Topics in Soliton Theory (North-Holland Mathematics Studies)

TOPICS IN SOLITON THEORY NORTH-HOLLAND MATHEMATICS STUDIES 167 (Continuation of the Notas de Matematica) Editor: Leo...

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TOPICS IN SOLITON THEORY

NORTH-HOLLAND MATHEMATICS STUDIES 167 (Continuation of the Notas de Matematica)

Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester New York, U S A .

NORTH-HOLLAND -AMSTERDAM

LONDON

9

NEW YORK

TOKYO

TOPICS IN SOLITON THEORY

Robert W. CARROLL Department of Math ematics University of lllin ois at Urbana- Champaign Urbana, IL, U.S.A.

1991 NORTH-HOLLAND - AMSTERDAM

LONDON

NEW YORK

TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 21 1, 1000 AE Amsterdam, The Netherlands Distributors for the U S A . and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y. 10010, U.S.A.

ISBN: 0 444 88869 1 Q 1991 ELSEVIER SCIENCE PUBLISHERS B.V. All rights reserved.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher.

No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Printed in The Netherlands

V

PREFACE

When s o l i t o n theory, based on w a t e r waves, plasmas, f i b e r o p t i c s , e t c . was d e v e l o p i n g i n t h e 1960-1970 e r a i t seemed t h a t perhaps KdV, and a few o t h e r equations, were r e a l l y r a t h e r " s p e c i a l " i n t h e s e t o f a l l i n t e r e s t i n g PDE. As i t t u r n s o u t , a l t h o u g h i n t e g r a b l e systems a r e s t i l l " s p e c i a l " ,

t h e mathe-

m a t i c a l i n t e r a c t i o n o f i n t e g r a b l e systems t h e o r y w i t h v i r t u a l l y a l l branches o f mathematics and w i t h many c u r r e n t l y d e v e l o p i n g areas o f t h e o r e t i c a l phys i c s , l e a d s one t o t h e c o n c l u s i o n t h a t one cannot T h i s book, a l t h o u g h l o n g and a t t i m e s " i n t r i c a t e " ,

not s t u d y

t h i s area.

b a r e l y touches on t h e ma-

t e r i a l p r e s e n t l y a v a i l a b l e , and concentrates, s t a r t i n g w i t h 17, on developi n g t h e theme o f t h e t a u f u n c t i o n .

I have e x t r a c t e d from many sources ( w i t h

r e f e r e n c e s g i v e n ) and t h e r e a r e a b o u t 1000 references; a t t i m e s I have f o l lowed v e r y c l o s e l y t h e " d e f i n i t i v e " o r v e r y i l l u m i n a t i n g e x p o s i t i o n o f c e r t a i n authors.

I had o r i g i n a l l y planned t o i n c l u d e s e c t i o n s o n

6

methods,

mu1 t i d i m e n s i o n a l i n v e r s e s c a t t e r i n g (dromions and a l l t h a t ) , n X n systems on t h e l i n e , i s o s p e c t r a l H a m i l t o n i a n flows, R m a t r i c e s and quantum groups, Toda l a t t i c e s and o t h e r examples, r e c u r s i o n operators, forced nonl i n e a r systems, formal c a l c u l u s o f v a r i a t i o n s , e t c . f i r s t and I r a n o u t o f space.

Somehow o t h e r t h i n g s were w r i t t e n

There a r e however a number o f books a l r e a d y

i n p r i n t o r f o r t h c o m i n g which cover many o f these t o p i c s ( c f . [AB5,12;BE4; C1; BL3; CA1; CR6; DX1; DK4; F2; KN1 ,2; NE1; N02; TO1 ;DS1; Z8;MKHl

I ).

We remark how-

ever t h a t KdV and KP equations a r e t r e a t e d e x t e n s i v e l y , w i t h some m a t e r i a l on NLS and AKNS systems, and i n f o l l o w i n g t h e t a u f u n c t i o n theme one i s l e d t o conformal f i e l d t h e o r y (CFT), s t r i n g s , and o t h e r t o p i c s i n p h y s i c s . There i s o f course a gap between mathematical r i g o r and f o r m a l i z a t i o n on one side, and a h o p e f u l l y meaningful d i s p l a y o f p h y s i c a l and geometrical i n g r e d i e n t s , w i t h formulas showing how t h e y f i t t o g e t h e r (sometimes h e u r i s t i c a l l y ) , on the other side.

I n many cases extreme mathematical r i g o r and f o r m a l i z a -

t i o n i s inappropriate, unavailable, o r a c t u a l l y counterproductive i n discuss i n g mathematical-physical

t h e o r i e s c u r r e n t l y under development.

I n certain

vi

ROBERT CARROLL

cases such v i r t u a l l y a x i o m a t i c f o r m a l i z a t i o n i s a v a i l a b l e and i n s t r u c t i v e and we have i n c l u d e d some such m a t e r i a l .

I n general we have o p t e d f o r t h e

o t h e r p o i n t o f view however; s i n c e so many t h i n g s a r e i n t e r a c t i n g w i t h each o t h e r we p r e f e r r e d t o l o o k a t t h e r e l a t i o n s and i n t e r a c t i o n s .

Thus we t r y

t o s k e t c h ( f o r a nonspecial i s t r e a d e r ) how v a r i o u s p h y s i c a l and mathematical ideas and themes a r e r e l a t e d , b u t t h e r e i s no a t t e m p t t o l o o k a t a l l poss i b l e cases o r f o r m u l a t e axioms and b e s t p o s s i b l e hypotheses (which would be silly).

Hence t h e book w i l l s u r e l y seem u n p o l i s h e d and u n d e f i n i t i v e and

t h e r e w i l l be p l e n t y o f m a t e r i a l t o keep c r i t i c s happy.

I n p h y s i c s my under-

s t a n d i n g i s v e r y l i m i t e d and i t comes m a i n l y from a s s o c i a t i n g p h y s i c a l conc e p t s w i t h m a t h e m a t i c a l l y meaningful s t r u c t u r e s .

The meaning f o r me l i e s i n

t h e mathematics b u t i t i s enormously h e l p f u l t o use p h y s i c a l ideas a s a s c r e e n i n g and s e l e c t i n g mechanism f o r d e v e l o p i n g and a p p r e c i a t i n g mathemat i c a l structure.

I n f a c t t h e r e a r e so many p o s s i b l e mathematical s t r u c t u r e s

t h a t i t i s t o me v e r y i m p o r t a n t t o ask why I s h o u l d be i n t e r e s t e d whenever one a r i s e s ; t h e p h y s i c s h e l p s me t o c a r e .

Somebody ( n o t

I ) should w r i t e a

book o f t h e t y p e " S t r i n g theory, o r conformal f i e l d theory, f o r t h e mathemat i c i a n " ( t o go w i t h books l i k e " D i f f e r e n t i a l geometry, o r topology, f o r t h e physicist"). We have l i m i t e d o u r i n c u r s i o n s i n t o physics, and indeed o f t e n i n t o t h e c h o i c e o f mathematical technique i t s e l f , by u s i n g t h e t a u f u n c t i o n as a g u i d i n g theme.

I n f a c t t h e t a u f u n c t i o n , a r i s i n g from v a r i o u s p o i n t s o f view, can

be c o n s i d e r e d as t h e dominant theme o f S7-11,

13-15, and 17-22.

Thus f o r ex-

ample we show how t a u f u n c t i o n s appear i n CFT and s t r i n g theory, and a f t e r a s k e t c h i n t o basics, s e l e c t o n l y m a t e r i a l i n these a r e a s which serves as a v e h i c l e t o develop s t r u c t u r e r e l a t i v e t o t a u f u n c t i o n s .

We f u r t h e r l i m i t

t h e p h y s i c s by n o t t a l k i n g about s u p e r s t r i n g s o r superanything (even super t a u f u n c t i o n s ) and by n o t going i n t o quantum groups, knots, b r a i d groups, e t c . (about which we d o n ' t know v e r y much anyway).

Also, r e l a t i v e t o a p p l i e d

mathematics, we do n o t r e a l l y d i s c u s s w a t e r waves o r a n y t h i n g e l s e v e r y phys i c a l i n t h e b e g i n n i n g s e c t i o n s on KdV and i n v e r s e s c a t t e r i n g . haps c u r i o u s l y ,

However, p e r -

I t h i n k o f t h i s as a book i n a p p l i e d mathematics.

l e c t u r e d here a t Urbana-Champaign,

I have

more o r l e s s s u c c e s s f u l l y , on t o p i c s i n

§ l - 7 t o mixed c l a s s e s o f graduate engineering, a p p l i e d mathematics, and phy-

v ii

PREFACE s i c s students, and w i t h l e s s success p a r t s o f §8,9,17,20,21

were a l s o cover-

ed; t h i s l a t t e r m a t e r i a l was however m a i n l y o f i n t e r e s t t o s t u d e n t s who had some background i n quantum f i e l d t h e o r y .

One would a l s o l i k e t o assume t h a t

some pure mathematics s t u d e n t s would be i n t e r e s t e d i n l a r g e chunks o f t h e m a t e r i a l (more i m p o r t a n t , I w i s h one c o u l d e l i m i n a t e a r t i f i c i a l b a r r i e r s between Dure and a p p l i e d mathematics, and perhaps t h i s book w i l l p o i n t i n t h a t direction).

In any event, i f one can c l a i m t h a t

i n t h e new C o u r a n t - H i l b e r t , p l i e d ma t hemati c s

KASll i s the f i r s t chapter

t h e n t h e p r e s e n t book can s u r e l y be c a l l e d ap-

.

Some d e f e c t s I have noted, b u t n o t c o m p l e t e l y solved, sage" i n a g i v e n paragraph o r s u b s e c t i o n .

i n v o l v e f i r s t t h e "mes-

Sometimes t h e message i s s i m p l y

t o show how v a r i o u s t h i n g s a r e r e l a t e d o r t o b u i l d up a theme toward t h e n e x t paragraphs.

T h i s can appear u n s a t i s f a c t o r y a t t i m e s i n terms o f con-

c l u s i o n s b u t a d v e r t i s i n g and m a r k e t i n g s t r i k e me as more t h a n u s u a l l y t a s t e l e s s when t h e m a t e r i a l i s i n t r i n s i c a l l y i n t e r e s t i n g i n i t s e l f .

This problem

i s perhaps exacerbated b y t h e f a c t t h a t I am w r i t i n g i n a s o r t o f n a r r a t i v e s t y l e , w i t h s p a c i n g by paragraphs and remarks, r a t h e r than i n a theorem-proof format.

There a r e o f course some theorems and p r o o f s as such b u t most r e -

s u l t s a r e embedded i n t h e t e x t .

Many m a t t e r s a r e b e i n g presented i n t h e

s p i r i t o f d i s p l a y i n g a g r e a t v a r i e t y o f i n g r e d i e n t s and showing how t h e y a l l f i t t o g e t h e r ; i t t h e r e f o r e seems i n a p p r o p r i a t e t o l a b e

something a theorem

when i t i s s i m p l y one o f many i n t e r e s t i n g r e l a t i o n s i n some development o f i d e a s (why s h o u l d t h e r e be a punch l i n e ? ) .

We t r y t o

n c l ude t h e e s s e n t i a l

f e a t u r e s and g e n e r a l l y t h e d e t a i l s can be f i l l e d i n as a n o t so h a r d e x e r c i s e ( i f they a r e missing).

We have t r i e d t o d i s p l a y

n separated t e x t

l i n e s enough m a t e r i a l so as t o make t h e book e a s i e r t o read t h a n say CC17 b u t we s t i l l f i n d i t e x p e d i e n t t o use 6 dark symbols as d i s p l a y i n d i c a t o r s i n t h e f o l l o w i n g order: -.-A.

.*-....

*****A*.-**.

*A*--*.

*A

.

*A*&+.

b

which a r e used

+ ** *A .*-*. A*

fi

T h i s makes t h e t e x t dense a t t i m e s and

we t r y t o c o n c e n t r a t e heavy d e n s i t y i n areas o f more p e r i p h e r a l i n t e r e s t o r i n the midst o f long calculations.

We have used a remark f o r m a t t o separate

o r punctuate t o p i c s , u s u a l l y w i t h t h e t o p i c s i n d i c a t e d , and have c r e a t e d 3 more o r l e s s n a t u r a l l o o k i n g chapters; Chapter 1 i s more c l a s s i c a l and emphasizes a n a l y s i s , Chapter 2 i s newer m a t e r i a l and employs more algebra, and

viii

ROBERT CARROLL

-

is used i n various ways; i n Chapter 3 i s based on physics. The symbol p a r t i c u l a r i t sometimes means "corresponds t o " . There will be frequent rep e t i t i o n o f ideas, constructions, d e f i n i t i o n s , e t c . t o make reading more continuous and sometimes w i t h s l i g h t l y a l t e r e d notation ( t o make l i t e r a t u r e references more a c c e s s i b l e ) . Connections o f notation a r e made when needed. In p a r t i c u l a r , various material on Riemann surfaces, bundles, sheaves, Lie theory, schemes, e t c . comes i n concentrated doses a t various places, o f t e n repeated ( i t a l s o appears i n the appendices). This is d e l i b e r a t e , i n an a t tempt t o generate f a m i l i a r i t y ; w i t h a l i t t l e patience one will see t h a t such continued b r i e f exposure t o c e r t a i n presumably new ideas will gradually make the ideas seem natural and understandable. A t l e a s t this has been our experience; t h e expert can o f course s k i p such m a t e r i a l .

We hope the book will make a v a i l a b l e to nonexperts, students, and s c i e n t i s t s , e s p e c i a l l y those who a r e somewhat i s o l a t e d geographically or c u l t u r a l l y , some o f t h e ideas and techniques o f " s o l i t o n mathematics" ( i n a broad s e n s e ) . Parts o f t h e book can be used a s classroom material a s indicated above (e.g. §1,2,6,7,8 and p a r t s o f 9,10,11,12,13,18,19,23). Although 13-5 do contain a l o t o f d e t a i l i n places they a r e b e t t e r s u i t e d a s survey m a t e r i a l . 91417 and 20-21 would be d i f f i c u l t a s classroom material f o r various reasons; t h e r e is o f t e n a l o t o f d e t a i l b u t these a r e hopefully just "windows" i n t o c e r t a i n a r e a s . They a r e more i n t h e s p i r i t of "physics f o r t h e mathematic i a n " and hopefully will provide access to some a r e a s of c u r r e n t research i n mathematical physics. The s e c t i o n s were w r i t t e n i n t h e o r d e r 9,10,11 ,3,4,5, 1,2,6,7,8,14,13,20,17,21,18,19,12,16,15,23 and t h i s may cause some referencing ahead, b u t t h e r e seems t o be no severe problem. The appendices were a l s o w r i t t e n a t d i f f e r e n t times and have d i f f e r e n t f l a v o r s . For example t h e l a s t p a r t o f Appendix A was written a t a time when I a n t i c i p a t e d i n s e r t i n g much more material on isospectral Hamiltonian flows e t c . . Thus i t contains additional material on momentum maps, coadjoint orbits, e t c . which was used only marginally i n the text; we have simply retained this material s i n c e i t has independent i n t e r e s t and i n f a c t i t is used enough i n t h e t e x t to justif y i t s inclusion. Let us remark f i n a l l y t h a t t h e r e was a period i n t h e era 1960-1975 say when p h y s i c i s t s seemed t o be speaking a d i f f e r e n t language from mathematicians. Now t h e r e is more o f a dialogue b u t t h e beginner needs

PREFACE

ix

many p o i n t s o f c o n t a c t and a minimal d i c t i o n a r y t o make any progress.

We

have t r i e d t o p r o v i d e such an e n t r y i n t o c e r t a i n a r e a s o f p h y s i c s (and mathem a t i c s ! ) b u t emphasize a g a i n t h a t we have o n l y s c r a t c h e d t h e s u r f a c e .

One

i s urged t o r e a d j o u r n a l s such as CMP, LMP, NPB, PLA, PLB, e t c . and t h e v a r i -

ous c o l l e c t i o n s o f papers which appear w i t h t i t l e s l i k e " S u p e r s t r i n g s 89".

I would 1 i k e t o acknowledge s t i m u l a t i n g c o n v e r s a t i o n and/or o t h e r r e w a r d i n g c o n t a c t w i t h many people o v e r t h e l a s t 3-4 y e a r s r e l a t i v e t o t h e s u b j e c t o f I n p a r t i c u l a r l e t me mention

s o l i t o n mathematics and my r e l a t e d t r a v e l s .

( w i t h a p o l o g i e s f o r any i n a d v e r t e n t o m i s s i o n s ) :

M. A b l o w i t z , R. Beals, E.

Belokolos, M. Berger, M. B e r g v e l t , M. B o i t i , B. B o j a r s k i , J. Bona, F. Calogero, Y. Chow, A. Degasperis, R. Delanghe, S . D e L i l l o , J. Donaldson, N. E r c o l a n i , A. Fokas, A . Friedman, V. Gerdzikov, R. G i l b e r t , T. G i l l , A. Grsnbaum, W. V.

Haboush, M. Hazewinkel, D. Holm, T. Kano, D. &up,

Korepin, N. KOstov, M. Lapidus, P. Lax, J.E.

B. Konopelcenko,

Lee, J.H. Lee, B. L e v i t a n ,

Y. L i , S. Manakov, H. McKean, T. Miwa, K. Mukherjee,

Y. Komura, P. Newton,

F. N i j h o f f , Y. Nutko, S . Oharu, J. Palmer, 0. Pashaev, F. P e m p i n e l l i , Ch. Psppe,

J. Rabin, M. Ramachandran, A. Ramm, R. Rao, L. Raphael, J. Ryan, D.

S a t t i n g e r , J. Saut, N. Sauer, T. Seidman, J. S e r r i n , W. Strauss, T. Suzuki, J. S z m i g i e l s k i , H. Tanabe, A. tenKroode,

P. Tondeur,

C. Tracy,

N. Kenmochi,

Y. Tsutsumi, S . Ukai, S. Venadides, M. k'adati, G. Wilson, Lo Yang, W. Zachary,

J. Zagrodzinski, X. Zhou, and N. Zobin.

I would a l s o l i k e t o thank

P r o f e s s o r L. Nachbin f o r i n c l u d i n g t h i s book i n h i s N o r t h H o l l a n d Mathematics S e r i e s and Drs. A. Sevenster o f E l s e v i e r Science Publishers, who has c o n t r i b u t e d much t o s c i e n t i f i c p u b l i s h i n g o v e r t h e y e a r s by h i s deep u n d e r s t a n d i n g o f m a t t e r s s c i e n t i f i c and i n t e r c u l t u r a l . s t r e n g t h and a g r e a t t r a v e l i n g companion.

My w i f e Joan has been a source o f

F i n a l l y i n view o f t h e paramount

f a c t t h a t t h i s s u b j e c t has developed i n i t s p r e s e n t form because o f t h e i n t e r a c t i o n and c o l l a b o r a t i o n o f people from many c o u n t r i e s and o r i g i n s ,

I

would l i k e t o d e d i c a t e t h e book t o such people who speak across i n t e r n a t i o n a l boundaries i n t h e s e r v i c e o f a r t and science.

This Page Intentionally Left Blank

xi

TABLE OF CONTENTS

PREFACE

CHAPEER I.

V

KdV and KP; ANALMCZC MEEH0Dk

1- Znverse scattering 2. KdV m the fine 3, Problems i n mechanics and Hill's equatim 4, 0n the geumetry of KBU 5, f i n i t e ame patentials 6- Hamiltmian t h e a q f a r KdV 7. Determinant methads f a r KdV and KP; tau functims CHAPEER 2,

CHAPEER 3,

kWEEW AND ALGEBRAIC MEEH0Da3 8- 0 r b i t s a f the uacuum 9. AKNS systems 10, kame Lie thearetic methaas 11, Ehe Hirota bilinear identity 12, Algebraic curves and KP 13. Zntroductary Sata theary PH!@Xk 14. Hulanumic quantum f i e l d s 15, Zsing m a e l and Base gas 16, kume remarks an 2 - ~quantum gravity and KaU 17. Canfarmal f i e l d theary 18. Mure an canfarmaf f i e l d theary and tau f unctinns 19, Mure m Kriceuer data, Crassmannians, curves, etc20. Remarks anktrrings 21. m r e an strings, Riemann surfaces, and tau functriuns

1

12 21 33 50

58 70

99

109 138

152 165 182

205 227 249 255 267 283 295

311

xii

ROBERT CARROLL R e m a r k s on tau functians, C a u c h y - R i e m a n n operators, ana aeterminant bunales Quantum inverse scattering

326 332

DZFFERENCZAC CE0RECR1J AND ECBIIENCAR1J HAMZCC0NNZAN CH€0R1J

341

R Z r m A " 3URFACES AND ACGBRAZC CURVI23

369

22. 23. APPENDIX A. APPENDIX 3. REFERENCS

397

ZND?iX

421

1

CHAPTER 1 KdV AND KP; ANALYTIC METHODS

1. ZNVERSE SCACCERRZNG. There are many possible ways to begin a discussion of soliton mathematics and we a r b i t r a r i l y choose the idea of inverse scattering. This will give an introduction t o one main theme of particular int e r e s t in PDE a n d applied mathematics. The idea i s that (with suitable hypotheses) solutions of the Korteweg-deVries (KdV) equation (*) u t - 6 u u x t = 0 correspond t o "potentials" u ( x , t ) for the inverse scattering probuxxx 2 lem gXx u6 = - k $ for the 1 - 0 Schrodinger equation. Why this should be so a n d what i t means i s a long story involving many areas of mathematics and phys i c s .

-

Consider (A) $xx - 3 = m ( c f . here [AB5;ClY13,23, W e are generally n o t interested in best

REmARK 1.1

(90% S0ClICL0W AND SeAECERZNG DACA). 2 -k2$ with u real and say (1 t 1x1 ) l u ( x ) l d x <

lI

24;CAl;CDl;DSl;DIl;Fl;KYl;NTl1 ). hypotheses on u here b u t in applied problems of course t h i s would be import a n t ( c f . [ CD1 ;CN1-5;DIl;F1,2;KP2,3;TB2]). Frequently we will suppose u E $ (Schwartz space of Coo functions decreasing as x + fm, along with a l l derivatives, f a s t e r t h a n any polynomial). For the moment assume there are no 2 bound s t a t e s ( i . e . L solutions of (A)) a n d observe t h a t as x fm, u 0, so one must have ( J o s t ) solutions f + ( k , x ) + exp(kikx) as x +m (asymptoti-+

-+

-+

2 cally the equation i s $,, = - k $1. For real k the Wronskian W(f+(k,x), f -+ ( - k , x ) ) = f2ik where W(f,g) = f ' g - f g ' . One can e x p l i c i t l y construct f -+ via i t e r a t i v e procedures from (1 . l )

f,(k,x) = e i k x +

Lm CSink(s-x)/klu(s)f,(k,s)ds;

f - ( k , x ) = e-ikx

t

{Sink(x-s)/klu(s)f-(k,s)ds

-+ +-. To see how the i t e r a t i o n proceeds take {Sink(sf n ( k , x ) , fo = exp(ikx), a n d f n ( k , x ) = x ) / k l u ( s ) f n - l ( k , s ) d s . One notes ( c f . [CDl;Cll) t h a t ISinzl c e x p ( I I m z ( )

Then f+(k,x)exp(Tikx)9 1 as x f, and-write f,(k,x) =

1;

-

2

ROBERT CARROLL

{clzl/(l

+ IZl)}

so, t h i n k i n g o f x

exp(-xImk) w i t h

w i t h lkl(s-x)/Cl + I k l ( s - x ) ) 5 I k l s / ( l + I k l s ) (exercise) 2 0 f o r convenience, one has f o r Imk 2 0, I f o ( k , x ) l 5

1 fl ( k , x ) l

/,”

5

[ u f s ) ~ x p ( - s I m k ) ~ c s / ( l k+l ls ) k x p ( ( s - x ) I m k ) d s

< cQ(x)exp(-xImk) where Q ( x ) = Ix s l um (s)l/(l+lkls)ds

Ikls)).

(note

Q’

=

slul/(l +

The n e x t stage i n v o l v e s

(1.2)

5

If2(k,x)l

c

l u ( s ) l c Q ( s ) e- s Imk Ics/(l+lkls)le(S-X)lmkds

2 2 which i s bounded by c exp(-x1mk)Q ( x ) / 2 .

n This leads t o Ifn(k,x)l 5 (c ; fn converges a b s o l u t e l y and u n i Qn(x)/n! )exp(-xImk) and hence t h e s e r i e s 1

f o r m l y o n compact s e t s i n Imk

2 0 f o r a l l x 2 0.

e s t i m a t e s w i l l h o l d f o r x < 0 and t h u s f+(k,x) f u n c t i o n f o r Imk

20

and a l l x

f

One checks t h a t analogous

i s d e f i n e d as an a n a l y t i c

R w i t h If+(k,x)exp(-ikx)j

(c^

( n o t e exp(cQ)

i s bounded a l o n g w i t h Q ( x ) and r e c a l l t h a t a s e r i e s o f a n a l y t i c f u n c t i o n s o f k c o n v e r g i n g u n i f o r m l y on compact s e t s i n k determines an a n a l y t i c f u n c t i o n ) .

