N = 2 superconforma l algebr a o f interes t fo r reasons discussed , the n ther e ar e analogou s exactl y solubl e minima l models . I n fact, the y ca n b e indexe d b y th e positiv e integer s P e Z an d hav e centra l charge s cp = 3P/(P + 2) . Again , eve n thoug h thes e value s o f th e centra l charg e ar e les s than th e desire d valu e o f nine, w e can tak e tenso r product s t o yiel d thi s value . (I n fact, a s w e shal l discuss , i t i s no t quit e adequat e t o simpl y tak e a tenso r product . Rather w e nee d t o tak e a n orbifol d o f a tenso r product. ) I n thi s wa y w e can buil d internal N — 2, c = 9 theorie s tha t hav e th e virtu e o f bein g exactl y soluble . I t i s worthwhile to emphasize that, a s in the Landau-Ginsburg case , these minimal mode l constructions d o no t hav e an y obviou s geometrica l interpretation ; the y appea r t o be purel y algebrai c i n construction . In th e previou s paragraph s w e have outline d thre e fundamenta l an d manifestl y distinct way s o f constructin g consisten t strin g models . A remarkabl e fact , whic h 3

Usually th e kineti c term s ca n onl y b e define d i n thi s implici t form—o r i n th e slightl y mor e detailed bu t n o mor e explici t manne r o f fixe d point s o f the renormalizatio n grou p flow, a s w e shal l discuss later .

6

BRIAN R . G R E E N E

will play a crucial rol e in our analysis , i s that thes e thre e approache s ar e intimatel y related. I n fact, b y varying certain parameter s w e can smoothly interpolat e betwee n all three . A s w e shal l see , a give n conforma l fiel d theor y o f interes t typicall y lie s in a multidimensiona l famil y o f theories relate d t o eac h othe r b y physicall y smoot h deformations. Th e paramete r spac e o f suc h a famil y i s known a s it s modul i space . This modul i spac e i s naturall y divide d int o variou s phase region s whos e physica l description i s mos t directl y give n i n term s o f on e o f th e thre e method s describe d above (an d combination s thereof ) a s wel l a s certai n simpl e generalizations . Thus , for som e rang e o f parameters , a conforma l field theor y migh t b e mos t naturall y described i n term s o f a nonlinea r sigm a mode l o n a Calabi-Ya u targe t space , fo r other range s o f parameter s i t migh t mos t naturall y b e describe d i n term s o f a Landau-Ginsburg theory , whil e fo r ye t othe r value s th e mos t natura l descriptio n might b e som e combine d version . W e will se e that physic s change s smoothl y a s we vary th e parameter s t o mov e fro m regio n t o region . Furthermore , i n som e phas e diagrams ther e ar e separat e region s associate d wit h Calabi-Ya u sigm a model s o n topologically distinc t spaces . Thus , sinc e physic s i s smoot h o n passin g fro m an y region int o an y othe r (i n th e sam e phas e diagram ) w e establis h tha t ther e ar e physically smoot h spacetim e topolog y changin g processe s i n strin g theory . The phenomeno n o f physicall y smoot h change s i n spatia l topolog y i s on e ex ample o f th e wa y i n whic h classica l mathematic s an d strin g theor y differ . I n th e former, a chang e i n topolog y i s a discontinuou s operatio n wherea s i n th e latte r it i s not . Mor e generally , man y pivota l construct s o f classica l geometr y naturall y emerge i n th e descriptio n o f physica l observable s i n strin g theory . Typicall y suc h geometrical structure s ar e foun d t o exis t i n strin g theor y i n a "modified " form , with th e modification s tendin g t o zer o a s th e typica l lengt h scal e o f the theor y ap proaches infinity . I n thi s sense , th e classica l geometrica l structure s ca n b e viewe d as specia l case s o f thei r strin g theoreti c counterparts . Thi s ide a encapsulate s wha t is mean t b y th e phras e "quantu m geometry" . Namely , on e ca n see k t o formulat e a ne w geometrica l disciplin e whos e basi c ingredient s ar e th e observable s o f strin g theory. I n appropriate limitin g cases, this discipline should reduce to more standar d mathematical area s suc h a s algebrai c geometr y an d topology , bu t mor e generall y can exhibit numerou s qualitatively differen t properties . Physicall y smoot h topolog y change i s on e suc h strikin g qualitativ e difference . Mirro r symmetry , th e phenom enon i n whic h tw o Calabi-Ya u manifold s whic h ar e distinc t fro m th e viewpoin t o f classical geometr y (the y are , i n fact , topologicall y distinct ) ar e identical fro m th e viewpoint o f quantu m geometr y a s the y giv e ris e t o isomorphi c strin g models . I n the followin g discussio n ou r ai m shal l be to cove r som e of the foundationa l materia l needed fo r a n understandin g o f quantu m geometr y o f strin g theory . All o f th e discussio n abov e ha s it s technica l root s i n propertie s o f th e N = 2 superconformal algebra . Hence , i n th e nex t sectio n w e shal l discus s thi s algebra , its representatio n theory , an d certai n othe r ke y propertie s fo r late r developments . We shal l als o giv e som e example s o f theorie s whic h respec t th e N = 2 algebra . In Sectio n 3 w e shal l broade n ou r understandin g o f suc h theorie s b y studyin g ex amples whic h ar e smoothl y connecte d t o on e anothe r an d henc e for m a famil y o f N = 2 superconformal theories . Namely , w e shal l discus s som e simpl e aspect s o f moduli space s o f conforma l theories . I n Sectio n 4 we shal l furthe r discus s som e o f the example s introduce d i n Sectio n 2 and poin t ou t som e unexpecte d relationship s between them . Thes e result s wil l b e use d i n Sectio n 5 t o discus s mirro r symme try. I n Sectio n 6 w e shal l appl y som e propertie s o f mirro r symmetr y t o establis h

ASPECTS O F QUANTU M GEOMETR Y 7

that strin g theor y admit s physicall y smoot h operation s resultin g i n th e chang e o f spacetime topology . These lecture s assum e som e familiarity wit h th e essentia l feature s o f conforma l field theory . Th e reade r uncomfortabl e wit h thi s materia l migh t wan t t o consult , for example , [42] . Whe n discussin g certai n importan t bu t wel l know n backgroun d material i n th e following , w e will conten t ourselve s wit h givin g referenc e t o variou s useful revie w articles rather than givin g detailed references t o the original literature . 2. Th e N = 2 superconforma l algebr a Good reference s fo r th e materia l i n thi s sectio n ar e [54] , the revie w article s o f [40,60,67] reference s therein . 2.1. Th e algebra . I t ha s lon g bee n know n tha t th e consistenc y o f string the ory demand s tha t w e describ e it s groun d state s i n term s o f tw o dimensiona l con formal field theorie s o f particular centra l charg e (dependin g o n th e loca l symmetr y algebra o f the particular strin g bein g studied). Fo r superstrings , thi s central charg e must b e fifteen. A typical setting for studyin g superstrin g theor y i s to realize a central charg e o f fifteen vi a M 4 x {anT V = 2 , c = 9 superconformal field theory } wher e by M 4 w e mea n Minkowsk i space—o r mor e precisely , th e c = 6 superconforma l field theory o f fou r fre e boson s an d thei r superpartners . Actually , thi s formulatio n would yield a theory whose Hilbert spac e is larger tha n tha t o f the string; rather , w e should wor k i n ligh t con e gauge i n which a total centra l charg e o f twelve i s realize d by th e abov e constructio n wit h M4 replace d b y th e c = 3 free conforma l theor y o f two fre e (transverse ) boson s an d thei r superpartners . A s discussed , th e restrictio n to N = 2 theories i s not fundamentall y require d bu t i t doe s give rise to a number o f important propertie s suc h a s perturbativ e stability , spacetim e supersymmetr y an d enhanced analyti c control . Fo r thes e reasons , w e shal l focu s exclusivel y o n N = 2 , c = 9 superconformal theories . To begi n ou r quantitativ e stud y o f thes e theories , let s first writ e dow n th e su perconformal algebra . A s with more familiar algebra s base d o n compact Li e groups, the superconforma l algebr a ca n b e expresse d i n term s o f th e (anti)commutator s o f its generators . Unlik e th e cas e o f a compac t Li e algebra , though , ther e ar e a n infinite numbe r o f generator s i n thi s case . I t i s especiall y convenien t t o writ e th e algebra i n term s o f th e operato r product s o f it s generators . Thi s contain s th e same informatio n a s the (anti)commutator s o f the mode s o f the generators , bu t fo r completeness w e will write both . Lets star t i n th e mor e familia r settin g o f the (N = 0 ) conforma l algebra . Fo r a given conforma l theory , thi s algebr a i s generate d b y th e stres s energ y tenso r T(z) whose operato r produc t wit h itsel f take s th e followin g for m

where c is the centra l charg e o f th e theory . T o write thi s i n term s o f commutators , we define th e mode s L n o f T(z) accordin g t o

(2.1) T{z)

= Y

JLnZ~

n 2

-.

n

Then, th e operato r produc t expansio n abov e implie s (2.2) [L

n,

Lm] = (n - m)L n+m +

T^n(n 2 - l ) 5

m+n>0.

8

BRIAN R . GREEN E

To extend thi s algebr a to the N = 1 superconformal algebra , w e include a n ad ditional generato r G(z) whic h i s the (worldsheet ) superpartne r o f the stres s energ y tensor T(z) an d ha s conforma l weigh t 3/2 . Th e operato r produc t o f T wit h itsel f i s unchanged an d th e algebr a i s thus determine d b y th e operato r produc t expansio n above an d th e followin g tw o equations :

+ G^ ± J ca n be writte n [L n , Gf] = (n/ 2 — s)Gn+ s fo r an y relevant choic e o f s (integra l o r half-integral) . With thes e notationa l conventions , le t u s now define : L'n = L n-\- rjr]J n + -77 2 = ty 1 + i\I> 2. A s * i s meant t o be th e superpartne r o f X an d w e ar e familia r tha t th e equation s o f motio n fo r X show tha t i t split s int o th e su m o f a lef t movin g (holomorphic ) an d righ t movin g (anti-holomorphic) part , w e ca n mor e precisel y introduc e a lef t movin g comple x fermion ^(z) (wit h comple x conjugat e ^*(z) , an d a righ t movin g fermio n X(z) (with comple x conjugat e A*(z) . Th e actio n fo r thi s theor y ca n b e writte n i n th e familiar for m (focusin g jus t o n th e holomorphi c part ) (2.43) S

d2z {dXdX* + ^d^ +

= f

^d^).

Our clai m i s tha t thi s theor y ha s N = 2 superconformal symmetry . I n fact , mor e precisely, i t ha s thi s symmetr y i n bot h th e holomorphi c an d anti-holomorphi c sec tors an d henc e has what i s usually denote d (2 , 2) worldsheet supersymmetry . Thei r are man y way s t o see this , th e mos t direc t o f whic h i s t o construc t th e generator s of th e N — 2 algebra directl y fro m th e fields definin g th e theory . Thus , th e reade r is urge d t o chec k tha t (2.44) T(z)

= -dXdX* +

(2.45) G+(z)

= \rdX,

(2.46) G~(z)

= yl>dX\

(2.47) J(z)

=

^

^i/>*dil> + ^

W ,

,

do i n fac t hav e th e correc t N = 2 superconformal operato r product s give n earlier . This theory has central charge 3 (in both th e holomorphic an d anti-holomorphi c sectors) comin g fro m tw o bosoni c degree s o f freedo m ( c = 2 ) an d tw o fermioni c degrees o f freedo m ( c = 1 ) . Let s explicitl y wor k ou t th e variou s chira l rings . A s discussed, quit e generally , a field satisfyin g h = ±Q/2 i s a chira l o r antichira l primary. Notic e tha t th e fields ^,^*, A an d A * al l satisf y thi s relatio n (o n bot h

BRIAN R . G R E E N E

18

the holomorphi c an d anti-holomorphi c sides) . Furthermore , appropriat e product s such a s ipX which has (ft , ft) = ( | , \) an d (Q , Q) = (1 ,1 ) an d henc e i s also a (chiral , chiral) = (c , c) ring field. Bearin g in mind that n o (chiral, chiral) fiel d ha s conforma l weights (ft , ft) greater tha n (c/6 , c/6) (wher e c her e i s th e centra l charge , equa l t o three i n thi s case) , w e se e tha t w e hav e exhauste d full y th e (chiral,chiral ) ring . Hence th e (chiral,chiral ) i n thi s exampl e consist s o f { l , ^ , A,^A}. Similarl y th e (antichiral, chiral) = (a,c) rin g consist s o f {1 ,^* , A,'0*A} , an d th e othe r tw o ring s are gotte n fro m thes e b y comple x conjugation . Although a ver y simpl e theory , thi s exampl e doe s pla y a ke y rol e i n strin g theory. Namely , we build string theory with four extende d dimensions , a s discussed , by th e constructio n M 4 x { c = 9,i V = 2 conformal theory } wher e M 4 reall y refer s t o a c = 6,7 V = 2 free superconforma l theor y (fre e becaus e o f the restrictio n t o flat spacetime). I n light con e gauge, the latter become s a c — 3 , N = 2 free theor y - that is, th e theor y jus t constructed . Hence , ou r exampl e correspond s t o th e extende d part o f spacetim e i n strin g theory . 2.5.2. Nonlinear sigma models. Ou r secon d exampl e i s tha t o f a n N = 2 su perconformal nonlinea r sigm a model . I n reality , thi s i s nothin g bu t a simple , ye t rich, generalizatio n o f th e previou s fre e fiel d theor y example . Namely , w e includ e more bosoni c fields , mor e fermioni c field s (th e partner s t o th e bosons ) an d w e n o longer requir e th e theor y t o b e free . Rather , w e imagin e tha t th e boson s ar e co ordinates o n a "targe t space " whic h migh t b e a curve d Riemannia n manifol d wit h nontrivial metric . (I n th e previou s case , on e ca n thin k o f X a s a coordinat e o n th e flat manifol d C 1 wit h trivia l Euclidea n metric. ) Th e fermion s ca n the n b e viewe d as section s o f th e (pullback ) o f the tangen t bundl e o f the targe t space . Concretely , we can writ e th e actio n fo r suc h a theor y a s (2.48) S=£^

f

d

2

z \^ 9(^(X)dzX»dzX» +

g^{^D- zr +

A^A" )

where g^ i s th e metri c o n th e targe t manifol d an d R^vpcr is it s Rieman n tensor . (We note tha t w e have no t explicitl y writte n th e ter m involvin g th e antisymmetri c tensor field. ) W e no w se t abou t t o determin e unde r wha t condition s thi s theor y has (2 , 2) superconforma l symmetry . Let s begi n wit h (2,2 ) supersymmetry . I n general, thi s theor y doe s no t posse s thi s symmetry , bu t a s show n i n [74 ] i f th e target manifol d i s a complex Kdhler manifol d the n i t does . Th e easies t wa y t o see thi s i s to not e tha t th e actio n (2.48 ) ca n b e explicitl y writte n i n a superspac e formalism. A s this i s discussed i n th e lecture s i n [31 ] , we will not bothe r t o d o thi s here i n detai l bu t simpl y writ e th e N — 2 superspace versio n o f (2.48) : (2.49) S=

- ^f

d

2

zdA6K{X\Xi),

where th e X % ar e chira l superfield s whos e lowes t component s ar e th e bosoni c coor 2 dinates abov e an d g^ = (2i/7r)d K/(dXidX^). What abou t conforma l invariance ? I n genera l th e actio n (2.48 ) i s no t confor mally invariant . A direc t wa y t o se e thi s i s t o calculat e th e ft functio n fo r th e metric g viewin g i t a s a couplin g "constant " i n thi s two-dimensiona l theory . Th e well known resul t (ignorin g th e dilato n an d antisymmetri c tenso r fields ) i s that th e

ASPECTS O F QUANTU M GEOMETR Y

19

(3 function, t o lowes t order , i s proportional t o the Ricci tenso r o f the target man ifold. Thus , w e can achieve conforma l invarianc e b y choosing ou r target manifol d to hav e a metric wit h vanishin g Ricc i tensor. Thi s i s a highly restrictiv e constraint . Our conclusio n is that (2,2 ) superconformal symmetr y implie s that th e target man ifold mus t b e complex, Kahle r an d admit a metric o f vanishing Ricc i tensor. Thes e conditions ar e the definin g propertie s o f Calabi-Yau manifolds. Thus , a nonlinea r sigma mode l wit h a Calabi-Ya u targe t spac e give s u s anothe r metho d o f buildin g (2, 2) superconformal field theories . How d o we construct th e (c , c) an d (a , c) fields i n these theories ? Th e answe r to thi s i s quite beautifu l an d goes back t o the work o f Witten i n his paper [68] . As we wil l no w discuss, thes e fields ar e closel y associat e wit h cohomolog y group s o n the targe t Calabi-Ya u space . To understan d thi s result , w e will approac h i t i n the manner employe d i n [68] taking int o accoun t tha t ou r targe t spac e i s a comple x Kahle r manifold . A s emphasized b y Witte n [68] , states whic h hav e nonzer o momentu m i n suc h a theor y necessarily hav e nonzer o energy . Thus , i n our effort t o understand th e zero-energ y modes—that is , Ramond groun d state s fro m whic h w e can get the (c , c) an d (a , c) rings by spectral flow—we should restric t attentio n t o zero momentum modes . Th e latter mode s are those which have no spatial dependenc e on the worldsheet. Hence , even befor e w e quantize th e theory, w e can simply dro p th e spatial dependenc e of the fields i n the action , thereb y effectivel y reducin g ou r mode l t o supersymmetri c quantum mechanic s o n a Kahler manifold . To analyz e thi s theory , first not e tha t th e Majorana-Weyl fermion s i n our action, restricte d t o their constan t mod e components (assumin g tha t th e fermions are periodic, i.e . in the Ramond secto r i n the common strin g parlance ) satisfy :

(2.50) Wrf}

=

{\V} = 0; W^} =

g l\

and similarl y fo r th e A fermions. W e see , therefore , tha t th e Kahle r structur e supplies u s wit h a natura l polarizatio n tha t allow s u s t o thin k of , say , the i\P as creation operators and the ^ a s destruction operators . Fo r ease of present notation , we wil l tak e th e opposit e conventio n fo r th e A fermions . Namely , w e will tak e A-7 to b e creation operator s an d A J to be destruction operators . W e will come bac k t o this poin t shortly . Consider no w the world sheet supersymmetr y operato r Q. Mor e precisely, sinc e we have (2,2 ) world shee t supersymmetry , ther e ar e two left movin g an d two right moving supersymmetr y operator s Q L , I ? QL,2, QR,I, QR,2- W e will focus o n the lef t moving operator s fo r mos t o f our discussion. I n term s o f fields, we have t o lowes t order, QL,I = 9^u^d zXv an d QL, 2 = 9^i?^d zXv'. I n th e N = 2 languag e w e have developed , thes e tw o operators aris e fro m takin g th e contou r integra l o f the worldsheet supercurrent s G +(z) an d G~(z) an d henc e w e write QL,I — G$ an d QR,I = GQ . Also , d zXv i s 7r^, the momentum conjugat e t o X v, an d hence ca n be thought o f as the functiona l derivativ e 8/8X v'. Whe n restricte d t o the zero mod e sector thi s become s th e ordinar y (covariant ) derivative , V^ wit h respec t t o X v. Thus, i n this zer o mod e approximatio n w e have (2.51) G

Q = VV

and (2.52) G

+ =

D

20

BRIAN R . GREEN E

Now, carryin g o n wit h ou r interpretatio n o f th e Ferm i zer o mode s i n term s of creatio n an d annihilatio n operators , let s choose a Foc k vacuu m |0 ) fo r ou r zer o mode secto r o f the Hilber t spac e o f state s suc h tha t ) = A f|0)=0.

(2.53) ^ | 0

Then, a genera l stat e ca n b e writte n

(2.54) |$

> = X X " ^ . . . ^1. . .\Hh •

• • V'-IO),

r,s

where we also sum over all repeated indices . Let s focus our attention on the buildin g blocks o f suc h states , namel y thos e wit h fixe d value s o f th e integer s r an d s. Suc h a stat e ha s U ( 1 ) L X U ( 1)R charge s (—r , s). Because of the anti-commutatio n propertie s o f these Fermi operators, thi s stat e is completel y antisymmetri c unde r interchang e o f an y tw o holomorphic , o r an y two anti-holomorphi c indices . Therefore , w e se e tha t th e spac e o f suc h stat e i s isomorphic to the space of (r , s) forms o n M. Now , a s i n previou s sections , let s demand tha t thi s stat e b e annihilate d b y th e tw o supercharges , i.e . b y GQ an d Go". Suc h a state , a s w e hav e described , wil l li e i n th e Ramon d groun d stat e an d hence wil l b e relate d t o (c , c) an d (a , c) field s b y appropriat e spectra l flow. Fro m (2.51) an d (2.52 ) (an d usin g the anti-commutatio n relation s o f the Ferm i fields ) w e see tha t th e former , actin g o n suc h states , i s isomorphi c t o th e operato r 8 actin g on th e correspondin g differentia l form , an d th e latter , actin g o n suc h states , i s isomorphic t o th e operato r 8^ actin g o n th e correspondin g differentia l form . Thus , demanding thes e operator s annihilat e th e stat e i s mathematicall y equivalen t t o finding harmonic (r , s) form s o n M . Therefore , w e hav e explicitl y show n tha t th e Ramond groun d state s i n suc h theorie s ar e i n on e t o on e correspondenc e wit h th e elements o f cohomolog y o n M . We have not ye t complete d ou r analysis , a s the reader ma y hav e noted, becaus e of th e arbitrar y choic e mad e i n (2.53) . Namely , whic h operator s ar e goin g t o b e interpreted i n term s o f creatio n vs . destructio n operators . Whe n deciding , say , between ij) % an d ^ 2 , eithe r choic e i s equivalent ; it s i s just a matte r o f convention . However, afte r makin g suc h a conventiona l choice , th e distinctio n betwee n A * and X1 is now on e o f content . Thus , i n additio n t o (2.53 ) w e shoul d als o conside r ) =A i |0> = 0 .