A s i m i l a r a n a l y s i s w i l l produce an a n a l y t i c bounded f u n c t i o n f - ( k , x ) e x p ( i k x ) ( Q z ) f o r Imk l a r g e (Imk (1.3)

2 0 (exercise).

We see a l s o d i r e c t l y from (1 .l)t h a t f o r I k l

0 ) one has

-

If+(k,x) A

ce

e

-sImk

ikx

I

5 ( l / l k l ) r {c

ds 5 ( c c ^ / l k l ) e

> 0 a g a i n f o r convenience (x -

-

-xImk

ds

one makes s u i t a b l e adjustments f o r x < 0 ) .

S i m i l a r l y , t h i n k i n g o f x < 0 f o r example (where I x - s I =

> 0

151

-

1x1) and Imk

with l k l large

(1.4)

If-(k,x)

-

e

-ikx

I

51: I c l k l l s l / ( l + l k l l s l ) } e Imk ”-”( l u ( s ) I / I k / )

One can now determine many i n t e r e s t i n g i n t r i n s i c f e a t u r e s o f (A) based on F+(k,x) f o r k r e a l . . F i r s t n o t e t h a t e.g. f,(-k,x) f, f (k,x) and f+(-k,x) a r e l i n e a r l y independent f o r k # 0, =

f

-

Further, since one can w r i t e

-

JOST SOLUTIONS

3

= -cZ2(-k), and some r o u t i n e c a l c u l a t i o n y i e l d s f o r k r e a l . cl1(k) 7 2 ==1/T, , cij(-k) = c'..(k), and Ic121 2 = 1 + lcllI 2 One w r i t e s cZ1 = c12 %1' 'J = RL/T, and cll = R/T, where T i s a t r a n s m i s s i o n c o e f f i c i e n t and t h e R, c22 RL a r e r e f l e c t i o n c o e f f i c i e n t s ( i l l u s t r a t e d below). A l s o t h e n o t a t i o n sll =

.

s Z 2 = T, s12 = R L y and sZ1 = R i s common.

An easy c a l c u l a t i o n g i v e s now

2 i k ; cZ2 = W ( f - ( - k , x ) , f + ( k , x ) ) / 2 i k

so v i a t h e a n a l y t i c i t y o f f,- f o r Imk 2 0 we see t h a t f o r k # 0, Imk 2 0, c12 i s a n a l y t i c . However a p r i o r i cll and c Z 2 may n o t be a n a l y t i c anywhere. Note a l s o from (1.1) t h a t one can w r i t e e.g.

It f o l l o w s from (1.5) t h a t as x

(1.8)

cll

S i m i l a r l y one o b t a i n s c,

LL

=

= 1

-

+

cllexp(ikx)

c 1 2 e x p ( - i k x ) so

- w h i l e c12 = cZ1

+

( 1 / 2 i k ) f I eiksu(s)f-(k,s)ds

l / Z i k ) i Z exp(iks)u(s)f+(k,s)ds

( 1 / 2 i k ) i I exp(-iks)u(s)f+(k,s)ds. +

n,

= ( 1 / 2 i k ) l z e- kSu(s ) f -(k,s)ds;

c1 2

as I k l

f-

+ my

T h i s y i e l d s cll

1 ( n o t e e.g.

cZ2

n,

and cZ1 = 1

-

= o ( l / k ) and c Z 2 = o ( l / k )

(l/Zik)/f

exp(iks)uexp(iks)

ds and t h e i n t e g r a l t e r m goes t o 0 by Riemann-Lebesgue). We can a l s o g e t a more p r e c i s e e s t i m a t e f o r cZ1 i n t h i s s p i r i t v i a c21 = 1

-

( l / Z i k ) L I u(s)ds + o ( l / k ) .

since i n p a r t i c u l a r x

-+ ?m

G e n e r a l l y one must be c a r e f u l w i t h a s y m p t o t i c s

or Ikl

-+

m

w i l l b o t h be o f i n t e r e s t ; however a t

t h i s s t a g e no d i f f i c u l t i e s a r e p r e s e n t .

OF XACCERZNG DACA).

REmARK 1.2 (FlIRCffER ANAI;#W cZ1

c12 may become i n f i n i t e as k

a l l cases s12

-+

-1 and sZ1

+

+

-1 as k

if

0 so sl1(0) +

The formulas show t h a t = 0 i n t h a t case w h i l e i n

0 ( c f . (1.8)).

G e n e r a l l y t h e sij

are

n o t continuous a t k = 0 i f o n l y (1 + I x l ) l u ( x ) l d x < - a n d t h a t i s one r e a 2 son we assume (1 + x ) l u ( x ) l d x < m ( c f . [CDl;CN1-5;DIl;F1,2;LCll f o r de2 2 tails). It i s a l s o i m p o r t a n t t o n o t e t h a t IT} + l R I 2 = 1 = I T [ + ~ R L I '

/f

4

ROBERT CARROLL

w i t h TE + R L i = 0.

= sZ2 i s meromorphic f o r Imk > 0

The f u n c t i o n T = sll

w i t h a t most a f i n i t e number o f s i m p l e p o l e s k

= i B . on t h e i m a g i n a r y a x i s j J (see below) w h i l e I s 1 2 ( k ) l = I s Z 1 ( k ) l < 1 f o r a l l r e a l k # 0. I f 1sl2(0)1 = 1 t h e n n e c e s s a r i l y sl2(0) = sZ1(O) = -1 (which occurs f o r " g e n e r i c " smooth

u as above

-

-

check t h i s v i a (1.8) when t h e i n t e g r a l s do n o t v a n i s h as k

see [AB5,10,11 1 f o r d e t a i l s ) .

-f

0

T h i s f a c t i s i m p o r t a n t i n showing t h a t ,

g e n e r i c a l l y , s o l u t i o n s u o f KdV o b t a i n e d b y i n v e r s e s c a t t e r i n g cannot be obt a i n e d from a r a p i d l y decaying s o l u t i o n o f t h e m o d i f i e d KdV (mKdV) e q u a t i o n 2 vt + 6v vx + vxxx = 0 v i a t h e M i u r a t r a n s f o r m a t i o n u = - v 2 i v x (cf.[[AB5]

-

-

more o n t h i s l a t e r ) .

= 0 (Imko > 0

-

L e t us check here t h e p o l e s o f T.

Thus l e t c 1 2 ( k o )

12,

n o t e from lc1212 = 1 + lcll

c12 cannot v a n i s h f o r k r e a l ) . 2 -2 Then f o r y = f + ( k o , x ) one has W(y,y)lz = ( y ' y - y i ' ) l : = 0 = - ( k o ko)* 2 l y / 2 d x . Note from (1.5) t h a t e.g. f+ = y Q e x p ( i k o x ) E L near m f o r 2 Imko > 0 and near - m y f+ % c Z 2 e x p ( - i k o x ) E L when c12(ko) = 0; hence (ko = 2 Therefore ko l i e s on t h e i m a g i n a r y a x i s a+iB) aB_/.f IyI dx = 0 so a = 0.

-

iI

and we w r i t e ko = i B , fi > 0 ( n o t e B = 0 has been excluded). Next we show t h a t such ko a r e s i m p l e p o l e s o f T and compute t h e r e s i d u e . From (1.5)-(1.6) t12 = /dk and p i c k a zero ko = iQ o f c1?. .1 2 (1/2iko)(W(f+,f- 1 + W(f+,F-)) = (1/2iko)(cllW(f+,f+) + cZ2W(f-,f 1. To 2 c a l c u l a t e t h e Wronskians one n o t e s e.g. from fl; + k f, = uf, t h a t Dx(f;(ko, Let

El,

= dc

-

Differentiating x)f+(k,x) f;(k,x)f+(ko,x)) = (k2 k 2o ) f + ( k o , x ) f + ( k , x ) . t h i s w i t h r e s p e c t t o k, and s e t t i n g k = ko, one g e t s upon i n t e g r a t i o n i-

(1.9)

W ( f + ( k o , x ) , ~ + ( k o y x ) ) = -2ko!;

Note W i n (1.9) vanishes as x

-f

m

A s i m i l a r c a l c u l a t i o n gives

and hence t h e formula f o r

l/iZ

z,,

above

We can a l s o w r i t e ( f r o m f ( k ,x)f-(ko,x)dx. mo 2 = c 11 f + ) 5 1 2 ( k0 ) = -icll(ko)im f+(ko,x)dx, which shows t h a t Cl2 # 0 and

y i e l d s i y = l/612(ko) =

-

> 0.

f o r Imk,

W ( f ~ ( k o Y x ) , ~ ~ ( k o y =x ) 2) k o i 2 f:(ko,s)ds f

f:(koys)ds

+

hence ko i s a s i m p l e zero. Thus t h e bound s t a t e e i g e n f u n c t i o n s fk(ko,x) (1.10)

(ll f+(ko,x)dx)-' 2

= mR = YC

(k

11

i n v o l v e n o r m a l i z a t i o n terms = -icll(ko)/~12(ko);

0

SCATTERING DATA

5

Note t h a t f o r real u , Y i s r e a l . As f o r the number N o f poles of T i n Imk > 0 one knows t h a t this i s expressed v i a N = ( 1 / 2 a i ) l C (612/c12)dk where C i s t h e real a x i s closed by a semicircular a r c a t contribution from t h e a r c will vanish since c

12

( s e e e.g. [ TC1 I ) .

m

-

1

%

The l / l k ( and E12/c12

l / l k 1 2 . S e t t i n g c12 = Ic121exp(i$(k)) one gets N = (1/2a)($(m) - $ ( - m ) ) < m (Levinson's theorem). Another way t o see t h a t N < m i s simply t o look a t

c12

%

1 f o r l a r g e l k l , c12 f 0 f o r k r e a l , so only a f i n i t e number of zeros

i n Imk > 0 a r e possible ( r e c a l l zeros of a n a l y t i c functions such a s c12 cannot accumulate i n a f i n i t e region). We r e f e r next t o a c l a s s i c a l construction i n complex a n a l y s i s (Poisson-Jen-

sen formula, e t c .

-

c f . [ TC1 1) in order t o w r i t e ( c 1 2 ( i P j )

0)

I t i s i n s t r u c t i v e (and necessary) t o consider o t h e r kinds of p o t e n t i a l s u , such as 6 functions, b u t we do not dwell on t h i s here ( c f . LCN1-5;FZ;LCl; KP2,31). The p r o p e r t i e s of c i j , fky e t c . can change r a d i c a l l y of course so one must be c a r e f u l . More d e l i c a t e problems involving p o t e n t i a l s u w i t h a r e discussed i n [CN1-5;DIl;F2]. only iz (1 + 1xl)luldx < RENARK 1.3 (EHE GEI;FAND-I;EI1ZC~AN-NR~EAN~ (GLl!t> ZQllACZ0N)- There a r e now 2 magical items which a r i s e . The f i r s t i s simply inverse s c a t t e r i n g i t s e l f whereby a b e a r t i f u l mathematical machine developed by Gel fand, Levitan, and MarEenko, among o t h e r s , permits one t o recover t h e potential u i n ( A ) from knowledge of R,T. The second i s t h e r e l a t i o n of u ( x , t ) as a s o l u t i o n of KdV t o u ( x , t ) a s a t-dependent potential i n the inverse s c a t t e r i n g problem f o r We deal f i r s t w i t h t h e GLM equation a s such ( c f . [ C1-3;5,9,11-13,15, (A). 16,23-41;CDl;Gl;KYl;LT1;3;MRl I ) . First take (1.5) i n t h e form ( 0 ) T(k)f-(k, x ) = Rf+(k,x) + f + ( - k , x ) ; Tf+(k,x) = RLf-(k,x) + f - ( - k , x ) and assume f o r simp l i c i t y t h a t t h e r e a r e no bound s t a t e s .

Denoting t h e Fourier transform by f

one knows t h a t t h e r e e x i s t s a t r i a n g u l a r kernel K ( x , S ) ( K ( x , S ) = 0 f o r x > 5 ) such t h a t

(1.12)

f+(k,x)

=

ejkx +

IxmK(x,S)eikSdS = FCS(x-E)

+ K(x,S)I

W e w r i t e g ( x , y ) = 6(x-y) + K(x,y) and note t h a t (1.12) follows b a s i c a l l y

ROBERT CARROLL

6

(since

from Paley-Wiener t y p e a n a l y s i s o r Hardy space i d e a s a p p l i e d t o f, i s bounded and a n a l y t i c f o r Imk > 0 ) .

f+(k,x)exp(-ikx) [ C1 ,20,23,24;CDl

;DM1 ;HD1 ;SUll

T h i s i s discussed i n

f o r example and o t h e r c o n s t r u c t i o n s o f K a r e

In a l s o p o s s i b l e u s i n g Riemann f u n c t i o n s and PDE ( c f . [C23,24;MR1 ;LT1,3]). 2 t h e p r e s e n t circumstances f+(k,x) e x p ( i k x ) E L i n k on l i n e s p a r a l l e l t o

-

t h e r e a l a x i s (Imk > 0 ) and K(x,y) w i l l have v a r i o u s p r o p e r t i e s (e.g. 2 E L ) which we do n o t d w e l l on here.

K(x,*)

Now we f o l l o w [ C1,131 ( c f . a l s o [ALl;NTl I)and n o t e t h a t (1.13)

( 1 / 2 ~ i ) 1 1Tf-(k,y)f+(k,x)dk

= 6(x-y)

T h i s i s a so c a l l e d completeness r e l a t i o n i n physics and can be d e r i v e d a s follows.

(E = k

2

-

Consider a Green's f u n c t i o n f o r D2

,

R e c a l l W(f,(k,x),f-(k,x))

Imk > 0).

+

u + k2 = L i n t h e form

= 2 i k / T from ( 1 . 6 ) and e v i -

Work i n t h e complex E p l a n e (as-

y ( L = Lx o r L ). Y suming no bound s t a t e s f o r convenience) and n o t e t h a t R x has a d i s c o n t i n u i t y

d e n t l y LR = 0 f o r x

-

a t y = x; namely ARx = Rx(x > y ) Rx(x < y ) = W ( f + , f - ) ( T / 2 i k ) 2 f o r Jl E C: ( C f u n c t i o n s w i t h compact s u p p o r t ) one has

I =

(1.15)

+

1:

Jl(x)LRdx =

Iy

Hence

= 1.

+

JlLRdx = ILR = $(y) Yx y-

Thus LR = 6 ( x - y ) which determines a Green's f u n c t i o n R. Now l e t 5

E

Coy 0 = (D

R/E = O(E-3/2)

for

IEl

2

-

u)S, and n o t e t h a t R = R(x,y,E)

-

+ (R/E,e

I n t e g r a t e around a l a r g e c o n t o u r his(y) =

1i m

dE '-{El=r

IEl

-

i n the

E p l a n e and p i c k s up t h e c i r c u l a r a r c \El

+

0 < k <

Q

and k2

-

i E 'L

= y i n t h e E plane t o get

t

JS(x)R(x,y,E)dx

Next draw a c o n t o u r C which c u t s o u t a r e c t a n g l e i E

with

Q

= s (y ) = (R,Lc)

(1.16)

2

I n distribution pairings (f,g) / f g d x we 2 = (R,(D2 u ) s ) + k ( R Y E ) . Hence S(y)/E = ( R , S )

large.

have (c,LR) ).

where E = k

--

< k < 0.

1' = y a t t h e ends.

By a n a l y t i c i t y

Not!

k

2

GLM EQUATION

7

(1 .17)

Now

-$ dER(x,y,k

2

m

-iE)

%

-Jo (T-/-Zik)f:f;Zkdk

=

0

(l/i)Lm

Tf,f

dk so (1.16)-

(1.17) y i e l d (1.13) ( T - ( k ) = T ( - k ) ) . The completeness r e l a t i o n (1.13) a l l o w s one now t o develop f o r m a l l y a number o f i n t e r e s t i n g r e l a t i o n s i n v o l v i n g kernels. F,f

=

{z

f(x)f+(k,x)dx

I f one t h i n k s o f t r a n s f o r m s

f o r example ( w i t h s u i t a b l e f ) t h e n a formal i n v e r s i o n

would i n v o l v e ( 1 / 2 n ) i z Tf-(k,y)F,fdk

=

if

f ( x ) s ( x - y ) d x = f(y).

S i m i l a r l y as

an a d j u n c t t o (1.13) one expects (when uniqueness p r e v a i l s ) a formula ( 6 ) (1/2r)!I

f,(k,x)T(m)f-(m,x)dx

= 6(k-m) and we r e f e r t o [ C1,13,23-261

d i s c u s s i o n o f t h i s ( w i t h d e t a i l s p r o v i d e d i n some s i t u a t i o n s ) . such as F,f

for a

Transforms

a r e e x t e n s i o n s o f t h e F o u r i e r t r a n s f o r m o f course and i t i s na-

t u r a l t o have corresponding t h e o r i e s i n s u i t a b l e spaces o f f u n c t i o n s and distributions.

For convenience we can assume f

E

f o r e ) f o r such t r a n s f o r m s and one notes t h a t T f " a d j o i n t " wave f u n c t i o n t o f,.

8 (Schwartz space as be(not f - ) i s the natural

I n f a c t the natural kernel r e l a t i o n ( f o r

completeness purposes) i s

( c f . [C1,181 and (**) below).

We remark a l s o t h a t f o r many arguments i n t h e

KdV development f o r example i t i s c o n v e n i e n t and a p p r o p r i a t e t o t a k e u E 8 . F i r s t ( c o n t i n u i n g t o assume no bound s t a t e s

-

i.e.

no p o l e s o f T f o r Imk

>

V

0 ) we can produce a formal s p e c t r a l r e p r e s e n t a t i o n f o r ~ ( x , y ) = 6 ( x - y ) t K(x,y) which maps e x p ( i k y )

(1.19)

V B(x,Y)

= (1/2n)lf

Then e v i d e n t l y (;(x,y), theory. (1.20)

-f

f,(k,x)

i n t h e form

f,(k,x)e

-ikYdk

exp(imy)) =

{

f,(k,x),a(k-m))

= f,(m,x)

by F o u r i e r

V

For a n i n v e r s e t o 6 we c o n s i d e r ;(y,x)

= ( 1 / 2 n ) / I T(k)f-(k,x)eikYdk

so t h a t by ( 6 ) above (~(y,x),f,(m,x))

= (T(k)exp(iky),a(k-m)/T(m))

= eiky

8

ROBERT CARROLL

formally.

= ~ ( x - Y )+ L ( y , x ) w i t h L t r i a n g u l a r ( L ( y , x )

We w r i t e a l s o ?(y,x)

= 0 f o r x < y).

The t r i a n g u l a r i t y can be proved d i r e c t l y from p r o p e r t i e s o f

T f - ( c f . t C1,23,241)

o r simply by n o t i n g t h a t v

7

i s the kernel f o r the i n -

verse o f a t r i a n g u l a r o p e r a t o r based on 6 and hence preserves t h e t r i a n g u l a r i ty. Now u s i n g (1.5) o r ( a ) f o r x < y we have 0 = ;(y,x)

(1.21) But (l/Zv)l; K(x,y)

= ( 1 / 2 v ) 1 1 eiky{ft(-kyx) = (1/2n)I;

exp(iky)f,(-k,x)dk

+ Rf+(k,x)ldk

exp(-iky)f,(k,x)dk

f o r x < y ) and hence f o r x < y K(x,y)

(1.22)

= - ( 1 / 2 v ) l z Re i k y f,(k,x)dk

On t h e o t h e r hand (1.21) says a l s o t h a t f o r general x,y. (l/Zv)L;

V

R e x p ( i k y ) f + ( k , x ) d k so f o r y < x (where B(x,y)

(1.23) (;(y,x)

(=

= i(x,y)

L(y,x)

y(y,x)

= E(x,y)

+

= 0 ) we o b t a i n

= ( 1 / 2 n ) l 1 ReikYf,(k,x)dk

L(y,x) f o r y < x ) .

pressed i n (1.22)-(1.23)

T h i s r e l a t i o n between t h e k e r n e l s K and L ex-

w i l l have c o u n t e r p a r t s i n v a r i o u s o t h e r s i t u a t i o n s

as w e l l ( c f . [ C 1 8 1 and 5 9 ) .

Next m u l t i p l y (1.12) by (1/2n)Rexp(iky),

y > x,

and i n t e g r a t e t o g e t t h e Maryenko (M) e q u a t i o n

0 = K(x,y) + F(x+y) t

(1.24)

Ix K(x,S)F(c+y)dS; m

R e c a l l t h e r e i s no d i s c r e t e spectrum here; cluce t h i s situation. 23,24;CDl;MRl

F(z) =

1:

Reikzdk/2n

F(z) w i l l be m o d i f i e d below

to in-

I n o r d e r t o connect K w i t h u now one method ( c f . [ C1,

1) i s t o combine (1.1) f o r f, w i t h (1.12) i n v o l v i n g

f, and K.

T h i s produces a c o m p l i c a t e d i n t e g r a l e q u a t i o n i n v o l v i n g u and K which r e duces n i c e l y a l o n g t h e diagonal t o y i e l d ( e x e r c i s e ) (*) u ( x ) = -2DxK(x,x) This i s then t h e essence o f i n v e r s e ( = 2DxL(x,x), f o l l o w i n g (1.22)-(1.23)). F i n d R by experiment, c o n s t r u c t F as i n s c a t t e r i n g ( w i t h no bound s t a t e s ) : (1.24),

s o l v e t h e M e q u a t i o n i n (1.24) f o r K, and compute u v i a (+).

We

n o t e t h a t (1.24) i s a Fredholm i n t e g r a l equation; i t w i l l be discussed a t g r e a t l e n g t h l a t e r i n t h e book.

GLM EQUATION When t h e r e i s a d i s c r e t e spectrum ( i . e .

9

T has a f i n i t e number o f s i m p l e

p o l e s i n Imk > 0 ) one can f o r m a l l y t a k e a c o n t o u r C s t a r t i n g a t - - t i 0 , s i n g o v e r t h e p o l e s o f T, and e n d i n g up a t - + i O . e r a l form o f (1.24) (k,x)

t f,(-k,x)

Then we d e r i v e a more gen-

( i n a d i f f e r e n t way) as f o l l o w s .

w i t h f,

pas-

W r i t e T f (k,x)

= Rf+

expressed v i a (1.12) i n terms o f K; m u l t i p l y by

( 1 / 2 n ) e x p ( i k y ) f o r y > x, and i n t e g r a t e o v e r C t o g e t (1.25)

0 = ( 1 / 2 s ) l c Tf-(k,x)eikYdk

( n o t e (1/2n)JC e x p ( i k ( y - x ) ) d k

= K(x,y) t F ( x t y ) t

= 6 ( x - y ) ) where F ( x t y ) = (1/2a)JC R e x p ( i k ( x +

y ) ) d k w i l l i n v o l v e some r e s i d u e terms. (1.25)

K(x,S)F(Sty)dS

Note a l s o t h a t t h e f i r s t i n t e g r a l i n

i s zero by v i r t u e o f a n a l y t i c i t y and growth f o r y > x

from above by a l a r g e s e m i c i r c l e .

Now r e c a l l R = cll/c12

e.g.

= Tcll

close C

may n o t be

u has compact support, i n

a n a l y t i c i n general b u t momentarily assume e.g.

w i l l be a n a l y t i c ( e x e r c i s e ) .

which case t h e cij

-

T w i l l have p o l e s a t a f i n -

i t e number o f p o i n t s k = i B j i n Imk > 0 w i t h i y = l / l ( i B . ) and mRj = j j 12 J y.c (ifi.) = -ic,l(iB.)/?12(iB.) as i n (1.10). Thus ( 1 / 2 s ) I c R e x p ( i k ( x + y ) ) J 11 J J J dk (1/2n)JC R e x p ( i k ( x t y ) ) d k = r e s i d u e s so

-il

-

(1.26)

F ( x t y ) = ( 1 / 2 n ) f I Reik(x+Y)dk

+

1N m 1

.e-Bj(xty) RJ

Thus i n t h e presence o f bound s t a t e s , f o r i n v e r s e s c a t t e r i n g one needs t o know R p l u s t h e p o s i t i o n s B . and t h e n o r m a l i z a t i o n c o n s t a n t s m f o r the J Rj bound s t a t e s .

M e q u a t i o n from t h e l e f t u s i n g ( m ) f - ( k , x ) = exp(-ikx) + K (x,S)exp(-ikS)dS b u t we emphasize t h a t K- and K- i n (1.18) a r e q u i t e d i f f e r e n t c r e a t u r e s . K- i s good f o r completeness i d e a s w h i l e KT i s a p p r o p r i a t e f o r M e q u a t i o n s . I n f a c t K-(x,y) = L(y,x) = L (x,y) ( T % One can e q u a l l y w e l l develop a

lt

transpose h e r e ) .

To see how t h s works n o t e from (1.20) b y F o u r i e r t r a n s -

form t h a t (*+) T ( k ) f - ( k , x ) I mx L(y,x)exp(-iky)dy

=

/f

= exp(-ikx

6(x-y) t L(y,x))exp(-iky)dy t

LaX

K-(x,y)exp(-iky)dy.

= exp(-ikx) t

Thus K-(x,y)

w e l l d e f i n e d as L(y,x)

and we n o t e t h a t (1.18) makes sense a s x

lows.

e x p ( - i k x ) we want somehow

Since f-(k,x)

( T - l ) e x p ( - i k x ) as x

fa Rexp(iky)f,(k,x)dk

-m

-+

-+ -m.

--

lt K - ( x , y ) e x p ( - i k y ) d y

To see how t h i s works w r i t e K-(x,y)

f o r y < x from (1.23).

-+

L e t x,y

+

--

is

as f o l -f

= (1/2s)

( y < x ) and con-

10

ROBERT CARROLL a

s i d e r ( r a t h e r crudely) K-(x,y) % ( 1 / 2 n ) j a exp(iky)ITf-(k,x) - f + ( - k , x ) l d k % a, ( 1 / 2 ~ ) _Texp(ik(y-x))dk /~ - g(x,y) % ( 1 / 2 n ) _ / I Texp(ik(y-x))dk - b(x-y) (note a l s o g(x,y) = 0 f o r y < x ) . = F(Texp(-ikx))(y)

Further Yx(y) =

_/ITexp(ik(y-x))dk

so (1/2r)( Yx(y),exp(-iky))

%

%

(fT)(y-x)

Texp(-ikx) and t h i s means

Yx(y) a c t s l i k e Tb(x-y) so K-(x,y) % (T-l)b(x-y). T h i s is h e u r i s t i c b u t could be polished up u s i n g t e s t functions. W e note from (1 + K)-' = 1 + L = 1 + K-T now t h e r e l a t i o n (operators a n d kernels a r e used interchangably when no confusion can a r i s e and 1 is used f o r I or f o r 6 function convolution) (1 + K T ) ( l + K ) = 1 ; K(x,y) + K-(y,x) +

(1.27)

K(C,y)K-(C,x)dC = 0

f o r x 5 y. There a r e a l s o d i r e c t r e l a t i o n s between completeness and GLM co equations ( c f . [ C1,13,23-251). For example one can w r i t e o u t (1/2n)ico Tf( k , y ) f + ( k , x ) d k i n terms of ( 1 . 1 2 ) , u s i n g (*), so t h a t everything i s expressed in terms of K and F (assume no bound s t a t e s ) . Then t h e M equation ( 1 . 2 4 ) applied twice, implies (1.13).

The kernel K - does not f i t nicely i n t o

a M equation b u t K - from ( w ) w i l l . T h u s formally use ( b u t with Tf, now) t o get f o r y < x now

(0)

again a s i n (1.25)

A

0 = (1/2n)Ic Tf+(k,x)e-jkydk = F(x+y) + K-(x,y) +

(1.28)

n

Again F will generally involve some residue terms and from (1.10) (1.29)

IN

? ( z ) = (1/211)lf RLe-ikZdk + 1 m L j e B j z

REWW 1.4 (VARlAClbW bN GCUI). Another i n t e r e s t i n g d e r i v a t i o n o f t h e M equation can be obtained via minimization procedures as i n [ C1,12,13,331 . T h u s note f i r s t t h a t E ( k ) = R ( - k ) f o r k real so t h a t when T ( k ) = f ( - k ) one m -has -la m Rfdk Lm Rfdk. Assume no bound s t a t e s and w r i t e t h e completeness r e l a t i o n (2.13) a s b(x-y) = ( 1 / 2 1 ~ ) j zI f + ( - k , y ) + Rf+(k,y))f+(k,x)dk. For s u i t a b l e f ( e . g . f E):C define again F+f = = ( f ( x ) , f + ( k , x ) ) . As a k i n d of Parseval formula one can w r i t e

;+

GLM EQUATION (1.30)

(1/2a)fI T?-(k)i+(k)dk =

(1/2a

11:

T(f(x),f-

(

11

f(x),g(x))

=

( k y x ) ) (g(Y), f+(k,y) )dk

-

A

Now f o r f r e a l ?+(k) = f + ( - k ) so one w r i t e s IIfl12 = (l/Za)L: (I(f+,f)/' R ( f + , f ) 2 dk i s r e a l . Consider R ( f + , f ) 2 )dk and by remarks above A = (1.31)

( e iky

-

f+(k,y)

+ qK(y,c)eikCdS)2Rdk;

( 1 / 2 x ) C leiky

-

f+(k,y)

f

T1 = (1/2a)l: ZZ =

and one wants t o m i n i m i z e E = El

(1.32)

-

-

+ (1/211)$

2q K ( y , t ) ( F ( t + y )

A

E = A

A

where T = El

z

+

21" Y

-

leikY

K(y,S

f+(k,y)12dk

/o

dy T

4'

= .-1 A J

-

+

-

+ (1/21~)1: (eiky

f+(k,y))

Adding t h e terms we f i n d

F(S+y)dS +

A

duce a t r a c e i n t e g r a l

K ( y Y S ) ~ ~ d d 5i k( Y e

/ym K ( Y ~ S ) K ( Y , ~ ) F ( S + ~ ~ ~ C ~ T I ;

+

t o make sense. + P 2 i s assumed I

.$ '

K(Y,C)K(~,TI)~: Reik(S+q)dkdSdrl

5 ) here. (1.33)

+ (1/

f+(k,y))2Rdk

+ K+(y,S))dS

= (1/2a)ff

K runs o v e r some s u i t a b l e c l a s s o f Now w r i t i n g o u t t h e terms we have

] f o r motivation.

El = ( 1 / 2 n ) 1 1 (eiky

f+(k,y))eikSRdk

Z 2 as

K(y,S)eikSd5l2dk

We d e f e r a d i s c u s s i o n o f p h i l o s o p h y here and r e f e r

real t r i a n g u l a r kernels. t o [ Cl,12,13,24,33;DYl

f

q

+

K(Y,S)K(Y,~)(G(S-TI)

f

F(S+n))dSdo

It i s now c o n v e n i e n t t o i n t r o -

n (1.33) ( o r i n (1.31)) and o m i t t i n g

i,'

-

2 now we

(2 2 ) d y . Note here t h a t TrK % K(x,x)dx and want t o m i n i m i z e M(K) = T TrKF 'L d x q K(x,S)F(E+x)dS w i t h TrKF = TrF*K* = TrFK* f o r F symmetric.

/o

Also TrKK* = n e l form).

i,' I," K(x,S)K(x,S)dSdx

( s i n c e KK*

Hence we want t o m i n i m i z e

%

.

1" K(x,S)K(y,S)dS max ( x y 1

i n ker-

12

ROBERT CARROLL

M ( K ) = T r W F + FK* + K ( l + F)K*I

(1.34)

o v e r a s u i t a b l e c l a s s o f admissable t r i a n g u l a r k e r n e l s K .

One performs a

v a r i a t i o n a l argument based on t h e known e x i s t e n c e o f a m i n i m i z i n g k e r n e l K+. Thus s e t K = K+ + dl, ?l admissable, d i f f e r e n t i a t e M(K+ + E

= 0.

Em)

in

and s e t

E,

One o b t a i n s 0 = 2 T r I ( K + ( 1 + F) + F)M*3 f o r a l l admissable M and con-

s e q u e n t l y K+(1 + F ) + F = 0 which i s t h e M equation.

The r e s u l t i s t h e r e -

f o r e t h a t the M equation i s the minimizing c r t e r i o n (Euler equation) f o r

M ( K ) and t h i s c h a r a c t e r i z e s t h e k e r n e l K., There i s a l s o an i m p o r t a n t f a c t o r i z a t i o n (imp i c i t i n t h e form o f which says y ( x , y ) (1.35)

=

(1 + K - ) ( y , x )

= (1 + K+)(1

(1 + K-)(y,x)

x - c ) + F(€,+x))

and

y)

or

+ F)(y,x>

To see t h i s s i m p l y w r i t e o u t i n s p e c t r a l terms

(assuming no bound s t a t e s ) .

(F(S,x)

=

'b

= F(x + 5 ) )

(1.36)

(

;(y,s),S(x-s)

( 1 / 2 1 ~ ) / Reim(S+x)dm) (1/211)/ eikx(f+(-k,y)

=

+ F(C,x))

= (

( 1 / 2 n ) l e-ikSf+(k,y)dk,S(x-~)

( 1 / 2 n ) l e-jkxf+(k,y)dk

+ Rf+(k,y))dk

=

+ ( 1 / 2 ~ l ) l Rf+(k,y)eikxdk

( 1 / 2 n ) / e i k x Tf-(k,y)dk

=

+ =

;(x,y)

L e t us emphasize here t h a t f o r KdV (and ( w i t h y ( x , y ) = (1 + K - ) ( y , x ) ) . AKNS systems) t h e r e i s an u n u s u a l l y " t i g h t " r e l a t i o n between K- and K+; f o r KP t h e s i t u a t i o n i s d i f f e r e n t .

equations i n [C1,13,23-25] situations. 2.

There a r e many more formulas i n v o l v i n g GLM

f o r example, b o t h i n h a l f l i n e and f u l l l i n e s i -

We w i l l encounter such equations i n v a r i o u s r o l e s .

KdlJ ON &HE CINE.

We w i l l d i s p l a y many p o i n t s o f view now, w i t h n o t h i n g

d i s t i n g u i s h e d as " t h e fundamental s t a r t i n g p o i n t " , h i s t o r i c a l l y o r conceptually.

I n f a c t t h e " u l t i m a t e meaning" o f a l l t h i s i s v e r y m u l t i f a c e t e d ,

s i n c e KdV i s so v e r y i n v o l v e d w i t h many areas o f mathematics and physics. This d i v e r s i t y and r i c h n e s s w i l l be i n c r e a s i n g l y v i s i b l e as we go along.Perhaps t h e s i m p l e s t approach i s t o l o o k a t wave f u n c t i o n s $ ( x , t ) (2.1)

$xx + q ( x , t ) $ = LIL

2

- k IL; ILt = B$ = -4J/,,,

-

6qILx

satisfying

-

3qX$

LAX EQUATIONS N o t a t i o n v a r i e s here.

We use L = D

2

13

-

+ q i n s t e a d o f D2

u a t t i m e s and t h e

KdV e q u a t i o n a r i s e s by r e q u i r i n g c o m p a t a b i l i t y o f t h e equations i n (2.1) i n t h e form (*) qt + 6qqx + qxxx = 0 o r ut

u

= K(u)

-

-

6uux + u xxx = 0 (sometimes w r i t t e n c f . [ AB5;Cl ;DS1 ;LC1 ] f o r elementary d i s c u s s i o n o f KdV and see

t e.g. [ AB1,9;GDl;LX1-31

One can express t h i s i n t h e l a n -

f o r history etc.).

Consider L6 = X$

guage o f Lax p a i r s (L,B) and " i s o s p e c t r a l i t y " as f o l l o w s . and 6,

= B$ w i t h t h e requirement t h a t A t

T h i s i s a c t u a l l y somewhat

= 0.

m i s l e a d i n g here s i n c e t h e r e i s no v i s i b l e " c o n t r o l " on t h e continuous spect r u m b u t one t h i n k s g e n e r a l l y o f t h e spectrum r e m a i n i n g f i x e d .

Actually,

t h e " i s o s p e c t r a l m a n i f o l d " i s determined b y T(k,t)

f i x e d i n t and R(k,t)

( o r r e a l l y phase R ) v a r i e s ( c f . [ C16-19;E3;Mc1,6-11

I).

The i s o s p e c t r a l mani

f o l d i s more v i s i b l e i n t h e f i n i t e zone s i t u a t i o n discussed i n 83-5, where Riemann s u r f a c e s and a1 g e b r a i c curves p r o v i d e some geometry. L e t L6 =

REMARK 2 - 1 (LAW [email protected]'I0N$)-

prisingly). LB.

$t = B6 and X t = 0.

-

+ 6txx

u6,

= $xxt

-

I n an obvious

(U), = 6xxt - ut$

= Lt$ + LGt = hGt;

n o t a t i o n we w r i t e now Thus i n p a r t i c u l a r Lt$

X6,

utrl

-

u6,

F u r t h e r Lt6 + LB$ = XB$ = BX$ = BL6 so

SO Lt

(A)

-

u$,.

%-ut (not sur-

-

Lt = [ B,L] = BL

T h i s i s c a l l e d a Lax e q u a t i o n and i s a s p e c i f i c form o f zero c u r v a t u r e Now i s o s p e c t r a l e v o l u t i o n i s connected w i t h

e q u a t i o n t o be t r e a t e d l a t e r .

t h e i d e a o f t h e L ( t ) = Lu b e i n g u n i t a r i l y e q u i v a l e n t ( i n some H i l b e r t space) Then f r o m L(0)rl

Thus suppose ( a ) L ( t ) U ( t ) = U ( t ) L ( O ) where UU* = U*U = I . (0,x)

= h(O)$(O,x) i t follows that 6(t,x)

L(t)U(t)$(O,x) X(t)

=

K(u).

= U(t)L(O)$(O,x)

= U(t)$(O,x)

= U(t)h(O)$(O,x)

= h(0)6(t,x).

Consequently

X(0) and eigenvalues X ( 0 ) would be i n t e g r a l s o f t h e e q u a t i o n ut F u r t h e r , w r i t i n g B = UtU* one o b t a i n s from

-

LUt = UtL(0) o r Lt = UtL(0)U* (0)).

satisfies L(t)rl =

LUtU* = [ B,L]

(0)

=

t h e e q u a t i o n LtU +

( s i n c e UtL(0)U*

= UtU*L

from

We w i l l r e t u r n t o t h i s p o i n t o f view l a t e r i n a H a m i l t o n i a n c o n t e x t .

We w i l l see l a t e r t h a t t h e r e a r e v a r i o u s o p e r a t o r s B g i v i n g r i s e t o h i g h e r KdV f l o w s i n a h i e r a r c h y p i c t u r e .

REmARK 2.2 (CIEIE EU0LLlC10N OF SPECCRAL DACA)e v o l u t i o n o f s p e c t r a l data. $

=

6$ f o r l a r g e 1x1, where u

t f+(k,x)

L e t us examine now t h e t i m e 2 u6 = - k 6 and

Consider a s o l u t i o n $ o f rlxx %

where f+ 'L e x p ( i k x ) as x

0, so t h a t rlt -+ m

( n o t e f,

%

-4rlxxx.

-

Suppose rl = h ( k , t )

i t s e l f cannot s a t i s f y 6,

= BJ,

14

ROBERT CARROLL

3 First a s x + m y $ t = htexp(ikx) = -4$ xxx = - 4 h ( i k ) exp(ikx) so 3 3 h t = 4ik h and thus h h(kY0)exp(4ik t ) . Next a s x - m y f, % ~ ~ ~ e x p ( - i k x ) + ~ ~ ~ e x p ( i k so x )w , r i t i n g o u t $t = -4GxXx we g e t as x

.+

a).

-+

3 ( k Y 0 ) e x p ( 8 i kt ) , c12(k,t) = c I 2 ( k , 0 ) , and m R j l3 ( i e j y t ) = m .(if3 O)exp(8tBj ). T h i s is very remarkable s i n c e now e . g . T RJ jy 3 3 ( k , t ) = T(k,O), R ( k , t ) = R(k,O)exp(8ik t ) , and mRj = mRj(iBj,0)exp(8Bjt) allows one t o apply inverse s c a t t e r i n g techniques f o r t 2 0. Indeed suppose Similarly ( 6 ) c l l ( k , t ) = c

u o ( x ) i s known, from w h i c h one can obtain ( i n p r i n c i p l e ) T ( k , O ) , and m R j ( i B j ' 0 ) (along w i t h t h e B j ) . Then compute F(x+y,t) a s in (1.26) u s i n g R ( k , t ) and m ( i 6 t ) . This introduces a parameter t i n t o t h e Rj j' M equation (1.25) which equation one solves f o r K(x,y,t) (y > x ) . Then u(x,O)

=

R(k,O),

u ( x , t ) = - 2 D x K ( x , x , t ) will s a t i s f y the Cauchy problem f o r KdV, with i n i t i a l data u ( x , O ) = u o ( x ) prescribed. In p a r t i c u l a r t h i s allows one t o solve a -+ nonlinear PDE by l i n e a r techniques. The map u ( x , O ) -+ ( T , R , m R j , B j ) ( k y O ) ( T , R , m R j , B j ) ( k , t ) -+ u ( x , t ) is o f t e n r e f e r r e d t o as the inverse s c a t t e r i n g transform (IST). More s p e c i f i c a l l y , assume no bound s t a t e s and look a t t h e maps (from (1.8) and (1.22) diagonalized)

(2.3)

2ikR/T =

1:

u(s)e-iksf-(k,s)ds; u ( x ) =

DxlIReikxf,(k,x)dk/n

T h i s i s a k i n d o f nonlinear Fourier transform s i t u a t i o n . Indeed i f u i s small w i t h T % 1 , R small, f, % exp(ikx), f - plr exp(-ikx), e t c . one has apm EikRexp(2ikx)dk w i t h 2 i k R % -jm u(x)exp(-2ikx)dx (or proximately u % 2 i k R = ( F u ) ( 2 k ) ) . Thus (2.3) e s s e n t i a l l y reduces t o the Fourier transform and one knows t h a t i n general (say w i t h no bound s t a t e s ) 2 i k R / T S has nice "Fourier" properties ( c f . [ TBZ] f o r a study o f t h e map u S(u)). Halfline

(1/~)11

-f

IST pairs analogous t o ( 2 . 3 ) a r e developed i n [ C1,3,5;F09,10]. A t f i r s t sight this phenomenom appears q u i t e s p e c i a l . Surely i t cannot apply t o many i n t e r e s t i n g nonlinear PDE? B u t i n f a c t many very important and i n t e r e s t i n g nonlinear PDE f a l l i n t o the c l a s s o f " i n t e g r a b l e systems"

SOLITONS

15

which have s i m i l a r properties. Moreover t h e r e a r e many o t h e r f a r reaching and deep a s p e c t s of v i r t u a l l y a l l a r e a s of modern mathematics w h i c h get i n volved w i t h i n t e g r a b l e systems and t h e a p p l i c a t i o n s i n physics go i n t o s t r i n g theory, conformal f i e l d theory ( C F T ) , s t a t i s t i c a l physics, 2-D quantum grav i t y , e t c . A t t h e applied mathematics level t h e r e i s some evidence toward t h i n k i n g o f some i n t e g r a b l e PDE a s a r i s i n g from the next approximation a f t e r l i n e a r i n dealing with problems i n mechanics, f l u i d dynamics, e t c . , b u t we will not go i n t o this.

REmARK 2.3 ($01;ZtbNS, C0WERVED Q,UN&I&ZE$, AND OARZBLlti tRANkFBRI‘IAtI0W). The so c a l l e d s o l i t o n s i n KdV theory a r i s e from t h e poles of T ( k , t ) = T(k,O) and a s an example suppose T has two poles a t iB1 and i B 2 w i t h B2 > B1 (and R = 0).

(2.4)

Then ( c f . [ LCl]) one finds 2 2 B2CSCh 2 2YZ

-

81CSCh 2 2~1

-

B1 Cothyl 3

u ( X , t ) = -2(B2-B1) (B2Cothy2

2

where y1 = B ~ X- 48,t3 + 61, y 2 = B2x - 4B2t + 62, and 6 i = +log{(mi(0)/2Bi) ( 8 2 - ~ , ) / ( ~ 2 + ~(mi 1 ) )‘L m L i ) . S e t t i n g A = Tan-’(B1/B2) one f i n d s t h a t a t 2 times long before or a f t e r i n t e r a c t i o n ( 2 . 4 ) i s represented by 2 sech type s o l i t o n s i n t h e form (2.5)

2 2 u ,I, -2B1Sech (y,

k

A)

-

2 2 2B2Sech (y,

T A)

2

We note t h a t u = Sech x i s not a s u i t a b l e potential f o r t h e inverse s c a t t e r i n g theory developed above. Nevertheless one can determine R and T a s before, b u t having some d i f f e r e n t properties ( s e e [LClI f o r e x p l i c i t formul a s ) . Similarly 6 function p o t e n t i a l s give r i s e t o s c a t t e r i n g data having d i f f e r e n t p r o p e r t i e s . For s i t u a t i o n s l i k e t h i s t h e r e a r e ( o f t e n formal) analogues o f M equations and recovery formulas f o r p o t e n t i a l s (see e.g. [ LC1 ] f o r many examples and c f . [ F 2 ] f o r so c a l l e d f i n i t e d e n s i t y s i t u a t i o n s where t h e theory i s completely developed and r i g o r o u s ) . One must be careful here b u t very o f t e n t h e formal machinery works. We r e f e r t o [ABl;LCl; DDZ;N02;WHl 1 and references t h e r e f o r more examples and physical discussion. Now we comment b r i e f l y here on another f e a t u r e o f KdV i n t h e form u t - 6uux + uxxx = 0 f o r example, namely t h e existence of an i n f i n i t e number of con-

16

ROBERT CARROLL

served q u a n t i t i e s . This will be t r e a t e d systematically l a t e r and f o r now we simply note d i r e c t l y t h a t (under s u i t a b l e hypotheses)

DtlI udx = l:

(2.6)

Dx(3u2

-

u x x ) d x = 0;

%D,LIu 2 d x = 1:

Dx(2u2

rl

-

u u x + L,u:)dx

= 0

...

2 Thus q u a n t i t i e s l i k e udx, if u dx, will be conserved and this has profound s i g n i f i c a n c e in t h e Hamiltonian theory ( c f . 16).

We i n d i c a t e next an i n t e r e s t i n g transformation developed by Darboux in d i f f e r e n t i a l geometry and used in changing p o t e n t i a l s by Crum, S a b a t i e r , Faddeev, ( c f . [ C24;CDl;CBl;DBl;DIl;F3;SA2]). The basic idea here i n i t s a p p l i c a t i o n t o KdV is t o s t a r t from (+) ( D 2 - u)$ = - k 2$ w i t h possibly dis2 Crete spectra a t k = i B j . Let B1 i ... < BN say and k2 = - 8 . w i t h no spec2 2 j 2 - j 2-J 2 trum f o r k < - B N . Let be any s o l u t i o n o f ( D - u)$ = B0$ where -Bo < -f(. Set = u - 2 ( l o g r ) " and consider t h e equation ( B ) ( D 2 - z)q = - k 2q . One checks t h a t i f $ i s t h e "general" s o l u t i o n of ( + ) then q = JI' - $(TI/;) i s t h e general s o l u t i o n of (m). Moreover t h e potential will involve a n , u additional eigenvalue ko = iBo w i t h corresponding eigenfunction q = l / $ . To check this note t h a t 2(logF)" = $''/T- T(l/$j" so (0' - Y ) ( l / F ) = (l/;)" 4 u - ((C'/r) + ? ( l / ~ ) " ) l / ~B:/=6-; hence q = l/;corresponds to k = i B 0 9 a s a s s e r t e d . Simila& ( a f t e r some c a l c u l a t i o n ) one checks t h a t ( D - ? ) ( $ I = - k 2 ( $ ' - $(C'/F))( e x e r c i s e - note (?'/T)' + (?'/;)I - u = p0). 2

...

-

JI(F/?))

T h i s can be seen i n a somewhat broader perspective a s follows ( c f . [ C24; DB1;LCl I ) . One can study t h e r e l a t i o n between t h e two equations

when cp = JI' + A$ (more complicated r e l a t i o n s can a l s o be t r e a t e d ) . P u t t h i s expression f o r q i n t o (2.7) a n d equate t h e c o e f f i c i e n t s o f $ and $ ' t o 0; one g e t s (2.8)

2A' +

q-

q = 0; A"

+ A(?-

q)

-

q'

=

0 ry

Eliminating ( F - q ) a n d i n t e g r a t i n g one gets A' - A ' + q = h (an i n t e g r a t i o n c o n s t a n t ) . This i s a R i c a t t i equation f o r A a n d can be l i n e a r i z e d via A =

CONSERVATION LAWS h

d

c*

so t h a t

N

+ qJI =

NCI

17

u - 2(10&))".

W (corresponding t o q

- u and

-

2

Furc ther, q-y= -2(log?)" corresponds t o u = One can e a s i l y cons t r u c t examples now of adding a s o l i t o n o r generating bound s t a t e s d i r e c t l y but we omit d e t a i l s ( c f . [ DSl;DIl;LCl;SA2]). -$I/$

$"

REmARK 2.4 (#Ul!lPC0CZCS,

C0WERYACZBN

=

A =

B 0 above).

LAW, AND IltZURA ERAWF0Rm). Let us in-

d i c a t e a preliminary approach to some asymptotic expansions f o r KdV ( c f . [ AB5;DSl;LCl I ) . Assume the potential q o r u = - q is s u i t a b l e (e.g. q E 17 i s o f t e n used b u t f u r t h e r rigorous a n a l y s i s of t h e s e matters would be useful ) . Then write f (k,x) = exp(-ikx + g ( k , x ) ) f o r l a r g e \ k \ , Imk > 0, and note t h a t we a r e doing asymptotics i n k ( x asymptotics a r e a l s o relevant and one should take c a r e w i t h any double l i m i t i n g procedure our c a l c u l a t i o n s here a r e only formal b u t c f . [AB5;LC1 I ) . Now $(k,-m) = 0 and from the equation (D2-u)f- = - k 2 f - we obtain a Riccati equation

-

(2.9)

-

g"

2ik4'

t

(g')

2

=

u

One t r i e s t o obtain a solution via an asymptotic s e r i e s 4' =

(2.10)

1;

gn(x)/(2ik)"'

-

N

By asymptotic s e r i e s one means t h a t a s / k \ (Imk > O),lim($'(k,x) - Lo ($,(x)/(2ik)"+'))kN = 0 f o r any N. Putting (2.10) i n t o (2.9) we obtain $Jo = -f

- u , $1 =

$A

=

- u ' , and

..