(2.55) ^*|0

Then, w e consider state s o f th e for m (2.56) |$

) = £ & £ ; ; ; £ A , , • . • K^ Jl •

• • tf' |0> ,

r,s

where A * = gijX^. Thes e state s hav e U(1 ) L X U ( 1)R charge s (r , s). Th e sam e analysis a s before show s these state s t o b e in on e to on e correspondenc e wit h (0 , s) forms takin g value s in A rT wher e T i s the holomorphi c tangen t bundl e o f M. Now , applying the condition s that th e supercharge s annihilat e suc h a state show s it t o b e harmonic, that is , a member o f the Dolbeaul t cohomolog y grou p # ? ' s ( M , A r T ) . T o make th e notatio n symmetric , th e (r , s) form s w e previously foun d ca n b e though t of a s lyin g i n th e cohomolog y grou p H^ 8 (M , Ar T*) wher e T * i s th e holomorphi c cotangent bundle . In our subsequen t discussio n o f these models, we shall be focusing ou r attentio n on certai n subset s o f th e (c,c) an d (a , c) rings . Mor e specifically , w e shal l b e looking a t element s whos e lef t an d righ t charge s hav e equa l absolut e values . Th e

A S P E C T S O F Q U A N T UM G E O M E T R Y

21

reader shoul d chec k that th e (c , c) states o f this for m arise , afte r spectra l flow , fro m H^p(M,ApT) an d th e (a,c) state s similarl y aris e fro m H° B'P(M, A*T*), fo r p = 0, 1,2,3. Let u s als o not e tha t an y elemen t i n H~ s (M, A r T) ca n b e associate d wit h a n (3 — r , 5) harmoni c for m o n M vi a contractio n wit h th e holomorphi c (3,0 ) for m Q, whic h al l Calabi-Ya u manifold s hav e b y virtu e o f th e trivialit y o f th e canonica l bundle. An importan t an d interestin g questio n i s to not onl y addres s th e geometrica l realization of the (anti)chira l primary fields, a s we have done, but als o to understan d the geometrica l interpretatio n o f the rin g structur e amongs t thes e fields . W e wil l not discus s thi s now , bu t wil l retur n t o it later. 2.5.3. Landau-Ginsburg models. Landau-Ginsbur g effectiv e field theorie s hav e played a prominent rol e in many area s o f physics. Usin g som e simple reasoning, w e shall shortly see that the y can be put t o great us e in the present setting . First , le t u s see how one can us e Landau-Ginzburg theor y t o construct N = 2 superconformall y invariant fiel d theories . Our basi c ingredien t i s a field theoretic realizatio n o f th e chira l an d antichira l representations o f the N = 2 superconformal algebra . Toward s thi s end , w e in troduce differentia l operato r realization s o f the N = 2 supersymmetry generator s G^ 1 / 2 an d G_ 1 /2- W e do thi s in a standard superspac e formalis m vi a

and similarl y fo r it s comple x conjugate . I n this representation , a chiral superfiel d 3> = $(2, z,Q±,6±) i s one tha t satisfie s (2.58) £>+

$ = V+$ = 0 .

In terms o f fields o f this type , we can buil d a n N = 2 supersymmetric quantu m field theory b y takin g a n actio n o f th e for m (2.59) S

= fi,... , ^ n]/(d^j W ( $ i , . . . , 3> n)). Concretely, thi s i s al l polynomial s i n th e chira l fields modul o relation s o f th e for m d^j W = 0 . Fo r instance , i f w e have a theor y wit h a singl e field $ appearin g i n W to th e n t h power , th e (c , c) rin g ha s element s {1 , , 2,..., n_2}. I n [54 ] it i s also shown tha t th e (a , c) rin g i n suc h theorie s i s trivial, consistin g onl y o f th e identit y element. Later , w e shall discus s orbifold s o f these theorie s fo r whic h bot h th e (c , c) and (a , c) ring s ar e nontrivial . 2.5.4. Minimal models. I n th e contex t o f nonsupersymmetri c (T V = 0 ) confor mal field theory , i t i s well know n tha t th e necessar y condition s fo r unitar y highes t weight representation s o f th e Virasor o algebr a constrai n th e value s o f th e centra l charge an d th e conforma l weight s o f the primar y fields o f th e theor y a s follows : (2.60) c

>1 ,h

>0

or (2.61) c

=1 ,

— ,h ra(ra+l)' '

((m + l)p — mq) — 1 "" 4 m ( r a + l ) 1 < p < m — 1 ,1

p,q(m)J = PiqK

< q < p.

In th e latte r case , thes e condition s ar e sufficien t a s wel l a s necessar y an d th e corresponding theories, labeled by the value of m, ar e called minimal models. Notic e that ther e ar e a finite numbe r o f primar y fields; thi s i s i n contras t t o th e fac t tha t for c > 1 we necessaril y hav e a n infinit e numbe r o f primar y fields. Thi s fact , i n conjunction wit h the conformal War d identities , allow s for the complete and explici t solution o f thes e theories . Tha t is , we can explicitl y calculat e correlatio n function s in thes e theories . For the supersymmetric cases , the situation i s quite similar. Fo r either i V = 1 or N = 2 we again hav e a n infinit e sequenc e o f unitar y theories , labele d b y a positiv e integer p , whic h hav e a finite numbe r primar y fields (wit h respec t t o th e extende d chiral algebr a whic h include s th e supercurrents) . Thes e theorie s ar e als o calle d minimal models. Fo r N = 1 , the minimal model s have c = 3( 1 —8/[(p+2)(p+4)])/2 while fo r N = 2 the y hav e centra l charg e c = Sp/(p + 2) . W e wil l retur n t o a discussion o f th e N = 2 minimal model s late r i n thes e lectures . 3. Familie s o f N = 2 theorie s 3.1. Margina l operators . I n tw o dimensiona l field theory , a n operato r o f conformal weigh t (h,h) i s sai d t o b e irrelevan t i f h + h > 2 , relevan t i f h + h < 2 5 Actually, a mor e correc t statemen t i s tha t th e onl y renormalizatio n suffere d b y th e super potential arise s fro m wavefunctio n renormalization . I f th e superpotentia l i s quasi-homogeneous , as w e hav e required , thi s renormalizatio n i s absorbe d b y a n overal l rescalin g tha t i n effec t leave s the superpotentia l unchanged .

23

ASPECTS O F QUANTU M GEOMETR Y

R

FIGURE

1 . Th e modul i spac e o f conformal theor y o n a circle.

and marginal if h+h — 2. This terminology arise s from studyin g what happen s if we deform a given original theory by such operators, and then allow the renormalizatio n group t o flow u s t o a n infrare d fixed point . I f th e operato r ha s h + h > 2 it will hav e n o effec t o n the theor y a t it s infrare d fixed point—th e R G drive s it s coefficient t o zero. I t is like addin g a nonrenormalizable highe r dimensio n ter m in four dimensions . Suc h operators , b y dimensional analysis , ar e suppressed b y some energy scal e EQ and hence a t lo w E (th e infrared limit ) the y ar e suppressed. I f h + h < 2 , suc h a n operator , vi a above reasonin g ca n be dominan t an d have a significant effec t o n the properties o f the theor y i n the infrared limit—i n fact , the y can mak e th e theory trivia l i n infrared limit . Of mos t interes t t o our present stud y ar e operators wit h h + h = 2 , and in particular, w e will stud y spinles s operator s wit h h = h = 1 . Thes e operator s deform a give n conforma l field theor y t o a "nearb y conforma l field theory " o f the same centra l charg e an d thereby generat e a family o f conformal field theorie s all continuously relate d t o one another. The simples t exampl e o f this is the case of N = 0, c = 1 conformal field theor y which ca n be realized, say , by a free boso n o n a circle of radius RQ:

sRo = Id with X ~ X + 2ITR 0. NOW SRQ b y this operator : (3.1) S

,

2

zdxBx

consider th e (1 ,1 ) operato r O = dXdX. W e can defor m

Ro

->S R = SRo + ejd

2

zO(z,z).

This i s just (3.2) S

R

= (1 + e) f d 2z dXdX.

Now, lettin g X = y/1 + eX thi s become s

(3.3) S

R=

Id

2

zdXdX,

with X ~ X + 27ri?o(v / TTe). So, the marginal operato r O = dXdX ha s the effect o f changing th e radius of the targe t spac e circle . Th e family o f conforma l field theorie s thereb y generate d consists o f c = 1 conformal field theorie s o f a free boso n o n a circle wit h arbitrar y radius R. (I n fact , i t ca n be show n tha t theorie s wit h radiu s R an d 1 /R ar e isomorphic s o need onl y conside r R > 1.) The Modul i Spac e of such theorie s i s thus: Actually, on e should not e tha t i n an arbitrar y conforma l field theor y no t all (1,1) operator s ca n be used to deform th e theory i n the manner described . Rather , some (1 ,1 ) operators ceas e to be (1,1) operators afte r perturbatio n o f the origina l theory. Suc h operator s ca n therefore no t be used t o move around th e moduli spac e as the y spoi l th e conformal symmetry . Th e collection o f operators whic h continu e

BRIAN R . G R E E N E

FIGURE

2 . Schemati c drawin g o f the modul i spac e o f a n N = 2 theory.

FIGURE 3 . Th e tw o sector s o f a n N = 2 moduli space . to b e (1 ,1 ) afte r perturbatio n b y an y othe r (i n th e collection ) ar e sai d t o b e truly marginal an d i t i s these operator s upo n whic h w e focu s ou r attention . By th e superconforma l War d Identitie s [32 , 33] i t ca n b e show n tha t amon g these operator s i n th e theorie s w e study are : 1. Le t G (c,c) rin g wit h h = h=\^Q — Q — 1 . The n defin e 0 b y §G~{z)(w,w) = 4)(w,w). $ ha s h=\ + \ = l, h=\', Q = 0 an d Q = 1 . The n le t $(i,i)(w,w ) = § dzG~(z)(j)(w,w). $(i,i ) ha s h = h = 1 , Q = Q = 0-$(i,i ) i s a trul y margina l operator . 2. Le t (j> G (a, c) rin g wit h h = h = | an d Q = - 1 Q = 1 . Followin g above , let *(_!,! ) = f G+(*i ) f G-(z 2) = ( G + 1 / 2 G : i / 2 0 ) ( w , w ) . $ ( _ M ) has fc = /i = 1 , Q = 0 , Q = 0 . I t i s truly marginal . We wil l focu s o n thes e trul y margina l operator s $(i ) i)$(_i j i) an d thei r (lowe r component) superpartner s i n th e (c , c) an d (a , c) rings . 3.2. Modul i spaces . I n analog y wit h ou r drawin g i n Figur e 1 fo r th e on e dimensional modul i spac e fo r a fre e boso n o n a circle , wit h th e radiu s o f this circl e being parameterize d b y th e modul i spac e variable i? , we can dra w th e multidimen sional modul i spac e fo r a continuousl y connecte d famil y o f N = 2 superconforma l field theories, schematically , a s Each poin t i n thi s modul i spac e correspond s t o on e N = 2 superconforma l theory. W e ca n defor m an y give n theor y int o anothe r b y usin g th e trul y mar ginal operators . I t ca n b e show n that , a t leas t locally , th e conforma l field theor y Zamolodchikov metri c o n thi s modul i spac e i s bloc k diagona l betwee n th e 3>(i,i) type an d $(-1 ,1 ) -type margina l operator s an d henc e w e ca n thin k o f modul i spac e as bein g a metri c produc t (a t leas t locally ) o f th e schemati c form .

ASPECTS O F QUANTU M GEOMETR Y

FIGURE

FIGURE

25

4 . Kahle r an d comple x structur e deformation s o f a torus .

5 . Geometrica l interpretatio n o f the CF T modul i spac e i n 2 .

One o f ou r mai n goal s i s t o full y understan d thi s pictur e i n detail , especiall y when w e ar e startin g fro m a n initia l theor y realize d a s a n N = 2 superconforma l sigma mode l o n som e Calabi-Ya u space . So , a s a start , i f w e buil d a n N = 2 superconformal fiel d theor y a s a nonlinea r sigm a mode l o n a Calabi-Ya u targe t space M , ca n w e giv e a geometri c interpretatio n t o th e tw o type s o f margina l operators (tha t w e expect t o exis t i f c > 3 so /i max = c/ 6 > \)1 Well, we already described th e way in which (c , c) fields correspond t o harmoni c (2,1) form s an d ho w (a , c) fields fields correspon d t o harmoni c (1 ,1 ) forms . Let's look at th e corresponding $(-i,i ) margina l operator . T o lowest order , th e (a,c) field ca n b e writte n a s bijX 1 ^ wit h b^ bein g a harmoni c (1 ,1 ) for m o n M . Using th e ma p betwee n (a , c) fields an d margina l operator s give n above , w e ma p this t o G^ 1 ,2GZ1 /2 (bijX 1 ^)- T o lowes t orde r again , thi s i s bijdX ldX^. This operator , a s i n th e cas e o f th e circle , deform s th e "size " o f M , i.e. , th e Kahler for m igijdX 1 A dX^ o n M . Similarly , th e $(i,i ) defor m th e "comple x struc ture" o f M—i.e., th e shape of M. I n case the reader i s not familia r wit h thes e idea s of deformin g th e Kahle r for m o r th e comple x structur e o n a Calabi-Ya u manifold , the followin g figure illustrate s thes e deformation s i n th e cas e o f a one-dimensiona l Calabi-Yau manifold , namel y th e torus . Th e origina l toru s i s draw n wit h a soli d line an d it s deforme d version s ar e draw n wit h dotte d lines . A Kahle r deformatio n leaves th e shap e (th e angl e betwee n th e cycles ) fixed, bu t change s th e volume . A complex structur e deformatio n leave s th e volum e fixed, bu t change s th e shape . We have thu s foun d tha t correspondin g t o th e tw o types o f operators tha t ca n deform ou r N = 2 superconformal theor y t o a nearby theory , ther e ar e two geomet rical operation s tha t defor m a Calabi-Ya u manifol d t o a nearb y manifold , withou t spoiling th e Calabi-Ya u conditions . Thus , w e hav e a geometrica l interpretatio n o f figure tw o i n th e for m This i s th e geometrica l interpretatio n o f N = 2 superconforma l field theor y moduli spac e tha t wa s ascribe d t o the m fo r som e time .

26

BRIAN R . G R E E N E

We will retur n t o this pictur e repeatedl y an d se e that th e geometrica l counter part t o the conforma l field theory modul i spac e i s deficient i n a number o f ways. I n other words , Figur e 4 is quit e incomplet e a s i t presentl y stands . Eradicatin g thes e deficiencies wil l lea d u s t o a numbe r o f r e markabl e mathematica l an d physica l features. As a presage to that discussion , le t us note one fact: Th e distinction between th e $(i,i) an d $(-i,i ) typ e margina l operators , a t th e leve l o f conforma l field theor y is rathe r trivia l bein g jus t th e sig n o f a U(l ) charge . Th e distinctio n betwee n their geometrica l counterparts—harmoni c (1 ,1 ) form s an d harmoni c (2,1 ) forms — is comparativel y dramatic . Thes e tw o type s o f form s ar e mathematicall y quit e different. Thi s i s a strang e asymmetr y t o whic h w e shal l return . 4. Interrelation s betwee n variou s N — 2 superconforma l theorie s At th e en d o f Sectio n 2 we described thre e genera l way s o f constructing N = 2 superconformal field theories : nonlinea r sigm a models , Landau-Ginsbur g theorie s and minima l models . Ou r goa l now is to try t o understand th e relationship betwee n these theories . W e wil l procee d b y finding pairwis e relations . 4.1. Landau-Ginsbur g theorie s an d minima l models . Conside r no w a Landau-Ginsburg theor y wit h a singl e chira l superfiel d # : S= I

d 2z d 46 K(9, tf) +

( J d 2z d 26 W(9) + c.c .

Take W(^f) — \I/ P + 2 . I n [54 ] th e author s calculat e th e centra l charg e cp o f thi s Landau-Ginsburg theor y a t it s infrare d fixed poin t t o b e

Recall tha t th e c < 3, N = 2 superconformal theorie s are the minima l models . Thus, w e lear n tha t a Landau-Ginsbur g theor y o f thi s particula r for m is , a t it s conformally invarian t infrare d fixed poin t (assumin g suc h a poin t exists ) th e P t h minimal model , MMp. (I n fact , t o b e a bi t mor e precise , i t i s th e P th minima l model wit h th e s o calle d A-modula r invariant. ) Fo r mor e informatio n supportin g this statemen t o f equivalenc e an d emphasi s o n subtletie s whic h arise , th e reade r should consul t [69] . Thus , the Landau-Ginsburg theorie s with the simplest possibl e superpotential give s th e Lagrangia n realizatio n o f th e minima l models . 4.2. Minima l model s an d Calabi-Ya u manifolds : A conjecture d cor respondence. Conformall y invarian t nonlinea r sigm a models with Calabi-Ya u tar get space s hav e centra l charg e c = 3d wher e d i s th e comple x dimensio n o f th e Calabi-Yau space . A s w e hav e discussed , th e MMp hav e centra l charg e c = 3P/(P + 2 ) < 3 . Thus , fo r d > 1 i t woul d no t see m tha t ther e coul d b e an y connection betwee n thes e type s o f conforma l theories . However , give n a collectio n of r conforma l theorie s wit h centra l charg e Q , i = 1 , . . . , r, on e ca n buil d a ne w conformal field theory—the tensor product theory —with centra l charge c = Y^=i c iThe Hilber t spac e o f thi s theor y i s th e tenso r produc t o f th e Hilber t space s o f th e constituent model s an d th e energ y momentu m tenso r take s th e for m r

(4.1) T

= ^1 0l0---0T

i

0l"-0l.

27

ASPECTS O F QUANTU M GEOMETR Y

In fact , sinc e the operation o f orbifolding b y a finite discret e grou p doe s not chang e the central charg e of a conformal theory , a quotient o f the abov e tensor produc t wil l also hav e centra l charg e c = YH=i c i- Applyin g thi s t o th e minima l models , w e se e that i f we choose a collectio n o f integers Pi, i = 1 , . .. , r suc h tha t X][= i 3Pi/(Pi 42) = 3d , the n th e tenso r produc t o f thes e conforma l theorie s an d orbifold s thereo f will hav e centra l charg e 3d. I n principle , then , ther e migh t b e som e relationshi p between th e MMp combine d i n thi s manne r wit h Calabi-Ya u sigm a models . The first evidenc e that ther e is such a relationship was found b y Gepner [38,39] . As w e discussed , i t ha s lon g bee n know n tha t a strin g theor y o f th e for m w e ar e studying, M\ x {a n N = 2 superconforma l field theor y o f c = 9} , ha s spacetim e supersymmetry i f and onl y if the superconformal theor y ha s odd integra l U ( 1 ) L an d U ( 1 ) R charg e eigenvalues . Followin g thi s lead , Gepne r considere d a tenso r produc t of minimal model s with Y^i=i 3Pi/(Pi + 2) = 9 orbifolded ont o a spectrum wit h od d integral U(l ) charges . Tha t is , h e considere d [MMp 1 x • • • x MMp r] |u(i)projected > which w e denot e b y ( P i , . . . , P r ) . H e the n compare d th e symmetrie s an d massles s spacetime spectr a o f o f a particula r cas e (Pi , P 2 , . . ., P5) = (3,3,3,3,3 ) wit h tha t of the best studie d Calabi-Ya u sigm a model with c = 9 given by the vanishing locu s of z\ + z\ + • • • 4 - z\ i n CP 4 . H e foun d thes e dat a t o b e essentiall y identica l (u p to som e additiona l massles s particle s i n th e minima l mode l formulatio n whic h wer e expected t o genericall y becom e massiv e unde r smal l perturbations) . I t wa s furthe r shown i n [32 ] that th e Yukawa coupling s (whos e form w e will discuss ) a s compute d in th e minima l mode l formulatio n an d i n th e Calabi-Ya u formulatio n fo r thi s an d a coupl e o f othe r example s als o agreed . Thes e result s gav e additiona l suppor t t o Gepner's conjectur e tha t th e minima l mode l constructio n yield s conforma l theorie s interpretable a s nonlinear sigm a models on particular Calabi-Ya u manifolds . Again , this i s quit e surprisin g a s th e minima l mode l formulatio n doe s no t appea r t o hav e any geometrica l content . This conjecture wa s put o n firmer foundatio n i n the work s of [48,55 ] an d mor e recently [70] . Eac h o f thes e give s a procedur e fo r identifyin g whic h Calabi-Ya u manifold shoul d correspon d t o a give n minima l mode l construction ; th e pape r o f [48] give s a heuristi c pat h integra l argumen t establishin g a direc t lin k betwee n th e two types o f constructions an d [70 ] provides a rigorous argumen t uncoverin g a ric h phase structur e (als o foun d i n [6]) . We no w briefl y revie w thes e argument s establishin g a lin k betwee n (orbifold s of) tenso r produc t minima l mode l construction s an d Calabi-Ya u sigm a models . 4.3. Argument s establishin g minimal-model/Calabi-Ya u correspon dence. Eac h o f th e paper s [48 , 55, 70] make s us e o f th e accepte d isomorphis m between th e minima l mode l theor y a t leve l P an d th e Landau-Ginzbur g theor y of a singl e chira l superfield 6 X wit h superpotentia l W = X p + 2 , describe d above . In particular , sinc e Hamiltonian s (an d henc e Lagrangians ) o f tenso r produc t the ories add , w e immediatel y lear n fro m ou r previou s discussio n tha t .

28

BRIAN R . GREEN E

We reemphasize that th e explicit for m of the kinetic terms consistent wit h conforma l invariance ca n no t generall y b e writte n down , bu t shoul d b e though t o f a s bein g determined b y th e fixed point s o f a renormalizatio n grou p flow. We no w specializ e t o th e cas e o f r = d + 2 . A simpl e bu t crucia l poin t [48 ] t o note i s tha t th e minima l mode l conditio n o f Ylj=i 3Pj/(Pj + 2 ) = 3d implie s tha t £>, th e leas t commo n multipl e o f th e Pj - f 2 , i s equa l t o X ^ i i D/(Pj + 2) . Thi s implies tha t i f we interpret th e superpotentia l W i n (4.2 ) t o b e a n equatio n i n th e weighted projectiv e space 7 W^P^ j D , (wit h uji = I/(Pi + 2) ) the n i t i s wel l defined (homogeneou s o f degre e D) an d it s vanishin g locu s satisfie s th e conditio n that D equal s th e su m o f th e weight s o f th e projectiv e space 8 an d henc e give s ris e to a Calabi-Ya u manifol d o f complex dimension d. A t th e moment, though , nothin g in ou r discussio n justifies ascribin g suc h a n interpretatio n t o W. Followin g [48 ] we can, however , giv e a t leas t a heuristi c justification . We consider (4.2 ) an d not e tha t sinc e the Kahle r potentia l i s irrelevant w e may choose i t a t will , an d i n particula r w e may choos e i t ver y small . T o a first approx imation, i n fact , w e ma y ignor e it . Th e pat h integra l representin g th e partitio n function o f th e theor y no w become s (4.3) fpXx]

... [DX

r]

ex p (i f d 2z d 26 (W(X 1 ,... X

r)

+ c.c. ) J .