2 ) / I t 4uu', , so T h i s gives i n p a r t i c u l a r $2 = - u " + u , o3 = - u t h a t $,, is a polynomial i n u and i t s d e r i v a t i v e s . Further a s x -+ m, f - k , x ) exp(ikx) 3 c12 + c l l e x p ( 2 i k x ) and f o r Imk > 0 t h e l a s t term tends t o 0.

(n 2 1).

Hence f-exp(ikx) = exp(4) -+ c12. Consequently $ ( k , - ) t e g r a t i n g we get ( r e c a l l c12 = 1/T) (2.12)

log T =

-1;

1/(2ik)"+l

=

log c12(k) a n d

n-

gndx

T h i s formula will have numerous important consequences l a t e r . In t h i s d i r = ection l e t us r e c a l l the Poisson-Jensen formula (1.11) and remember t h a t

18

ROBERT CARROLL

-

R(k) i s small a s I k ( + (Imk > 0 ) . Actually, i f t h e r e a r e no bound s t a t e s f o r example one shows t h a t u E S implies 2 i k R / T E S ( c f . [ TB2,3]). T h u s formally i t makes sense t o expand (1.1 ) i n t h e form ( l k l (2.13)

log T ( k ) =

(1 / 2 n i ( c f . [ DS1;LCl

I).

-1;

c

~ /kZn+' ~ ;+

+

m,

Imk > 0 )

~

111 k 2 n l og (1 - I R

We r e f e r t o 16 f o r comparison o f t h e s e r i e s (2.13)-(2.12).

Consider next some elementary f a c t s about t h e Miura transformation o r MiuraGardner-Kruskal (MGK) method and v a r i a t i o n s ( c f . [ CR1;DS1;GD1;KK1;WlY4,7-9; MM1 ,2;FD2;DDl;KGl;WA2,3]).

The matter has been discussed i n a modern s p i r i t

i n [ W1,4,7,8,91 f o r example and one knows t h a t t h e conserved q u a n t i t i e s a r rived a t by such MGK procedures agree w i t h those obtained from the GelfandO i k i i approach ( c f . [ G2-4,7,12;DK21).

W e simply give here some c l a s s i c a l

formulas and r e f e r t o [W3-5,7-91 f o r elegant approaches.

We mention again

t h a t t h e a l g e b r a i c techniques and the results a r e "seductive" but t h e r e remain various d i s t u r b i n g f a c t s connected t o inverse s c a t t e r i n g .

In p a r t i c u l a r

s o l u t i o n s u of KdV obtained by inverse s c a t t e r i n g cannot a r i s e via t h e Miura 2 transformation u = -v - i v x from r a p i d l y decreasing s o l u t i o n s of mKdV,vt + 2 6v vx vxxx = 0. The reason is t h a t generically R(0) = -1 f o r KdV while t h e corresponding r e f l e c t i o n c o e f f i c i e n t f o r t h e mKdV eigenfunction problem +

s a t i s f i e s IR(0)l < 1 ( c f . again [ AB5,10,11 1 ) .

We will not pursue t h i s par-

t i c u l a r matter however. In any case one can l e g i t i m a t e l y ask about t h e c l a s s of s o l u t i o n s t o which t h e expressions f o r conserved q u a n t i t i e s apply. Such questions i n f a c t a r i s e frequently i n t h e whole theory.

Often some o f

the algebra does n o t obviously apply to inverse s c a t t e r i n g s o l u t i o n s (see e.g. 511 where we develop t h e Hirota formula f o r inverse s c a t t e r i n g solut i o n s ) . One o f t e n " c r e a t e s " some algebra o u t of asymptotics a s i n (2.13). A l t e r n a t i v e l y t h e r e a r e various genuinely a l g e b r a i c procedures involving e.g. Kac-Moody (KM) algebras ( c f . 58,lO) where the v a r i a b l e k is b u i l t i n t o t h e

Then one makes various i d e n t i f i c a t i o n s o f c o e f f i c i e n t s o f comparable s e r i e s f o r example. Sometimes t h e r e s u l t i n g i d e n t i f i c a t i o n l e a d s t o c o r r e c t formulas f o r s o l u t i o n s n o t i n t h e domain of t h e d e r i v a t i o n . Sometimes i t i s not c l e a r why t h e i d e n t i f i c a t i o n s a r e algebra a s an indexing parameter.

MIURA METHODS

19

sensible a t a l l b u t they lead t o correct formulas for some class o f u. There i s need f o r more precision i n a l l t h i s and we will t r y t o p o i n t o u t , and occasionally indicate solutions for, problems o f this nature.

Now from the derivation above i t i s certainly legitimate to identify coeff i c i e n t s i n (2.12) and (2.13) which would lead to asserting t h a t @2n+l = 2n+l = The domain o f definition of b o t h f m q 2 n + l d x = 0 while @ 2 n / ( 2 i ) -co formulas i s the same and k i s a spectral variable i n a n asymptotic expansion. On the other hand l e t us look a t the Miura transform and variations. First note t h a t our KdV i n the form ut - 6 u u x t uxxx = 0 corresponds t o u = - v 2 ivx with vt + 6v 2vx + ;xxx = 0. Changing t -+ -t and u -+ q = -u we get q = t 2 If one puts a factor 6qqx t qxxx w i t h q = v + iv X and vt = 6v v x + vxxx. o f 46 here, i.e. w = v 2 + i46vx, then one can deal w i t h wt = wwx + wxxx a n d vt = v 2v x + vxxx ( c f . [ DS1 I ) . Let us follow [ DSl I where details are spelled o u t . Consider the interpolation equation (2.14)

vt

=

(E

2 2

v / 6 + v ) v X + vxXx

Then for E = 0 one has KdV a n d for E m a rescaling v ( ~ / 4 6 ) vleads t o 2 v t = v v x .+ vxxx' One should note however t h a t the symmetries o f KdV a n d mKdV a r e different (KdV i s Galilean invariant b u t not mKdV). Note also t h a t 2 3 from ( 2 . 1 4 ) DtLZ vdx = if D X ( € v /18 + $v2 t v x x ) d x = 0 i f v , vx, etc. 0 a s x &=. One i s interested in conserved quantities which do not a r i s e from total derivatives, i.e. nontrivial conserved quantities. Observe that a KdV solution w arises from w = E2 v 2/6 + v + iEvX i n (1.14) and hence t h i n k o f v = V ( W , E ) w i t h a n expression -+

-f

-+

-f

(2.15)

1"0 E n v n ( w ( x , t ) )

v(x,t)

(see [ DS1 1 for heuristics a n d discussion). w, setting v - ~ = v - ~= 0, t o get (2.16)

Thus w

w

=

1m0

E

n

P u t (2.15) i n the formula for

( v n + iDxvn-l

vo a n d , setting coefficients o f

E

equal t o 0 one has

20

ROBERT CARROLL

f o r n > 0.

Note t h e s i m i l a r i t y t o (2.11) here upon s e t t i n g n = k + l ; i . e .

+ iDxvk + (1/6)$-’ ~ ~ - = ~0. - The~ o nvl y d~i f f e r e n c e i n f a c t a r i s e s ‘k+l t h r o u g h o u r use o f w here i n s t e a d o f u. Working o u t t h e f i r s t terms g i v e s v2 = -(1/6)w 2 wxx, v3 = - i D x ( ( l / 3 ) w 2 + wxx), v4 = ( 1 / 3 ) now v1 = -iwxi ((l/6)w3 %wx) + Dx(t,w 2 2 + wXx), ( c f . [ D S l I ) . It t r a n s p i r e s h e r e t h a t

-

-

t h e odd powers o f

...

i n v o l v e t o t a l d e r i v a t i v e s and t h u s i n v o l v e t r i v i a l con-

E

served q u a n t i t i e s whereas t h e even powers have n o n t r i v i a l terms, namely, v 0 w, v2 = -(1/6)w 2 + Q, , v4 = ( 1 / 3 ) ( ( 1 / 6 ) w 3 - %wx) 2 + (Q, meaning Dx( 1).

=

Q

Note i n (2.6) we have a s i m i l a r beginning.

The q u e s t i o n o f comparing (2.11)

and (2.17) w i l l be discussed l a t e r , a t l e a s t h e u r i s t i c a l l y .

A p r i o r i there

i s no reason t o b e l i e v e t h a t t h e s o l u t i o n s d e s c r i b e d i n (2.11) l i e i n t h e domain o f d e r i v a t i o n o f (2.17).

R ~ R 2.5 K (LAW e)PERACBRS)-

We g i v e here an h e u r i s t i c d i s c u s s i o n o f Lax op-

e r a t o r s f o l l o w i n g [ DS1 1 which w i l l be expanded l a t e r r a t h e r e x t e n s i v e l y . Thus i n Remark 2.1 we discussed B = -40; - 6qDx - 3qx r e l a t i v e t o L = D2 + q and qt + 6qqx +,,,q

= 0.

Here f o r convenience we use (assume w r e a l )

Now t h i s i s o n l y one o f a w i t h wt = wwx + wxxx a r i s i n g from Lt = [ B , L l . whole h i e r a r c h y o f KdV f l o w s . One can e n v i s i o n more g e n e r a l l y

(a p o s s i b l e a d d i t i v e c o n s t a n t i s s i m p l y o m i t t e d ) .

From Remark 2.1 i t i s na-

t u r a l t o t h i n k o f B = - i H f o r example ( c f . a l s o 96) w i t h U = e x p ( - i H t ) f o r U* = e x p ( i H t ) , U t U* = - i H = B . Then B i s i s a n t i h e r m i t i a n ( i . e . B* = iH* = i H = -B) and t h i s l e a d s t o odd 2 o r d e r d i f f e r e n t i a l expressions D ( i n an L c o n t e x t D* = -D), and thence t o

m a l l y w i t h H s e l f a d j o i n t so t h a t ut = -iHU,

(2.19).

Then one can examine ( f o r L g i v e n ) t h e b r a c k e t

I B,Ll and r e q u i r e

t h a t i t be a H e r m i t i a n m u l t i p l i e r o p e r a t o r (corresponding t o Lt = wt/6). W r i t i n g o u t [B,,L] (2.20) where K,

[B,,LI

t h i s requires f i r s t =

K,,,(W) +

1:

ojcj(W)oj

i s m u l t i p l i c a t i v e ( t h e l a s t t e r m i s t h e general form o f a H e r m i t i a n

LAX OPERATORS

21

o p e r a t o r w i t h C = CS). Secondly requiring (2.20) t o be m u l t j p l i c a t i v e proj vides m equations which determine a l l b . ( w ) uniquely. For m = 0 one gets J For m = 1 we [ Bo,Ll = aD,LI = awx/6 which f o r a = 1 corresponds t o wt = wx. get B = a D3 + a (Dw + w D ) and some c a l c u l a t i o n gives a 3 = 8al which f o r al 3 1 = 8 y i e l d s (2.18). One can continue and compute t h e higher Bm e x p l i c i t l y ( c f . 96 f o r more on t h i s ) b u t here we show a d i f f e r e n t point of view. Thus consider a formal expression (a

%

ax

=

Dx)

( i n t e r p e r t a t i o n of a - ’ will be g i v e n l a t e r ) . One formally determines the a n = L = a 2 + w/6. Then Lm+’ = LL2(2m+1) = LmL’ = ( a 2 t ~ / 6 ) ~ . by s e t t i n g (L’)‘ L4 can be w r i t t e n out formally a n d we denote by ”L: t h e p a r t of t h e r e s u l t 0 (Lm+li contains terms with deg(3) i n g formal expression where deg(a) is + + L”’and [ Lm+’,L] = 0 ( s i n c e f o r L’ = P t h i s Evidently Lm+’ = Lm+’ < 0). m+4 is [ P2m+1,P2] = 0 ) . Hence one has I Ly+’,Ll = -[L- , L l and s i n c e the l e f t (resp. r i g h t ) s i d e has o r d e r 1. 0 ( r e s p . 5 0 ) , both s i d e s must be multiplicat i v e . In p a r t i c u l a r we can consider Bm = a Lm+’ a s a possible component 3 p t Indeed f o r Bo = aoL one has L; = a + ao(w) and t h e o f a Lax p a i r (B,,L). computation (L’ ) = L y i e l d s in p a r t i c u l a r ao(w) = 0 and al ( w ) = w/12 (ex+

...,

t

e r c i s e ) . Hence L’= a + (1/12)wa-l + and L: = a . W i t h u0 = 1 we obt a i n Bo = a a s before ( a f t e r (2.20)). Going t o t h e next term one finds L4 = a + (i/i~)wa-~ (i/z4)w a-‘ + w i t h L ~ / * = a 3 + ( i / 8 ) w + gwa + o(a-’) 3 x-l ) ( e x e r c i s e ) . Consequently L”: = a + (1/8)(aw + w a ) + o(a = a 3 + (1/8)

...,

( a w + w a ) and w i t h a1 = we have our standard B1 f o r KdV as in (2.18). will see l a t e r t h a t t h i s method i s very productive t h e o r e t i c a l l y . 3, PR03EW IN IEC€WICS AND

HZLC’S EQllACI0N.

We

We go t o LAC1 ,S;ADZ;AOZ;CQl;

DI2;DU3;EZY3;FL2,6;GAl ;HOl;ICl ;JOl;KR5;KVl ;LX1 ,2;MC1-9;ME1 ;MO1-6;MGl ; P L Z ; RT1 ,Z;SG1,3;VElY2J i n order t o provide a background f o r many dominating ideas i n sol i t o n mathematics ( f i n i t e zone p o t e n t i a l s , f i n i t e dimensional mechanical problems, resolvant expansions, a1 gebraic curves, Riemann surfaces, e t c . ) . Without such a background connection many of t h e algebrogeometric techniques o f s o l i t o n mathematics would seem even more mysterious.

As usual we do not t r y t o function a s a h i s t o r i a n a n d will o f t e n simply

22

ROBERT CARROLL

s t a t e r e s u l t s w i t h o u t proof ( r e f e r e n c e s a r e however i n d i c a t e d ) . mainly follow

M01-51,

We w i l l

e x p e c i a l l y [ M021, here s i n c e t h i s p r o v i d e s an e x c e l -

l e n t summary o f t o p i c s and r e s u l t s ( f u r t h e r r e f e r e n c e s appear a t t h e end o f O c c a s i o n a l l y more d e t a i l i s p r o v i d e d b u t more o f t e n l e s s and we w i l l

53).

pursue some ( n o t a l l ) o f t h e t o p i c s i n more d e t a i l l a t e r .

T h i s s e c t i o n (and

§ 4 ) a r e t o be considered as p r i m a r i l y m o t i v a t i o n a l ; t h e idea i s t o p r o v i d e some background p e r s p e c t i v e .

F i n i t e zone p o t e n t i a l s a r e t r e a t e d more sys-

t e m a t i c a l l y i n 54,5; a s y m p t o t i c s o f t h e r e s o l v a n t a r e discussed a t v a r i o u s o t h e r places, and f o r t h i s one r e f e r s t o

G2-9,12;DK1-41

for details.

REWRK 3.1 (HA1IIIC&0NNZAN UECC0R F E D $ AND LAX EQMEZ0U). L e t us say t h a t a

1,” (aH/api)a/aqi

H a m i l t o n i a n v e c t o r f i e l d XH

ift h e r e e x i s t

n integrals

F.

1

-

(aH/aqi)a/api

i s integrable

H I = lln(aF./aqi) J (aF./ap.)(aH/aqi) = 0 ( b ) IF.,F 1 = 0 and ( c ) t h e dF. a r e l i n e a r (aH/api) J l J k J l y independent i n some r e g i o n ( c f . Appendix A ) . A few examples a r e ( 1 ) H = n p 2j + wiq: ( w . > 0 ) d e s c r i b e s a system o f n o s c i l l a t o r s w i t h F = p i + E

J

-

C

s a t i s f y i n g (a) I F

jy

+Il

j

-

w?q? ( 2 ) H = .llpy2 l / l q l ( K e p l e r problem i n Rn) has commuting i n t e g r a l s J J 2 [p.q p . q . ) f o r 2 5 k 5 n ( 3 ) The Calogero system i n F1 = H and Fk = < J l j J 1 v o l v e s n p a r t i c l e s o f equal mass (say 1 ) on t h e l i n e r e p e l l i n g each o t h e r

&

-

1

V(qi-q.), V(x) = w i t h an i n v e r s e c u b i c f o r c e . Thus ’q’ = -aU/aqj, U = 2 j J b / l x 1 2 , and H = p . t U(q). The i n t e g r a l s i n i n v o l u t i o n here a r e d e t e r J mined as eigenvalues o f t h e m a t r i x (zeros on t h e diagonal i n t h e second

+I

matrix)

( 3.1)

L(q,p)

= diag(pj)

+

i ( ( ( l q 6 j k ) / ( qj - 9 k ) ) )

4.)

i s u n i t a r i l y equivalent t o L(0) I n fact L ( t ) = L(q(t),p(t)) (with pj = J v i a a u n i t a r y m a t r i x U ( t ) such t h a t U ( 0 ) = U ( t ) - ’ L ( t ) U ( t ) (see below). Cons e q u e n t l y t h e eigenvalues o f L ( t ) a r e i n t e g r a l s o f m o t i o n and so a r e t h e k symmetric f u n c t i o n s o f t h e e i g e n v a l u e s g i v e n by Fk(q,p) = T r L ( k = I , ..., n).

That t h e Fk commute w i l l be shown l a t e r i n a more general c o n t e x t .

In

o r d e r t o c o n f i r m t h e u n i t a r y equivalence o f L ( t ) and L ( 0 ) one can ask f o r B skew symmetric such t h a t dU/dt = B ( t ) U ( t ) w i t h U(0) = I . equivalence r e q u i r e s U - l ( L t

-

Then u n i t a r y

BL t LB)U = 0 which i s a Lax e q u a t i o n .

Re-

v e r s i n g t h e argument, i f one can f i n d B = -B* such t h a t Lt = CB,LI f o l l o w s

HAMILTONIAN FIELDS from

j

=

- a U / a q . we will have unitary equivalence. J

23 I t turns o u t t h a t

Thus t h e equations $' = - a U / a q . give r i s e t o i s o will s a t i s f y Lt = [ B , L ] . j J spectral deformations of L and we r e f e r t o [ M01-4 1 f o r f u r t h e r discussion. In t h i s example L and B were apparently guessed a t f i r s t . R€CNRK 3.2 (C0WCRAZhlED m0CZ0N AND CHE C. hlEl.ImAEQN PR03Cfin). Given a Hamiltonian H suppose one has motion constrained t o a submanifold M C R Z n via G1(x) = = GZr(x) = 0 where one assumes det((G.,G 1 ) # 0 ( j , k = 1 ,..., J k 2r) so t h a t M will be symplectic ( e x e r c i s e - c f . Appendix A ) . If a system 1 x = JHx = JVH, J = ( 0- I o) ( I = I n ) a s i n Remark A32 defines a vector f i e l d X H tangent t o M then t h e motion i s natural on M and this s i t u a t i o n i s described via 0 = X G = -IH,Gj>for j = 1 , ...,2r ( c f . Remark A31). Generally

...

H j

this will not hold and one wants t o define a new "constrained" vector f i e l d X i = X H - Ifr AjXGj on M , based on H = H - Ifr A j G j , where t h e AJ. ( x ) a r e deA.{G.,G 1 = 0 (XK termined on M so t h a t X; i s tangent t o M via IH,GkI J J k on M does not depend on t h e extension of A . ( x ) o f f M ) . Generally X i may not J be i n t e g r a b l e b u t i f X H is i n t e g r a b l e w i t h commuting i n t e g r a l s F . (1 5 j 5 J n ) and M is given by F1 = = Fr = 0 = G = ... = Gr w i t h d e t ( { F i , G j l ) f 1 0 for 1 5 i , j 5 r , then take Gj+, = F . and t h e F . r e s t r i c t e d t o M will be J J i n t e g r a l s o f t h e constrained system. Indeed, s e t t i n g = H - 1 ; ( AJ. F .J, + * V . G . ) with 0 = {H,FkI = 1' u . { G . , F I we see t h a t v = 0 on M and hence H = J J 1 J J k, j H - 1 ; h j F j w i t h A J. determined via {H,GkI = 0 ( k = 1 , ...,r ) will give r i s e t o an i n t e g r a b l e vector f i e l d :X = X H - 1,r AjXF;(since evidently X$Fk) = 0 ) . Y

...

T h i s s i t u a t i o n can be i l l u s t r a t e d with the C. Neumann problem of finding t h e

motion o f a mass point w i t h a l i n e a r force (quadratic p o t e n t i a l ) constrained t o a u n i t sphere. Thus one i s looking f o r equations = -Aq + vq w i t h say A a symmetric matrix and v i s t o be chosen so t h a t l q I 2 = 1 and necessarily ( q , p ) = 0. Take F = F1 = %(1qI2 - 1 ) and G1 = ( p . 9 ) as c o n s t r a i n t s with H 2 2 2 = UAq,q) + I p I - ( 9 . p ) ) . Then F1 i s an i n t e g r a l a n d IF1,G1l = 1 . # 0. For t h e constrained Hamiltonian H = H - AIF1 we want hl = IH.G1) = ( A q , q ) so t h e d i f f e r e n t i a l equations a r e

24

ROBERT CARROLL

Thus f o r 2 l q l = 1.

V

- lplz we

= A1

have a c o n s t r a i n i n g f o r c e v - q and

6=

-Aq

+ vq on

RfiltARK 3.3 (BE0LWZC FC0U ON AN ELLZPd0ZI)). We mention n e x t t h e geodesic T flow on M:
... < an and x

eigenvalues 0 < a1 <

q.

The d i f f e r e n t i a l e q u a t i o n s w i l l be

v = \ A-1 XI2 (A-’x’,x’)

d 2 x/ds2 =

(3.4)

%

2 2 + Indeed, n o t e t h a t v i s determined v i a 0 = %d /ds ( A - l x , x ) = (A-’x,x”) -1 2 (A-’x’,x’) = -VIA + (A-’x’,xf). Consider t h i s system as a r i s i n g from a 2 c o n s t r a i n e d t o T*M d e f i n e d by ( A - l q , q ) = 1 w i t h f r e e Hamiltonian H =

XI

$1~1

(A-lq,p)

= 0 n e c e s s a r i l y (we use q o r x i n t e r c h a n g a b l y ) .

-

3.2 one works w i t h H = L,lpI2 Al((A-lq,q) - f l A - l q l - 2( A - 1 p,p) and A 2 = IA-1ql-2(A-’q,p) w r i t t e n as H = % l p l 2 + (v/2)$o(q,p) $,(q,p)

= ((A-’q,q)

- l)(A-’p,p)

-

-

#A-

1

Then from Remark

1 ) - h+A-’q,p) w i t h X1 = ( e x e r c i s e ) . This can be r e q,p)

( A - l q , p ) , a2n d

2

where

-1

u = IA ql

the l a s t term i n

-2

and

H i s drop-

c

ped s i n c e i t vanishes o n T*M a l o n g w i t h i t s d e r i v a t i v e s (so H = % l p l

2

+

?iH*

= p and $ o ( q y p ) ) . Then t h e c o n s t r a i n e d system y i e l d s equations dq/ds = u P which agrees w i t h (3.4). dp/ds = -H = -u(AA-lp,p)A-lq q Now t h e extended system X r a n d @,(q,p) haveconsiderable geometrical meaning

(see [M01,2,5]

f o r d i s c u s s i o n ) and i t t u r n s o u t t o be g e o m e t r i c a l l y r e v e a l -

i n g t o i n v e s t i g a t e t h e c o n f o c a l q u a d r i c s QZ: Q,(q,p)

= ((21

-

A)-’q,p)

(9,

ever ( r e f e r r i n g t o [ M01 ,2,5]

= M).

+ 1 = 0 (z

Q,(q,p)

+ a k ) where

We w i l l n o t d w e l l on t h e geometry how-

f o r t h i s ) b u t what happens i s t h a t v a r i o u s

g e o m e t r i c a l q u a n t i t i e s (and mechanical q u a n t i t i e s ) a r i s i n g i n t h i s c o n t e x t

w i l l have o t h e r i n t e r p e r t a t i o n s l a t e r ( c f . Remark 3.8).

The development

here w i l l a l l o w one t o compare v a r i o u s such q u a n t i t i e s w i t h ” c o r r e s p o n d i n g ” (-D 2 + q)JI = XJI w i t h f i n i t e zone p o t e n t i a l s . Thus w r i t e now 2 Qz(q) = Qz(qyq) and s e t $,(q,p) = (1 + Q,(q))Q,(P) Q,(q,p) (so $,(q,P)

objects f o r

-

agrees w i t h o u r p r e v i o u s n o t a t i o n ) . l i n e s through q t o

51).

9,

One checks t h a t t h e cone o f t a n g e n t

i s g i v e n by {p; $,(q,p)

One can w r i t e ( p a r t i a l f r a c t i o n s )

= 01 ( e x e r c i s e

-

c f . [M01,2,

NEUMANN PROBLEM

25

One checks t h a t r e l a t i v e t o t h e Poisson b r a c k e t {F,G}

- (aF/api)(aG/aqi), 2 1; Fk = I p I a l s o

=

1;

(aF/aqi)(aG

/api)

tF.,F I = 0 and { $ ,$ I = 0 ( c f . [M01,2,41). Since J k ZI 2% commutes w i t h t h e Fk t h e Fk a r e commuting i n t e g r a l s f o r #u

t h e system determined b y

H

= + / p I 2 + W$,(q,p)

(note

0,

=

- 1(Fk/ak)

so +o

commutes w i t h t h e F . ) . The r e s t r i c t i o n s o f Fk t o M = 9, a r e t h e n i n t e g r a l s J f o r t h e geodesic problem ( n o t e on M y (F./H.) = - $ ~ ~ ( q , p )= 0 b u t g e n e r i J J c a l l y one has n-1 commuting independent i n t e g r a l s on M).

1;

REItlARK 3.4 (c0NNEcEZ0W C0 EHE NEl.lMA" PR03i;ER AND RIrmANN SURFACES).

We 2 c o n t i n u e t o f o l l o w [MOl-51 and n o t e now t h a t i n t h e H a m i l t o n i a n H = + l p l 2 t +u$o(qyp) one has { l p l y $ o } = 0 and t h e v e c t o r f i e l d s X 2 and X40 can IPI be discussed s e p a r a t e l y . The system (*) q j = ( a / a p . ) $ and p j = -(a/aqj)C0

,

J

O

can be i n t e r p e r t e d as i s o s p e c t r a l d e f o r m a t i o n w i t h L = L(q,p) q 0 q)Pp ( l p l > 0 ) where ( q 0 q I i j

= q . q . and (PPlij 1 J

t h e p r o j e c t i o n o n t o t h e orthogonal complement o f p.

= 6ij

-

= P (A P pipj/lp12

is

Thus Lp = 0 (so A = 0

i s an e i g e n v a l u e w i t h e i g e n v e c t o r p ) and f o r B one chooses B = - ( (

qipj Hence

-

) ) t o g e t L' = [ B , L l a g r e e i n g w i t h (*) ( c f . [MOl;AdPI). = 0 ) a r e i n t e g r a l s f o r (*) and one

q.p.)/aiaj J 1

t h e eigenvalues hk o f L, 1 5 k 5 n-1 (A

n

finds that

(3.6)

2

(IPI /z)det(z-L)/det(z-A)

Therefore A,

,...,An,l

and I p I

2

= $,(q,P)

=

,$ Fk/(z-"k)

can be viewed as f u n c t i o n s o f t h e Fk and w i l l

consequently a l s o commute. The l e a v e s o f t h e " f o l i a t i o n " Fk ck can a l s o n be d 2 f i n e d v i a l p I 2 = ck, Aj(q,p) = B~ ( j = l,...,n-l) where t h e B . a r e J Thus f o r d i s t i n c t B . t h e l e a v e s a r e zeros o f c k / ( z - a k ) ( c f . (3.6)). J 2 = Ck; $ B , = = $Bn-, = 03 and t h e s e (q,p) c o r r e s g i v e n b y {(q,p);

ll

1;

IPI

11"

...

pond t o l i n e s q t sp which a r e common tangents t o t h e q u a d r i c s

...,n-1

( c f . [M01,2] f o r t h e geometry).

For f l o w on

9,

@IBj ,j

= 1,

( c f . ( * ) ) t h e r e must

be a n o t h e r 0 e i g e n v a l u e (besides t h a t c o r r e s p o n d i n g t o p), say = Bn-l 2 = 0; t h e n hly An-2 and I p I a r e n-1 commuting i n t e g r a l s on T*M.

...,

One can now r e l a t e t h i s t o t h e C. Neumnn problem o f Remark 3.2. panding $z we g e t

Indeed ex-

ROBERT CARROLL

26

(3.7 2 t 2H(q,p)/z t ... where H i s the C . Neumann This Hamiltonian o f Remark 3 . 2 . Comparing w i t h ( 3 . 5 ) one has H = L,l: a k F k ( p , q ) 2 (note p and q a r e interchanged in F k ) w i t h l q l = 1; F k ( p , q ) Hence the F k ( p , q ) defined i n ( 3 . 5 ) (interchanging p and q) a r e integra s f o r t h e Neu2 : Fk commutes with the Fk the constrained mann problem and s i n c e I q l = 1 system is a l s o i n t e g r a b l e . The geometric background of t h i s r e l a t i o n between t h e Neumann problem and geodesic flow i s based on the Gauss map and we

r e f e r t o [M01,2;KBl 1 f o r d e t a i l s .

<

Now one considers t h e i n t e g r a t i o n of t h e Neumann problem = - A q + vq, v = 2 ( A q , q ) - I q l . The functions ( p z ( 4 , q ) = 1 ; F k ( t , q ) / ( z - a k ) a r e i n t e g r a l s and 2 we consider s o l u t i o n s on i n t e g r a l manifolds F k ( 4 , q ) = c k w i t h I q l = 1 = 1,” c k . Further f o r s i m p l i c i t y assume the r a t i o n a l function 1 c k / ( z - a k ) has

,...,

n-1 d i s t i n c t real r o o t s B1 ,.. . , B n - l so t h a t a1 anyB1,..., 8n-1 a r e d i s One can take w i t h o u t l o s s of g e n e r a l i t y a1 < B1 ... < B , - ~ < a . Recall = 0 and consider the flow in M = { ( q , 4 ) = 0; F k ( { , q ) = n ckYk = 1 ,n}. One uses e l l i p t i c coordinates on the sphere ( p l pn-l) n 2 defined a s t h e zeros of Q Z ( q ) = 1 q / ( z - a j ) . Recall here Q,(q) = 1 j ( (zI-A)-’q,q) and, r e f e r r i n g t o [ M01,2 1 f o r perspective a n d d e t a i l s , one can w r i t e m(z) = nln- 1 ( z - I J ~ ) ,a ( z ) = det(z1-A), a n d Q z ( q ) = m ( z ) / a ( z ) so t h a t 2 q2 = m ( a j ) / a ’ ( a . ) ( l q l = 1 i s used h e r e ) . Note t h a t the q . can be deterj J J mined ( u p t o s i g n ) from the p by the residue formula just i n d i c a t e d . Furj 2 t h e r in order t o compute from $ z ( { y q ) = Q z ( q ) ( Q z ( { ) + 1 ) - Q z ( 4 , q ) we 2 n-1 ( z - 6 . ) (8. a r e have $ z ( 4 y q ) = -Qz({,q) f o r z = uj-, and setting b ( z ) = n1

-

t i n c t and r e a l .

,...

,...,

4,

r o c t s of p

,j

M).

’

6.

1,” c k / ( z - a k ) ) one

j = l,...,n-1

J

has Q z ( q , q ) = (-$z({,q))’

(recall t h a t $z(4,q) =

T h u s one has n-1 equations f o r

*

The

p.

J

1;

4, plus

Fk(Gyq)/(z-ak) (

q,:)

J

= (-b(z)/a(z))’ for z = =

l1n c k / ( z - a k ) on

= 0, which will determine

a r e thus parameters o n M .

One notes here a connection t o Riemann surfaces and h y p e r e l l i p t i c curves (which will be developed below following [M01,21 and l a t e r i n more g e n e r a l i t y following [ OUly8;E2;FL2;MC1-ll ] and a host of other authors t o be i n d i cated).

The idea (explicated l a t e r i n §4,5) is t h a t f o r R ( z ) = a ( z ) b ( z ) t h e

equations f o r

IJ

j

( a r i s i n g from equations f o r 9 . ) will have the form 6 J

j,l

=

27

HILL’S EQUATION n-1 n - j - 1

c1 t h e Jacobi =

pk

pk/2(-R(vk))’

t

( j = l,...,n-l)

and such formulas a r e r e l a t e d t o

I,”-’

map o f t h e Riemann s u r f a c e w2 = -4R(z) g i v e n b y s = (pkyw~) t j ?odd n- j - 1 z dz/2(-R(z))’ taking a d i v i s o r class (pky2(-R(pk))2) t o a p o i n t s E

Cn-’/r where r i s t h e p e r i o d l a t t i c e ( t h e d i f f e r e n t i a l s i n v o l v e d here a r e d i f f e r e n t i a l s o f t h e f i r s t k i n d (DFK)

-

The d i f -

c f . Appendix B and §4, 5).

+ s.(O) which l i n e a r i z e s o r s = 6j,lt J Further i n t h i s context o f d i s t i n c t a.,B J k t h e s o l u t i o n s w i l l be q u a s i p e r i o d i c (see Remark 3.6) w i t h a t most n-1 f e r e n t i a l equations become

-

6jyly

t h e f l o w on t h e Jacobi v a r i e t y . frequencies.

REIRARK 3.5 (HILL’S EQllACZ0N AND SPECCRAL ZW0RI1ACI0N).

We w i l l g i v e now a

s k e t c h o f some m a t e r i a l i n v o l v i n g t h e d i f f e r e n t i a l o p e r a t o r L = - D

2

+

q for

q ( r e a l ) a l m o s t p e r i o d i c o r p e r i o d i c and i n d i c a t e connections t o t h e mechani

c a l problems o f Remarks 3.1-3.4

( t h e a l m o s t p e r i o d i c case i s v e r y complica-

t e d and we do n o t d w e l l o n t h i s ) .

For background m a t e r i a l see here [AC1,3;

AD2;BIlY2;BC1;DI1 , 2 ; D U 1 y 3 ~ E 2 ~ D K 2 ~ F L 2 ~ G A 1 ; H O l y 3 ; J O l ; K V 1 ; L X l y 2 ; L N l ; M C Z - 9 ~ M N 2 ~ MO1-5;TBl

1.

There a r e many omissions o f d e t a i l here, some o f which i s i n

f a c t s u p p l i e d elsewhere i n t h e book, b u t t h e main idea i s t o supply perspective.

For b a s i c ideas o n Green’s f u n c t i o n s , second o r d e r d i f f e r e n t i a l equa-

t i o n s , e t c . c f . [ AD1;Cl ,20;CIl;DPl;GQ1;G2;H02;JOl;LT1-1O;MG1;PS1].

Now f o r

2

L d e f i n e d on Co(R) t h e r e e x i s t s a unique s e l f a d j o i n t e x t e n s i o n which we c a l l L.

The spectrum o(L) i s on t h e r e a l a x i s and f o r q p e r i o d i c (which we

assume now u n l e s s o t h e r w i s e s t a t e d ) u c o n s i s t s o f a g e n e r a l l y i n f i n i t e numb e r o f i n t e r v a l s (band spectrum) w i t h no p o i n t eigenvalues. gives the existence o f a n o n t r i v i a l s o l u t i o n o f +(x,A) = exp(w{A))p(x,A)

(A)

where p has t h e same p e r i o d

Floquet theory

L$ = he o f t h e form

Q as q so t h a t 9(x+Q,

A ) = j ~ ( h ) @ ( x , h ) , p = exp(w(A)Q) (p i s c a l l e d t h e Floquet m u l t i p l i e r ) .

one w r i t e s (A)

9 as y ’ = A(x)y, y = (,,),

mental m a t r i x s o l u t i o n Y(x,A)

0 1 A = (q-A o ) y

w i t h Y(0,A)

If

t h e n t h e r e i s a funda-

= I. Since detY = 1 t h e eigen-

values o f Y(Q,A) ( r o o t s o f Y = 1 ) have p r o d u c t 1 and can be designated by -1 One w r i t e s p + 11-l = TrY(Q,A) = A(h) (which i s e n t i r e ) and f o r I m A pyv

+ 0,

.

v(A) i s a n a l y t i c .

For Imh > 0 one can a r r a n g e t h a t I p ( A ) I < 1 and For

t h e r e a r e branch p o i n t s where p(A) = +1 (double eigenvalues o f Y(Q,h)). 1-1 =

ii # ft lh e s i t u a t i o n i s u n s t a b l e ( h y p e r b o l i c ) and f o r

the situation i s stable ( e l l i p t i c ) .

Equivalently +A(A)

= 1, p

#

kl,

> 2 (hyperbolic) o r

28

ROBERT CARROLL

-2 < A ( A ) < 2 ( e l l i p t i c ) . The s e t { A E R ; - 2 2- A ( h ) 5 23 = u ( L ) c o n s i s t s of a s e t o f i n t e r v a l s (bands). The open i n t e r v a l s where ? A ( A ) > 2 a r e c a l l e d gaps and belong t o the resolvant s e t o f L. Since v ( A ) (not A ( A ) ) has a gen-

e r a l i z a t i o n t o the almost periodic case one works w i t h u ( A ) and considers a ( A ) = Imw(A) = (l/Q)Im logu(A).

f i e s a ( A ) -+ 0 f o r ReA ( c f . [ JOl]) t h a t i f

-+

--.

ImW($,7)

a i s harmonic f o r ImA

Working i n ImA =

Im($T'

-

&$I)

>

+ 0 and one speci-

0 with I u ( A ) l < 1 one can show > 0 for

x = 0 then $ ( x , A )

+0

for x 2 0 and

2

To s e e t h i s one notes t h a t OxW($,S) = 2iIm 1 $ 1 and - I m ( $ ' / $ ) = ImW($,T)/ ) $ I 2 ( W being purely imaginary). T h u s a ( h ) i s a p o s i t i v e harmonic function i n Imx > 0 w i t h a continuous extension t o ImA L 0 ; i t i s c a l l e d t h e r o t a t i o n

number.

In f a c t a ( L ) = union of closed i n t e r v a l s where . ( A )

is s t r i c t l y inIt

creasing and t h e gaps = union of open i n t e r v a l s where a ( A ) = c o n s t a n t . follows t h a t i n any gap j = a Q / n i s a nonnegative i n t e g e r .

iE

This r e s u l t s

N(x,A)/x ( A r e a l ) where N(x,A) i s t h e from t h e c h a r a c t e r i z a t i o n a(A)/r = number of zeros o f a ( r e a l ) s o l u t i o n $ ( t , A ) i n 0 5 t 5 x. Hence one l a b e l s the bands by b

REmARK 3.6

j

= {A E R; j

5 Q a ( x ) / r5 j + l l , a n d m ( b j )

(REk0l;VANC AND GREEN'S f l l N t C Z 0 N ) .

= length b

j'

The s e t A of almost periodic

( A P ) functions is defined a s the closure of t h e trigonometric polynomidls f(x) 1 cvexp(iAvx) i n t h e uniform topology over R . One defines M(f) = l i m ( l / x ) g f ( t ) d t ( f E A ) a n d a Fourier s e r i e s ( F S ) i s associated w i t h f X-toJ

via c A = M(fexp(-ixx)) so cA i 0 only f o r A = X v and the formal FS i s then 1 c A exp(ihvx). The A v a r e c a l l e d the frequencies of f and t h e smallest module over Z containing t h e A v is t h e frequency module M(f) ( = i n t e g e r , f i n i t e sum).

1 nvxvy

nv

I f f is periodic M(f) is generated by 2n/Q a n d i f

M(f) i s generated by f i n i t e l y many X u then f is c a l l e d quasiperiodic.

One

f i x e s now a countable frequency module M a n d w r i t e s A(M) f o r t h e s e t o f AP functions w i t h M ( f ) = A ( M ) . Then f o r g E A ( M ) ( 3 . 8 ) holds again a s well as t h e c h a r a c t e r i z a t i o n o f a ( A ) via N ( x , A ) .

Again a ( A ) i s monotone nondecreas-

ing and i f I i s a gap of L ( i . e . a n open i n t e r v a l in R / u ( L ) ) then a(A) i s constant i n I with 2 a ( x ) E M ( g ) . Viewing da a s a measure one has then o(L)

29

RESOLVANTS = s u p p da(A).

Now r e c a l l t h a t the Green's function G(x,y,A) i s t h e kernel

of the resolvant Rh = ( L - A ) - ' .

let $

a c o n s t a n t ) o f W = A$ such t h a t $+ E

RY1 I ) .

, $-

5 L (0,m)

be the s o l u t i o n s (unique u p t o and $-

E

L 2 (--,0)

( c f . [ LT2;

Then G(x,y,A) = @ + ( & A 111- ( Y , X )/W(11+,11-)

(3.9)

(x 2 Y )

*

2 Further f o r Imh 0, W($+,$+) = -2iImX< I$+/ d t and W($ - ,? - ) = 2iImX 2 111-1 d t so $+,11- have no zeros and m + ( x , h ) = $J(x,A)/$+(x,X) i s well defined. UsTng $- f o r IP in ( 3 . 8 ) One knows t h a t m+- and G ( x , x , h ) E A(M) ( c f . [ SF;]). e t c . one o b t a i n s a ( h ) = -ImM(m-) and f o r Imh > 0, -M(m ) = M(m+)

= M(-l/

PG(x,x,h)). To see this note l / G ( x , x , h ) = m- - m+ from (3.9) and m+ + m= DxlogG(x,x,X). Hence defining w ( h ) = -L,M(G-l(x,x,h)) (Imh > 0 ) one has Floquet o b j e c t s i n t h e Imw(h) = o(A) w i t h w holomorphic f o r Imh > 0 ( w , p

AP s i t u a t i o n ) . Next one can show t h a t dw/dA = M(G(x,x,X)) ( c f . [ JOl,G2]) we will derive this i n Remark 3.7). The termM(G(x,x,X)),which e x i s t s s i n c e G E A(M),is a substitute f o r t h e t r a c e o f Rh and one w r i t e s dw/dA = TrRh.

-1 Tr log(RA R A, ) = Tr l o g ( 1 + (A-A o ) R x A (note formally J d z / ( L - z ) = logI(L-Ao)/(L-X)l = log R R - l and one could A, A 0 use a n a l y t i c functional calculus t o j u s t i f y such c a l c u l a t i o n s - c f . [CZOI) Similarly f o r real X , A o one can w r i t e Integrating one o b t a i n s w ( X )

-

w(Xo)

=

The a n a l y t i c ~ ( h )(Imh > 0 ) s a t i s f i e s Rew(h) < 0 and Imw(A)/ImA > 0 f o r ImA

+ 0 and

can be determined from

c1

on R v i a w(z)

-

w ( 5 ) = ( l / n ) / I a(X)dX/

(A-Z)(A-C).

REFtl\RK 3.7

(C@NNECCl0@ CO K M ) .

d e r i v a t i v e s a l s o belong to A(M).

W e consider now g E A ( M ) such t h a t a l l The gradient o f a functional F ( g ) i s de-

fined v i a (3.11

so F

DEF(g+Eh)lE=O = M(VF(g)h)

2 + bq 3 ) implies O F = -2aqxx + 3bq 2 .

= M(aqx

In t h i s s p i r i t t h e Poisson

bracket is {F,G} = M ( V F a x v G ) ( c f . [DSl;GDl ]).and a Hamiltonian H ( q ) gives 3 r i s e t o qt = a x ( v H ( q ) ) . Thus H ( q ) = M(L,qc + q ) implies q t = -(Ixxx + WiX

30

ROBERT CARROLL

(we make no a t t e m p t t o m a i n t a i n t h e same s i g n s and c o e f f i c i e n t s i n KdV equasense t h a t {F,Hl G(x,x,A) w(A-q), and e.g.

I m X = 0, i s an i n t e g r a l F i n t h e

Now t h e F l o q u e t t e r m w(A,q),

t i o n s here).

= 0, and {w(A1.q),w(A2,q)3

JO1 1.

( I m A # 0 ) and we s k e t c h a p r o o f from [

a(X,q)

-

This f o l l o w s from vw =

= 0.

F i r s t n o t e w(A,q)

=

( s i n c e t h e d i f f e r e n t i a l e q u a t i o n i s 0" = (-A+q)$)

= a A-q)

i s continuous i n q i n t h e sup norm t o p o l o g y ( e x e r c i s e ) .

a(A,q)

Con

s i d e r (3.11) f o r F = w and use t h e v a r i a t i o n a l d e r i v a t i v e n o t a t i o n 6w/6q f o r vw.

D,w(z,q) S e t t i n g h = -1 one g e t s DEw(z,q-E)lE=O = D E ~ ( ~ + ~ , q ) l E == O

and wz = -Mx(Gw/Gq) -G(x,x,z).

(z

Consequently wz = M x ( G ( ~ , ~ , ~ ) ) when 6w/6q =

A).

For Gw/Gq w r i t e G(x,y)

f o r G(x,y,z,q)

-

f o l l o w s from t h e r e s o l v a n t i d e n t i t y R, (3.12)

GG(X,X) =

= -(l/W2){$:(x)l:

-ff

R,,

= ( A - Y ) R ~ R ~ . Thus

G(x,Y)Gq(y)G(y,x)dy

((y)Q(y)dy

+

and n o t e 6 R Z = -RZGqRz

=

-1:

2

G (x,y)bq(y)dy

=

$ F ( x ) g $;(y)dq(y)dyl

2 2 2 2 - 2 = -GG(X,X)/ZG (x,x) = %I$(x)l: $-Gqdy + JI; Ix JIt 2 2 dqdyl = (l/2W){-($+/$-)'lx Q-Gqdy + ($-/$+)'Ix- $+Gqdy) s i n c e ($+/$-I' = 2 -W/$and ($-/$+)' = :W$:/ Thus ( 0 ) 6(1/2G(x,x)) = -G(x,x)Sq = a x H where x 2 2 H(x) = (l/2W){-($+/$-)Lw $-dqdy + $+GqdyI = K(x,y)Gqdy ( K havT h e r e f o r e 6(1/2G(x,x))

(1L-/JIt)r

i n g t h e obvious form). M(G(x,x)Gq)

Take mean values i n

(0)

rl

now t o g e t M(G(l/G(x,x))

and one r e c a l l s t h e d e f i n i t i o n w ( A ) = M(-l/G(x,x)).

Next one expands t h e Green's f u n c t i o n as

t h a t Vw = 6w/6q = -G(x,x,A). (3.13)

G(x,x,A)

( c f . [ G2,ll

I)

%

=

It f o l l o w s

(l/Z(-A)'){l

+

1;

Gk(x)/Akl

where t h e Gk a r e f u n c t i o n s o f q and i t s d e r i v a t i v e s (e.g.

G1

The c o e f f i c i e n t s Gk can be determined by comparing c o e f f i c i e n t s i n = q/2). the R i c c a t i equation (3.14)

(G"

-

Z(q-A)G)G

-

%((GI)'

-

1) = 0

To check (3.14) we n o t e f o r a = $+ and 6 = $- ( n o r m a l i z e d t o W(a,B) = 1 ) one has G(x,x,A)

+

= aB SO G I =

2 a ' ~ ' . Hence (3.14)

= 0 as desired.

alB

t aB' and G" = a"B

-

+

aB"

becomes 2 a ~ a ' ~ '% ( a ' ~t a a ' )

2

+

2a'B' = Z(q-A)aB

+ ?.2

The expansion (3.13) i s based o n [G2,11

2

= %(l - W(a,B) )

1 where

one expands

FINITE BAND POTENTIALS

2

(-D

+ q

(3.15)

-

2 i n powers o f (-D - A ) - ’ .

A)-’

+

= -(-h)%l

w(A,q)

1;

31

We use (3.13) t o get

wk(q)/Ak}

where wk = M(Wk(q)) w i t h Wk(q) a polynomial i n q and i t s d e r i v a t i v e s (one uses here dw/dA = G(x,x,A)

and i n t e g r a t e s ) .

d i f f e r e n t A one g e t s a l s o I w j . w k l

Since t h e w(A,q) commute f o r

= 0 and i w

w(A,q)l = 0. E x p l i c i t c a l c u jy 2 2 l a t i o n s y i e l d w1 = %M(q), w2 = (1/8)M(q ), w3 = (1/16)M(%qx + q ), etc., so H = 16w3 corresponds t o KdV.

REi!MRK 3.8

(FINICE 3AND P0EENCIAC$ AND C0NNECCI0Nk CO CHE NELUllANN PR0BCEE).

Given 2Nt1 numbers A.

< Al

such t h a t o ( L ) = [ hoyhl

IU

c

... <

[ A2,A31

A Z N one wants now a l l r e a l p o t e n t i a l s q

. ..

U

X,q)

AZNym),

L b e i n g based o n q.

More p r e c i s e l y one t h i n k s o f G x,y,

These N+l i n t e r v a l s a r e c a l l e d bands.

0.

U [

r e a l on p ( L ) n R ( r e s o l v a n t s e t = p ( L ) ) and ImG(x,x,A,q)

> 0 f o r ImA >

has an a n a l y t i c c o n t i n u a t i o n t o t h e i n t e r i o r o f t h e

Assume G(x,x,A,q)

bands and i s p u r e l y i m a g i n a r y t h e r e w i t h G(x,x,A,q)

near X J J j T h i s i s a n a t u r a l s i t u a t i o n ( i s o l a t i n g a c l a s s o f problems) and %

y.(A-A.)-’

( y . f. 0 ) . J i n v o l v e s a Riemann s u r f a c e R o f genus N ( c f . §4,5 and Appendix B ) .

w i l l have G(x,x,A,q)

One a l s o

a n a l y t i c f o r ImA > 0 w i t h (3.14) v a l i d and G(x,x,A,q)

Now w r i t e a = A ( j = 1,2, ...,n=N+l) and B = j 2j-2 j n-1 and s e t a(A) = n ( h - a . ) ( 1 5 j 5 n ) w i t h b ( h ) = n1 J The f u n c t i o n (-b(A)/a(A))’ i s meromorphic on R and one chooses t h e

l/Z(-A)’as

x

+

-a.