In a patc h o f field space i n whic h X\ ^ 0 , w e can rewrit e th e pat h integra l (4.3 ) i n terms o f ne w variable s

(4-4) £ r = X i

; fc = j&si

Since (4.5) W(X

U

...,X r) =

W'fa,. ..,&• ) = £ i W " ( l , & , . . . , £ P),

(4.3) become s (4.6) j[Vh\...

[V^

2

r}Jexp(ijv

zV2 0 £ i ( W ' ( l , 6

, ...&•) + c.c.)) ,

where J i s the Jacobia n fo r th e chang e o f variables , an d i s given b y J = X", wit h

(4.7) "

= ^r(i-E« N

2= 1

If v = 0 , the n J i s a constant , an d th e integratio n ove r £ i i n (4.6 ) yield s a delt a function 8(W'(1 ,^2 • • • 4>))> constraining th e remainin g fields to li e on the manifol d manifold9 W = 0 . Proceedin g i n lik e fashio n wit h th e othe r fields, w e ca n cove r field spac e wit h patches , i n eac h o f whic h w e obtai n a simila r result . W e note , though, tha t th e chang e o f variable s w e hav e use d t o simplif y th e pat h integra l "We recall that weighte d projectiv e spac e WP^^.^ Wd+2 consist s of d +2 homogeneou s coordi nates ( z i , . . . , ^d+2 ) subjec t t o th e equivalenc e relatio n ( z i , . . . , ^ + 2 ) ~ (A™ 1 21 ,..., \ Wd+2Zd+2) for an y nonzer o comple x numbe r A . 8 The reade r shoul d consul t [48 ] fo r a discussio n o f wh y thi s ensure s tha t th e resultin g spac e is Calabi-Yau . 9 We hav e ignore d th e fermioni c component s o f th e superfleld s here . Carryin g the m throug h the calculatio n yield s a delt a functio n constrainin g the m t o li e tangen t t o th e manifol d parame terized b y th e bosoni c coordinates , a s expecte d b y supersymmetry .

ASPECTS O F QUANTU M GEOMETR Y

29

is no t one-to-one . I n fact , upo n inspectio n w e se e tha t & ar e invarian t unde r th e transformation (4.8) Xi

- + e 2™*Xi.

The Xi ar e thus naturally interprete d a s homogeneous coordinates on WPp^ D(jJ . However, becaus e o f thi s invariance , th e mode l w e have show n t o b e equivalen t t o propagation o n th e manifol d W = 0 in thi s projectiv e spac e i s not th e theor y (1 .2 ) but rathe r th e quotien t o f this b y th e transformatio n (4.8) . Sinc e th e charg e o f Xi is uji, this i s precisel y th e quotien t b y go = e 2nlJ° require d t o obtai n a consisten t (spacetime supersymmetric) strin g vacuum [38,39,66] , as we discussed i n Section 2. Many propertie s o f th e resultin g mode l ma y b e extracte d fro m th e superpotentia l alone, a s discusse d i n [54] . A n importan t rol e i n thi s equivalenc e wa s playe d b y the fac t tha t th e Jacobia n fo r th e transformatio n t o homogeneou s coordinate s wa s constant. Fo r r = 5 thi s i s simpl y th e conditio n tha t th e centra l charg e c = 9 . It als o turn s ou t t o b e precisel y th e conditio n tha t th e hypersurfac e W = 0 hav e vanishing firs t Cher n clas s [48] . There ar e a fe w relevan t remark s w e should make . 1. Th e connectio n betwee n Landau-Ginsbur g theor y an d Calabi-Ya u sigm a models ha s bee n mos t easil y achieve d b y first deformin g awa y fro m th e con formally invarian t theorie s (b y manipulatin g th e Kahle r form) , finding a n isomorphism an d the n allowin g th e renormalizatio n grou p t o appropriatel y adjust kineti c term s t o reestablis h conforma l invariance . Nothin g i n ou r argument assure s tha t th e resultin g kineti c ter m i n th e Calabi-Ya u sigm a model wil l b e sufficientl y "large " t o ensur e tha t sigm a mode l perturbatio n theory will be valid. Hence , such a Calabi-Yau sigm a model would only trul y be define d vi a analyti c continuation . Thi s i s precisely wha t happen s i n th e more rigorous approach t o relating Landau-Ginzbur g theorie s to Calabi-Ya u sigma model s discusse d i n th e nex t section . 2. W e have made a number o f specializations i n our discussions . First , w e have focused attentio n o n th e cas e o f r , th e numbe r o f minima l models , bein g equal t o d + 2 . Second , w e hav e restricte d attentio n t o th e A-invariants . Although w e shall no t discus s i t i n detai l here , bot h o f those specialization s can b e substantiall y relaxed . Although compelling , the argumen t o f [48 ] has some obvious deficiencies. Para mount amongs t thes e is the manipulation o f the kinetic energy terms in the Landau Ginzburg action . A s show n i n th e appendi x o f [48 ] on e ca n wor k wit h a mor e conventional kineti c term , althoug h th e argumen t doe s becom e a bi t cumbersom e and delicate . More recently , Witte n ha s reexamine d thi s correspondenc e an d foun d a mor e robust an d satisfyin g argumen t whic h als o point s ou t a numbe r o f importan t sub tleties. W e no w briefl y revie w thi s approach . Ou r discussio n wil l b e limite d t o th e simplest possibl e cas e o f a Calabi-Ya u hypersurfac e wit h ft 1 '1 = 1 , i.e . th e quinti c hypersurface. Mor e genera l complet e intersection s ar e treate d i n [70 ] an d fro m th e viewpoint o f tori c geometr y i n [3] . Witten begin s wit h ani V = 2 supersymmetric gaug e theor y (calle d th e "linea r sigma model" ) wit h gaug e grou p U(l ) an d actio n (4.9)

& — ^kineti c + b\y H " gauge + ^ F / - D t e r m ?

30

BRIAN R . G R E E N E

where th e onl y term s whos e precis e for m w e nee d t o explicitl y writ e ar e (4.10) S

w=

2

zd26W{P,Su...,S5),

[d

where W i s th e superpotentia l o f th e theory , P, Si, ... , S$ ar e chira l superfield s whose U(l ) charge s ar e — 5 , 1, . . ., 1 respectively , an d W take s th e U(l ) invarian t form (4.11) W

= PG(S

U...,S5).

G i s a homogeneou s transvers e quinti c i n th e Si an d SFi-Dterm i s th e supersym metric Fayet-Illioupoulo s D-term . Th e importan t poin t fo r ou r presen t discussio n is that th e bosoni c potentia l o f the theor y take s th e for m (4.12) U

= |G( S i )| 2 + W 2 £ | gf +

±>& +

2 M2 f c N

2

+ 25|p| 2 ),

with

(4.13)J

D = -e 2(£|Si|2-5|p|2-A

In this expressio n lowe r cas e letters represen t scala r component s o f the correspond ing capita l lette r superfield s an d a i s a scala r field comin g fro m th e twiste d chira l multiplet whos e presenc e i s quite importan t bu t shal l no t pla y a central rol e i n ou r discussion. Ou r goa l is to study th e classical ground state s o f this theory fo r variou s choices o f the paramete r r. It turn s ou t tha t ther e ar e tw o qualitativel y differen t answer s dependin g upo n the sig n o f r . Let s stud y th e tw o possibilitie s i n turn . First, let s tak e r > 0 . Minimizin g th e D-ter m i n U the n implie s tha t no t al l Si can vanish . Assumin g G t o b e a transvers e quinti c polynomia l the n implie s tha t not al l \dG/dsi\ vanis h an d henc e minimizing U forces p = 0 . Furthermore , wit h a t least on e Si nonzer o w e lear n tha t a = 0 an d finally, minimizin g U furthe r implie s G = 0 an d 2

(4.14) EN

=r-

i

We are not quit e don e in ou r identificatio n o f the groun d stat e becaus e no t al l suc h configurations ar e distinc t du e t o th e gaug e symmetr y o f th e model . Rather , w e have th e U(l ) identification s (4.15) (s

u...,s5)~(e

ie

si,...,eies5).

How can we picture the meaning of the conditions we have found? A t first sight , th e fields si , . . . , 55 live i n C 5 . Th e combine d constraint s (4.1 4 ) an d (4.1 5) , however , take u s from C 5 t o CP 4 . T o see this not e tha t th e equivalenc e relatio n o f projectiv e space ( z i , . . . , 25) ~ A(zi,... , z^) ca n b e use d t o pick out on e representative o f eac h class. W e ca n d o thi s b y enforcin g condition s o n th e coordinate s whic h uniquel y pick ou t a valu e o f A for a give n choic e o f initia l coordinate s (21 ,.. . ,25) . Notic e that (4.1 4 ) an d (4.1 5 ) d o precisel y thi s an d henc e allo w u s t o interpre t th e Si a s living i n CP 4 . Thus , th e othe r conditio n o f G = 0 yields th e vanishin g o f a quinti c polynomial i n CP 4 —the familia r quinti c Calabi-Ya u hypersurface . Thus , fo r r > 0 the fields are constraine d t o li e on thi s Calabi-Ya u manifol d an d henc e ou r origina l U(l) gaug e theor y reduce s t o thi s Calabi-Ya u sigm a model . W e note , a s discusse d

ASPECTS O F QUANTU M GEOMETR Y

31

in [70 ] tha t th e origina l Lagrangia n ha s othe r fields no t presen t i n th e Calabi-Ya u sigma model whose masses ar e determined b y the valu e of r. Thus , fo r r ^ 0 , thes e fields play n o role and henc e i n this limi t w e actually ar e recoverin g th e Calabi-Ya u manifold no t onl y a s th e groun d state , bu t als o a s governin g th e effectiv e quantu m field theory. I n the infrare d limi t an y nonzero mass field drops out an d henc e in th e conformal limi t w e as well regain th e conformall y invarian t nonlinea r sigm a model . Lets als o not e tha t r determine s th e "size " o f th e Calabi-Ya u manifol d fro m (4.14). I n thi s sense , th e variabl e r ca n b e though t o f a s determinin g th e Kahle r modulus o f th e theory . W e haste n t o add , though , tha t a s w e le t th e theor y flow to th e infrare d an d simultaneousl y integrat e ou t th e massiv e degree s o f freedom , the valu e of r wil l change a la the Wilso n renormalizatio n group . Hence , the actua l value of the Kahler modulu s a t th e infrared fixed poin t wil l in general be determine d by r bu t wil l no t b e equa l t o it . Th e paramete r r i s ofte n calle d th e "algebraic " coordinate o n th e modul i space . I t i s th e natura l variabl e fo r th e linea r sigm a model. Th e valu e of the Kahle r modulu s r a t th e infrare d fixed poin t i s often calle d the "sigma-model " coordinat e a s i t i s the natura l variabl e fro m th e latte r poin t o f view. Fo r mor e discussio n o n thes e point s se e [7,70] . Having discussed th e cas e of r > 0, lets move on to the cas e of r < 0. Reasonin g exactly a s w e di d above , w e find tha t al l o f th e Si must vanish , p i s constrained t o be \J— r/5, an d ther e i s an unbroke n Z 5 symmetry grou p (becaus e p ha s charg e 5) . Hence, unlike the cas e r > 0 , the vacuum stat e i s not a n extended space , but rathe r is unique: geometricall y i t i s a point . Furthermore , fro m th e for m o f the potential , the Si ar e massles s fluctuations abou t thi s vacuu m state . Th e configuration space , therefore i s C 5 /Zs. Now , from ou r earlier discussion, thi s is an orbifold of a Landau Ginsburg theor y sinc e the latter ca n be described a s a theory wit h a unique vacuu m state wit h som e numbe r o f massles s fields. Th e Z 5 identification s comin g fro m th e unbroken gaug e grou p give s ris e t o th e state d orbifolding . W e not e tha t thi s Z 5 action i s in fac t nothin g bu t th e actio n o f e 27TtJo an d henc e constitute s th e require d U(l) projectio n tha t w e have discussed earlier . I t play s exactly th e sam e role as th e required identifications , fro m th e nonlinea r chang e o f variable, i n th e pat h integra l argument give n previously . Thus, b y varying th e paramete r r i n (4.9 ) i t ha s bee n show n [70 ] that w e inter polate between a linear sigm a model on a Calabi-Yau spac e an d a Landau-Ginzbur g orbifold. B y allowin g th e renormalizatio n grou p t o act , w e thereby interpolat e be tween th e conformall y invarian t limit s o f thes e tw o type s o f theories . W e migh t re-emphasize her e th e importan t poin t tha t th e reaso n w e shoul d tak e \r\ t o b e large i n eac h regim e i s t o suppres s th e massiv e excitation s whic h woul d caus e th e theory obtaine d t o diffe r fro m a linea r sigm a mode l o r a Landau-Ginzbur g model . After w e flow to a n infrare d fixed poin t b y th e renormalizatio n group , though , an y initial nonzer o mass , n o matte r ho w small , become s effectivel y infinite . Thus , s o long a s w e ultimatel y flow t o a n infrare d fixed point , th e siz e o f |r | ca n b e arbi trarily small . A s we mentioned, fro m (4.1 3 ) w e see that th e actua l valu e o f r, fo r r positive set s th e overal l siz e o f th e ambien t projectiv e space—i.e . r determine s it s Kahler form ; b y restrictio n t o th e Calabi-Ya u hypersurfac e r determine s it s Kahle r form a s well . Fo r r negative , it s actua l valu e set s th e expectatio n o f twis t fields in th e Landau-Ginzbur g theor y [70] . Hence , i n th e languag e o f Sectio n 3 r ma y be though t o f a s a Kahle r modul i spac e paramete r an d th e modul i spac e (fo r thi s simple discussion ) consistin g o f R divide d int o tw o region s r > 0 and r < 0. Physi cally, th e forme r regio n ha s a point (th e "dee p interior point" ) wit h r bein g infinit e

32

BRIAN R . GREEN E Calabi-Yau "Large Radius Limit'

Sigma-Model Regio n

Landau-Ginsburg Orbifold Regio n

Landau-Ginsburg Enhanced Symmetry Point FIGURE

6 . Th e Kahle r modul i spac e fo r th e exampl e discussed .

corresponding t o a n infinit e volum e Calabi-Ya u space . Th e latte r regio n o f r < 0 contains it s own deep interior poin t o f r bein g negativ e infinit y whic h we have iden tified a s the Landau-Ginzbur g orbifol d point . (I t i s at thi s point , fo r example , tha t the theor y ha s a n enhance d quantu m symmetr y [66,70]. ) W e cal l th e firs t regio n of th e modul i spac e th e Calabi-Ya u sigm a mode l regio n an d th e secon d regio n th e Landau-Ginzburg orbifol d region . The lef t han d sid e of Figure 6 shows the r modul i space . I t i s just th e rea l line . Now, a s i s wel l known , strin g theor y instruct s u s t o complexif y th e variabl e r b y combining i t wit h a n antisymemtri c tenso r fiel d r —> t = b + ir. A special featur e o f b is that shift s o f its valu e b y a n integer , i n th e domai n o f larg e r , d o no t effec t th e theory, an d henc e the natura l comple x variabl e to us e is w = e 27r2(6_Hr). I n thi s wa y we can map the sigm a mode l regio n of the modul i spac e to the uppe r hemispher e of a sphere . On e can giv e similar argument s fo r th e Landau-Ginsbur g regio n [2,6,70 ] and i n this wa y obtain a natural compactificatio n o f the complexified modul i space , as show n i n th e righ t han d sid e o f Figure 6 . W e see that thi s modul i spac e consist s of tw o region s o r "phases " an d i t ca n b e show n tha t ther e i s n o obstructio n t o smoothly varyin g th e comple x paramete r t t o mov e fro m on e phas e t o th e other . Points i n th e firs t regio n correspon d t o Calabi-Ya u sigm a model s wit h Kahle r class determine d b y th e precis e locatio n o f th e point ; point s i n th e secon d regio n correspond t o Landau-Ginzbur g orbifold s wit h valu e o f th e twis t fiel d bein g deter mined b y th e locatio n o f th e point . A s w e try t o mov e fro m th e firs t regio n t o th e second (o r vic e versa) conforma l perturbatio n theor y abou t th e dee p interio r poin t in regio n on e (o r regio n two , goin g th e othe r direction ) break s down . I n th e sigm a model regio n thi s i s merel y th e statemen t tha t i f th e Calabi-Ya u get s t o small , the expansio n paramete r (a') 2/r get s bi g an d perturbatio n theor y will b e invalid . However, usin g the result s o f [6 ] and [70 ] we know that th e conforma l theorie s cor responding t o almos t al l point s i n th e modul i spac e ar e perfectl y wel l defined, eve n if a perturbativ e understandin g o f the m break s down , an d henc e w e ca n smoothl y continue ou r journe y alon g suc h a pat h i n th e modul i space . A s a matte r o f con vention, i f a theory ca n be describe d vi a conforma l perturbatio n theor y aroun d on e of ou r dee p interio r points , the n w e categoriz e i t a s belongin g t o th e sam e typ e o f

ASPECTS O F QUANTU M GEOMETR Y

33

theory a s thi s dee p interio r point . Thi s justifie s th e name s w e hav e give n t o th e two region s above : point s withi n th e regio n i n th e r > 0 secto r o f Figur e 5 ar e called Calabi-Ya u sigm a model s whil e thos e i n a simila r regio n i n th e othe r secto r are calle d Landau-Ginzbur g orbifol d theories . O f course , i f w e allo w fo r analyti c continuation, the n w e ca n mak e sens e o f a perturbativ e expansio n abou t th e dee p interior poin t o f th e r > 0 secto r fo r essentiall y an y poin t i n th e modul i space , even wit h r < 0 . Thus , i n thi s sense , w e ca n eve n thin k o f th e dee p interio r poin t in th e Landau-Ginzbur g orbifol d secto r a s bein g (th e analyti c continuatio n o f a ) Calabi-Yau sigm a mode l wit h a particula r (identifiable ) Kahle r class . I n term s o f the paramete r r , w e see that thi s specia l choic e seem s t o requir e a negativ e Kahle r class, o r mor e precisely , a n analyti c continuatio n t o a negativ e Kahle r class . W e should note , though , tha t i n [7 ] i t wa s show n tha t th e physic s o f th e situatio n implies tha t physica l radi i r (an d thei r analyti c continuations ) whic h aris e fro m integrating ou t massiv e mode s i n th e linea r sigm a model , ar e nontrivia l function s of the r paramete r whic h appea r t o alway s b e nonnegative. Thus , i n this sense , on e can interpre t th e resul t o f [70 ] t o sa y tha t a Landau-Ginzbur g orbifol d conforma l model (o f th e typ e considere d here ) i s equivalen t t o th e analyti c continuatio n o f a conformally invarian t nonlinea r sigm a mode l o n a Calabi-Ya u spac e t o a particula r (and identifiabl e [7,66,70] ) "smal l an d positive " valu e o f th e Kahle r class . We see , therefore , tha t thi s relativel y rigorou s argumen t lay s ou t quit e clearl y the relationshi p betwee n Calabi-Ya u conforma l field theorie s an d conformall y in variant Landau-Ginzbur g orbifol d theories . The y ar e different "phases " o f the sam e overarching theor y (4.9 ) relate d b y differen t value s fo r th e paramete r r (mor e pre cisely, t). In ou r discussio n o f [70 ] w e hav e limite d ou r attentio n t o theorie s wit h on e Kahler modulus , an d henc e on e r parameter . Fo r Calabi-Ya u manifold s wit h a n /i 1 ' 1 dimensiona l Kahle r modul i space, there will be ft 1 '1 r parameters , r i , . . . , r hi,i. The modul i spac e will agai n naturall y divid e itsel f int o phas e regions , howeve r th e structure wil l typicall y b e fa r mor e ric h tha n th e tw o phas e region s foun d i n th e one dimensiona l setting . W e ca n agai n describ e theorie s i n eac h regio n i n term s o f the mos t natura l interpretatio n o f thei r correspondin g dee p interio r point . Ther e will typically b e a Landau-Ginzbur g orbifol d region , numerou s smoot h Calabi-Ya u regions (wit h th e variou s Calabi-Ya u space s bein g birationall y equivalen t bu t pos sibly topologicall y distinct) , Calabi-Ya u orbifol d region s an d region s consistin g o f hybrids o f these . Fo r detail s th e reade r shoul d se e [70 ] an d [6] . 5. Mirro r manifold s In ou r discussio n t o thi s point , w e hav e see n tha t ther e ar e certai n abstrac t properties o f (2,2 ) superconforma l theorie s that ca n b e realized b y a variety of field theoretic constructions . I n particular , w e have see n tha t ther e i s a nic e correspon dence between geometrical construct s i n the nonlinear sigm a model formulation an d their abstrac t conforma l field theoreti c counterparts . W e hav e seen , fo r instance , that i n genera l ther e ar e two types o f marginal operator s i n a n abstrac t c = 9 (2,2 ) superconformal theor y an d tha t thes e correspon d t o th e tw o geometrica l way s o f deforming a Calabi-Ya u manifol d withou t spoilin g th e Calabi-Ya u conditions . We note , however , tha t ther e i s a n uncomfortabl e asymmetr y betwee n th e ab stract conforma l field theor y descriptio n an d th e geometrica l realization . Namely , the tw o kind s o f conforma l field theor y margina l operator s diffe r onl y i n a rathe r

34

BRIAN R . GREEN E

trivial way : th e conventiona l sig n o f a U(l ) charge . O n th e othe r hand , thei r geo metrical counterparts differ fa r more significantly: th e cohomology groups H 1 (M , T) and H l(M, T*) ar e vastly differen t mathematica l objects . I t i s surprising tha t suc h a pronounced geometrica l distinction finds such a trivial conformal field theory man ifestation. Thi s led the authors of [33] and [54 ] to speculate on a possible resolution : if fo r eac h Calabi-Ya u manifol d M ther e wa s a secon d Calabi-Ya u M corresponding to the same conformal field theory bu t wit h th e associatio n o f H l(M,T) an d Hl(M,T*) t o conforma l field theor y margina l operator s reversed relativ e t o tha t of M , th e asymmetr y woul d b e resolved . Eac h conforma l field theor y margina l op erator woul d the n b e interpretable geometricall y eithe r a s a Kahler o r a s a comple x structure deformation , provide d on e choose s th e Calabi-Ya u manifol d realizatio n judiciously. Although a n interestin g idea , a t th e tim e o f thes e speculation s ther e wa s n o evidence fo r th e existenc e o f suc h pair s o f Calabi-Ya u manifold s correspondin g to th e sam e conforma l field theory . Subsequently , tw o simultaneou s independen t developments change d this . I n [23 ] the author s generated , vi a computer , numerou s Calabi-Yau manifold s embedde d i n weighte d projectiv e fou r space . Th e dat a s o generated consiste d almos t completel y o f pair s o f manifold s (M , M) satisfyin g dim H l(M,T) = d i (5.1) _ dim H\M,T*) = d i

mH

l

(M,T*),

mH

l

(M,T).