( j = l,...,n-l=N)

h2j-1

(A-8.). J branch i n Imh > 0 w h i c h has a p o s i t i v e i m a g i n a r y p a r t i n t h e bands when approached from above. A,q)

x

+

Set t h e n r ( x , A )

= I’(x,A,q)

= -2(-b(A)/a(A))’G(x,x,

which i s s i n g l e valued i n C w i t h s i m p l e poles a t a -a).

(3.16)

Consider r(x,A)

1= -2(-b/a)’G(x,x,A)

=

j

(and

r

%

as

I1n r (x)/(A-a.) j J

r

i s p o s i t i v e i n t h e bands [ a B . ] ( j 5 n-1) and i n [ a n y m ) so r . ( x ) > 0. j yJ J One proves now t h a t i f (3.16) h o l d s w i t h G = G(x,y,A,q) then there e x i s t

JI.

2

o f LJ/. = a.$. such t h a t r = Ilj. J J J J To see t h i s r e p r e s e n t J/,,Ilas _ l i n e a r combinations o f 2 normalized s o l u t i o n s

real solutions

+1y$2

o f L+ = A$ w i t h W ( I $ ~ , + ~ ) ( O ) = 1 ( q ( 0 ) = 1,

$,’(O)

= 0, e t c . ) .

Then

ROBERT CARROLL

32

2 a r e e n t i r e in A a n d from G = $+$-/W($ty$-) one gets r = A . 4 + 2B j 2 3 1 j l 2 2 41$2 t C . 4 (where X i s replaced by a.) a n d one wants A C - B j > 0. This J 2 J follows from (3.14); indeed f o r r one has $ ,$

(3.17)

2(r" - z(q-x)r)r-

2 = 4b/a

(rl)

Then some calculation using (3.16) (exercise

(Aj+ + Bj$2)2 ( f o r A . 4 0 ) a n d thus A j A . = 0 one takes $ j =JC$2.

>

- c f [ M021)-Lleads

0 with $

j

=

Aj2(Aj+

t o r j = A;' + B 0 ). If j 2

J

Now t h i s i s connected with the C . Neumann problem as follows. We have a ren 2 A - ~ as x -a one gets 1 = l a t i o n r ( x , A , q ) = 1 $ (x)/(X-a.) and since r n 2 1 j J = 0 o r $'! = -a$. + q$. so x = $ . ( t ) can be in1, $ j . Further (L-a.)$ J j J J J J j J terperted a s the component of a vector satisfying = -Ax + q ( t ) x f o r A = diag(a.) and constrained by In x2 = 1 . Recall also now Qz, $z, e t c . from Q ,

1

J

Remarks3.3-3.4 and write (3.18)

a,($)

=

c1n

-f

j

2

Qj/(A-aj) =

r;

$A($',$) = ( 1 + Q A ( $ ' ) ) Q A (-$ )Q f ( $ ' , $ )

The $A correspond t o integrals of the mechanical problem a n d one obtains immediately from (3.18) Q , ( $ ' , J / )

=

In1 $~j . $ ' / ( A - a . ) = W'. Another J

differentia-

tion gives 1 + Q,($') = +(I"' - Z(q-A)r)a n d (3.17) yields then 4A($'y$) =

b/a. Hence x = ($l , . . . , $ n ) corresponds t o a solution of the mechanical problem w i t h = b/a so 3, l i e s on the invariant manifold defined by = 0. Summarizing one can s t a t e (see [ M O Z I f o r the proof of the 4 ($I,$) Bi 1as t statement ) If G(x,y,A,q) s a t i s f i e s the hypotheses indicated then G ( x , x , L h 2 X , q ) = - % ( - a / b ) * l l g j ( x ) / ( A - a . ) where $ = ( $ l , . . . , $ n ) i s a solution of the

CHE0)RZTLI 3.9.

y.

J

+ q$. with $A($',$) = b/a and (viaasymptoJ j , J 1 7 1 8 ~ aj. Further i f $ . s a t i s f i e s the probticsx.) q ( x ) = 21: aj$5 J lem ( N ) with $ A ( $ ' y $ ) = b/a t h e n q defined by ( 0 ) i s the potential of a n operator L with band spectrum as above. C . Neumann problem ( N )

= -ad

c1

To i n t e r p e r t the e l l i p t i c coordinates pk o f Remark 3.4 one writes ( 6 ) a,($) 2 n-r $ / ( A - a . ) = n , ( A - p k ) / a ( A ) a n d hence the p k correspond t o zeros of J 1 j They depend on x a n d a r e r e s t r i c t e d t o the gaps (since G ( x , x , G(x,x,A,q). =

In

NEUMANN PROBLEM

33

A , q ) = 0 i n t h e bands).

One will have t? < u . ( x ) i a j + l ( r e c a l l a. < B~ < j- J ( n = N+1) w i t h bands [ a . , B . I and [a,.,=)). The poJ J t e n t i a l q can be expressed as q ( x ) - a1 = n-l (“k+l t?k - 2 V k ) ( c f . CMC3I). To s e e this one compares c o e f f i c i e n t s o f X i n the expansion o f ( 6 ) and uses ( 0 ) ( e x e r c i s e ) .

a2 <

p2 <

... <

t?n-l

< an

$

Some continuation of c e r t a i n themes i n t h i s s e c t i o n i s developed i n §4,5. Some o f the themes sketched here have been extended and refined considerably i n various d i r e c t i o n s and we will not t r y t o cover this. For example i n [AC1,3,5-81 tine develops t h e idea o f p r o v i d i n g a systematic l i n k between f i n i t e dimensional i n t e g r a b l e systems, flows i n loop algebras, and spectral considerations, via the use o f momentum maps, e t c . This builds upon e a r l i e r work o f various authors and i s r e l a t e d t o o t h e r contemporary work o f many people. We simply p r o v i d e here a reference l i s t t o some papers of this nature which seem important (some o f these a r e a l s o mentioned a t various o t h e r places i n t h e book). We had o r i g i n a l l y intended t o do more w i t h this s u b j e c t b u t o t h e r t h i n g s were w r i t t e n f i r s t and we ran o u t o f space. Some relevant constructions such a s momentum maps a r e however i n Appendix A; most o f t h e s e items a r e used a l s o a t o t h e r places i n the book so t h e i r inclusion i s n a t u r a l . T h u s l e t us mention ( w i t h apologies f o r any i n a d v e r t e n t omissions) [ AC1-8;AD1-4;DHl-5;DI3-5;CUl; FL2,9; IC1 ;KUlY4;KOS1;LU1; MAW1 ;RE1 -1 0; RT1 ,2 ; SD1; SM1-5; SY1-3; SCAl ;VM1 ,2;W3,7,10;OY1 ;WE1 ;WS1-5 ;ZEl ;ZU1 1.

REmARK 3.10.

ON CHE GE0mECRg 0F K&J The t i t l e here is from [ E3;MC7,8,10] and we will i n d i c a t e some glimpses o f their program based on [ MCl-ll;E3,4] a s well a s some o t h e r points o f view and c o n t r i b u t i o n s from various sources. Mainly however t h i s s e c t i o n sketches h e u r i s t i c a l l y some e a r l i e r work on Hill ‘ s operator and h y p e r e l l i p t i c curves based on [ MC3-5;TBl I ( c f . a l s o [ LV1,4,5; KP1 I ) and then goes i n t o geometrical aspects h l a [MUl,Z;MCll;PE2,4;FL2,61 e t c . ( f u r t h e r references a r e c i t e d a s we go along). Certain a s p e c t s o f t h e geometry o f s c a t t e r i n g ( t h e Grassmannian formulation) a r e t r e a t e d i n 911. Here we emphasize t h a t the presentation i s h e u r i s t i c and motivational. We c o l l e c t a minimal amount o f information i n order t o hopefully give a more o r l e s s e x p l i c i t p i c t u r e of what goes on and via connections t o §3,5 i t should be coherent. One should be prepared t o accept ( p r o v i s i o n a l l y ) 4.

34

ROBERT CARROLL

various assertions, b u t references t o details a b o u n d ; the main point here i s to see the picture. In terms o f detail we are more interested i n picking u p the theory a t the level e.g. of l i n e bundles over Riemann surfaces (RS) b u t one must see how t h i s a ri se s and what i t means ( cf . Remark 4.3). Thus a l t h o u g h we have great respect for analytic detail (and have spent a l o t o f time on i t in the past) t ha t i s n o t our choice o f subject material here ( o r in §3,5) and we accordingly only se le c t stepping stones and p a t h s portraying many connecting ideas a n d leading t o various perspectives. Thus e.g. in Remark 4.3 one sees t h a t " a l l " information a b o u t a f i n i t e gap potential q ( x ) i s encoded i n the Baker-Akhiezer ( B A ) function or i n a line bundle tx over a RS; hence the detailed study of (generic) q ( x ) i s less urgent ( a n d has been done - with references and some essential features provided). Some information a b o u t KdV flows, linearization i n the Jacobian, e t c . will be covered in more detail from diffe re nt points of view elsewhere in the book. Only a brief sketch of [ E31 i s given since i t i s very technical and we do n o t r e al l y discuss the general philosophy of [ E3;MC7-10 1. For consistency w i t h i n sections (based on a certain body of reference material) we may occasionally use s l i g h t l y di ffe re nt notation in differ ent sections; differ ent formulas representing the same objects will n o t always be shown t o be equivalent and t h i s i s l e f t as an exercise. Repetition of ideas, definitions, e t c. i s deliberate and di ffe re nt points o f view ar e developed deliberately. The BA function i s introduced heuristically here and will be discussed more thoroughly i n many places l a t e r . Our eventual program of relating everything t o everything e lse via t a u (or theta) functions i s n o t complete (and may n o t always be the most productive approach) b u t i t does provide a unifying theme. Generally there i s however a stimulating lack of completeness in the whole theory and i t should be emphasized t h a t there i s s t i l l much to do a n d understand. RfClARK 4.1

(SPECCRAI; ZW0RI3ACZ0N AND RZECIANN $llWACE$)

-

We go here t o [ MC 2

3-5;MR2;TBl ] for background material. One considers Q = - D + q , q E C y = 1 real smooth functions of period 1 ( a n d assume /o q ( S ) d S = 1 for convenience). 2 For complex periodic q see [BI1,2]. If one considers Q in L ( - m , m ) with domain c," then Q 'L L i n 53 a n d has spectrum consisting generally of a n infini t e number of intervals [ h o , h l ] U [ x 2 , X 3 ] LJ ... as in Remark 3.8. Write

RIEMANN SURFACES

35

here -- = A -1 < A 0 < A 1 <- h2 < A 3 5 A 4 < A 5 5 ... t m . The i n t e r v a l s (X2j-l,A2j), j = Oyl,... a r e i n t e r v a l s o f i n s t a b i l i t y where no s o l u t i o n i s bounded while i n t h e i n t e r v a l s of s t a b i l i t y (A2j,A2jtl), j = 0,l ,... every s o l u t i o n is bounded b u t none i s o f period 1 o r 2 . The points A. < A1 5 h 2 < A 3 5 ... a r e c a l l e d the periodic spectrum. Here t h e principal s e r i e s A. c X 3 5 A 4 < A, 5 As < is t h e spectrum o f Q a c t i n g on f € L12 w i t h D ( Q ) = 2 H1 (periodic boundary conditions on ( 0 , l ) ) and t h e complementary s e r i e s A1 2 5 A 2 < A 5 5 A6 < f i l l s o u t the spectrum o f Q i n i t s a c t i o n on L2 (periodic boundary conditions on (0,Z)). I t turns o u t t h a t A. i s a simple eigenvalue (EV) w i t h eigenfunction (EF) o f period 1 while f o r i = 1 , 2 , . . . , A2i-l o r hZi is a simple o r double EV according a s A2i-1 < A 2 i o r h2i-1 = h Z i w i t h EF o f period 1 o r 2 according t o whether i = 1,3,5 ,... o r i = 2,4,6 The EF f 2 i - l and fZi f o r AZim1 o r h Z i r e s p e c t i v e l y have p r e c i s e l y i roots in a period 0 5 x 5 1 . The same c l a s s i f i c a t i o n of points may a l s o be described via A ( A ) = y l ( l , A ) t yi(1,A) where y1 ( r e s p . y,) s a t i s f y Qy = hy w i t h y ( 0 ) = I , ~ ‘ ( 0 =) o ( r e s p . y ( 0 ) = 0, y ’ ( 0 ) = I ) . In f a c t ~ A ( x ) / > 2 i n i n t e r v a l s of i n s t a b i l i t y , la(A)\ < 2 i n i n t e r v a l s o f s t a b i l i t y , A ( A ) = 2 on the p r i n cipal s e r i e s , and A ( A ) = -2 on the complementary s e r i e s . The roots u n ( n 2 1 ) o f y 2 ( 1 , u ) = 0 coincide w i t h EV of Q a r i s i n g from EF vanishing a t 0 and 1 ( t i e d spectrum) and s a t i s f y A. < Al 5 ul 5 A 2 < X 3 5 u2 5 X4 < .... Note 1 2 t h e normalized EF corresponding t o un i s Y ~ ( x , L I ~ ) /y2(6,vn)dE)’ (& where 1 y2dg 2 = N n = y ; ( l , u n ) i 2 ( l , u n ) ( - means d / d v ) . Note a l s o Q extends t o be s e l f a d j o i n t i n L 2 w i t h e.g. D ( Q ) = H 2 .

...

...

,....

/o

Now i n studying t h e problem o f determining t h e potential g i v i n g r i s e t o t h e same isospectral s i t u a t i o n A. < h l 5 A 2 < A 3 5 ... i t will be useful and appropriate to deal w i t h c e r t a i n Riemann s u r f a c e s . The H i l l ’ s s u r f a c e S f o r q is found by c u t t i n g 2 copies o f the Riemann sphere along the i n t e r v a l s of i n s t a b i l i t y and a t t a c h i n g a l l the lower ?ips on one sphere t o the corresponding upper l i p s on t h e o t h e r , and vice versa. S is the Riemann s u r f a c e of say s(A) = ( - ( h - a ~ ) ( X - A ~...( ) A - A ~ ~ ) (purely ) ~ simple spectrum A: w i t h f i n i t e genus). A model of S w i t h canonical homology basis i s shown i n (4.1) ( c f . Appendix B and 55 f o r any missing terminology o r d e f i n i t i o n s - note f o r i l 2 l u s t r a t i o n we use a model where s(A) = ? ( A - 4)’).

36

ROBERT CARROLL

T h i n k here of the shaded regions as holes and cf. [COl;FAl;MIlI a n d Appendix B for discussion etc.; the arrows are often reversed.

Now the i n f i n i t e genus g case requires a l o t o f technical details so we work w i t h the case o f 2 n t l (simple) EV k: ( O : i : 2 n ) satisfying k: < A; < ... < which i n volves a sphere with n handles (genus g = n ) . In this situation the rest o f the spectrum will be double and i t turns out that the simple spectrum determines the double by explicit formulas ( c f . [ FIl;FL7;GAl;MC2-4;HO2;TBll ) . Note here t h a t A(A) i s a n entire function o f order L2 and i s determined v i a 2 u, 2 4 - A ( A ) = c I I , ( l - x / x ~ ) n (l-M+i) (n over the double spectrum - assume em0 0 0 p t y f o r i l l u s t r a t i v e purposes). There will now be terms wi E [ k 2 i - 1 , A 2 i l i = l y . . . , n , and i t i s known t h a t A ( A ) can be recovered from the ( p n Y N n ) . In fact there i s a 1-1 map from potentials q t o I(pn,Nn)I ( c f . [ BC1;FIl; KP1;LNl;TBll) a n d thus i n describing the isospectral class Q o f potentials q h a v i n g the same A ( A ) one can deal w i t h a Hill's surface o f f i n i t e genus (lp,li), g = n . To portray S one t h i n k s e.g. of expanding the cuts (-myk:), 0 0 a n d ( A ~ , A ~ )(sample case) a n d gluing them together as indicated i n the form ( c f . MI1 I )

(4.2)

0 0- $) -

RI EMANN SURFACES

37

The " d o u g h n u t " can be represented i n t h e form

c_s> -

(4.3)

A2

and t h e r e is a l s o a torus p i c t u r e (described below) o f t h e form (4.4) hl

OXOX4

... <

uon can occupy any position i n C: [ A , , x , I x ... x [ x ~ ~ - ~ , A&]; t h e norming constants determine the s i g n o f s(uk) (s(A) = (-(A-A:) 0 0 (X-hl) ...(A-Ain))') so, on a c i r c l e Tk a s i n (4.4) based on [X2k-l,h2,$, one puts vk on t h e upper or lower l i p according t o whether t h e sign i s plus or minus r e s p e c t i v e l y . There i s then a 1-1 map o f t h e i s o s p e c t r a l c l a s s 9 having t h e same A ( X ) onto the torus T = n; Tk. There i s a l s o in f a c t a 0 + 1; + x ~ - ~2 u L-( x ) )~. Here u i ( x ) r e f e r s ( t r a c e ) formula q ( x ) = :A + t o a t r a n s l a t e d potential q(*+x) so q(O+x) = q ( x ) and q(0) = :A + 1 ; NOW

ul0

<

0

0

0

(~02~

hik-l

-

2~:)

i s t h e background formula ( c f . [MC2,4;TB1] and see a l s o [ DU2;

N O 3 1 f o r a d i f f e r e n t approach, along w i t h remarks a f t e r ( 4 . 6 ) below). (pE,Nk)

formula. q(-),

Thus

determine q ( x ) and one should be careful i n i n t e r p e r t i n g t h e t r a c e The u y ( x ) a r e "contrivedu here; they a r e determined by t h e function

not q ( x ) a t x.

The same s i t u a t i o n will prevail as n

-+ m

with

suit-

a b l e technical adjustments ( c f . [ FIl;MC3,5;KPl;MEl;VEl,21). The upshot i s now t h a t one can study this isospectral c l a s s v i a the t o r u s T o r via t h e Riemann surface S ( % Hill I s s u r f a c e ) of t h e h y p e r e l l i p t i c curve based on s(X) = (-(A-A:)(A-AY) ... (A-AZn)) 0 %.

REmARK 4.2 (EHE MEL-JAC031

mP).

Consider now s(A) = (-(A-A:)(A-Ay)

...

(A-Ain))'which determines t h e t o r i Tk and a Riemann surface S. One can use general theory as in Appendix B o r s p e c i f i c formulas such as (4.5) below

38

ROBERT CARROLL

t o a t t a c h t h e i s o s p e c t r a l problem t o an a s s o c i a t e d Riemann s u r f a c e ( R S ) . 2 2 One notes t h a t polynomials w P2n+l ( A ) o r w = P2n+2(h) b o t h l e a d t o RS o f genus g ( C f . [ DU1 I ) . L e t now M * RS; T n Tk ((11, Y s ( p k ) ) % (Pk,SgnS(Uk))) and l e t oi

Q

0 X2i-1

M.

w i t h pi L e t Ai,Bi

%

(!J~,s(P~)% ) a v a r i a b l e p o i n t on Ti

so p =

be t h e homology b a s i s i n d i c a t e d i n (4.1) and l e t

( pl,...p ) = Aj-'dX/s(X) denote t h e d i f f e r e n t i a l s o f t h e f i r s t k i n d (DFK c f . Apj pendix B ) . There i s a necessary and s u f f i c i e n t p e r i o d i c i t y c o n d i t i o n f o r

-

w

...,

Xo (i= 0, 2n) t o be t h e s i m p l e spectrum o f a H i l l ' s o p e r a t o r Q o f p e r i l od 1, namely ( n = g; J A cycles % r e a l values) Q

where mi

= 1 -t t h e number o f double ( o r p a i r s o f s i m p l e ) EV t o t h e l e f t o f

h2i-1, i s t h e common number o f r o o t s p e r p e r i o d o f fiii-l and f 0Z 9 . T h i s = 0 ( j < g) (and = 2 f o r j g = n); here A.(w.) = I J 1 J AL 'j The r e a l Jacobi v a r i e t y JR o f S i s d e f i n e d as f o l l o w s . Set oi

means lmiAi(w.)

ZIP1 w..

a1-i J = h2i-1 a g a i n and r e f e r t o p = (pl,...,pn)

as a d i v i s o n where pk

(pk,s(uk))

%

1:

a p o i n t on T Then t h e Abel ( o r Abel-Jacobi) map &p: w j = x . deterk' J mines a p o i n t $ = (xl,...,xg) E Rg (the i n t e g r a t i o n path i s r e s t r i c t e d t o N

0

a c y c l e c o v e r i n g [X2i-l,X~il-

I f LR = p e r i o d l a t t i c e

b u t s t o p p i n g a t pi).

o b t a i n e d from c l o s e d i n t e g r a t i o n paths t h e n JR = Rg/LR i s a g dimensional r e a l t o r u s ( n o t e d i v i s o r s p as i n d i c a t e d

-

dual o f DFK = (DFK)' and JR % (DFK)'/LR).

i n real position The map Q

-t

-

belong t o t h e

JR i s 1-1 o n t o and

t h e i n v e r s e map JR + Q can be expressed v i a t h e t h e t a f u n c t i o n o f JR ( t h u s each {(pi,Ni))

corresponds t o one x

E

JR).

T h i s i s sketched below, a f t e r we

d i s c u s s t h e flows, and i n v a r i o u s ways a t o t h e r places i n t h e book.

One can

proceed i n many ways b u t we i n t r o d u c e now f o r Q = -D2 + q t h e f u n c t i o n $ ( t ) =

1;

exp(-hit) $(t)

(4.6)

as t

-t

DK2;BVl

%

T r ( e x p ( - t Q ) ) ( a c t i n g on f u n c t i o n s o f p e r i o d 2 ) .

%

(l/(nt)')L;

0, w i t h H-l

I).

(-t)mHm-1/(2m-3)..

= 1 and Hm =

&I

Im(q,q',

Then

.3.1

... )dx

( c f . [MC2,4;GDl;G2-4,11;

We w i l l d i s c u s s such formulas a l s o a t o t h e r places.

The k m i l -

t o n i a n f l o w s f o r KdV a r i s e from Xmq = Dx6Hm/6q and one has t h e customary 2 t a b u l a t i o n I. = q, I1 = Jiq , I2 = Jiq3 + %(q')', ( u p t o divergence terms

...

ABEL- JACOB1 MAP

39

-

assumed 0); f u r t h e r Xmq = (qD t Dq %D )6Hmm1/6q (Lenard r e l a t i o n ) . One 1 shows t h a t , w i t h XF(q) = (6F/6q)XqdxY t h e X, (1 5 m 5 n ) commute, t h e n ciXi, preserve t h e p e r i o d i c spectrum, and t h e Xm span f l o w s Xq = (i, X =

/o

c1

t h e t a n g e n t space t o Q a t e v e r y p o i n t (Q

'L

RS).

There i s a map Q

-f

J R con-

n e c t i n g KdV m o t i o n i n Q based o n { = X.q, i n t o s t r a i g h t l i n e m o t i o n on JR, -5 J x -F t tx o f c o n s t a n t speed. T h i s i s proved by computing X x = X, jy m j Jpi w and t h i n k i n g o f exp(tXm) a c t i n g on q ( c f . [MC2,4] n o t e t h a t 11; = 00' j pi(q) e t c . ) . The a c t i o n o f X1 f o r example on u s can be c a l c u l a t e d as f o l l o w s .

1

A

-

2

Recall s(A) = (-(A-A:)(A-Ay) t h a t Xluyo= f o r 1-1 = pi

(1:

-

0

2s(p!)/Il

5 f dx

(ui

= 1;

...(A-A&))'with Q = - 0 t q and we w i l l show 0 uj). To see t h i s l e t f ( x ) be t h e n o r m a l i z e d EF so ( Q - v ) 6 f

f ( x ) = ciy2(x,~g))

t

6 q f = u6f t 6uf

Then t a k i n g s c a l a r products we have ( ( Q - v ) 6 f , f ) t 2 Then which i m p l i e s ( 6 q f , f ) = ~ L Io r 6 d 6 q = f 1 1 2 1 1 ( f ( 0 ) = f ( 1 ) = 0 ) Xlpy = J (6p?//6q)q'dx = J f q ' d x = -J0 2 q f f ' d x -2/0 0 2 P (uff' t f"f')dx = -(f') and from b a s i c formulas i n v o l v i n g f = ciy2(x, (note

(6qf,f)

1; f 6 f d x

= 0).

= p(6f,f)

t

.

(6pf,f)

'

lo,

p s ) t h i s can be e v a l u a t e d t o be

2s(p;)/n

0

0

e v a l u a t e X Jpi w . f o r example w i t h A(pi) 0 j-1 1oOi J / S ( L I ~ )and Xl(pY) 2, X1(pi). (pi)

0

(pi-uj) = pi

(cf. [MC2,4;TBl I). Then t o

say one m u l t i p l i e s

w.(p?)