This, o f course, is a necessary conditio n fo r M an d M t o be identified wit h the sam e conformal field theor y a s above , howeve r i t i s fa r fro m sufficient . Tw o quantu m field theories ca n hav e sectors wit h th e sam e number o f fields, but ye t b e otherwis e completely unrelated . I n [45] , on th e othe r hand , a n explici t constructio n o f pair s of Calabi-Ya u manifold s M an d M satisfyin g (5.1 ) and corresponding to the same conformal field theory wa s given . Usin g th e fac t tha t H l{M,T) = iJ 1 ,:L (M) an d l H (M,T*) ^ H^^lM), on e ca n phras e (5.1 ) a s h1>1(M) = h

d1 1

- > (M),

(5.2) J hd-1>1{M) = h 1 >1 (M), where h ^(M) = di m H ^(M). I n particular , thi s implie s tha t th e Hodge diamond for M i s a mirror reflection throug h a diagona l axi s o f th e Hodg e diamon d fo r M . For this reason, we chose the term mirror manifolds fo r such pairs (M , M) [45] . Th e construction o f [45] , therefore, establishe d conclusivel y tha t mirro r manifold s exist . As of this writing, this is the only established constructio n o f mirror manifolds 1 0 an d hence shal l b e th e focu s o f th e presen t article . Ther e hav e bee n othe r conjecture d constructions o f mirror manifold s (includin g th e wor k o f [23 ] allude d t o above ) an d these ar e discusse d i n th e revie w pape r o f Berglun d an d Kat z [1 6] . l

l

5.1. Strateg y o f th e construction . Befor e discussin g the detail s of the mir ror manifol d constructio n o f [45] , w e no w briefl y describ e th e genera l strateg y o f our approach . Le t C be a conformal field theor y associate d wit h a Calabi-Yau man ifold M. C may b e though t o f a s th e nonlinea r sigm a mode l wit h M a s a targe t space or , somewha t mor e generally , C may b e thought o f as the equivalenc e clas s of 10 We not e tha t recen t wor k o f Aspinwal l an d Morriso n [1 0 ] ha s provide d som e alternat e constructions i n certai n cases .

ASPECTS O F QUANTU M GEOMETR Y

35

conformal theorie s (regardles s o f the detail s o f their particula r construction ) whic h are isomorphi c t o thi s nonlinea r sigm a model . Ther e ar e man y operation s on e ca n perform upo n C to generat e a ne w conforma l theor y C (o r a ne w equivalenc e clas s C o f conforma l theories) . Fo r instance , earlie r w e discusse d th e operation s o f de formation b y trul y margina l operator s whic h yiel d a ne w conforma l theor y fro m some chose n initia l conforma l model . I n general , no t al l operation s whic h tak e on e conformal theor y t o anothe r hav e a geometrica l interpretation . I n suc h a case , th e resulting theor y C ma y n o longe r b e associate d wit h a Calabi-Ya u manifold . I n other words , i n th e equivalenc e clas s C ther e nee d no t b e a conforma l theor y con structed fro m a nonlinea r sigm a model . A s ou r interes t her e i s i n th e geometrica l interpretation o f conforma l field theories , w e will hencefort h restric t ou r attentio n to operation s T takin g C — • C whic h hav e a functoria l geometri c realization . I n plain language , w e focu s o n T suc h tha t i f th e nonlinea r sigm a mode l o n M i s i n the clas s C then th e nonlinea r sigm a mode l o n T(M) = M i s in the clas s T(C) = C. Amongst th e operation s T whic h hav e thi s propert y i s the operatio n o f takin g the quotien t o f C by a discret e symmetr y grou p G , s o lon g a s G ha s a geometrica l interpretation a s a holomorphi c isometr y preservin g th e holomorphi c (d , 0) for m on th e associate d Calabi-Ya u space . I n commo n parlanc e thi s i s referre d t o a s orbifolding [34 ] by G. Equivalently , thi s operatio n amount s t o gaugin g th e discret e group G a s w e shal l discus s later . Fro m th e geometrica l viewpoint , orbifoldin g by th e grou p G ha s th e effec t o f yieldin g a ne w spac e T(M ) = M/G whic h i s characterized b y th e identificatio n o f al l point s x,y i n M whic h ar e relate d b y (5.3) x

= g{y) wit h geG.

If ther e ar e point s x i n M suc h tha t x = g(x), the n x i s calle d a fixed poin t an d M/G (usually ) acquire s a singularit y a t x. Now, let s imagin e tha t w e ca n find a n operatio n T whic h ha s a geometri c realization and suc h that : 1. r(C ) i s isomorphic t o C. Thi s woul d impl y tha t th e nonlinea r sigm a model s on M an d T(M) ar e isomorphic as conformal field theories. Suc h distinc t spaces M an d T(M) whic h nonetheles s giv e rise t o th e sam e conforma l field theory ar e know n a s string equivalent [46] . 2. Th e explici t ma p realizin g th e isomorphis m betwee n C and T(C ) i s changin g the sig n o f th e righ t movin g U ( 1 ) R charg e o f eac h operato r i n C. 3. Th e ma p betwee n operator s i n th e conforma l field theor y an d geometrica l constructs i n th e associate d Calabi-Ya u spac e (a s discusse d i n Sectio n 2 ) i s independent o f T. We clai m tha t i f suc h a n operatio n T meetin g condition s 1 - 3 ca n b e found , then M an d T(M ) = M woul d constitut e a mirro r pai r o f Calabi-Ya u spaces . To se e why , let s conside r C an d M a s describe d wit h th e property , say , tha t marginal operator s wit h U(1 ) L X U ( 1)R charge s (—1 ,1 ) ar e associate d wit h th e co homology grou p iJ 1 ,:L (M) an d margina l operator s wit h charge s (1 ,1 ) ar e associate d with th e cohomolog y grou p H d~1 ,1 (M). Now , let s appl y I \ B y propert y 1 , T ha s a geometrica l interpretatio n an d henc e w e ca n construc t a ne w Calabi-Ya u spac e T(M) = M correspondin g t o th e conforma l theor y T(C). B y propert y 3 , the coho mology group s H X,1 (M) an d H d~1 ,1 (M) correspon d t o th e margina l operator s i n T(C) wit h U ( 1 ) L x U ( 1 ) R charge s (—1 ,1 ) an d (1 ,1 ) respectively . B y propert y 2 ,

36

BRIAN R . GREEN E

we se e therefor e tha t th e cohomolog y group s H ljl(M) an d H d1 1, (M) ar e associ ated wit h margina l operator s i n C with U(1 ) L X U ( 1)R charge s (1 ,1 ) an d (—1 ,1 ) respectively. Hence , M an d M ar e string equivalent (the y correspond t o isomorphi c conformal fiel d theories ) an d the y satisf y (5.2) . Thi s mean s tha t the y constitut e a mirror pair . I n th e followin g section s we shall sho w that a suitabl e operatio n Y ca n in fac t b e constructed . Befor e doin g so , however , w e shoul d emphasiz e on e subtl e point. A s w e hav e discussed , th e N = 2 superconforma l fiel d theorie s w e stud y are ofte n par t o f continuou s familie s o f theories . W e le t T(C) denot e th e famil y of theorie s whic h ar e al l relate d t o C via deformatio n b y trul y margina l operators . Now, not e that i f we can fin d a n operatio n Y meeting condition s 1 - 3 fo r an y theor y C C .F(C), then suc h a n operatio n exist s fo r al l other theorie s i n T{C). Th e reaso n for thi s i s a s follows . Le t C an d C both belon g t o T{C) an d b e relate d vi a (5.4)

C =W(C),

where U denote s th e appropriat e margina l operato r deformation . Now , i f Y i s th e operation meetin g condition s 1 - 3 fo r C , the n th e operatio n x

(5.5) U~

oYoU

meets condition s 1 - 3 whe n actin g o n C. B y definition , U~ l i s th e invers e defor mation o f li compose d wit h explicitl y changin g th e sig n o f al l U ( 1 ) R eigenvalues . Thus, s o long a s we can find a n operatio n Y meetin g 1 - 3 fo r on e theory i n !F(C) we are assure d o f suc h a n operatio n fo r ever y theor y i n th e family . W e note , further , that ofte n time s onl y som e subse t o f al l o f th e theorie s i n F(C) wil l hav e a natu ral geometri c interpretatio n i n term s o f a Calabi-Ya u sigm a model . Th e meanin g of condition s 1 an d 3 fo r thes e theorie s i s tha t whe n a candidat e operatio n Y i s transported vi a (5.5 ) t o theorie s i n T(C) wit h a nonlinea r sigm a mode l interpreta tion (assumin g suc h point s exis t i n th e family) , the n condition s 1 an d 3 ar e me t there. (W e will discuss late r th e cas e i n which n o theory i n T{C) ha s a geometrica l interpretation i n term s o f a nonlinea r sigm a model. ) 5.2. Minima l model s an d thei r automorphisms . A s discussed i n the las t section, a n importan t ingredien t i n th e constructio n o f mirro r manifold s i s a n un derstanding o f the minima l mode l conforma l field theories . W e now turn t o a mor e detailed discussio n o f thes e models . The superconforma l primar y fields of the leve l P minima l mode l ar e labeled b y six integers, / , m , s , I, m, s an d ar e typicall y written 1 1 3> s'™' 'm (z, z). Th e meanin g of these indice s i s made clea r b y recallin g tha t a n N = 2 minimal mode l a t leve l P is isomorphic t o th e cose t o f a n SU(2 ) WZ W mode l a t leve l P b y a U(l ) subgrou p together wit h a fre e boson . Tha t i s (5.6) MM

P

= ^y

P

x

fre e boson .

In othe r words , w e remov e a fre e boso n a t on e radiu s b y dividin g ou t th e U(l ) and w e pu t a fre e boso n bac k a t a differen t radius . Now , primar y fields o f a n SU(2) WZ W mode l ar e labeled , i n part , b y thei r usua l SU(2 ) angula r momentu m quantum number s / , m wit h \m\ < / ; thi s i s th e meanin g o f th e indice s / , m (an d T, m) i n 3> s'™' ,rn(z,z), an d thes e indice s satisf y th e sam e inequality . I n fact , t o 11 We note , a s discusse d below , tha t thes e field s ar e not , strictl y speaking , superconforma l primary field s fo r al l value s o f th e s an d s indice s (whic h themselve s ar e define d modul o 4) .

37

ASPECTS O F QUANTU M GEOMETR Y

avoid dealin g with hal f integra l value s of spin, / and m ar e denne d t o be twice thei r SU(2) counterpart s an d henc e m ca n rang e fro m — / to I in step s o f two. Wit h thi s convention, th e valu e o f I ca n rang e u p t o P . Th e inde x s arise s a s a convenien t bookkeeping device . Namely , i t prove s convenien t t o spli t th e Verm a modul e buil t upon a give n superconforma l primar y field int o thos e state s whic h diffe r fro m th e primary field b y th e actio n o f a n eve n versu s a n od d numbe r o f supercurrent s G ±. In th e N S sector , w e tak e th e valu e o f s t o b e zer o o r two ; th e forme r referrin g t o states whic h diffe r fro m th e highes t weigh t stat e b y th e actio n o f a n eve n numbe r of supercurrent s (therefor e includin g th e bon a fide origina l primar y field) an d th e latter referrin g t o state s obtaine d b y the actio n o f an od d numbe r o f supercurrents . In the R sector, we define s to be one or three with these values playing an analogou s role (w e can equivalently replace three by — 1 since this index is only defined modul o four). Th e inde x s i s referred t o a s th e 'fermion ' number . More concretely , i f w e temporaril y ignor e th e s inde x an d al l o f th e anti holomorphic dependence , superconforma l primar y fields i n th e N S secto r ca n b e labeled $z, m wit h conforma l weight s ft = Z( Z + 2)/[4( P + 2) ] - ra 2/[4(P + 2) ] an d U(l) charg e Q = m/(P + 2 ) (an d similarl y fo r th e suppresse d anti-holomorphi c sector). Th e chiral primar y fields, i.e . thos e fo r whic h G^_ x,2 also annihilate s th e corresponding state, have m = / and the antichiral primary fields (GZ 1 /2 annihilate s the correspondin g state ) hav e m = — L I n th e Ramon d (R) sector , ou r primar y fields can b e writte n a s \I/ Z m wher e ± refer s t o whethe r th e stat e i s annihilate d b y GQ o r b y GQ . (Ramon d state s annihilate d b y bot h o f thes e operator s for m th e Ramond groun d stat e an d hav e ft = P / 8 ( P + 2). ) Then , th e conforma l weight s o f such state s ar e h = l{l + 2)/[4(P + 2) ] - (m ± 1 ) 2 /[4(P + 2) ] + 1 / 8 an d thei r U(l ) charges are Q = ( r a ± l ) / ( P - f - 2 ) ± | . Th e Ramon d groun d state s are \k'+ z and \P Z~_/. Now let s reintroduc e th e s index . W e defin e 4>g' m t o be : $i,m for 5 = 0 ; \P* m fo r 5 = 1 ; \£j~ m for s = — 1; G^Q^m fo r s = 2 , an d th e inde x s i s defined modul o four , so thi s i s a complet e list . Now , th e las t definitio n migh t see m ambiguous , bu t i n fact w e will onl y us e thi s notatio n t o describ e th e Verm a modul e (modul o fermio n number a s discussed) H ls'm t o which these state s belon g and bot h G ±^irn li e in th e same Verm a modul e (an d hav e th e sam e fermio n numbe r mo d 2) . Now, give n suc h a field $ ls,rn(z, z), w e conside r th e characte r x* s'm define d b y (5-7) X

l

m,s{T,z,u) =

e-

2

™Kni

2-KIZJQ 27rir(Lo

—c/24)

We reemphasiz e tha t th e trac e i s taken ove r a projectio n H lm s t o definit e fermio n number (mo d 2 ) o f a highes t weigh t representatio n o f th e (right-moving ) N = 2 algebra wit h highes t weigh t vecto r 3>q ;)S;g,s(0), wit h fermio n numbe r 1 assigned t o the superpartner s G ± o f th e energy-momentu m tensor , a s discusse d i n th e las t paragraph. Our goa l is to put togethe r suc h chiral characters i n a modular invarian t wa y t o construct a consisten t partitio n function . T o d o so , w e mus t first understan d ho w the character s transfor m unde r modula r transformations . Thi s wa s worke d ou t i n [38,39] an d th e result s are : (5-8) x L ( r + l ) = e x p ( 7 T 2

BRIAN R . G R E E N E

38

(5-9) xU-i/r) =

7 f7

V

^

r

q + W + i)

^y £ t

Z

^ Z'+ (e 2 7 r i n i / < ? 1 $i,..., e27rin*/qss),

for arbitrar y integer s ( n i , . . . ,n s ) suc h tha t X^= i n j/(±j m a n integer . Establishing thi s clai m require s a calculatio n tha t ca n b e foun d i n [45 ] [43] . For ou r discussio n her e w e simpl y not e tha t i t i s a familia r fac t i n conforma l field theory tha t successiv e quotient s o f a theor y ca n und o eac h othe r i f th e subsequen t quotients ar e quantum version s o f the previou s one . Al l we have here i s an exampl e of this phenomenon . Th e U(l ) projectio n ha s th e effec t o f undoing thos e quotient s of th e theor y tha t d o no t respec t (5.22) . Thus, w e have show n tha t {Pl

(5.23) (P

u...,Ps)=

'-G'Ps)

with th e isomorphis m betwee n th e tw o theorie s bein g a reversa l i n th e sig n o f al l U ( 1 ) R eigenvalue s o f th e fields i n th e lef t han d sid e relativ e t o th e righ t han d side . Since this operatio n o f orbifolding i s independent o f the Kahle r modulu s o f th e theory, i t i s trivia l t o transpor t i t t o a smoot h Calabi-Ya u region . Th e actio n o n the Calabi-Ya u manifol d M , (5.24) z

fi+2 + . . . ^ +

in the weighted projectiv e spac e WP^~,} p , is given b y (5.25) (*x,..

. ,* a ) - ( e

2

2ND

2

=0, ,,p +2 N wit h arbitrar y Kahle r for m

™1 / * * ! , . . . 9 e2*in'/«-za) ,

for arbitrar y integer s ( n i , . . . ,n s ) suc h tha t S j = i n jlclj IS a n integer . Thi s condi tion, whic h define s G , i s interpretabl e i n th e Calabi-Ya u regio n a s th e conditio n o f preserving th e holomorphi c d-for m O on M.

42 BRIA

N R . GREEN E

TABLE 1 . Orbifold s o f theor y (3,3,3,3,3) . Theory

£=(3,3,3,3,3) or inP4

ft2-1 h1'1 101 1 49 5 21 1 21 17 17 21 1 21 5 49 1 101

Symmetries [0,0,0,1,4] [0,1,2,3,4] [0,1,1,4,4], [0,1 ,2,3,4 ] [0,1,1,4,4] [0,1,3,1,0], [0,1 ,1 ,0,3 ] [0,1,4,0,0], [0,3,0,1 ,1 ] [0,1,2,3,4], [0,1,1,4,4], [0,0,0,1,4]

X -200 -88 -40 -8 8 40 88 200

Now, this operation o f orbifolding b y G meets conditions 1 and 2 of Section 5.1: since it is true a t th e minimal model point i t i s true everywhere i n the moduli space , as discussed . Thi s operatio n als o meet s conditio n 3 a s ca n mos t quickl y b e see n in th e followin g way : conside r a conforma l field A associate d wit h a geometrica l harmonic for m a& such tha t A an d henc e a A ar e invarian t unde r th e actio n o f G (there always will be at leas t on e such field: th e restriction o f the Kahler clas s of the ambient projectiv e spac e to M). Then , i n the theory based on M/G, A and a A agai n correspond. Thi s implie s tha t i f margina l operator s o f charge s (1 ,1 ) an d (1 ,-1 ) are associate d t o element s of H 1 (M , T*) an d H 1 (M , T) respectivel y (o r vice versa), then th e sam e associatio n hold s i n th e theor y base d o n M/G. Namel y margina l operators o f charge s (—1 ,1 ) an d (1 ,1 ) ar e associate d t o element s o f H 1 (M/G, T*) and H l(M/G,T) respectivel y (o r vic e versa) . Thi s i s s o becaus e th e associatio n of conformal fields t o geometrica l cohomoiog y ca n onl y tak e tw o possible form s (a s explicitly noted) . On e established associatio n o f a conformal field and a geometrica l harmonic for m distinguishe s betwee n thes e tw o possibilities . Sinc e w e hav e show n that i n bot h M an d M/G w e hav e a t leas t on e identica l association , w e ar e done . Hence, ou r operatio n meet s conditio n 3 as well . Thus, w e hav e show n tha t th e Calabi-Ya u hypersurfac e M give n b y (5.24 ) (for arbitrar y choic e o f Kahle r form ) ha s mirro r give n b y M/G. Le t u s not e tha t following ou r discussion o f the phase structure o f the moduli spac e of these theories , it i s mor e appropriat e t o sa y th e following . Le t M b y a Calabi-Ya u manifold . I t belongs to a moduli spac e of conformal theories . Conside r M/G. I t i s a Calabi-Ya u space (i t i s no t smooth ) whic h als o belong s t o a famil y o f conforma l theories . W e have show n tha t fo r eac h poin t i n th e first modul i spac e ther e i s a correspondin g point i n the secon d modul i spac e givin g rise to a n isomorphi c theory . Thus , mirro r symmetry i s mor e precisel y a statemen t o f pair s o f families o f conforma l theories . 5.5. E x a m p l e s . I n thi s sectio n w e giv e a fe w example s whic h illustrat e th e construction o f th e las t section . Th e followin g tw o table s sho w mirro r pair s o f theories constructe d vi a the orbifoldin g operatio n above . Th e colum n 'symmetries ' denotes th e grou p actio n b y whic h w e quotient . Fo r instance , i n th e first table , [0,0,0,1,4] indicate s tha t w e take th e quotien t b y th e Z 5 actio n (5.26) (zi,Z2,23,24,25)

->

(zi,z

2,z3laz4,a

4

z5),

where a i s a fifth roo t o f unity . Mirro r pair s resid e i n symmetri c position s i n th e tables wit h respec t t o th e horizonta l axi s throug h th e center .

ASPECTS O F QUANTU M GEOMETR Y 4

3

TABLE 2 . Orbifold s o f theor y (3,8,8,8) . Theory

Symmetries

P = (3,8,8,8 ) or

A + zf + • • • + zf + z\ = in ^ 2 4 i , i , i , 5

[0,0,5,5] [0,2,2,6] [0,0,1,9] [0,0,2,8] [0,1,2,7] [0,5,4,1] 0 [0,8,1,1] [0,5,5,0], [0,8,1 ,1 ] [0,5,5,0], [0,1 ,9,0 ] [0,0,1,9], [0,8,0,2 ] [0,0,1,9], [0,1 ,1 ,8 ]

h?+ ft1-1 145 99 47 39 37 29 17 15 13 11 3 1

1 3 11 15 13 17 29 39 37 47 99 145

X -288 -192 -72 -48 -48 -24 24 48 48 72 192 288

5.6. Implications . Havin g reviewe d th e initia l speculation s an d subsequen t work whic h establishe d th e existenc e o f mirro r symmetry , w e woul d no w lik e t o turn t o a discussio n o f the implication s o f this phenomenon , a s well a s som e recen t work applyin g mirro r symmetr y t o interestin g an d explici t examples . Let M an d M b e mirror Calabi-Ya u manifolds eac h corresponding to the confor mal fiel d theor y K. Conside r a (non-vanishing ) thre e poin t functio n o f conforma l field theor y operator s correspondin g t o (2,1 ) form s o n M . Mathematically , thi s correlation functio n i s given b y th e simpl e integra l o n M [65 ] (5.27) /

uj

ahc

l^ A

b {bj) A bi k) A T (wit h T suc h a holomorphic curve) , 7r m i s a n m-fol d cove r P 1— > P 1 an d u^ = u o 7rm. J refer s t o the Kahle r for m o n M .

44

BRIAN R . G R E E N E

Now, since both (5.27 ) and (5.28 ) correspond to the same conforma l field theory correlation function , the y mus t b e equal ; henc e w e hav e [45 ] (5.29) /

uo

abc

b^ A

b[j) A Uck) A u = f b^

A bU) A 6 (fe)

JM JM

+ J2 e/r 1 Ja...iB**1 ...4s,

where V . kjij = d. Differen t choice s fo r th e constant s a^^... ^ correspon d t o different choices fo r th e comple x structur e o f th e underlyin g Calabi-Ya u manifold . There ar e tw o importan t point s worth y o f emphasi s i n thi s regard . First , no t all choice s o f th e o^^... ^ giv e ris e t o distinc t comple x structures . Fo r instance , distinct choices o f th e a ^ . . . ^ whic h ca n b e relate d b y a rescalin g o f th e Zj o f the for m Zj —> XjZj wit h Xj £ C * manifestl y correspon d t o th e sam e comple x structure (a s the y diffe r onl y b y a trivia l coordinat e transformation) . Th e mos t general situatio n woul d requir e tha t w e consider a ^ ^ ^ ' s relate d b y genera l linea r transformations o n th e Zj's . Second , no t al l choice s o f a ^ . . . ^ giv e ris e t o smoot h Calabi-Yau manifolds . Specifically , i f th e a^ 2 ...i 5 ar e suc h tha t P an d dP/dzj have a commo n zer o (fo r al l j ) , the n th e spac e give n b y th e vanishin g locu s o f P is no t smooth . Th e se t o f al l choice s o f th e coefficient s a^ 2 ...i 5 whic h correspon d to suc h singula r space s compris e th e discriminant locus o f th e famil y o f Calabi Yau space s associate d wit h P. Th e precis e equatio n o f th e discriminan t locu s i s generally quit e complicated ; however , th e onl y fac t w e nee d i s tha t i t form s a complex codimensio n on e subspac e o f th e comple x structur e modul i space . Fro m the viewpoin t o f conforma l fiel d theory , th e nonlinea r sigm a mode l associate d t o points o n th e discriminan t locu s appear s t o b e il l denned . Fo r example , th e chira l ring become s infinit e dimensional . I t i s a n interestin g an d importan t questio n t o thoroughly understan d whethe r ther e migh t b e som e wa y o f makin g sens e o f suc h theories. Fo r the presen t purposes , though , al l we need t o know is that a t wors t th e space o f badly behave d physica l model s i s complex codimensio n on e in the comple x structure modul i space . W e illustrat e th e for m o f th e comple x structur e modul i space i n Figur e 9 . 6.2.4. Implications of mirror manifolds: Revisited. Locall y th e modul i spac e of Calabi-Yau deformations i s a product spac e of the complex and Kahler deformation s (in fact, u p to subtleties whic h will not b e relevant here , we can think o f the modul i space a s a globa l product) . Thus , w e expect , a s i n Figur e 5 , ( 6 . 5 ) A^CF

T = ^ c o m p l e x structur e X A^Kahle r structure ?

ASPECTS O F QUANTU M GEOMETR Y

49

with .M(... ) denotin g th e modul i spac e o f (...) • Pictorially , w e ca n paraphras e this b y sayin g tha t th e conforma l field theor y modul i spac e i s expecte d t o b e th e product o f Figur e 8 and Figur e 9 , whic h i s a mor e robus t versio n o f Figur e 5 . This, i n fact , wa s the pictur e whic h ha d emerge d fro m muc h wor k ove r th e las t few year s an d wa s generall y accepted . Th e adven t o f mirro r symmetry , however , raised a seriou s puzzl e relate d t o thi s descriptio n (a s firs t observe d i n [8]) . Le t M and M b e a mirro r pai r o f Calabi-Ya u spaces . A s w e discusse d before , suc h a pai r correspond t o isomorphi c conforma l theorie s wit h th e explici t isomorphis m bein g a chang e i n sig n of , say , th e righ t movin g U(l ) charge . Fro m ou r descriptio n o f the modul i space , i t the n follow s tha t th e modul i spac e o f Kahle r structure s o n M should b e isomorphi c t o th e modul i spac e o f comple x structure s o n M an d vic e versa. Tha t is , both M an d M correspon d t o the sam e famil y o f conformal theorie s and henc e yiel d th e sam e modul i spac e o n th e lef t han d sid e o f (6.5) . Therefore , the righ t han d sid e of (6.5 ) mus t als o be th e sam e fo r bot h M an d M. Th e explici t isomorphism o f mirro r symmetr y show s thi s t o b e tru e wit h th e tw o factor s o n th e right han d sid e o f (6.5 ) bein g interchange d fo r M relativ e t o M . The isomorphism o f the Kahler moduli space of one Calabi-Yau and the comple x structure o f its mirror i s a statement whic h appear s t o b e in direct conflic t wit h th e form o f Figur e 8 an d tha t o f Figur e 9 . Namely , th e forme r i s a bounde d domai n while th e latte r i s a quasi-projectiv e variety . Mor e concretely , th e subspac e o f theories whic h appea r possibl y t o b e badl y behave d ar e th e boundar y point s i n Figure 8 (wher e th e metri c o n th e associate d Calabi-Ya u fail s t o mee t (6.1 ) ) an d the point s o n th e discriminan t locu s i n Figur e 9 . Th e forme r ar e rea l codimensio n 1 whil e th e latte r ar e rea l codimensio n 2 . Therefore , ho w ca n thes e tw o space s b e isomorphic a s implie d b y mirro r symmetry ? 6.2.5. Flop transitions. A s the puzzl e raise d i n th e las t sectio n wa s phrase d i n terms o f thos e point s i n th e modul i spac e whic h hav e th e potentia l t o correspon d to badl y behave d theories , i t prove s worthwhil e t o stud y th e natur e o f suc h point s in mor e detail . W e wil l firs t d o thi s fro m th e poin t o f vie w o f th e Kahle r modul i space o f M. Consider a pat h i n th e Kahle r modul i spac e whic h begin s dee p i n th e interio r and move s toward s an d finall y reache s a boundar y wal l a s illustrate d i n Figur e 1 0 . More specifically , w e follo w a pat h i n whic h th e are a o f a P 1 ( a rationa l curve ) o n M i s continuously shrun k dow n to zero , attaining th e latte r valu e on the wall itself. The questio n w e ask ourselve s is : doe s thi s choic e fo r th e Kahle r for m o n M yiel d an il l define d conforma l theor y an d furthermore , wha t woul d happe n i f w e tr y t o extend ou r pat h beyon d th e wal l wher e i t appear s tha t th e are a o f th e rationa l curve woul d becom e negative ? (W e not e th e linguisticall y awkwar d phras e "are a of a curve" arise s sinc e we are dealing with comple x curve s whic h therefore ar e rea l dimension two. ) As a prelud e t o answerin g thi s physica l question , w e not e tha t precisel y thi s operation i s well known and thoroughl y studie d fro m th e viewpoint o f mathematics. Namely, i n algebrai c geometr y ther e i s an operatio n calle d a flop i n which th e are a of a rationa l curv e i s shrun k dow n t o zer o (blown down) an d the n expande d bac k to positiv e volum e (blown up) i n a "transverse " direction . Typicall y (althoug h no t always) thi s operatio n result s i n a chang e o f the topolog y o f the spac e in which th e curve is embedded. Thus , when we say that th e blow n up curve has positive volum e we mean positive with respect t o the Kahler metri c on the new ambient space . Tha t

50

BRIAN R . G R E E N E

FIGURE

FIGURE

1 0 . A pat h t o th e wall .

1 1 . A topology-changing path .

is, th e flop operatio n involve s firs t followin g a pat h lik e tha t i n Figur e 1 0 whic h blows th e curv e down , an d the n continuin g throug h th e wal l (a s i n Figur e 1 1 ) b y blowing th e curv e u p t o positiv e volum e o n a ne w Calabi-Ya u space . Th e latte r space, M' als o ha s a Kahle r con e whos e complexificatio n i n th e exponentiate d w\ coordinates is another bounde d domain . Thus , the operation of the flop corresponds to a pat h i n modul i spac e beginnin g i n th e Kahle r con e o f M , passin g throug h on e of its walls and landin g i n the adjoining Kahle r con e of M'. Althoug h M an d M / 1 5 can be topologically distinct , thei r Hodg e numbers ar e the same ; they diffe r i n mor e subtle topologica l invariant s suc h a s th e intersectio n for m governin g th e classica l homology ring . Mathematically , the y ar e sai d t o b e topologicall y distinc t bu t i n the sam e birationa l equivalenc e class . The mathematica l formulatio n o f what i t mean s t o pas s to a wall i n the Kahle r moduli space has led us to a more detailed framework fo r studying the correspondin g description i n conforma l field theory . W e se e tha t fro m th e mathematica l poin t o f view, distinc t Kahle r modul i space s naturall y adjoi n alon g commo n walls . W e can rephras e ou r initia l motivatin g questio n o f tw o paragraph s ag o as : doe s th e 15

To avoi d confusion , w e not e tha t th e mirro r t o M i s calle d M , no t M'

ASPECTS O F QUANTU M GEOMETR Y

51

operation o f flopping a rationa l curv e (an d thereb y changin g th e topolog y o f th e Calabi-Yau unde r study ) hav e a physica l manifestation ? Tha t is , does a pat h suc h as tha t i n Figur e 1 1 correspond t o a famil y o f wel l behaved conforma l theories ? This i s a har d questio n t o answe r directl y becaus e ou r mai n too l fo r analyzin g nonlinear sigm a model s i s perturbatio n theory . Th e expansio n parameter s o f suc h perturbative studie s ar e o f th e for m y/a jR wher e R refer s t o th e se t o f Kahle r moduli o n th e targe t manifold . Now , whe n w e approac h o r reac h a wal l i n th e Kahler modul i space , a t leas t on e suc h modul i field R i s going t o zer o (namel y th e one which set s the siz e of the blow n down rationa l curve) . Hence , sigma model per turbation theor y break s dow n an d w e are har d presse d t o answe r directl y whethe r the associate d conforma l theor y make s nonperturbativ e sense . This situation—on e i n whic h w e requir e a nonperturbativ e understandin g o f observables o n M —is tailo r mad e fo r a n analysi s base d upo n mirro r symmetry . Perturbation theor y break s dow n o n M becaus e o f th e degenerat e (o r nearl y de generate) choic e o f it s Kahle r structure . Not e tha t al l o f ou r discussio n coul d b e carried throug h fo r an y convenien t (smooth ) choic e o f it s comple x structure . Vi a mirror symmetry , thi s implie s that th e relevan t analysi s fo r answerin g th e questio n raised tw o paragraph s ag o should b e carrie d ou t o n M fo r a particula r for m o f th e complex structure (namely , tha t whic h i s mirror t o th e degenerat e Kahle r structur e on M ) bu t fo r an y convenien t choic e o f th e Kahle r structure . Th e latter , though , determines th e applicabilit y o f sigm a mode l perturbatio n theor y o n M. Thus , w e can choos e thi s Kahle r structur e t o b e arbitraril y "large " (tha t is , distan t fro m any wall s i n th e Kahle r cone ) an d henc e arrang e thing s s o that w e can completel y trust perturbativ e reasoning . I n othe r words , b y usin g mirro r symmetr y w e hav e rephrased th e difficul t an d necessaril y nonperturbativ e questio n o f whethe r con formal field theor y continue s t o mak e sens e fo r degeneratin g Kahle r structure s i n terms o f a purel y perturbativ e questio n o n th e mirro r manifold . This latter perturbativ e questio n i s one which is easy to answe r and , i n fact, w e have alread y don e s o in ou r discussio n o f the comple x structur e modul i space . Fo r large value s o f the Kahle r structur e (again , thi s simpl y mean s tha t w e are fa r fro m the wall s of the Kahle r cone ) th e onl y choice s o f the comple x structur e whic h yiel d (possibly) badl y behave d conforma l theorie s ar e those which lie on the discriminan t locus. A s note d earlier , th e discriminan t locu s i s comple x codimensio n on e i n th e moduli spac e (rea l codimensio n two) . Thus , th e comple x structur e modul i spac e is, i n particular , pat h connected . An y tw o point s ca n b e joine d b y a pat h whic h only passe s throug h wel l behaved theories ; i n fact , th e generi c pat h i n th e comple x structure modul i spac e ha s th e latte r property . Thi s i s the answe r t o ou r question . By mirror symmetry , thi s conclusio n mus t hol d fo r a generic path i n Kahler modul i space an d henc e i t woul d see m tha t a topolog y changin g pat h suc h a s tha t o f Figure 1 1 (by a suitabl e smal l jiggle a t worst ) i s a physicall y wel l behaved process . Even though th e metri c degenerates , th e physic s of string theor y continue s t o mak e sense. W e ar e alread y familia r fro m th e foundationa l wor k o n orbifold s [34 ] tha t degenerate metric s ca n lea d t o sensibl e strin g physics . No w w e see tha t physicall y sensible degeneration s o f othe r type s (associate d t o flops) ca n alte r th e topolog y of th e universe . I n fact , th e operatio n bein g described—deformatio n b y a trul y marginal operator—i s amongs t th e mos t basi c an d commo n physica l processe s i n conformal field theory .

52 BRIA

N R . GREEN E

^ c o m p l e x W ^enlarge

d Kahler ^

II?

^enlarged Kahler^ ) ^ c o m p l e x ^

)

FIGURE 1 2 . Th e conforma l field theor y modul i space .

To summariz e th e pictur e o f modul i spac e whic h ha s emerge d fro m thi s dis cussion w e refe r t o Figur e 1 2 . Thi s i s th e pictur e whic h replace s th e ol d an d incomplete versio n o f Figure 4 . Th e conforma l field theor y modul i spac e i s geomet rically interpretabl e i n term s o f th e produc t o f a comple x structur e modul i spac e and a n enlarge d Kahle r modul i spac e .A/fenlarge d Kahler - Th e latte r contain s numer ous complexifie d Kahle r cone s o f birationall y equivalen t ye t topologicall y distinc t Calabi-Yau manifold s adjoine d alon g commo n walls 1 6 . Ther e ar e two such geomet ric interpretations , vi a mirro r symmetry , wit h th e role s o f comple x structur e an d Kahler structur e bein g interchanged . Thi s i s als o indicate d i n Figur e 1 2 . We should stress that fro m a n abstract poin t o f view this is a compelling picture. Although w e do no t hav e tim e o r spac e t o discus s i t here , th e augmentatio n o f th e Kahler modul i spac e i n th e manne r presente d (and , mor e precisely , a s w e wil l generalize shortly ) gives i t a mathematica l structur e whic h i s identical t o tha t o f the comple x structur e modul i spac e o f its mirror . I n th e importan t cas e o f Calabi Yau's whic h ar e tori c hypersurfaces , bot h o f thes e modul i space s ar e realize d a s identical compact toric varieties. Hence , the picture presented resolve s the previou s troubling asymmetr y betwee n th e structur e o f these two spaces which ar e predicte d to b e isomorphi c b y mirro r symmetry . 16

T h e unio n o f suc h region s constitute s wha t w e cal l th e "partiall y enlarged " Kahle r modul i space. Th e enlarge d Kahle r modul i spac e include s additiona l region s a s w e shal l mentio n shortly .

53

ASPECTS O F QUANTU M GEOMETR Y

Although compelling , w e hav e no t prove n tha t th e pictur e w e ar e presentin g is correct . W e hav e foun d a natura l mathematica l structur e i n algebrai c geometr y which if realize d b y th e physic s o f conforma l field theor y resolve s som e thorn y issues i n mirro r symmetry . W e hav e no t established , a s yet , whethe r conforma l field theor y make s us e o f thi s compellin g mathematica l structure . I f conforma l field theor y doe s avai l itsel f o f thi s structure , though , ther e i s a ver y precis e an d concrete conclusio n w e ca n draw : ever y poin t i n th e (partially ) enlarge d Kahle r moduli spac e o f M mus t correspon d unde r mirro r symmetr y t o som e poin t i n th e complex structur e modul i spac e o f M . Thi s implies , o f course , tha t an y an d al l observables calculate d i n the theories associated t o these corresponding point s mus t be identicall y equal . Let' s concentrat e o n th e thre e poin t function s w e introduce d earlier i n (5.29) . A s w e discussed , i f w e choos e a poin t i n th e Kahle r modul i space fo r whic h th e instanto n correction s ar e suppressed , th e correlatio n functio n approaches th e topologica l intersectio n for m o n th e Calabi-Ya u manifold . Fo r eas e of calculation , w e shall stud y th e correlatio n function s o f (5.28 ) i n thi s limit . Thi s analysis will be similar to that presente d i n [9] although in this case in the (partially ) enlarged Kahle r modul i space , ther e i s no t a singl e uniqu e "larg e radius " poin t o f the sor t w e ar e lookin g for . Rather , ever y cel l i n th e (partially ) enlarge d modul i space supplie s u s with on e such point. Sinc e these cell s are the complexifie d Kahle r cones o f topologicall y distinc t spaces , th e intersectio n form s associate d wit h thes e large radiu s point s ar e different . I f th e modul i spac e pictur e w e ar e presentin g i n Figure 1 2 i s correct , the n ther e mus t b e point s i n th e comple x structur e modul i space o f th e mirro r whos e correlatio n function s exactl y reproduc e eac h an d ever y one o f thes e intersectio n forms . Thi s i s a precis e an d concret e statemen t whos e veracity woul d provid e a stron g verificatio n o f th e pictur e presente d i n Figur e 1 2 . In th e nex t sectio n w e carry ou t thi s verificatio n i n a particula r example . 6.2.6. An example. I n thi s sectio n w e briefl y carr y ou t th e abstrac t progra m discussed i n the last fe w sections i n a specific example . W e will see that th e delicat e predictions jus t discusse d ca n b e explicitl y verified . We focu s o n th e Calabi-Ya u manifol d M give n b y th e vanishin g locu s o f a degree 1 8 homogeneous polynomia l i n the weighte d projectiv e spac e P | 6 6 3 2 u a n d its mirro r M . Fo r th e forme r w e ca n tak e th e polynomia l constrain t t o b e (6.6) zl

+ z\ + z% + z% + z\ s + a 0z0z1 z2z3z4 =

0,

where th e Zi ar e th e homogeneou s weighte d spac e coordinate s an d a o i s a larg e and positiv e constan t (whos e value , i n fact , i s inconsequentia l t o th e calculation s which follow) . Th e mirro r t o thi s famil y o f Calabi-Ya u space s i s constructe d vi a the metho d o f [45 ] b y takin g a n orbifol d o f M b y th e maxima l scalin g symmetr y group Z 3 x Z 3 x Z3. A stud y o f th e Kahle r structur e o f M reveal s tha t ther e ar e five cell s i n it s (partially) enlarge d Kahle r modul i space , eac h correspondin g t o a sigm a mode l on a smoot h topologicall y distinc t Calabi-Ya u manifold . I n eac h o f thes e cell s there i s a larg e radiu s poin t fo r whic h instanto n correction s ar e suppresse d an d hence th e correlatio n function s o f (5.28 ) ar e jus t th e intersectio n number s o f th e respective Calabi-Yau's . W e hav e calculate d thes e fo r eac h o f th e five birationall y equivalent ye t topologicall y distinc t Calabi-Ya u space s an d w e record th e result s i n Table 3 . T o avoi d havin g t o dea l wit h issue s associate d wit h normalizin g fields i n the subsequen t discussion , i n Tabl e 3 we have chose n t o lis t ou r result s i n term s o f

54

BRIAN R . GREEN E

TABLE 3 . Ratio s o f intersectio n numbers .

Resolution {D\D4){DlDl) (PlDi)(DlDi) (D2D3D4)(D2D3D4) (D2D3D4)(HDl) (D%D4)(HD2D3) (D2D3D4)(HD21) (D2,D4)(HD2D3)

Ai

A2

A3

A4

A5

-7

0/0

0/0

oo

9

2

4

0

0/0

0/0

1

1

1

0

0/0

2

1

00

0/0

0

ratios o f correlatio n function s fo r whic h suc h normalization s ar e irrelevant . (Th e Di an d H ar e divisor s o n M , correspondin g t o element s i n H l(M, T* ) b y Poincar e duality.) Following th e discussio n o f th e las t section , ou r goa l no w i s t o find five limi t points i n th e comple x structur e modul i spac e o f M suc h tha t appropriat e ratio s o f correlation function s yiel d th e sam e result s a s i n Tabl e 3 . T o d o so , w e not e tha t the mos t genera l comple x structur e o n M ca n b e writte n (6.7)

~3 9 W = zl + z\ + z% + z\ + .18 z\° + a 0z0z1 z2 z 3z4 + a\z^z\

+ CL2Z3Z4 + CL3Z3Z4 + G k ^ £ 3 Z4 — 0 .