J

1

'L

Thus r o u g h l y t h e KdV flows a r e i n t e g r a t e d v i a t h e a d d i t i v e c o o r d i n a t e s x as The t h e t a f u n c t i o n

a n g l e v a r i a b l e s and t h e spectrum A; as a c t i o n v a r i a b l e s . i s now d e f i n e d a s @ ( x ) =

1 exp(2~rix.n)exp(-sC(n))

w i t h sum o v e r t h e dual

l a t t i c e LR o f p o i n t s n € Rg such t h a t x - n E Z f o r p e r i o d s x

E

LR and C(n) i s

t h e q u a d r a t i c form based on t h e p e r i o d m a t r i x C = BA* formed from A = ((Ai

-

( w . ) ) ) and B = - i ( ( B i ( w . ) ) ) ( c f . Appendix B). One knows o ( x ) > 0 on JR and J J 2 a r e s u l t o f [ I T 4 1 y i e l d s q ( x ) = -2Dx logo(; t x;~) where x" (Gly...y;n) €

R

and

?1

i s a v e c t o r t a n g e n t t o JR i n t h e d i r e c t i o n c o r r e s p o n d i n g t o Xlq = i q ' ( i . e . x" t x? = (ii t xxl) r e c a l l Xmq as d e f i n e d e a r l i e r v i a Ho = /qdx, 1 2 I n f a c t one has X q = 0, H1 = +q! dx, Xlq = q ' , e t c . ) . J

-

0

(4.7)

i;(

Q

'L 2,

(;i)

d i r e c t i o n corresponding t o Xi). E

JR and q ( 0 ) depends 1-1 on

We emphasize a g a i n t h a t { ( u j , N i ) I

x".

For p e r i o d i c problems,theta

t i o n s , e t c . see a l s o 15 and [DU1-3,5,6,8;KR1-13;LTl;N02-4].

func-

40

ROBERT CARROLL

RECMRK 4.3

(3A Fl.INCCZ0W

AND LZNE 3UNDtEs).

We follow here [ BD1 ;EN1 ;CF1-3; we simply s t a t e a few r e s u l t s and

E2;FL2,6;GNl;KR2,3,5,8;KVl;MC2;MU2;SG1,21;

do not s t r i v e f o r elegance or g e n e r a l i t y here ( c f . §5,12,18,19,21 f o r more

on t h i s theme).

... t

Take e.g. L = D

2

t

q ( x ) and l e t B = DZg+l t U ~ ( X ) D ~ t~ - ~

operator of minimal odd order commuting with L . One knows t h a t such a B ( i f i t e x i s t s ) is unique ( u p t o a constant m u l t i p l i e r ) U ~ ~ + ~ ( be X )t h e

and L , B s a t i s f y a n i r r e d u c i b l e polynomial equation B2 = P ( L ) where P ( A ) = n:gtl ( h - a k ) . This c h a r a c t e r i z e s the f i n i t e gap operators L via [ L , B l = 0. We assume P has no repeated f a c t o r s so t h e a f f i n e curve R : = P(A) (or RS) is nonsingular. This excludes s o l i t o n o r r a t i o n a l p o t e n t i a l s b u t includes periodic and quasiperiodic ones.

Then (following mainly [ E 2 ;F L Z I

here) t h e r e is a unique common s o l u t i o n J , ( x , p ) ( = BA function) of LJ, = XJ,,

BJ, = vJ/, w i t h p = (X,P) E R, determined via the conditions ( A ) For x E C is meromorphic on R w i t h poles independent o f x - t h e poles a r e to l i e i n some nonspecial d i v i s o r 6 = 6 l t.. . t 6 on R = t h e p r o j e c t i v e one 9 point completion R U { A II = m l (nonspecial means t h a t t h e only meromorphic functions on w i t h poles only i n & a r e constant - see [ SQ1 ] f o r d e t a i l s and near O,J,

A

6

c f 55 f o r f u r t h e r d i s c u s s i o n ) ; note here 6 i s given i n advance. ( 6 ) J , ( O , p ) = 1 . ( C ) Near m,J,(x,p)exp(-kx) = 1 t O ( 1 , k ) where k = JA. ( D ) For fixed p !--6,

x

-f

J , ( x , p ) i s holomorphic in a neighborhood ( N B H ) of 0.

Then f o r L a f i n i t e gap operator t h e common EF

of LJ, = A $ , BJ, =

J,

PJ,

can be

normalized t o be a BA function and conversely any BA function determines a unique f i n i t e gap operator L = D2 t q ( x ) . The poles 6 1 y . . . y 6 of 9 a r e in9 dependent o f x b u t t h e zeros p 1 ( x ) , . . . , p ( x ) vary w i t h x. Now l e t t ( 6 ) be 9 t h e l i n e bundle corresponding to t h e nonspecial d i v i s o r 6 ( c f . 55 and AppenA

dix 6 ) . This means t h a t f o r IUal a covering of R one picks fa i n Ua such (when 6 i E Ucl of c o u r s e ) . The t r a n s i t h a t fa has zeros only a t 6 1 y . . . , 6 g = f /f then determine t ( 6 ) ( i . e . f B = g f connects t i o n functions g aB B 0. a8 a excluding a l l 6 k a n d l e t s local c o o r d i n a t e s ) . Here one picks a NBH Um of

-

= 1 i f neif, = 1 . Next one defines a holomorphic l i n e bundle LA via g ' aB t h e r c1 or B = m while g',,(p) = exp(-xJA) in UcznUrn. Then LA has 0 Chern

c l a s s and corresponds to the d i v i s o r zeros

-

poles on R of any BA function

associated t o R ( i . e . any function defined by (A)-(0) above w i t h w i t h poss i b l y d i f f e r e n t 6 of the type i n d i c a t e d ) . Set now t, = t(&) BI t; (BA bundle)

LINE BUNDLES Then a (holomorphic ( A ) - ( D ) above.

41

s e c t i o n u o f C x d e f i n e s a BA f u n c t i o n w i t h p r o p e r t i e s

The t r a n s i t i o n f u n c t i o n s o f C x a r e h i n l o c a l coordinates u

= h

#

a)

Thus away from

a,

= fs/fa

at3

and ham = exp(-xdA)

a

(a,B

B at3 a' a g / f B = ua/fa = F ( d e f i n i t i o n ) i s meromorphic w i t h poles a t most i n 6 w h i l e

so F = ",/fa = umexp(xJA) ( i . e .

a t ->urn = exp(-xJA)ua/fa exp(xJA) near 6,

5

A t x = 0, F

m).

1 a t x = 0, and

f

t h e (unique) BA f u n c t i o n d e s c r i b e d above. p ( x ) i s t h e zero d i v i s o r o f

PI,

holomorphic

1 ( s i n c e 6 i s n o n s p e c i a l ) so F has p o l e s

e x p ( x d A ) ( l t ...) a t

Q

F

m;

hence F, d e f i n e d v i a u, i s

Note t h a t C i = C ( v ( x ) - 6 ) where

4 ; hence tx 2 C ( 6 ) B t ( v ( x ) - S ) and t h i s i m p l i e s

PI, C ( p ( x ) ) (v(x) t i e d spectrum i n Remark 4.2 c u l a t i o n s w i t h 1 i n e bundles and d i v i s o r s ) . Q

-

c f . Appendix B f o r c a l -

One sees t h a t a l l i n f o r m a t i o n about a f i n i t e gap p o t e n t i a l q ( x ) i s encoded ( o r i n Lx), and J/ can be used t o s t u d y connections between L and

i n J/(x,p)

t h e C. Neumann problem f o r example.

We c o n t i n u e now w i t h some m a t e r i a l from

[ E2;FL2] which p r o v i d e s an i n s t r u c t i v e use o f technique and l i n k s some i d e a s .

l i n e s through 0 i n discussed i n §3. R e c a l l ( c f . a l s o Appendix B ) t h a t Pn cn+l ; a l i n e has t h e form u = Aa , w i t h n o t a l l a = 0, g i v e n i n homogenej j j R e c a l l a l s o t h a t a d i v i s o r on a complex ous c o o r d i n a t e s as { a :...:an}. Q

m a n i f o l d M (e.g.

M

... t

sum A = P1 t

PI,

!c o m p l e x i f i e d -

- .. . -

-

t h i s i s discussed l a t e r ) i s a formal

where each P i'Q.J r e p r e s e n t s a potent i a l l o c a t i o n o f a p o l e o r zero o f a meromorphic f u n c t i o n on M. One w r i t e s L(A) =

Pn

Q,

Q,

a l l meromorphic f u n c t i o n s on M w i t h poles a t most i n P1,

zeros a t l e a s t i n Ql,...,Qm. L e t

fo,...,fr

t u a l l y one wants t o deal w i t h some v e r s i o n o f t h e map i,: {fo(m):

... : fPA(m)}

s o r s A,A'

..., Pn

and

be a b a s i s o f L(A) and even-

M

-+

Pr:

(which i s n o t defined i f a l l fi v a n i s h a t m ) .

m

+

Two d i v i -

on R a r e c a l l e d l i n e a r l y e q u i v a l e n t i f t h e r e i s a meromorphic

f u n c t i o n whose d i v i s o r o f zeros and poles i s A - A ' .

The s e t o f e q u i v a l e n c e

c l a s s e s o f d i v i s o r s o f degree d i s c a l l e d Picd(R^) ( P i c a r d v a r i e t y ) and t h e p o l e d i v i s o r s 6 = 61 t

($) etc.).

o f t h e BA f u n c t i o n range o v e r P i c g ( $ ) .

In

2 Jac(?) = JC(R), t h e complex Jacobi v a r i e t y ( u s u a l l y one t h i n k s

f a c t Picg(:) o f Pico(;)

... +A 6 g

2

J(i) and t h i s

Indeed l e t A1

i s e q u i v a l e n t v i a (4.8) below s i n c e 6-60 E Pico

,...,A 9 and B1 ,..., Bg

be a b a s i s o f homology ( c f .

w h i l e w . represents a = B.-B. = 0 and A:B. = 6 j 1 J i J ij J normalized b a s i s o f holomorphic d i f f e r e n t i a l s ( / w = 6ij; i n t e g r a l o v e r Ail. j

Appendix B ) w i t h Ai-A

42

ROBERT CARROLL

F i x '6

E

P i c g ( ? ) and l o o k a t t h e Abel map w i t h base p o i n t '6

t o J = Cg/L where L i s t h e p e r i o d l a t t i c e o f t h e w . (see J 15 and Appendix B f o r more on t h i s ) .

T h i s takes Picg(;)

REmARK 4.4 (3A AND CHE NElllilANN PR03CETl).

Now one augments t h e BA bundle

(we f o l l o w [ E2;FLZI b u t see [ KV1;SGl I f o r a general e x p o s i t i o n ) .

Take a

nonspecial d i v i s o r Z o f degree g w i t h Z + 6 nonspecial and l o o k a t L ( 6 ) I

L(Z) 3I

2 L(Z+6) B LA.

C;

P i c k g+l a p p r o p r i a t e holomorphic s e c t i o n s t o form -8

a (column) v e c t o r BA f u n c t i o n from 55 dimL(D) = g+l f o r

D

Q

J,

w i t h e n t r i e s J,.(x,p) (1 5 j 5 g+l - n o t e A J $ w i l l have poles i n Z+6. One t h e n

Z+6).

f i n d s commuting d i f f e r e n t i a l o p e r a t o r s L and B (now ( g t l ) X (g+1) m a t r i c e s ) A

o f which J, i s a common EF and [ L,Bl = 0 corresponds t o t h e Neumann system. We w i l l g i v e some i n d i c a t i o n o f t h i s and r e f e r t o [ E2;FLZl f o r d e t a i l s and F i r s t a c o n s t r u c t i o n i n [ E2] i s o f p a r t i c u l a r i n t e r e s t i n t h e

references.

geometry o f KdV as expounded i n [ E3;MC1,Zy6-1l

I.

Thus i f $ i s a BA f u n c -

6

t i o n c o r r e s p o n d i n g t o L(6) ILA = C x l e t p, For W(f,g)

= f'g

-

Then x has poles 6

E R-6-m

and

T:

( J , p ) -+ (A,-p).

-

f g ' d e f i n e X ( X , P ) = W($(x,p),J/(x,rp,))/(A(p)

+

p,

and behaves l i k e $(x,tp,)exp(kx)/k

at

X(P,)).

( k = 41);

m

a l s o t h i s procedure shows how t o c o n s t r u c t f u n c t i o n s h a v i n g such p r o p e r t i e s . Further s e t t i n g y(x,p)

= x(x,p)/$(x,~p,)

one sees t h a t T(x,p)

= y(x,p)/?(O, 2 p ) i s a g a i n a BA f u n c t i o n w i t h c o r r e s p o n d i n g p o t e n t i a l G(x) = q ( x ) + 2D X

I n o t h e r c o n t e x t s t h i s r e s u l t i s r e f e r r e d t o as a B i c k l u n d l o g J,(x,p,). o r Crum-Darboux t r a n s f o r m a t i o n , and goes back t o [ BN1 ] i n t h e p r e s e n t context.

Further i f 6

,r a r e

t h e p o l e d i v i s o r s f o r $;,

and i n f a c t any two d i v i s o r s 6 , 6 ' E transferences,

i . e . ~ ( 6 1 )= ~ ( 6 )+

Picg(;)

1;

A(P,.)

($4i s

Next f o r t h e Neumann problem r e c a l l t h e s i t u a t i o n ...,g+

1, w i t h 1;"x?

1, 1Y''x.i

then

A(F) =

A(6 ) + A(p,)

can be r e l a t e d by a t most g such

= 0, and q =

a transference a t p*).

y . + q x j = hjxj, j = 1, + x- 2. ( c f . 53). One

2 1,g + i h.x. J J

J J j J can express t h i s v i a BA f u n c t i o n s as f o l l o w s ( r e c a l l L$ = A$ e t c . ) . Let L 2 = Dt + q ( t ) ( t r e p l a c e s x here) be a f i n i t e gap o p e r a t o r and hly...,hg+l be

a subset o f t h e f i n i t e branch p o i n t s o f p2 = n(A-a.) J t h e r e e x i s t c o n s t a n t s P ~ , . . . , P ~ + ~ such t h a t x . ( t ) = J

( 1 5 j 5 2g+1). Jp.J,(t,A.)

J

J

Then

satisfies

BA FUNCTIONS

18”x?(t)

J dicated.

= 1.

43

D i f f e r e n t i a t i n g this equation we g e t the Neumann system in-

To see how this goes one looks a t q ( t , p ) =

a r e independent s o l u t i o n s of

3; +

(A,-P)). = p (recall T: (A,LI) cancel) and is i n v a r i a n t under

!b(t,Tp)

so t h a t

q,!b

qy = hy except a t branch points where

TP

T

$v i s meromorphic on R ( t h e exponentials so i t is a r a t i o n a l function o f h alone.

-

A(Pj(t)))/(X

-f

Write t h e r e f o r e g e n e r i c a l l y (4.9)

!b(t,p)q(t,p) =

q

(A

-

h(Sj))

(where r . ( t ) and 6 . represent t h e zeros and poles of !b). Let ft. = Yy (XJ J h ( S.))dA/p and h = p ( p ) / q + ’ (A-X .) so h has poles a t the X and zeros a t t h e J J j remaining branch points a l , . . . , a say. Therefore w = h$@ = n7 ( X - X ( p . ( t ) ) ) / 9 J +1 II? ( A - h . ) has simple poles a t X 1 y . . . ~ h g + l ~and m. Set p = Residue(hn) a t J j 2 A . and s i n c e $ = q a t branch points we get Residue w ~ . $ ( t , h . ) ‘ = x . with J J J J = 1 . Note a l s o Res(w) = -1 a t m. Since 1 Residues = 0 one has 1 p j $ ( t , h j ) ‘ t h a t the expansion of $qM2 = If (X-h(r .(t)))dA/na+’ (A-X .) in p a r t i a l fracJ t i o n s gives 1 a + ’ x ; d ~ / ( h - ~ . ) ( w i t h 1 xZJ= 1 ) . This expresses t h e use of e l j J l i p s o i d a l coordinates i n 13 in t h e language o f BA functions. Finally we i n d i c a t e the r o l e o f the vector BA map.

Let { a l , .

-

..,a 9 ,-I

U (Al

...,x g+l ) be a p a r t i t i o n o f branch p o i n t s o f the type just considered and take h a s above w i t h zeros Z + = 1 ; a i + m and poles a t t h e X Then t h e j’

.

functions X . ( t , p ) = Jp . W ( $ ( t , p ) ,!b(t,A j ) ) / h ( p ) (A (p)-A j ) ( j = 1 ,.. , g + l ) have J J poles in 6 + Z and behave l i k e J p . $ ( t , A . ) e x p ( k t ) = x.exp(kt) a t m ( i . e . they J J J ) T E cg+1 The X . define X ( t , p ) = ( X l y . . . X a r e s e c t i o n s o f C(6+Z) IBI g+l J or {X1: ... : X I E Pg. Further ( c f . [ E 2 ; F L Z I ) t h e r e e x i s t ( g + l ) X ( g + l ) 9 +1 a (j = 1 ,. . . , g + l ) and polynomimatrices L , B with e n t r i e s depending on x

’;.

jyxj.

a l l y on h such t h a t

=

BX, LX = - h A X , and L = [ B , L I, which i s equivalent Some f e a t u r e s of a l l t h i s become changed and c l a r i -

to t h e Neumann system. f i e d i n generalization and we r e f e r t o [ SG1-31 f o r this.

RERARK 4.5

(n0RE ON CHE KdO-NElIMANN CBNNECEZBN- W e rephrase t h e KdV-Neu-

mann connection now in s t i l l another way following [ BEA1;MCll ;MUlY2;PE2,4, 4 2g+1 51. Take again R a h y p e r e l l i p t i c curve corresponding to r 2 = A (”ai) = ; Pi on R = R - m w i t h P ( X ) ( a i d i s t i n c t ) . Given a nonspecial d i v i s o r 6 = 1

7

Pi = ( h i y ~ i ) define 3 polynomials (following Jacob?)

,

ROBERT CARROLL

44

MA)

(4.ioj

=

19 ( A - A ~ ) ; V ( A )

=

17 u i

U(A)/U~(X~)(A-A~)

a n d W ( A ) such t h a t UW t V2 = P ( A ) . Then ( c f . [MU1 I ) ( A ) The a f f i n e subv a r i e t y of C3gt1 given by c o e f f i c i e n t equations i n UW + V 2 = P w i t h U , W monic o f degree g , g t 1 and deg(V) 5 g-1 ( R ) is a 0 divisor

,

i s isomorphic t o Jac(R)/o (0C Jac

- cf [MU11 ; we discuss t h i s l a t e r ) ( B ) The expressions

define vector f i e l d s o n Jac(R) which f o r s u i t a b l e choices o f p will span t h e tangent space t o Jac(R).

The KdV vector f i e l d s (associated to t h e local

a r e t h e r e f o r e l i n e a r combinations o f any basis o f 6” w i t h P c o e f f i c i e n t s depending on t h e choices o f p’s and on P ( A ) (C) W i t h s u i t a b l e 2g+1 i n s e r t i o n of an x v a r i a b l e 25 s a t i s f i e s KdV where 5 = -1; h i t ai (cf. parameter A-’)

+I1

here Remark 4.1 and t h e formula f o r q ( x ) a f t e r ( 4 . 4 ) ) . Further connections t o t h e Neumann system a r i s e now a s follows ( c f . [ PEZ]). Give C2m the symplectic s t r u c t u r e based on w =

1 dqi

A d p i and f i x al

,...,

Then ( r e f e r r i n g t o Remarks 3.3-3.4 w i t h H k n, F k ( p , q ) , a k n, a k , . . . ) a,,,. 2 2 ( A ) The HamiltoniarsHk = q k t I j = k ( q k p j - q j p k ) / ( a -a ) commute and a r e a k j complete s e t preserving t h e spectrum o f t h e m X in matrix L ( q , p ) = d i a g ( a . ) J

+ q

Evolutions a r e given via L = [ L , B l f o r a s u i t a b l e B ( c f . 13). ( B ) 2 The system constrained t o M = q i = 1 , 1 q i p i = 0) i s t h e C . Neumann sys8 q.

€1

tem.

e

Now l e t M = M/group generated by involutions ( q i y p i )

-f

and

(-qi,-pi)

def i ne (4.12)

U(p,q) =

(IT q2i / ( A - a i ) T

w ( p , q ) = (1 t

(A-ai); V ( p , q )

I1m p2i / ( A - a i ) q

=

i(1;

P

qipi/(X-ai)n(h-ai) 1

(h-ai) I\

i s isomorphic t o Jac(i;)/e where R i s t h e Then each integral manifold o n smooth h y p e r e l l i p t i c curve of genus g = m-1 w i t h a f f i n e equation UW + V 2 = p2

( a s described above b u t now f o r U , V , W of (4.12)).

The constrained flows

GEOMETRY OF KdV

Hw

projected to

45

commute w i t h t h e KdV f l o w and a r e equal t o ~

~

where 6 ~

,

6

i s g i v e n i n (4.11) f o r (ak,o) = pk and Ck = 4in(ak-ai)-l (see [ MU2;MClll a, f o r more about a l l t h i s and c f . a l s o PE2,41). We do n o t t r y t o s o r t o u t here a l l o f t h e correspondences and connections i n t h e v a r i o u s p o i n t s o f view so f a r i n §3,4.

T h i s m a t e r i a l i s t o be regarded as " c l a s s i c a l " back-

ground f o r m a t e r i a l o f v a r i o u s s o r t s t o be presented l a t e r i n more d e t a i l ; a l s o v a r i o u s n a t u r a l e x t e n s i o n s r e f e r r e d t o e a r l i e r a r e n o t t r e a t e d here a t all.

Many d e t a i l s , h i s t o r i c a l m a t t e r s , p e r s p e c t i v e s o f v a r i o u s kinds, e t c .

have been o m i t t e d b u t we hope t h a t a good idea o f t h e i n t e r a c t i o n s and some connections emerges (never mind a l i t t l e mystery, which i s s t i l l p r e s e n t i n t h e whole theory, e.g.

i n t h e r e l a t i o n s between quantum g r a v i t y and KdV d i s -

cussed i n §16 f o l l o w i n g [ DG3;DJl ;MF5,6;WT5]).

RfFllARK 4.6

(PjEQIIECRg O f KdU @PCHE I LINE).

m a t e r i a l i n [ E3,4;MC1,6-11].

We s k e t c h here n e x t some o f t h e

L e t us d i g r e s s a moment t o r e c a l l t h a t t a u

f u n c t i o n s a r i s e i n v a r i o u s ways (many o f which a r e covered l a t e r ) .

They

come a l g e b r a i c a l l y from Kac-Moody (KM) a l g e b r a s and o r b i t s o f t h e vacuum, from Gra s sma nn ia n cons idera t io ns , d ir e c tl y from de t e r m i na n t c o n s t r u c t i o n s , v i a t h e t a f u n c t i o n s , v i a c o r r e l a t i o n f u n c t i o n s i n quantum f i e l d theory, e t c . One can argue t h a t t h e t a u f u n c t i o n i s t h e b a s i c o b j e c t i n much o f s o l i t o n t h e o r y and once i t i s i n t r o d u c e d v i a determinants i n t h e s c a t t e r i n g case as i n 911 one has a h o l d o n t h e a l g e b r a and geometry o f t h e s c a t t e r i n g s i t u a t i o n (we t h i n k here o f KdV s i t u a t i o n s w i t h no d i s c r e t e spectrum and g e n e r i c potentials q E

9

-

no f i n i t e gap s t r u c t u r e i s i n v o l v e d ) .

The o n l y o t h e r

c l a s s i c a l i n t r o d u c t i o n o f a l g e b r a i n such cases comes v i a a s y m p t o t i c s i n s u i t a b l e h a l f planes by means o f which a H a m i l t o n i a n h i e r a r c h y can be i s o l a t e d and t h e v a r i o u s KdV f l o w s generated; a c t u a l l y t h e r e a r e o t h e r ways o f g e t t i n g a t t h e H a m i l t o n i a n and b i h a m i l t o n i a n s t r u c t u r e as i n 56.

Now t h e r e

i s another way t o g e t a t t h e geometry o f t h e s c a t t e r i n g s i t u a t i o n based on [ E3,4;MC1,6-11

I.

The idea i s t o extend t h e f i n i t e band Riemann s u r f a c e

t h e o r y and t h e a s s o c i a t e d t h e t a f u n c t i o n s f o r p e r i o d i c problems t o general potentials q

E $

as above.

This i s a general program begun i n [ E3,4;MClY6-

11 ] ( c f . a l s o [ VEl,2]) w i t h a p p l i c a t i o n s t o o t h e r c l a s s e s o f p o t e n t i a l s as w e l l , and we o n l y deal w i t h t h e s c a t t e r i n g case here. o m i t t e d b u t we hope t o c a p t u r e some o f t h e f l a v o r .