We wil l describ e thes e limi t point s b y parameterizin g th e comple x structur e a s ^ = s Ti fo r rea l parameter s s an d ri an d w e sen d s t o infinity . Th e limi t point s are therefor e distinguishe d b y th e rates a t whic h th e a^ approac h infinity . Ou r task, therefore , i s t o find appropriat e value s fo r th e Ti (i f the y exist ) suc h tha t w e obtain mirror s t o th e five larg e radiu s Calabi-Ya u space s o f th e las t paragraph . The techniqu e w e us e t o d o thi s i s t o describ e bot h th e comple x structur e modul i space o f M an d th e enlarge d Kahle r modul i spac e o f M i n term s o f toric geometry . This description , a t a fundamenta l level , make s i t manifes t tha t thes e tw o modul i spaces ar e isomorphic . W e do no t hav e tim e t o presen t suc h analysi s here—rather , we refer th e reade r t o [6] . Fo r the presen t purpos e w e note tha t a direct outcom e of this analysi s i s a predictio n fo r five choice s o f th e vecto r ( r o , . . . ,7*4) which shoul d yield the desire d mirrors . A s we have discussed, a sensitive tes t o f these prediction s is t o calculat e th e mirro r o f th e ratio s o f correlatio n function s i n Tabl e 3 (usin g (5.27) an d th e metho d o f [1 9] ) fo r eac h o f thes e comple x structur e limit s an d se e if we get th e sam e answers . W e have don e thi s an d w e show th e result s i n Tabl e 4 . Note tha t i n th e limi t s goe s t o infinit y w e ge t precisel y th e sam e results . (Th e ipi are element s o f H X(M,T).) This, i n conjunction wit h the abstract an d general isomorphism w e find between the complex structure modul i space of a Calabi-Yau an d the enlarged Kahler modul i space o f it s mirro r (usin g tori c geometry) , provide s u s wit h stron g evidenc e tha t our understandin g o f Calabi-Ya u conforma l field theor y modul i spac e i s correct . In particular , a s ou r earlie r discussio n ha s emphasized , thi s implie s tha t th e basi c operation o f deformatio n b y a trul y margina l operato r (fro m a spacetim e poin t o f view, thi s correspond s t o a slo w variatio n i n th e vacuu m expectatio n valu e o f a

ASPECTS O F QUANTUM GEOMETR Y 5

5

TABLE 4 . Asymptoti c ratio s of 3-point functions . Resolution Direction

2> ( 8 -f J2 a=1 (8 D ^ a)^a. Pro that th e loca l for m o f th e perio d ove r 8 is given b y (6.20) /"f

t = - L Y^(8 H7°)( /

J (in / Q

n

m thi s w e lear n

J + singl e valued .

Specializing thi s genera l expression , w e therefore se e (6.21) G

J = Y~- ( zJ ln (z J ) + (l>2

z

) (

ln

(J! M ) ) + s i n g l e valued .

By specia l geometry , thi s latte r expressio n determine s th e propertie s o f th e singu larity associate d wit h th e conifol d degeneratio n bein g studied . Thus , th e questio n we now seek to answe r is : i f we incorrectly integrat e ou t th e blac k hol e states whic h become massles s a t thi s conifol d point , d o w e reproduce th e for m (6.21 ) ? To addres s thi s issu e w e mus t identif y th e precis e numbe r an d charge s o f th e states tha t ar e becomin g massles s a t th e degeneratio n point . A s discusse d i n [64] , the countin g o f blac k hol e state s i s a delicat e issu e fo r whic h ther e i s a s ye t n o rigorous algorithm . I n [64] , one homolog y clas s i n H 3 degenerate d a t th e conifol d singularity an d i t wa s hypothesize d tha t thi s implie s on e fundamenta l blac k hol e state—the on e o f minima l charge—need s t o b e include d i n th e Wilsonia n action . In th e presen t example , though , w e hav e sixtee n thre e cycle s i n fiftee n homolog y classes in H 3 degenerating . I n [44 ] it was argued tha t thi s should impl y sixteen fun damental blac k hol e field s nee d t o b e include d i n th e Wilsonia n action . Physicall y speaking, th e blac k three-bran e ca n wra p aroun d an y o f th e sixtee n degenerat ing three-cycles , whic h a t larg e overal l radiu s o f th e Calabi-Ya u woul d b e widel y separated. I t thu s seem s sensibl e tha t eve n thoug h ther e ar e onl y fiftee n homolog y classes degenerating, w e actually get sixteen massless black hole states. Th e charge s of thes e state s ar e eas y t o derive . I f w e le t H a b e th e blac k hol e hypermultiple t associated wit h th e vanishin g cycl e 7 ° the n th e charg e o f H a unde r th e I t h U(l ) i s given b y Q aj = A\ fl7 a wher e w e write F^ a s the sel f dua l par t o f Y^i a 1 F\ ' wit h a1 dua l t o Ai. W e immediatel y lear n fro m thi s tha t th e blac k hole s state s hav e charges (6.22) Q

a

j = 8 aT, 1 < a < 1 5 an d Qf

=

- 1 , 1 < I < 15,

with al l othe r charge s zero . Thi s i s enoug h dat a t o determin e th e runnin g o f th e gauge couplings : 1

(6.23) r

1

7J

6

= —^Q?Q51 n(m a=l

a

)

62

BRIAN R . GREEN E

where th e mas s m a o f H a i s proportional t o Yli Qfe 1 • Usin g th e abov e charge s w e therefore hav e 1 1 (6.24) Tu = —8u ln(z

1 5 / \ ) + — I n I ]T z k I + singl e valued .

J

Integrating w e find therefor e (6.25) Gj

= ^-z JlnzJ +

l n

^ r f v) t

( X X) )

+ singl e valued

-

We not e tha t thi s matche s (6.21 ) an d henc e w e hav e show n tha t inclusio n o f th e sixteen blac k hol e solito n state s whic h becom e massles s cure s th e singularity . Having show n tha t a sligh t varian t o n Strominger' s origina l proposa l i s abl e t o cure th e singularit y foun d i n thi s mor e complicate d situation , w e no w com e t o th e main poin t o f th e discussion : 6.3.3.2. Wha t i s th e physica l significanc e o f nontrivia l homolog y relation s be tween vanishin g cycles ? T o addres s thi s questio n w e conside r th e scala r potentia l governing th e blac k hol e hypermultiplets . I t ca n b e writte n a s

(6.26) V

= J2E

I

a/3Ef,

where 16

(6.27) E'

a0

= ] T Q' aeaih^hf -

(a - > /3),

a=l

in whic h th e indice s satisf y I = 1 , . . . , 1 5 ; a, 3. 7 = 1 , 2 . Th e fields M an d /i 2 are th e tw o comple x scala r fields i n th e hypermultiple t H a. We consider th e possibl e flat direction s whic h thi s potentia l admits . Th e mos t obvious flat direction s ar e thos e fo r whic h (/ A ) = 0 wit h nonzer o value s fo r th e scalar fields i n th e vecto r multiplets . Physically , movin g alon g suc h flat direction s takes u s bac k t o th e Coulom b phas e i n whic h th e blac k hol e state s ar e massive . Mathematically, movin g alon g suc h flat direction s 15 x

isfc''M

x

/c=l

7

/ x

1

5

E

k

/c=l

gives positive volume back to the degenerated S 3 's and hence resolves the singularit y by deformation . The nontrivia l homolog y relatio n implie s tha t ther e i s anothe r flat direction . Since Q* a = A / D 7°, w e see tha t th e homolog y relatio n YL a=i 7° = 0 implie s ]C a Qa — 0 f° r au " I- Thi s the n implie s tha t w e hav e anothe r flat directio n o f th e form (/i l ) = v@ fo r al l a with v constant. I n fact , simpl y countin g degree s freedo m shows tha t thi s solutio n i s uniqu e u p t o gaug e equivalence . Wha t happen s i f w e move alon g thi s flat direction ? I t i s straightforwar d t o se e tha t thi s take s u s t o a Higgs branch i n which fifteen vectors multiplets pai r u p with fifteen hypermultiplet s to becom e massive . Thi s leave s ove r on e massles s hypermultiple t fro m th e origina l sixteen tha t becom e massles s a t th e conifol d point . W e see therefore tha t th e spec trum o f the theor y goe s fro m 1 0 1 vector multiplet s an d 1 hypermultiplet (ignorin g the dilato n an d graviphoton ) t o 1 0 1 — 1 5 = 8 6 vector multiplet s an d 1 + 1 = 2 hypermultiplets. No w precisely thes e Hodg e number s aris e fro m performin g th e othe r

ASPECTS O F QUANTU M GEOMETR Y

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means o f resolvin g th e conifol d singularit y (beside s th e deformation)—th e smal l resolution describe d earlier ! Hence , w e appea r t o hav e foun d th e physica l mecha nism fo r affectin g a smal l resolutio n an d i n thi s manne r changin g th e topolog y o f the Calabi-Ya u background . Although w e hav e focuse d o n a specifi c example , i t i s straightforwar d t o wor k out wha t happen s i n th e mor e genera l settin g o f P isolate d vanishin g cycle s satis fying R homolog y relations . Followin g ou r discussio n above , w e ge t P blac k hol e hypermultiplets becomin g massles s wit h R flat direction s i n thei r scala r potential . Performing a generi c deformatio n alon g thes e flat direction s cause s P — R vector s to pai r u p wit h th e sam e numbe r o f hypermultiples . Henc e th e Hodg e number s change accordin g t o (6.29) {h

2

\hn) ->

(h 21 - ( P - R), h 1 1 + R).

The Eule r characteristi c o f th e variet y thu s jumps b y 2P . So, i n answe r t o th e questio n pose d above : homolog y relation s amongs t th e van ishing cycle s giv e ris e t o ne w flat direction s i n th e scala r blac k hol e potential . Moving alon g suc h flat direction s take s u s smoothl y t o ne w branche s o f th e typ e II strin g theor y modul i space . Thes e othe r branche s correspon d t o strin g propaga tion o n topologicall y distinc t Calabi-Ya u manifolds . W e hav e therefor e apparentl y physically realize d th e Calabi-Ya u conifol d transition s discusse d som e year s ago — without a physical mechanism—in insightfu l paper s of Candelas, Gree n and Hubsc h [21, 22]. I n th e typ e I I strin g modul i spac e wit h thu s se e tha t w e ca n smoothl y go fro m on e Calabi-Ya u manifol d t o anothe r b y varyin g th e expectatio n value s o f appropriate scala r fields. There i s another aspec t o f these topolog y changin g transition s whic h i s worth y of emphasis. I n the Coulomb phase, the black hole soliton states are massive. A t th e conifold poin t the y become massless. A s we move into the Higgs phase some number of the m ar e eate n b y th e Higg s mechanis m wit h th e remainde r stayin g massless . Now, wit h respec t t o th e topolog y o f the ne w Calabi-Ya u i n th e Higg s phase, thes e massless degree s o f freedo m ar e associate d wit h element s o f if 1 ' 1 . Suc h states , as i s wel l known , ar e perturbativ e strin g excitations—commonl y referre d t o a s elementary "particles" . Thus , a massive black hole sheds its mass, becomes massles s and the n re-emerge s a s a n elementar y particle-lik e excitation . Ther e i s thu s n o invariant distinctio n betwee n blac k hol e state s an d elementar y perturbativ e strin g states: the y smoothl y transfor m int o on e anothe r throug h th e conifol d transitions . 7. Conclusion s In thes e lecture s w e hav e sough t t o giv e th e reade r som e understandin g o f th e emerging field o f quantu m geometry . Th e basi c philosoph y w e hav e followe d i s t o allow th e physics of N = 2 superconforma l field theor y (tha t is , th e physic s o f spacetime supersymmetri c strin g theory ) t o b e ou r guid e toward s th e correc t ge ometrical framewor k fo r describin g strin g theory . W e hav e see n tha t thi s analysi s has naturall y le d u s t o som e unexpecte d consequences . Foremos t amongs t thes e is th e realizatio n tha t on e underlyin g strin g mode l may , i n fact , hav e tw o distinc t nonlinear sigm a mode l realizations—tha t is , with tw o distinct targe t spaces . Whe n the explici t isomorphis m betwee n thes e two realizations involve s flipping th e sig n of one th e U(l ) charge s i n th e N = 2 superconformal algebra , w e cal l th e tw o targe t spaces mirro r manifolds . Fro m th e viewpoin t o f classical geometry , thes e tw o man ifolds ar e distinc t objects ; fo r instance , the y ar e topologicall y distinct . Fro m th e

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viewpoint o f quantu m geometry , a s w e hav e discussed , the y ar e identical . Thi s i s a prim e exampl e o f ho w classica l an d quantu m geometr y differ . W e hav e als o see n how an operation whic h is singular i n classical geometry ca n be smooth i n quantu m geometry. Namely , w e hav e foun d smoot h operation s i n th e latte r whic h chang e the topolog y o f the targe t spac e while, by fundamental definitio n i n classica l geom etry, topolog y chang e i s discontinuous. Finally , w e have also briefly mentione d tha t not onl y ca n on e smoothly chang e th e targe t spac e topology, bu t on e ca n smoothl y deform th e theories we have discussed int o "phases " whos e interpretation mos t nat urally involve s construction s distinc t fro m th e sigm a mode l method . Thes e model s can b e describe d i n term s o f geometrica l dat a (mos t notabl y fro m tori c geometry ) but th e geometr y doe s no t manifes t itsel f throug h a sigm a mode l realization . I n this sense , strin g theor y i s reall y forcin g u s t o broade n ou r understandin g o f th e way i n whic h geometrica l dat a determine s th e physic s o f fou r dimensiona l strin g theory. One can' t hel p feelin g tha t w e ar e catchin g glimpse s o f a ne w geometrica l discipline whos e propertie s reduc e t o thos e o f familia r classica l geometr y a t larg e distances, bu t otherwis e ca n profoundl y differ . Developin g thi s discipline—tha t is , developing quantum geometry—i s of utmost importanc e an d i s sure to deeply affec t our understandin g o f strin g physics . Acknowledgments I would lik e to than k P . Aspinwall , T . Chiang , J . Distler , M . Gross , Y . Kantor , D. Morrison , an d R . Plesse r fo r collaboration s whic h yielde d i n som e o f the result s described here . References 1. P . Argyre s an d M . Douglas , New phenomena in SU(3 ) super symmetric gauge theory, Nuclea r Phys. B 44 8 (1 995) , no . 1 -2 , 93-1 26 . 2. P . S . Aspinwall , The moduli space of N = 2 superconformal field theories, 1 99 4 Summe r School i n Hig h Energ y Physic s an d Cosmolog y (Trieste , 1 994) , , ICT P Ser . Theoret . Phys. , vol. 1 1 , World Sci . Publishing , Rive r Edge , NJ , 1 995 , pp . 352-401 . 3. P . S . Aspinwal l an d B . R . Greene , On the geometric interpretation of N = 2 superconformal theories, Nuclea r Phys . B 4 3 7 (1 995) , no . 1 , 205-227 . 4. P . S . Aspinwall , B . R . Greene , an d D . R . Morrison , The monomial-divisor mirror map, Inter n a l Math . Res . Notice s (1 993) , no . 1 2 , 31 9-337 . 5. , Space-time topology change: The physics of Calabi-Yau moduli space, Proceeding s of String s '9 3 Conference , Berkeley , 1 993 . 6. , Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory, Nuclea r Phys . B 41 6 (1 994) , no . 2 , 41 4-480 . 7. , Measuring small distances in N = 2 sigma models, Nuclea r Phys . B 42 0 (1 994) , no. 1 -2 , 1 84-242 . 8. P . S . Aspinwal l an d C . A . Lutken , Quantum algebraic geometry of superstring compactifications, Nuclea r Phys . B 35 5 (1 991 ) . no . 2 , 482-51 0 . 9. P . S . Aspinwall , C . A . Lutken , an d G . G . Ross , Construction and couplings of mirror manifolds, Phys . Lett . B 24 1 (1 990) , no . 3 , 373-380 . 10. P . S . Aspinwal l an d D . R . Morrison , String theory on KS surfaces, Mirro r Symmetry , II , AMS/IP Stud . Adv . Math. , 1 , Amer . Math . S o c , Providence , RI , 1 997 , pp . 703-71 6 . 11. , Topological field theory and rational curves. Comm . Math . Phys . 1 5 1 (1 993) , no . 2 , 245-262. 12. A . C . Avram , P . Candelas , D . Jancic , an d M . Mandelberg , On the connectedness of the moduli space of Calabi-Yau manifolds, Nuclea r Phys . B 46 5 (1 996 ) no . 3 , 458-472 .

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74. B . Zumino , ?, Phys. Lett . B 8 7 (1 979) , 203-? . F. R . NEWMA N LABORATOR Y O F NUCLEA R STUDIES , CORNEL L UNIVERSITY , ITHACA , N

14853, U.S.A .

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https://doi.org/10.1090/amsip/010/02 AMS/IP Studie s i n Advance d Mathematic s Volume 1 0 , 1 99 9

Introduction t o Enumerativ e Invariant s Shing-Tung Yau The firs t majo r conferenc e o n mirro r symmetr y wa s held i n Berkele y [27] . Th e conference wa s hel d i n respons e t o th e spectacula r discover y o f Greene-Plesse r an d Candelas et al Durin g tha t conference , a grea t dea l o f attentio n wa s give n t o the calculatio n mad e b y Candela s et al [3 ] a s ther e wa s discrepanc y betwee n th e calculation i n [3 ] and th e one made by algebraic geometers. Whe n th e disagreemen t was finall y understood , ther e wa s unanimou s belie f tha t th e numerica l calculatio n given b y mirro r symmetr y i s accurate . A substantia l amoun t o f wor k wa s devote d to exten d th e calculatio n t o othe r Calabi-Ya u manifold s beyon d th e quintic . Suc h calculations ha d le d t o importan t developmen t o f strin g theory . A t th e sam e time , many attempts were made to understand th e "numbe r o f rational curves" calculate d by using the technique of mirror symmetry. A s was explained b y Witten i n [26] , the "counting" o f rational curve s i s obtained b y a certain localizatio n schem e t o reduc e certain pat h integral s o n infinite-dimensiona l manifolds . Whil e suc h pat h integral s were related vi a conforma l fiel d theor y t o th e deformatio n o f complex structure s o f its "mirror" , th e precis e definitio n i s no t clear . Severa l alternat e definition s wer e proposed. On e i s b y Rua n [1 9 ] an d Rua n an d Tia n [21 ] . Th e othe r on e i s alon g more classica l line s vi a projectiv e geometry . I t i s therefor e rathe r confuse d t o cal l all thes e invariant s t o b e Gromov-Witte n invariants . I t i s perhap s fai r t o cal l th e definition arise d i n symplecti c geometr y t o b e Gromov-Ruan-Tia n invariant . Th e one arise d i n physic s t o b e Candelas-Greene-Plesser-Witte n invarian t an d th e on e arised i n projectiv e geometr y t o b e Schuber t invariant . I t i s fa r awa y fro m clea r that thes e thre e differen t invariant s ar e th e same . I t i s onl y recentl y tha t L i an d Tian proved , th e invariant s define d b y simplecti c mean s ar e th e sam e a s th e on e defined b y projectiv e means . Kefen g Liu , Bon g Lian , an d mysel f prove d tha t th e later invariant s ca n be identified wit h the numerica l quantit y produce d b y Candela s et al. via deformatio n o f comple x structur e o f th e mirro r (throug h th e calculatio n of period s o f algebrai c integrals) . Bot h paper s appeare d i n th e Asia n Journa l o f Mathematics 1 i n Decembe r 1 997 . I n th e following , w e shal l explai n i n mor e detai l the variou s definitions . Let X b e a smooth , compac t comple x manifold . After fixing a polarizatio n (i.e., a Kahle r form) , w e ca n for m th e spac e o f holomorphi c 2-sphere s i n X o f de gree d , whic h i s th e spac e o f degre e d holomorphi c map s / : P 1 — > X modul o th e 1991 Mathematics Subject Classification. 01 -02 , 1 4N1 0 , 53C1 5 , 1 4C1 7 , 58D27 . The Asia n Journa l o f Mathematics i s published b y Imnternationa l Press , www . i n t l p r e ss . com

x

© 1 99 9 America n Mathematica l Society , Internationa l Press , an d Centr e d e recherche s mathematique s

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S H I N G - T U N G YA U

automorphism grou p o f P 1 . Le t u s denot e thi s modul i spac e b y M o (X, d). Her e the subscrip t 0 stand s fo r th e genu s o f P 1 . Whe n X$ i s a quinti c threefol d i n P 4 , we usuall y choos e th e polarizatio n t o b e th e on e induce d b y th e hyperplan e lin e bundle o f P 4 . I t wa s firs t observe d b y Clemen s [4 ] that th e spac e a t th e first-orde r deformations a t a rational curv e in X§ i s identical t o (non-canonically ) th e obstruc tion spac e to deformation s a t thi s curve . Thi s prompte d hi m t o conjectur e tha t fo r general quintic s X5 , the spac e o f rationa l curve s i s discret e an d reduced . (Her e w e say Mo(X$,d) i s reduce d a t / G Mo(Xs,d) i f / ha s n o non-trivia l first-orde r de formations.) Thi s conjectur e i s stil l ope n thoug h som e partia l result s wer e know n recently [5] . Th e computatio n o f th e physicist s mentione d befor e wa s partiall y based o n th e assumptio n tha t Mo(-X"5,d ) ar e discret e an d reduced . Thi s suggest s that th e numbe r the y compute d shoul d b e certai n enumeratio n number s relate d t o the modul i spac e M 0 (X 5 ,d). This fit s int o th e framewor k o f Donaldson' s theor y o f topologica l invariant s o f smooth 4-manifold s develope d i n th e 1 980's . Th e mai n schem e o f Donaldson-typ e invariants ar e a s follows . Le t X b e a manifol d wit h som e auxiliar y structure . B y choosing a n appropriat e syste m o f equations , w e ca n for m th e modul i spac e o f solutions o f this syste m modul o equivalenc e relation . W e denot e thi s modul i spac e by M. M usuall y admit s a famil y o f tautological topologica l classes , sa y IJL:H*(X,Q)^>H*(M,Q).

Now assum e M support s a fundamenta l clas s [M] G i^2y(M, Q) a t degre e 2V. Le t ei, . . . , e n b e a basi s o f iJ*(X , Q). The n w e ca n for m a forma l powe r serie s (1) (tu...,t

n)

= / exp(tiei,---+t

nen),

J[M]

which ar e calle d th e Donaldson-typ e invariants . Nevertheless , i n mos t o f the case s studied, includin g Donaldson' s origina l wor k o n hi s invariant s o f 4-manifolds, M i s not compac t an d possibl y singular . Henc e M doe s no t suppor t a fundamenta l clas s in th e usua l sense . Fo r this , on e need s t o fin d a compactincatio n M o f M , whic h usually ca n b e constructe d b y addin g "singular " objects . Fo r instance , fo r modul i at Anti-Self-Dua l connections , th e Uhlenbec k compactincatio n wa s constructe d b y adding regular AS D connections couple d wit h concentrate d instantons . An d fo r th e symplectic invariant s define d b y Gromov , Rua n an d Ruan-Tian , th e compactinca tion i s the on e that include s (pseudo- ) holomorphi c map s whos e domain s ar e noda l curves. However , eve n afte r constructin g th e compactincation , eithe r th e choic e of its fundamenta l clas s [M] is no t apparen t o r th e extensio n o f the tautologica l clas s /i t o /x:tf*(X;Q)^#*(M;Q) is no t obvious . (Th e invariant s o f symplecti c manifold s fal l int o th e firs t categor y and th e Donaldso n invariant s o f 4-manifold s fal l int o th e secon d category. ) Th e technique develope d sinc e Donaldson' s pionee r wor k i s t o reinterpre t th e integra l (1) Sa s an enumerativ e proble m b y using Poincar e dualit y formally . Mor e precisely , given a se t o f cohomolog y classe s a i , . . . , a ^ G H2*(M, Z) , instea d o f evaluatin g the integra l a i U • • • Uafc, / which ma y no t mak e muc h sens e unles s th e fundamenta l clas s [M] i s found , on e constructs (formally ) thei r Poincar e dua l representative s V\ , . . . , Vk C M s o tha t

I N T R O D U C T I O N T O E N U M E R A T I V E INVARIANT S

71

Vi D • • • H Vk is discrete an d V\ D • • • fl Vk intersects transversally . The n on e define s JM

as th e su m of the algebrai c intersectio n number s (i.e. , countin g orientation) . Ap parently thi s approac h doe s no t requir e compactifyin g M. However , sinc e th e invariants s o defined hav e to be independent o f many choice s mad e durin g th e construction, th e compactificatio n M i s needed , loosel y speaking , t o mak e sur e tha t none o f the points i n V\ Pi • • • fl Vk escapes t o infinity whe n Vi is varying. Thi s was exactly ho w Donaldson constructe d hi s invariants i n the first place . Thi s wa s also the approac h use d b y Ruan an d Tian i n their constructio n o f symplectic invariant s for semi-positiv e manifolds . Al l the success reinforce s th e notion tha t on e can and should construc t Donaldson-typ e invariant s b y counting solutions to certain syste m of PDE. Since ou r goa l i s t o explai n th e Donaldson-typ e invariants , w e now turn ou r attention t o th e questio n o f enumeratio n o f rationa l curve s i n algebrai c varieties , constructing invariant s o f symplecti c manifolds . I n earlie r wor k o f enumeratin g rational curve s i n Calabi-Ya u threefold , physicist s use d th e ter m "th e numbe r o f rational curves " quit e liberally . Thoug h i t wa s not, a s ha s no t been , establishe d that fo r genera l X 5 C P 4 th e spac e o f rationa l curve s ar e discret e an d reduced , the physicist s nevertheles s proceede d a s i f the y wer e computin g th e "number " o f rational curve s i n X§ i n each degree . T o mathematicians, thi s se t o f numbers ca n be though t o f as the physicists' numbe r o f rational curve s i n a general quintic . Now w e switch t o a genera l smoot h projectiv e variet y X. Afte r fixing a homology clas s a G H2(X;Z) an d integer s g an d n , w e can for m th e modul i spac e consisting o f dat a {f:C—>X an d X i , . . . , x n € C}, where C ar e smooth genu s g curve s (i.e. , Rieman n surface s wit h conforma l struc tures), / ar e holomorphic map s suc h tha t /*([C] ) = a an d # i , . .. , xn ar e n-distinct points o f C. I t was known tha t thi s modul i space , denote d b y M9in(X, a ) , is quasiprojective an d admits tautologica l topologica l clas s /i: H*(X,Q) x

H*(X,Q) x

• •. x n H*{X,Q) - > H*(M

9in(X,a);Q)

defined b y /x(ei,..., e n ) = evi(ci ) U • • • U evn (e n ), where ev^ : M— » X ar e the evaluatio n morphism s define d b y evi(f) = f(xi). W e shall abbreviat e M g,n(X, a) t o M i f ther e i s n o confusion . (I f on e work s wit h a symplectic manifol d (X,u), the n on e can pic k a tame d almos t comple x structur e J an d for m th e moduli spac e o f similar object s wit h / bein g pseudo-holomorphic . In thi s setting , M i s still a well-formed modul i space. ) I t i s known tha t M g^n(X, a) usually i s not compact . Th e choic e o f its compactificatio n consist s o f al l map s / as befor e excep t tha t C i s allowe d t o b e singula r wit h a t wors t noda l singularity . (One als o need s t o impos e stabilit y conditio n o n / . Whil e w e shall no t elaborat e the compactificatio n usin g stable map s here, we like to point ou t that suc h concept s date bac k t o Sach-Uhlenbeck o n harmonic map s o n spheres. I t wa s put on a usefu l setting b y Parker-Wolf son, Ye , Schumacher, an d Kontsevich.) Wit h thi s done , the tautological clas s / i extends canonicall y t o fl: H*(X,Q) x

H*(X,Q) x

• • • xn H*(X,R) - > tf*(M;Q).

72

S H I N G - T U N G YA U

To understan d th e mai n difficult y i n definin g th e invariants , w e nee d th e no tion o f expecte d dimension . Give n a n elemen t i n M , it s neighborhoo d i s th e so lution spac e o f a n operato r F: V\ — > V2 (near 0 G Vi) whos e first-order variation s 6F: TQVI — • T0V2 is a Fredhol m operator . Le t v b e it s index . The n i n cas e 6F i s surjective, th e invers e function theore m implie s that F~ 1 (0) i s smooth o f dimensio n v nea r 0 . Followin g the convention , w e call v the expecte d dimension , o r the virtua l dimension of M. Th e virtual dimension can be computed usin g Atiyah-Singer inde x theorem. (A s a consequence , v doe s no t depen d o n th e element s o f M. ) Unlik e th e situation o f ASD-connection s o f 4-manifolds , fo r genera l X , th e dimensio n o f th e space M wil l be bigger than the expected dimension , an d M ma y not be dense in M. For a specia l clas s o f symplecti c manifolds , th e so-calle d semi-positiv e symplecti c manifolds whic h includ e al l Calabi-Ya u threefolds , Rua n an d Tia n [21 ] constructe d the perturbe d modul i spac e M g,n{X,a)^ b y considerin g al l map s / : C — > X a s before suc h tha t / satisfie s th e inhomogeneou s Cauchy-Rieman n equatio n

djf = £ This ha s th e advantag e tha t M^ i s smoot h an d ha s th e expecte d dimension , an d that it s complemen t i n it s compactificatio n ha s codimensio n a t leas t 2 . Therefore , for an y ei , . . . , e n G H2*(X,Z) s o tha t n

^2 de S ei =

dim

R M g,n(X, a)*,

T=l

one ca n pic k cycle s V\ , . . . , V n C X s o tha t thei r respectiv e homolog y classe s ar e the Poincar e dua l t o ei , . . . , e n . The n i f Vi , . . . , V n ar e i n general position , th e se t {/ € M fl ,„(X,a)« I f(xi) eV i,i =

l,...,n}

is discret e an d th e tota l numbe r (countin g orientation ) i s a topologica l invariant . Ruan an d Tian called this symplectic invariant o f (X, e^,..., e n ). On e needs to check that th e s o defined numbe r i s independent o f the choic e of the inhomogeneou s ter m £ an d th e P.D . representative s Vi. Thi s i s true becaus e M — M i s small. ) Summing up , th e Donaldson-typ e invariant s ar e topological invariant s formall y defined b y / /x(£i)U"-U/x(e

n ).

JM

However, du e t o th e technica l difficulty , th e approac h t o "evaluate " thi s integra l i s to find "Poincar e dual " E ^ C M of/i(e^ ) an d defin e (th e integral) t o be the algebrai c intersection numbe r #aig(Sin...nEn). Morally speaking , th e approac h def

Topological Invarian t = Enumeratio n Invarian t is based o n th e fac t tha t 1. th e modul i spac e M i s smooth an d ha s th e expecte d dimensio n an d 2. th e complemen t o f M i n M i s relatively small . 1 usuall y ca n b e achieve d b y perturbin g th e syste m o f equations , an d 2 requires a rough estimat e o f th e siz e o f th e singula r solution s t o th e perturbe d system . Th e main advantag e o f this approac h i s that i t allow s one to bypass th e task o f studyin g the loca l structur e o f th e solutio n spac e nea r a singula r solution , whic h i s usuall y very difficul t t o carr y out . However , i n cas e dim( M — M) , o r eve n wors e whe n

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73

d i m M i s bigger tha n wha t i s expected, the n thi s approac h fail s completely . Thi s is indeed the case for general symplecti c manifolds . A new approac h wa s pioneere d by L i and Tian [1 8] . The ne w approach avoid s the process of transforming th e topological invariant s to a n enumerative numbe r altogether . Instea d i t relie s o n finding cycle s [M] vir G if v (M;Q), calle d th e virtual modul i cycles , an d defines th e topological invariant s by / J[M}™

/x(ci)U---UM(e

n

).

Here v is the expected (real ) dimensio n o f M. To better understan d thi s approach , le t us look at a toy model. Le t Z C P3 be defined b y equations z^z\ = 0 an d z§z\ = 0. Here [ZQ, . . ., zs] is the homogeneous coordinat e o f P3 . Z i s the union o f the plane \zo = 0 | an d the doubl e lin e \zi = z\ — 0|. O f cours e th e expecte d (complex ) dimension o f Z is dim z P 3 - #|equations | = 3 - 2 = 1 , which differs fro m th e actual dimension of Z. Imagin e that w e need to define certai n Donaldson-type invariant s o n the "modul i space " Z. W e cannot tak e th e cycl e [Z] € if*(P 3 ;Z) sinc e i t has the wrong degree . Instead , w e need t o find the virtual cycle [Z] V1 T in H2(Z;Z) a s the choice o f the "fundamental " class . T o accomplis h this, w e rewrite th e defining equation s a s follows: Le t L b e the line bundl e o n P3 at degre e 3 . The n bot h zfizi, an d z§z\ ar e sections o f L. Le t V = L © L an d le t s b e the section o f V define d b y {z^z\,z§z\). The n Z i s the vanishing locu s o f s. Under thi s setting , the virtual cycl e [Z) vu, i s nothing bu t the Euier clas s e(v) of V, which i s 9[£] e # 2 ( P 3 , Z ) , wher e £ C P 3 is a line . W e now recall th e two differen t ways to see such identity : Th e topological metho d an d the algebraic method . The topological method i s the usual technique. On e perturbs s to a new section, say 5 , so that th e graph o f s is transversal t o the 0-section o f V. The n [Z}™ = e(v) = [r

1

(0)}eH*(F3,Z).

The algebrai c approac h i s base d o n the normal con e constructio n o f Fulto n an d MacPherson [1 0] . Le t t b e a comple x number . On e considers th e section ts o f V and it s graph T ts i n the total spac e o f V. B y letting t — • oo , we obtain th e limit r ^ = li m T ts t—*oc

that i s a cycl e (eithe r considere d a s an algebrai c cycl e o r an oriented current ) i n the tota l spac e of V. Not e tha t suc h limi t doe s exist becaus e s is algebraic. T^ ha s pure comple x dimensio n 3 , which i s the dimension o f Ts. Th e virtual cycl e [Z] vir then ca n be defined b y first pickin g a general smoot h sectio n T and then let

[Z]™ = [r

3 00nrv)eH2(F ,z).

Indeed, sinc e T ^ i s contained i n the total spac e o f V\Z, th e so defined cycl e i s in H2(Z,Z). I t is called th e localized Eule r clas s of V wit h th e section s. The constructio n o f the virtual modul i cycl e [M] vlr o f M f f ) n (X,a) follow s the same line , dependin g o n whether on e works wit h th e projective varietie s o r sym plectic manifolds . Th e first construction o f this cycl e [M] vlr fo r projective varietie s was don e b y J. Li and G. Tian, whic h wa s reported i n the Santa Cru z Conferenc e on Algebrai c Geometr y durin g th e Summer o f 1 995 . A n alternative constructio n

74

SHING-TUNG YA U

was foun d b y Behren d an d Fantech i [2] . Th e constructio n o f virtua l modul i cycle s for symplecti c manifold s wer e independentl y carrie d ou t b y J . L i and G . Tia n [1 8 ] and Fukay a an d On o [1 2] . Alternativ e construction s wer e foun d b y Rua n [20 ] an d Siebert [22] . We no w sa y a fe w word s o n Li-Tian' s constructio n usin g bot h algebrai c an d symplectic method . W e first mentio n th e algebrai c construction . Le t X b e an y smooth projectiv e variet y an d le t M f f ) n (X,a) b e th e modul i spac e o f stabl e map s as before . M 9in(X, a) i s a projectiv e scheme . Followin g th e algebrai c constructio n of localize d Eule r clas s o f the to y mode l just mentioned , the y constructe d a vecto r bundle E ove r M 9tn(X,a) an d a con e cycl e C i n th e tota l spac e o f V. Thi s i s th e analogue of the bundl e V \ Z an d th e con e Too in the toy model. The n the y procee d to defin e th e virtua l modul i cycl e [M] vir b y first pickin g a smoot h sectio n r o f E a t general positio n an d the n defin e [M]viT = [Cnr

v]eH*(M;Q).

We will not giv e any mor e detail o n their constructio n bu t t o mentio n tha t th e pai r (E, C) i s constructed base d o n deformatio n theor y o f stable maps . (Her e we ignor e the technica l issu e wher e som e stabl e map s i n M 9in(X,a) ma y hav e nontrivia l automorphism.) Their symplecti c constructio n o f th e virtua l modul i cycl e [M] vir proceed s a s follows. The y first embe d M int o a n ambien t spac e s o tha t i t i s define d a s th e vanishing locu s o f a sectio n o f a vecto r bundle . Le t W b e th e spac e o f al l dat a {f:C-*X an

d xi,...,x

n

eC}

as befor e suc h tha t / i s continuou s an d suc h tha t th e restrictio n o f / t o th e irre ducible component s o f thei r domain s i s smooth . (Not e tha t C ma y b e singular. ) Obviously, W i s a n infinit e dimensiona l topologica l space . A t eac h [/ ] G W, le t V / be th e spac e o f al l (0 , l)-form s

A°/(rTC) and le t V = U / e w ^ / - Ther e i s a canonica l sectio n s o f V — • W define d b y setting s(f) = 8(f) fo r eac h / G W. I t i s eas y t o se e tha t M C W i s exactl y the vanishin g locu s o f s. Shoul d W b e a Banac h manifold , V a Banac h vecto r bundle an d ds(f) : TfW — > Vf i s Fredhol m fo r eac h / G M C W, i t i s know n ho w to pertur b th e sectio n s t o obtai n a ne w sectio n s s o tha t [5 -1 (0)] gives ris e t o the desire d virtua l modul i cycle . I n thi s particula r case , W i s no t Banac h nea r M — M. However , W enjoy s certai n weakl y smoothnes s nea r M — M an d th e tuple [PV , V, S] form s a weakl y Fredhol m bundle . Suc h notio n (o f weakly Fredhol m bundle) wa s introduce d i n th e pape r o f L i an d Tia n [1 8] . Th e propert y o f bein g weakly Fredhol m bundl e i s sufficient fo r the m t o construc t th e virtua l modul i cycl e [M] vir G if*(W,Q), an d consequentl y define s th e invariant s fo r genera l symplecti c manifolds. It will b e interestin g t o exten d th e definition s t o quasi-projectiv e manifolds . The specia l cas e o f noncompac t Ricc i flat manifol d i s alread y ver y interestin g a s Vafa an d hi s co-author s hav e recentl y develope d loca l mirro r manifolds . References 1. K . Behrend , Gromov-Witten invariants, Invent . Math . 1 2 7 (1 997) , no . 3 , 601 -61 7 . 2. K . Behren d an d B . Fantechi , The intrinsic normal cone, Invent . Math . 1 2 8 (1 997) , no . 1 , 45-88.

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3. P . Candelas, X . C. de la Ossa, P . S. Green, an d L . Parkes , A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Essay s o n Mirror Manifolds , Internat . Press , Hon g Kong, 1 992 , pp . 31 -95 . 4. H . Clemens, Some results about Abel-Jacobi mappings. Topic s i n Transcendental Algebrai c Geometry (Princeton , N.J. , 1 981 /1 982) , Ann . o f Math. Stud. , vol . 1 06 , Princeto n Univ . Press , Princeton, NJ , 1 984 , pp . 289-304 . 5. , Curves on higher-dimensional complex projective manifolds, Proceeding s o f the International Congres s o f Mathematicians, vol . 1 , 2 (Berkeley, 1 986) , Amer . Math . S o c , Provi dence, RI , 1 987 , pp . 634-640 . 6. S . Donaldson, Polynomial invariants for smooth four-manifolds, Topolog y 2 9 (1 990) , no . 3, 257-315. 7. P . Delign e an d D. Mumford, The irreducibility of the space of curves of given genus, Inst . Hautes Etude s Sci . Publ . Math . 3 6 (1969), 75-1 09 . 8. G . Ellingsrud an d S . A. Str0mme, The number of twisted cubic curves on the general quintic threefold. Math . Scand . 7 6 (1995), no . 1 , 5-34. 9. W . Fulton, Intersection theory, Ergeb . Math . Grenzgeb . (3) , vol . 2 , Springer-Verlag, Berlin New York , 1 984 . 10. W . Fulton an d R . MacPherson, A compactification of configuration spaces, Ann . o f Math. (2) 139 (1 994) , no . 1 , 183-225. 11. W . Fulto n an d R . Pandhariparde , Notes on stable maps and quantum cohomology (t o appear) . 12. K . Fukaya an d K . Ono, Arnold conjecture and Gromov- Witten invariants (preprint) . 13. T . Johnsen an d S . Kleiman, Rational curves of degree at most 9 on a general quintic threefold, Comm. Algebr a 2 4 (1996), no . 8 , 2721-2753. 14. S . Katz , Rational curves on Calabi-Yau threefolds, Essay s o n Mirro r Manifolds , Internat . Press, Hon g Kong , 1 992 , pp . 1 68-1 80 . 15. M . Kontsevich , Enumeration of rational curves via torus actions, Th e modul i spac e o f curves (Texel Island , 1 994) , Progr . Math. , vol . 1 29 , Birkhauser , Boston , MA , 1 995 , pp . 335-368 . 16. M . Kontsevich an d Yu . Manin , Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm . Math . Phys . 1 6 4 (1994), no . 3 , 525-562. 17. J . Li.Algebraic geometric interpretation of Donaldson's polynomial invariants. J . Differentia l Geom. 3 7 (1993), no . 2 , 417-466. 18. J . Li and G . Tian , Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J. Amer . Math . Soc . (t o appear). 19. Y . Ruan, Topological sigma model and Donalds on-type invariants in Gromov theory, Duk e Math. J . 83 (1996), no . 2 , 461-500. 20. Y . Ruan, Virtual neighborhoods and pseudo-holomorphic curves, preprint . 21. Y . Ruan an d G . Tian, A mathematical theory of quantum cohomology, J . Differential Geom . 42 (1 995) , no . 2 , 259-367. 22. B . Siebert, Gromov-Witten invariants for general symplectic manifolds, (t o appear) . 23. G . Tian, Quantum cohomology and its associativity, Curren t Development s i n Mathematics , 1995 (Cambridge , MA) , Internat . Press , Cambridge , MA , 1 994 , pp . 361 -401 . 24. E . Witten, Topological sigma models. Comm . Math . Phys . 1 1 8 (1988), no . 3 , 411-449. 25. , Two-dimensional gravity and intersection theory on moduli space, Survey s i n Differ ential Geometr y (Cambridge , MA , 1 990) , Lehig h Univ. , Bethlehem , PA , 1 991 , pp. 243-31 0 . 26. , Mirror manifolds and topological field theory, Essay s o n Mirror Manifolds , Internat . Press, Hon g Kong , 1 992 , pp . 1 20-1 58 . 27. S . T. Yau (ed.) , Essays on mirror manifolds, Internat . Press , Hon g Kong , 1 992 . DEPARTMENT O F MATHEMATICS, HARVAR D UNIVERSITY , CAMBRIDGE , M A 021 38 , U.S.A .

E-mail address: yauQmath.harvard.ed u

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https://doi.org/10.1090/amsip/010/03 AMS/IP Studie s i n Advance d Mathematic s Volume 1 0 , 1 99 9

Compactified Modul i Space s o f Pseudo-Holomorphi c Curve s Thomas H. Parker ABSTRACT. Sequence s o f pseudoholomorphi c map s ma y fai l t o converg e be cause o f th e bubblin g phenomenon . However , on e ca n obtai n a limi t b y th e bubble tre e convergenc e schem e introduce d i n [9] . Thi s iterate s th e renormal ization procedur e o f Sacks-Uhlenbec k t o produc e bubble s o n bubbles . Thi s article contain s a n improve d proo f o f th e Bubbl e Tre e Convergenc e Theorem ; in particular , i t extend s th e resul t t o th e perturbe d holomorphi c maps . The latte r section s describ e ho w th e Bubbl e Tre e Convergenc e Theore m leads t o severa l way s o f compactifyin g th e modul i spac e o f perturbe d an d un perturbed holomorphi c maps , an d ho w thes e compactification s ar e use d i n th e definition o f symplecti c invariants , popularl y know n a s "Gromo v invariants" .

Introduction Over th e las t decade , holomorphi c map s hav e becom e a n importan t too l fo r studying th e structur e o f symplecti c manifold s (X,LJ). Th e basi c idea , du e t o Gromov i n 1 985 , i s t o introduc e a n almos t comple x structur e J an d stud y th e set o f J-holomorphi c curve s o n X. Gromov' s idea s wer e furthe r develope d b y Floer, McDuff , an d others . Recently , remarkabl e advance s hav e bee n mad e b y Kontsevich, Rua n an d Tian , an d Taubes . Thes e idea s ar e rapidl y evolvin g int o a new theor y fo r addressin g problem s i n symplecti c topology , Kahle r geometry , an d classical enumerativ e geometry . Here is the setup . Give n a compact symplecti c manifol d (X,w), on e can alway s choose an almost comple x structure J tame d b y u (se e (1 .1 ) below) . Then , fo r eac h Riemann surfac e E with comple x structure j , on e can consider th e maps / : E— > X that satisf y th e Cauchy-Rieman n equation s df oj = Jo df. Such map s ar e calle d pseudoholomorphi c map s an d thei r image s ar e pseudoholo morphic curves . Mor e generally , on e ca n conside r (J , v)-holomorphic maps . Thes e are maps, introduced b y Gromov and Ruan-Tian [1 1 ] , satisfying th e inhomogeneou s Cauchy-Riemann equations : (0.1) dfoj-Jodf

=

uoj

1991 Mathematics Subject Classification. 58G03 . The autho r i s partiall y supporte d b y th e N.S.F. . © 1 99 9 America n Mathematica l Society , Internationa l Press , an d Centr e d e recherche s mathematique s 77

78

THOMAS H . PARKE R

where v i s an appropriate perturbatio n term . Thes e "perturbed " holomorphi c map s are easie r t o wor k wit h because , b y varyin g v, on e ca n insur e tha t thei r image s ar e in 'genera l position' . To construct symplecti c invariants, we fix a genus g surface (E , j) an d a generi c u;-tamed J o n X , an d conside r th e 'modul i space '

Matg(V,X) of pseudoholomorphi c map s representin g a clas s a G Hz(X). I n som e case s on e can directl y relat e Ai t o X , thereb y obtainin g informatio n abou t th e symplecti c structure o f X. I n quit e genera l circumstances , on e can us e M t o defin e symplecti c invariants b y following , ste p b y step , th e constructio n o f the Donaldso n invariants . To carr y ou t thi s program , on e need s a compactificatio n o f th e modul i space . The usua l 'Uhlenbec k compactification ' i s no t sufficien t fo r thi s purpos e because , when takin g limit s o f sequences i n the Uhlenbec k compactification , on e losses trac k of some of the components of the limiting pseudoholomorphic curve . A n appropriat e compactification wa s described , somewha t vaguely , b y Gromo v i n hi s well-know n paper [3] . A wea k versio n o f thi s compactificatio n wa s prove d b y J . Wolfso n [1 6] . A complete proo f (fo r fixe d comple x structure) wa s given in [9] . (Th e modification s needed t o dea l wit h varyin g comple x structur e hav e bee n outline d b y Y e [1 7 ] an d Ruan-Tian [1 1 ]) . Th e constructio n involve s thre e set s o f arguments : (a) backgroun d P.D.E . result s o n pseudoholomorphi c maps , (b) a renormalizatio n schem e yieldin g a 'Bubbl e Tre e Convergenc e Theorem' , and (c) th e constructio n o f th e compactificatio n usin g th e Convergenc e Theorem . The pape r i s a n expositio n o f thi s construction . Th e firs t tw o section s contai n background results ; thes e exten d th e analyti c result s fro m [9 ] t o cove r perturbe d holomorphic map s an d als o includ e som e ne w 'Colla r Lemmas' . Th e renormaliza tion schem e presente d i n Sectio n 3 i s mor e direc t tha n give n i n [9] . Th e Bubbl e Tree Convergenc e Theore m i s state d an d prove d i n Sectio n 4 , an d the n applie d i n Sections 5 an d 6 t o construc t construc t th e 'Minima l Bubbl e Tre e Compactifica tion'. Sectio n 7 i s a brie f overvie w o f th e definitio n o f th e symplecti c invariants , and th e las t sectio n contain s remark s o n symplecti c invariant s fo r 4-manifolds . The Bubbl e Tre e Convergenc e schem e i s rather complicated . Mos t o f the com plication, however , i s in th e technicalities . Strippe d o f the details , th e schem e goe s as follows . One begin s wit h a unifor m C 1 estimat e fo r pseudoholomorphi c map s wit h small energ y (Theore m 2.2) . Combinin g thi s estimat e wit h th e coverin g argu ment o f Sacks-Uhlenbec k [1 3] , on e show s tha t an y sequenc e o f holomorphi c map s fn: E — > X wit h bounde d energ y ha s a subsequenc e tha t converge s locall y i n C k off finitely man y point s pi £ E (Propositio n 3.1 ) . A renormalization procedure , du e to Uhlenbeck , i s then use d t o produc e a "bubbl e map " fa : Sf— > X associate d wit h each pi. Th e ke y point i s to realize that ther e are , i n fact, man y bubble s associate d with eac h p i% T o ge t a t these , on e renormahze s th e origina l map s f n aroun d eac h Pi t o obtai n a sequence o f map s j ny. Sf — > X associate d wit h eac h pi. These , in turn , converg e of f secondar y bubbl e point s p^ G Sf. Renormalizin g again , on e obtains a ne w laye r o f bubbl e map s associate d wit h eac h o f th e p^ . Th e proces s terminates afte r a finit e numbe r o f steps . Altogether , w e fin d tha t th e sequenc e

COMPACTIFIED MODUL I SPACE S O F PSEUDO-HOLOMORPHI C CURVE S 7

[fail

[fad

P23\

M

[fl]

p \

9

/P22

/

Po]

FIGURE 1

.

{fn} decomposes , i n a natura l way , int o sequence s / n , / : S V 5 2 V 5 2 >X that converg e in L 1 , 2 n C ° . Th e domain s o f these maps for m a 'bubbl e tre e domain ' as shown i n the middl e o f Figure 1 ; the image s ar e the correspondin g configuratio n of curves , show n o n th e righ t (a s a n algebrai c geomete r woul d dra w it) . Thes e ca n be encode d i n th e 'bubbl e tree ' o n th e left . Thus the limit i s specified b y a formal tre e T, a base map fo G MA0 (£ , X) wit h bubble points pi G S, an d bubble maps // G MAX (SJ, X) wit h bubble points pj on e the bubbl e sphere s Sj . T o parameterize thi s dat a on e should includ e a unit tangen t vector a t eac h bubbl e point , an d i n th e en d divid e b y th e circl e actio n rotatin g th e tangent space s an d th e bubbl e spheres . Altogether , thi s define s a completio n M[fo](X,X)xSymk°S(TX)

MAC]}

x jj Mapff t (Sl„X) x

Sym* ' S(TS

( o\\k )*„ /I \° )

2

where [/ ] denote s th e homolog y clas s o f a ma p / i n th e bubbl e tre e an d k — ko + X^fcj . Thi s i s give s one Bubbl e Tre e Compactification ; others , includin g th e 'Minimal Bubbl e Tre e Compactification ' describe d i n Sectio n 5 , ar e obtaine d a s quotients o f this one . Th e C° convergenc e ensure s that w e have capture d th e entir e image. I t als o impose s constraint s o n th e bubbl e tree—a t eac h leve l th e image s o f the bas e an d bubbl e mus t intersect . Al l thi s i s described i n detai l i n Section s 4-6 . This pape r deal s onl y wit h compactification s o f the spac e o f map s fro m a fixed Riemann surface . Th e genera l case , wher e th e comple x structur e o n th e domai n i s not fixed, i s outlined i n [1 7 ] an d [1 2] . Readers avers e t o analyti c technicalitie s ar e advise d t o rea d Section s 1 and 3 quickly (skippin g Sectio n 2) , an d begi n readin g i n earnes t a t Sectio n 4 . I than k Jo n Wolfso n an d Eleny-Nicolet a Ione l fo r man y helpfu l discussions . 1. Perturbe d an d unperturbe d holomorphi c map s Let X b e a close d symplecti c manifol d o f dimensio n 2n wit h symplecti c for m UJ. A n almos t comple x structur e o n X i s a n endomorphis m J o f TX satisfyin g J2 = — Id; thi s reduce s th e Sp(2n ) fram e bundl e t o a U(n ) bundle . Sinc e U(n ) i s a deformatio n retrac t o f Sp(2n) , (X,LU) admit s a n almos t comple x structure , an d any tw o suc h structure s ar e homotopic . A n almos t comple x structur e J i s calle d uj-tamed i f UJ i s positiv e o n al l J comple x line s i n T X , i.e . i f (1.1) UJ(V,JV)>0

Vv

eTX,v^O,

THOMAS H . PARKE R

80

and i s calle d UJ-compatible if i t als o satisfie s (1 .1 ) an d (1.2) LJ(JU,

Jv) =

LU(U, v) fo r al l u, v e TX.

It i s eas y t o se e tha t th e space s o f al l smoot h a;-tame d an d ^-compatibl e almos t complex structures , denote d J(u) an d J c{w) respectively , ar e nonempt y an d con nected. The advantag e o f imposin g th e additiona l conditio n (1 .2 ) i s t o mak e contac t with (almost ) Hermitia n geometry . Eac h ^-compatibl e J determine s a Hermitia n metric Z i o n l , calle d th e J-compatible metric, define d b y (1.3) h(u,

v) = (j(u, Jv).

Note h i s positive-definite b y (1 .1 ) an d symmetri c b y (1 .2) . Now let E b e a closed Rieman n surfac e o f genus g with comple x structur e j . A map / : E — • X i s calle d j J-holomorphic i f (1.4) dfoj

=

Jodf.

This i s th e Cauchy-Rieman n equatio n fo r th e ma p / . I t i s a firs t orde r ellipti c system. W e wil l sometime s writ e (1 .4 ) a s djf = 0 ; whe n thi s i s don e on e mus t bear i n min d tha t th e operato r dj i s nonlinear (i t depend s o n th e valu e o f J a t the imag e o f / ) . Eac h solutio n / represent s a clas s a e H2(X, Z) ; we then cal l / a j J-holomorphic a-map an d th e imag e o f / a j J-holomorphic a-curve. To define invariant s o f a symplecti c manifol d (X,LJ), on e introduces a a;-tame d or cj-compatible almos t comple x structur e J o n X an d consider s th e 'modul i space ' (1.5) M

a(Z,X)

of j J-holomorphic map s representin g a clas s a. Th e invariant s ar e the n define d by followin g th e construction s o f Donaldso n theor y (se e Sectio n 7 below) . I n it s execution, thi s schem e run s int o technica l difficultie s o f severa l types . However , Ruan-Tian [1 1 ] observe d tha t man y o f thes e difficultie s ca n b e overcom e b y per turbing th e equatio n (1 .4) . Thes e perturbe d equation s ca n b e describe d i n tw o equivalent ways . 1. Directly , we replace (1 .4 ) by the inhomogeneous Cauchy-Rieman n equatio n (1.6) dfoj-Jodf

=

voj

(also writte n djf = v) wher e v i s chose n a s follows . Conside r th e projection s ExX

EX and th e bundl e H = H o m ^ T E , n^TX) ove r E x N. W e tak e v t o b e a n elemen t of the spac e Vj , j o f all smooth section s o f H whic h ar e anti-J-linear , tha t is , satisf y (1.7) u(j(v)) for al l tangen t vector s v G TE.

=

-J(u(v))

COMPACTIFIED MODULI SPACE S O F PSEUDO-HOLOMORPHI C CURVE S 8 1

2. Alternatively , give n / : E — • X , w e can conside r it s grap h F: E — > E x X by F(x ) = (a;,/(#)) . Introduc e a n almos t comple x structur e J' o n E x X b y

(1.8) J

J : T E 0 TX -> TE 0 TX.

'=r J

If J i s (j-tamed an d r i s an area form o n E, then J' i s tamed b y the symplecti c for m T+LU o n E x X. Th e zero in the matrix J' mean s that th e projectio n m : E x X— > E is J'j-holomorphic . The n (a) I f / : E — *A T i s a (jJ,v) perturbe d holomorphi c a-ma p the n it s grap h F i s an (unperturbed ) j J'-holomorphic ma p representing [E] +a G H2(T, x X, Z). (b) Conversely , i f F : E — • E x X i s a jJ'-holomorphi c ([E ] + cx)-ma p the n IT o F : T, —» E i s a degre e 1 j-holomorphic map . Thu s afte r precomposin g with a holomorphic automorphism o f (E, j ), F ha s the form F(x) = (x, /(#) ) where / i s a (j J, */ ) perturbed holomorphi c a-map . Thus passin g fro m / t o it s grap h give s a n equivalenc e (1.9) MaA*,

N)

= A4 [ E ] + a (E, E x iV )

between th e modul i spac e o f (jJ , z/)-holomorphi c a-maps , an d th e modul i spac e o f unperturbed j J'-holomorphic ([E ] -f a)-map s int o E x X . A ma p / : E — > X satisfyin g (1 .6 ) an d representin g a G #2(X, Z) i s calle d a (j J, is)-holomorphic map o r simply a perturbed holomorphic map. Sinc e maps in th e right-hand modul i spac e o f (1 .9 ) ar e injective , on e see s tha t thi s modul i spac e i s a manifold (cf . §6) . I n contrast, th e unperturbed modul i space (1 .5 ) is a manifold onl y removing th e subse t o f multiply-covere d maps . Thu s th e perturbatio n effectivel y smooths th e modul i space , makin g i t a muc h mor e tractabl e objec t t o dea l with . One o f th e aim s o f thi s articl e i s t o exten d th e result s o f [9 ] t o perturbe d holomorphic maps . I n mos t case s thi s ca n b e don e quit e simpl y usin g th e corre spondence describe d i n 2 above. W e will use thi s techniqu e i n severa l o f the proof s below. Before proceeding, i t is obligatory to insert a few words on the precise definitio n of th e variou s modul i space s o f holomorphi c maps . Fo r thi s i t i s mos t convenien t of wor k wit h Sobole v spaces . Fix , onc e an d fo r all , a n intege r s > 3 . Le t L s'p be th e Sobole v spac e whos e nor m i s th e su m o f th e L p norm s o f th e derivative s through orde r s. Th e Sobole v embeddin g theore m show s tha t an y L 5 ' 2 functio n o n E i s C 1 , an d an y l / s + n ~ 1 ' 2 functio n o n X i s C 1 . W e ca n the n complet e th e spac e Map(E,X) o f al l smoot h map s / : E — > X i n th e I/ s ' 2 norm , obtainin g a smoot h Hilbert manifol d Map s (E,X) (thi s constructio n i s standar d an d functorial—se e Palais [7]) . I n th e sam e wa y w e ca n us e th e l / s + n - 1 > 2 nor m t o complet e J(uo), JC{ C g whos e fibers ar e th e M a,j,j,iy Thes e large r moduli space s ar e use d i n th e literatur e ([3 , 1 2 , 1 4]) , bu t wil l no t b e discusse d here. There ar e thre e measure s o f th e 'size ' o f a ma p / : E —- > X int o a symplecti c manifold. Th e mos t natura l i s th e "symplecti c area "

J r 0 such tha t ch(v, v) < OJ(V, Jv) < Ch(v, v) fo r al l v G TX. Thu s fo r a j J-holomorphic a-curv e / (1.11) cAie*

h(f)


(X , h) i s conforma l in th e sens e tha t f*h i s i n th e conforma l clas s j. Henc e w e hav e (1.12) Are

a / l (/)

= £(/) .

The mos t basi c propert y o f perturbe d an d unperturbe d holomorphi c map s i s the followin g energ y bound . LEMMA 1 .1 . Suppose that (X,u) is a compact symplectic manifold. Fix an UJtamed almost complex structure J, a J-Hermitian metric h on X and a perturbation v. Then there is a constant C such that

(1 1 3 ) / {

\df\ 2du < lC^°^ when

U

M

}

' 7 s^ " \ C ( [ o ; ] , a ) + 2Area/ l (E) when for any Riemannian metric \x on E and any f G Ma,g,j,v

=0

v ^ 0

P R O O F . Firs t suppos e v = 0 . Fix p G E and a ^-orthogonal basi s {ei , e2 = je\} of T P E. Sinc e / i s j J-holomorphic (f*j = J/* ) an d J i s a;-tamed w e have , a t p ,

\df\2 = |/*ei| 2 + \fj ei\2 = 1

while f*v = {Fuj^e A

2|/*e 1 | 2 < cuif+euJf+a) =

c(/*(j)(ei,e 2 ),

2

e = (f*Lj)(e ue2)dfjL. Thu s [W\2d» =

c([w],a).

For v 7 ^ 0, thi s inequalit y applie s t o th e grap h F{x) = ( x , / ( x ) ) , whic h i s a jJ fholomorphic (a + [E])-ma p wit h \dF\ 2 = 2 + \df |2 . Th e resul t follows . •

COMPACTIFIED MODUL I SPACE S O F PSEUDO-HOLOMORPHI C CURVE S 8

3

When v = 0 poin t map s ar e jJ- holomorphic. Th e nex t lemm a show s that , for j J-holomorphic maps , ther e i s a n energ y leve l belo w whic h ther e ar e onl y th e trivial poin t map s ([1 3 , Theore m 3.3 ] give s a simila r statemen t fo r harmoni c o r a-harmonic maps) . W e continu e t o suppos e tha t (X,LJ) i s a compac t symplecti c manifold an d J i s a n almos t comple x structur e tame d b y LJ. LEMMA 1 .2 . There is a constant Bo > 0 depending only on J,u and h such that any smooth (unperturbed) j J-holomorphic map f: E — » X, with Area/ l (/) < BQ is a map to a point. P R O O F . Whe n Area^(/ ) i s sufficientl y small , (1 .1 1 ) show s tha t th e intege r ([a;], a) i s zero , an d the n (1 .1 3 ) implie s tha t df = 0 . •

2. Analyti c result s The holomorphi c ma p equation s (1 .4 ) an d (1 .6 ) ar e elliptic , an d fro m a n an alytic viewpoin t ar e quit e simila r t o th e harmoni c ma p equations . Th e theorem s in thi s sectio n describ e th e basi c analyti c propertie s o f holomorphi c maps . Thes e results ar e th e buildin g block s fo r th e renormalizatio n construction s develope d i n later sections . I n th e beginnin g thes e theorem s paralle l thos e o f harmoni c ma p theory, bu t a s w e proceed som e importan t difference s emerge . The followin g regularit y resul t wa s prove d i n [9 ] using standar d ellipti c theor y assuming unde r th e assumptio n tha t v = 0 . Replacin g / b y it s grap h F(x) = (x, /(#)) immediatel y give s th e sam e resul t fo r genera l v ^ 0 . T H E O R E M 2. 1 (Regularit y Theorem) . Suppose (E,j ) and (X,u) are smooth, J e J s+2 and v e V$ 2. Then any weakly (j J, v) -perturbed holomorphic map f: E — > X that is L 1 ,p for some p > 2 or is C a for some a > 0 lies in L s, 0 , depending on J, v and the conformal class of the metric on E , such that whenever l f-.Yi-^Xisa C (jJ,v) perturbed holomorphic map and D(2r) is a geodesic disk of radius 2r < ro with E(2r) = f D,2r\ e(/ ) < eo, then T H E O R E M 2.

| < - y / £ ( 2 r ) + C f/.

(2.1) s u p | # D(r) ?

When v = 0 we can take r o = 1 and C = 0 . P R O O F . Firs t conside r th e cas e v — 0. B y differentiatin g th e J-holomorphi c map equation s an d usin g a Bochner-Weitzenboc k formula , on e obtain s th e differ ential inequalit y [9 , Lemm a 2.2] :

A e ( / ) < C i e ( / ) + C 2 e 2 (/). Let po be th e poin t wher e th e maximu m o f th e functio n f(p) = p 2 su

p e(f)

D(2r-2p)

is attained . Se t e 0 = su

p e(f)

D(2r-2p0)

84

THOMAS H . PARKE R

and le t xo b e a poin t suc h tha t e(#o ) = ^o - I t follow s tha t e(f) < 4eo pointwis e i n the dis k D = JD(#O , A))- Se t A = 1 + 4eo - Switchin g t o th e metri c p! = A/x , D ha s radius R = poVX, an d e'(/ ) = X~ 1 e(f) < 1 pointwise o n D. Henc e b y (2.20 ) e' satisfies (A 7 — a)e' < 0 where a = C\ + C 2 . Th e proo f o f Trudinger' s Mea n Valu e Theorem ([2 , Theore m 9.20] ) show s tha t ther e i s a constan t C suc h tha t e\x0)X with length(/(7^) ) < e\ii has an associated homology class a and satisfies

(2.4)

Area(/(fi)) < C A(a) + J2 ^ngth 2 (/( 7i ))

where A(a) = (w, [a]} is the "symplectic area" of the homology class a. P R O O F . Usin g th e J-tame d metri c o n X , fix e i smal l enoug h tha t eac h bal l B(p, ei) i s convex . The n fo r eac h poin t p e X w e hav e UJ = d(3 p on B(p, ei ) b y th e Poincare Lemma . I n fact , (3 P can b e determine d b y integratin g outwar d fro m p , s o satisfies a boun d

(2.5)

|/?p(x)j < Cdist(p , x) \/x

e B(p, ei) .

Here C depend s o n p , bu t b y th e compactnes s o f X w e ca n find a unifor m con stant C. The hypothesis o f the theorem ensure s that th e image of each boundary compo nent 7 i lie s in a ball Bi o f radius length(/(7;)) . Choos e smoot h disk s Di C Bi wit h dDi = /(7i) . Th e homolog y clas s o f suc h disk s Di i s well-defined . Consequently , the close d surfac e

5 = /(fi)u|j A i

defines a homology clas s a tha t i s naturally associate d t o /(fi) . It s symplecti c are a is

=w

A(a) = UJ

JS Jf(to) so b y th e tame d conditio n w e hav e Area(/(fi)) < c [ UJ JfW L

+ 5Z / i

w

>

JDi

= c\A(a) + V I / w ^ ' ^\JDi

On th e othe r hand , UJ — d(5i o n eac h Di, s o b y (2.5 ) (2.6)

[ u\ = \[ &

JDi I 1 .7/(7 The resul t follows .

0

< sup|/? 2 |[length(/( 7 i ))] < C

length2(/(7 X with E{r/2, 2R) < e 2 satisfies

(2.7) diamf(A(r,R))