Most d e t a i l s w i l l be

46

ROBERT CARROLL

We go f i r s t t o IMC11 1 which g i v e s a n o t h e r p e r s p e c t i v e f o r t h e f u n c t i o n s U, V o f Remark 4.5. Thus t a k e now a v e r s i o n o f P ( h ) o f Remark 4.5 i n t h e form L + X ) I 4 = D ( p ) where p = (X,cc), p E R based on X i # ( A ) = II(-X)T ( h i h)(Xi

-

t < A1 <

...

-

+

A + = 0 < A< A < h < -. Consider a d i v i s o r p 'L (p, ,..., , o 1 9 9 p ) o f degree g o n R; t h e p r o j e c t i o n p A(p.): R -+ P1 " r e c o r d s " p i n t h e g j J polynomial U = lI ( X ( p 1. ) - h ) w h i l e ~ ( p i) s recorded by t h e i n t e r p o l a t i o n po1g 2 lynomial v = U I ~ u ( p i ) / u ' ( p i ) ( ~ - ~ ( p i ) ) . One s e t s uw + V' = u a g a i n (as i n

=

m

-+

We r e c a l l t h a t a d i v i s o r p

(4.10)) and W w i l l have degree g + l . pg) = pl...p

+ 1.

( pl,...,

N

i s i n r e a l p o s i t i o n when A(p.) E fh;,Ai One n o r m a l i z e s t h e 9 A+. w = tiij and s e t s oi = (Ai,O) b a s i s w = (w l,...,w ) o f OFK so t h a t ':AJ2 9, j w i t h p -+ f: w = x ( p ) (modulo p e r i o d s ) p r o v i d i n g a 1-1 map from d i v i s o r s

'

1;

W r i t e ( c f . Remark 4.1)

p i n r e a l p o s i t i o n t o JR = Rg/LR ( c f . Remark 4.2).

1;

q(x) =

+ -

(A; + Xi)

pansion a(;)

21; A(pi)

as a f u n c t i o n on JR w i t h a t r i g o n o m e t r i c ex-

= I z g t(n)exp(Znin-;)

( r e c a l l here t h e P;(X)

-

c o n t r i v e d from s h i f t i n g t h e argument i n q w i t h no x ) .

pg

here

t,

= DxA(pi)

=

0;

) and ;(-) 9 (2v(pi)w/U'(pi))

(w1,...,w

w =

-I9 I

;(p)

-2u(pi)/U'(pi)

JR and A(pi)

E

Q

w/dk a t

-,

E

k = l/(-A)').

1( u oi ) , u g

Q ,

A(pi)

-+

p(x) reflected i n

JR, namely,

+

2 -+

Note X1:(p)

= Residuem(u+W)w/U' = G(m),

t h e o r i g i n o f such equations f o r X

Q

e t c . ) consider

( n o t e 2u(pi)/U'(pi)

these g i v e r i s e t o a m o t i o n p

a s t r a i g h t l i n e motion a t c o n s t a n t speed o f ;(p) Q

Q

F o l l o w i n g [MC2,41 (as sketched i n Remark 4.2,

XIA(pi) t h e e q u a t i o n s on 1 = Residue(U- (u+W) a t pi); (w

2

i n Remark 4.1 a r e

=

xG(-)

Xll;

and see [MC2,4] f o r

( c f . a l s o remarks a f t e r

(4.6)). This f l o w a c t i o n thus produces a q u a s i p e r i o d i c q (z,x) = l z g ( ( n ) P exp(Zain-;)exp(Zain.G(m)x) w i t h frequency module G(m).Zg. A c c o r d i n g l y one considers the H i l l ' s operator Q = Q b e i n g f i x e d , and A(pi(x))

x,

2,

-

-D 2 + qps w i t h q (?,x) P

ui(x). OP

for A < Consider now t h e Green's o p e r a t o r G XY (A) = (Q-A)-' XY

(Q

-A

= -0

2

+ 4).

U/2u ( w i t h

Then, f o r U,V,

positive real radical

X1x(pi)(~-h(pi))-'

-

Q

U/2u.

-

5

We

1.

Thus X U = U ' = 2V s i n c e U ' = 1 = 2UIa (u(pi)/U1(pi))(A-A(pi))-' = 2V. Next, a

2 2 V = p t a k e s t h e form L,(U')2 G(x,x,A)

+

(Ai - A . 1)

c f . here [MU1 ] and Remark 3 . 7 ) .

l i t t l e argument ( c f . [ M C l l ] ) shows t h a t XIV

Q

-1;

W d e s c r i b i n g t h e moving p ( x ) , Gxx(A) =

s k e t c h some o f t h e computation from [ M C l l

-UI;

a function o f

-

t,UU"

+

= V ' = - W + (p-X)U so t h a t UW t

(q-h)U2 = ic

2 ( c f . here (3.14)

F u r t h e r one can w r i t e t h e BA f u n c t i o n a s

SO ,G ,

SCATTERING

47

e(x,p) = U4exp(,d( u ( p ) / U )

(4.13)

a n d Qe = A(p)e.

Divide e(x,p) now by (U(0))’so

t h a t e(0,p)

= 1 and then e

may be characterized for A ( p ) < 0 as the solution of Qe = A(p)e with e ( x , p ) = o ( 1 ) as

x

+ -m

( o r as x

-f

m)

depending on whether # ( p ) has + (or - ) sign. A

e i s also the function of rational character on R-m having singularity exp (x/k) ( k l/(-A)’) a t 00, fixed poles pi ( i = l , . . . , g ) , and a necessarily equal number of (movable) roots p i ( x ) ( c f . Remark 4.3). Hence the normaliQ

zed e can be written as

(4.14)

e ( x , p ) = eXGp

[e(;+xk(m)-/,P

w ) e ( ~ ) / e ( ~ + x ~ ( m ) ) ew)]( ~ - ~ ~ 2 +

... a t m with 9 A i w m = 0 and o ( 0 , 0 ) (recall 3; + x G ( m ) = 1’ IPik)w modulo periods). This a l l leads t o a gen0; and q E R pl...p era1 addition formula ( c f . [ BY1;FIl;ISl;MCll I ) . If p 9 (R hyperelliptic curve based on ~ ( h ) then ) q pl . . . p E m r l . . . r with po2 9 9 Generally for tential qr = q p - 2Dxlog e ( x , q ’ ) ( 9 ’ = ( A , - u ) for q = (A,u)).

where w,=

(A:-X)dX/p(p)

= -dk/k

Q

Q

Q

a n d q = q1 . . . q there results pq E p+q = rl 2 g and q r = q p - 2Dxlog w(e(xyq,l1,. .,e(x,q;)).

p = pl...p m

9

.

...r g t

g points a t

RZmARK 4.7 (GE0mECRtJ 0F dCACCERZNb), We do n o t discuss directly the situation of a hyperelliptic curve of i n f i n i t e genus ( c f . [ MC3;MEl 1) b u t sketch now very briefly the scattering geometry outlined in [ E31. One takes a singular l i m i t of Hill I s curves by considering q E d with approximations of 2 q ( x + n P ) (period P). This q by compactly supported q with Q, = - D + gives r i s e t o a H i l l ’ s curve X, usually of i n f i n i t e genus and i s encoded as points on real ovalc(cf. ( 4 . 4 ) ) . Then l e t P t m a divisor p1+p2+ ... with

lz

(Q,

-f

Q ) t o get a continuum of real ovals, one for each k in [ 0,m),

say,cov-

ering doubly a vertical s l i t k X { e ; l e l 5 Cos-’ I s l l ( k ) l > (the extremities -1 +Cos I s l l ( k ) l being branch points). The divisor of Q takes the form of a cross section assigning one point t o each oval. Thus p ( k ) = (phase(f-(0,k) f+(O,k)/sll(k),sgn loglf,(O,k)/f-(O,k)l) where f,- a r e Jost solutions a n d a l l projection of p ( k ) t o the terminology corresponds t o 51y2. The f i r s t term Q

s l i t 181 5 COS-’ I s l l ( k ) l and the second term i s sheet information. The curve Xm i t s e l f i s somewhat mysterious of course. The Abel map takes the form

48

ROBERT CARROLL

(4.15)

PH(sZ1 (k)/isll

+

);4'

(k)) =

dCosS/(Cos 2 5

( l / r ) / (dk'/(k'-k)&p;>l'!

-

dSinS/(Cos2S

-

Isll ( k ) I 2 ) '

Isll ( k ' ) I 2 ) ~

L

Here sgn p ( k ) r e g u l a t e s t h e s i g n o f ( ) ' and o ( k ) = C o s - l I s l l ( k ) l . o(k) i s the divisor

'L

sZ1 and PH(s2,/is11)

-

p(k)

moves i n s t r a i g h t l i n e s a t conJR Isz1; I S z 1 1 = ( l - I s l l I 2 1% <

s t a n t speed under KdV f l o w s (PH = phase).

Q

c o n s t a n t ; 0 5 k < -1. C o n t i n u i n g w i t h i d e n t i f i c a t i o n s one has ( c f . §7,11) (4.16)

ik(x+Y)s

o ( s Z 1 ) = d e t I l + (1/2s)l:

e

and r e c a l l t h a t one i s t h i n k i n g here o f sll

as f i x e d so 1sZ11 = ( l - I s l l (2 )%

i s c o n s t a n t ( a l s o t h e r e a r e no bound s t a t e s ) . sll

dk} 21

R e c a l l t h a t t h e KdV f l o w s f i x

so t h e t y p i c a l i s o s p e c t r a l m a n i f o l d f o r KdV i s determined by s l l f i x e d

and corresponds t o t h e map q t e r PH(s2,/is11) d i n a t e o f JR.

-+

sZ1 w i t h I s z 1 I f i x e d .

Thus PH(sZ1 ), o r b e t -

a f t e r a s u i t a b l e adjustment, i s t h e n a t u r a l a d d i t i v e coorOne has a l s o ( c f . 511 where a s l i g h t l y d i f f e r e n t n o r m a l i z a -

t i o n i s used) (4.17)

"( k- k ' ) / (k+k ' ) )/o( sZ1e

f+ = s 11ei kXIo(s21eZi

f- = e- k x I 0 ( sZ1 e2

from which p ( k ) can be recovered. and e (x,k) sll

= exp(ikx)f-/sll

'

2ik'x

( k+k ' )/ ( k- k ' ) ) / e( s 21 e

One t h i n k s here o f e+(x,k)

w i t h e+(k) = f+(o,k)/sll

)I; Zik'x)) = e-jkxf+/sll

and e ( k ) = f (O,k)/

as BA f u n c t i o n s ( c f . I l l f o r more i n t h i s d i r e c t i o n ) .

The KdV i n v a r i a n t

m a n i f o l d f o r f i x e d s l l i s determined v i a " a d d i t i o n " o f d i v i s o r s where a d d i n g 2 a p o i n t p = ( k 2 < 0 , k ) t o t h e d i v i s o r o f Q = - 0 + q corresponds t o Q Q 2 2Dx l o g f +-( x , k ) o r e q u i v a l e n t l y sZ1 ( k ' ) sZ1 ( k ' ) ( ( k - k ' ) / ( k + k ' ) ) s g n p ( c f . -+

-

-+

here [ S A Z I ) .

The d i v i s o r p ( k )

-

o(k), 0 < k <

mines a l i n e bundle o f p a i r s e, E C + H2+-(H2+ patched a l o n g R v i a e;

+ sZ1e+ = s 11e - *

m,

a s s o c i a t e d t o sZ1 d e t e r -

= Hardy space as i n §11),

This i s g i v e n i n terms o f 0 v i a

(4.17) ( f o r o ( s Z l ) # 0 ) . Now we w i l l g i v e a few comments and formulas f o l l o w i n g [ E 3 1 b u t r e f e r t o t h a t paper f o r d e t a i l s and a d d i t i o n a l p h i l o s o p h y .

The b a s i c s c a t t e r i n g

49

DIVISORS

i n f o r m a t i o n i s d i s p l a y e d i n more d e t a i l i n §2,11.

One t a k e s sZ1 E

I , s2,

( 0 ) = -1 so s

( 0 ) = 0, s 11 ' ( 0 ) $: 0 p o s i t i v e imaginary, 0 < lsllI < 1 e:se11 where, and s21 = - r e x p ( i 8 ) w i t h 0 < r 5 1, r E I even, and 8 r e a l , odd, and smooth.

One t h i n k s o f s i t u a t i o n s w i t h no d i s c r e t e spectrum.

The KdV f l o w s 3 , etc.

move e a l o n g s t r a i g h t l i n e s a t c o n s t a n t speed (X18 = 2k, X38 = 4k w i t h s u i t a b l e normalization).

For p = (k2,+)

spec(Q) ( Q = -D2+q) w i t h f = f, d i v i s o r o f Q ) i s d e f i n e d v i a A': (k+k'))sgnPs21(k').

-

w i t h k2 < 0 t o t h e l e f t o f

c o r r e s p o n d i n g t o sgnp, a d d i t i o n ( o f p t o t h e Q

-f

Q

2

-

2 D x l o g f ( x , p ) and APsZl(k')

With n o t a t i o n as i n 111, O(sZl)

= ((k-k')/

= det((l+prsZ1)lH2+)

1

= -D 2 + q(xtnP) P as i n d i c a t e d above one f i n d s by a s t r a i g h t f o r w a r d c a l c u l a t i o n t h a t t h e d i s -

and Q = - D 2

2DxlogO(s21exp(2ikx)). 2

Now w o r k i n g from Q

c r i m i n a n t o f Q i s A(k) = l ~ ~ ~ \ - ~ C o +s (a (kkP) ) , w i t h gaps (Ai,Ai) d e t e r P mined by A ( k ) = (-l)n ( i . e . kP + a ( k ) = nn + Cos-'lsllI. For t h e b a s i c d i f -

2

wm i n Remark 4.6) one uses w = idA/(A -1)' and 2 a s p i k e map based on [MR2] i s employed. One s t u d i e s y2(p,k ) = 0 and, s e t f e r e n t i a l s o f second k i n d

(%

t i n g B = PH(f-(O,k)f+(O,k)/s,,) t h a t t h e d i v i s o r o f Q,

w i t h y = loglf,(O,k)/f-(O,k)l

i s p ( k ) = (B,sgny),

0 < k

i m.

i t turns out

The argument and c a l -

c u l a t i o n h e r e a r e somewhat c o m p l i c a t e d and we r e f e r t o [ E31. r a t h e r c o m p l i c a t e d procedure y i e l d s (4.15). t a i n e d by l i m i t i n g procedures p

-+ m

A similar

Another i n t e r e s t i n g formula ob-

i s t h e t r a c e formula q ( 0 ) = - ( 4 / n )

10" B(k)kdk which i s a l i m i t i n g form o f

1 q(nP)

R m R K 4.8 (GIUICS 0F PER10DIC SICUACI0W).

cm (An +

t

-

An 2A(pn)). 1 We make a few comments here =

1:

One takes a p o t e n t i a l v ( x ) = v(x-np), v smooth, compactly P supported, and nonnegative. The H i l l ' s o p e r a t o r Q = -D2 + v i s considered P A t + 1; 2 ~ . ( x ) ( c f . Remark 4.1). Setting as i n [MC3] and v ( x ) = A. + P i J J b m + + A:) one w r i t e s t h e n v ( x ) = A. + (-l)j+lluj(x)l(u.-A 1' U I P j = %(Aj + J b tz P m k m J 0 (Pi-Uj)/lAi U.1 'IAi U j l and L,I11j(J(-1) (uk-Ao)4/(pj-!Jk)Ul (pi-Uk)/ about [ V E Z ] .

1"

-

1;

+

b

21 -

-

-

1%

-

1. Set a = pj+l-p.,c.(x) = (n/aj)(U.(x)-pj), 6j = (n/ IAi-uk I Ai-uk 1 ?):. and choose t h e j i n d i c e s j (Jp ) J such t h a t p -+J k2 as p -+ m. Then 2a.)( ' J j J j as p -+ m, a Q 2nk/p, 6j 2, 6 ( k ) = C o s - l I T ( k ) l ( T = t r a n s m i s s i o n c o e f f i c i e n t ) , j s j t ( k , x ) , sj c ' , and -+

(4.18)

-f

v ( x ) = - ( 4 / n ) l m <(k,x)kdk 0

50

ROBERT CARROLL

( v = l i m v ). D e f i n e e(mod 2n) and $ b y Cose(k,x) P sgn Sine = s g n ( S ' ) , Cosh$(k,x) = CosC(k,x)/Cos6(k), Sine = -Sinh?/TanG e(-k,x)

= -SinS(k,x)/Sin&(k),

sgn$ = -sgnS'.

and one extends t o n e g a t i v e k v i a $(-k,x)

= -e(k,x).

Then = $(k,x) and

It f o l l o w s t h a t

+ e(k,x)

and - ( l / n ) J I ( $ ( k ' , x ) / ( k - k ' ) ) d k ' i s independent o f x (P,C

-

2kx :p ( k )

-

c ( k ) + &n (mod ZIT)

a r e phases o f t h e l e f t r e f l e c t i o n and t r a n s m i s s i o n

I t i s stated i n [VEZIthat t h i s solution o f the inverse

c o e f f i c i e n t s o f v).

problem ( ~ ~ 5 .-+ 6 v, v i a a n i n t e g r a l e q u a t i o n i n v o l v i n g $ , e l d i f f e r s

from t h e

GLM s o l u t i o n ( c e r t a i n l y t h e technique i s d i f f e r e n t ) .

5. FZNZtE Z0NE PBCENtZAU.

We have g i v e n an i n t r o d u c t i o n t o f i n i t e zone

p o t e n t i a l s , H i l l I s curves, e t c . i n §3,4 t o p r o v i d e m o t i v a t i o n , h e u r i s t i c s , e t c . and g i v e now a somewhat more s y s t e m a t i c approach based on [ DU1-3,5-8; KR1-9,ll-13 I ( c f

RECHRK 5.1

. a1 so [ BA1,Z;

(eWECA FUNCCIOW).

882; BH1; BL1 ,2; FB1; GN1; IT1 -4 We e x t r a c t f r o m

I).

DU1,81 i n p a r t i c u l a r and

t h e r e w i l l be some r e p e t i t i o n o f d e f i n i t i o n s and c o n s t r u c t i o n s from Appendix B and o t h e r s e c t i o n s ( t o emphasize ideas and make c o n s t a n t recourse t o a " d i c t i o n a r y " unnecessary).

For c o n s t r u c t i o n s w i t h i n t h i s s e c t i o n we a d o p t

n o t a t i o n which may d i f f e r s l i g h t l y from o t h e r s e c t i o n s b u t t h i s s h o u l d cause no s e r i o u s i r r i t a t i o n .

Another reason f o r d o i n g t h i s ( s i n c e we p o i n t o u t

t h e comparisons w i t h o t h e r n o t a t i o n ) i s t o make more a c c e s s i b l e papers w r i t t e n i n d i f f e r e n t "schools".

We m a i n l y i g n o r e q u e s t i o n s o f p r i o r i t y and r e -

f e r t o t h e Russian l i t e r a t u r e f o r d i s c u s s i o n o f such m a t t e r s .

) ) = B w i t h n e g a t i v e d e f i n i t e ReB jk (Riemann m a t r i c e s ) one a s s o c i a t e s a t h e t a f u n c t i o n @ ( z ) = O ( z l B ) = @ ( z ) =

Thus f o r a symmetric g X g m a t r i x ( ( B

czg ni(

exp(VBn,n) + (n,z)) Bn,n)) i s used, P

Q ,

( c f . here e.g.

Remark B5 where exp(Z.rri( n,z) +

B; s i n c e b o t h n o t a t i o n s seem t o be common we do n o t

try t o "regularize" matters).

The l a t t i c e o f p e r i o d s i s A(B) = C h i n + Bm,

n,m E Zg} and e v i d e n t l y @(z+Znin + Bm) = exp(-% Bm,d - ( m , z ) ) @ ( z ) .

0 func-

t i o n s w i t h " c h a r a c t e r i s t i c s " as i n (B.4) a r e d e f i n e d v i a O(a,B)(z) = &g e x p ( V B(n+a),n+a)

+ (z+Zaig,n+a)).

One w r i t e s TZg = C g / A g ( B )

(Tg i s used

RIEMANN SURFACES

51

1 a l s o i n o t h e r s e c t i o n s ) and ds2 = - ( l / Z n ) l (ReB)imdzkdzm g i v e s a G h l e r metr i c on TZg ( c f . [ GR1 I ) .

1 h 1J . .dzi

d‘;

B

j

=

1 $i

8

R e c a l l t h a t a H e r m i t i a n m e t r i c expressed v i a ds2 =

Ti

Ti

i s K a h l e r i f w = i / 2 1 $i A

i s d c l o s e d (dw =

0 ) . I f a ,B a r e r e a l c o o r d i n a t e s i n RZg = C g where z = 2aiB t Ba t h e n t h e k j c o r r e s p o n d i n g form i s n = ( - i / 4 ~ 1 ) 1(ReB)-’dz A dTk = dBk A dak. Such jk j t o r i a r e A b e l i a n v a r i e t i e s w i t h p r i n c i p a l p o l a r i z a t i o n and can be r e a l i z e d

1;

as a l g e b r a i c v a r i e t i e s i n p r o j e c t i v e spaces ( c f . Remarks 84, 85 and see [ GR 1I for details).

The s e t Hr o f g X g Riemann m a t r i c e s i s a l e f t Siegal h a l f

H as i n Remark B4 (i.

space ( c f . Remark B5) and Mg = Sp(2gyZ)/{+11 a c t s o n e. here B

equivalent). (or A

gb d)

E

One has t h e n TZg(B) = T Z g ( B ’ ) w i t h t h e same

a

2si(aB

-f

-t

2 s i b ) ( c B t h i d ) = B ’ f o r M = (:

dix

-

9 and A

n o t e ?1 a r e

= H /M 9 9 .g below) p a r a m e t r i z e s t h e abel i a n t o r i TZg w i t h p r i n c i p a l p o l a r i z a t i o n

9 We r e c a l l a l s o i n p a s s i n g t h a t t h e moduli space 111

a.

M

B) can be embedded i n A - H /Sp(2gYZ) ( i : Rl + A

4

o f curves ( c f . Appen-

b y T o r e l l i ’ s theorem 9 9 9 g Jac(C)) and t h e S c h o t t k y problem was t o c h a r a c t e r i z e i ( m ) C A (see 9 9 c f . [DO1 I f o r T o r e l l i i n schemes). 512 f o r r e s o l u t i o n v i a KP (C

-f

-

R E E M K 5.2

(RZElNNN SLIRFACW, DZFFERENCZAW, ABEL-JAC03Z MP, ECC.).

Again

w i t h some r e p e t i t i o n a (compact) RS o f genus g, say S, i s t o p o l o g i c a l l y a sphere w i t h g handles and one chooses a s t a n d a r d homology b a s i s ai-a

= bij The l i n e a r space o f holomorphic d i f -

b . = 0, a . - b . = 6 (see Appendix B ) . J i J ij f e r e n t i a l s (DFK) w = f ( z ) d z has dimension g and one chooses a b a s i s w i t h

I w.

= 2ai6 and B = J w i s t h e p e r i o d m a t r i x ( o r Riemann m a t r i x B ) . ak J jk 2,jk b~ j The corresponding T ( B ) = J ( S ) i s c a l l e d Jac(S). The Abel map A: S J(S) -f

(A(p) = (Ai(p))

Then i s d e f i n e d as usual v i a Ak(p) = JP wk f o r po f i x e d . P, i = l,...’n, t o be t h e zeros and

( A b e l ’ s theorem) i n o r d e r f o r (PiyQi),

p o l e s o f a meromorphic f u n c t i o n o n S i t i s necessary and s u f f i c i e n t t h a t

1;

1

- 1:

nipi A(Pk) A ( Q k ) = 0. A d i v i s o r D = (D) = nk where f o r meromorphic f, D = ( f ) =

1

i s d e f i n e d as usual w i t h deg

1;

d i v i s o r c o r r e s p o n d i n g t o zeros Pi and poles Q k ) .

pipi

- 1:

qkQk ( p r i n c i p a l

The map A extends t o t h e

a b e l i a n group o f d i v i s o r s and D i s p r i n c i p a l i f and o n l y i f deg(D) = 0 w i t h A(D) :0.

Two meromorphic d i f f e r e n t i a l s a r e l i n e a r l y e q u i v a l e n t ( d i f f e r by

a p r i n c i p a l d i v i s o r ) and produce an e q u i v a l e n c e c l a s s KS ( c a n o n i c a l c l a s s ) w i t h deg(KS) = 29-2. Pi’ Pi > 0).

One says D

D ’ i f D-D’ i s p o s i t i v e ( i . e .

One w r i t e s L(D) = I f ; ( f )

- D I and f o r D

20

D-D’

=

1 ni

i n general p o s i -

52

ROBERT CARROLL

5 g (note L(D) c o n t a i n s con2 g. I n p a r t i c u l a r f o r D = 1 f o r n 5 g; those p o i n t s P f o r which

t i o n (nonspecial d i v i s o r ) dimL(D) = l i f d e g ( D )

-

s t a n t s ) w i t h dimL(D) = deg(D)

g + 1 f o r deg(D)

nP, P i n general p o s i t i o n , dimL(nP) =

t h e r e e x i s t s a meromorphic f u n c t i o n w i t h a s i n g l e p o l e o f m u l t i p l i c i t y < g a r e c a l l e d W e i e r s t r a s s p o i n t s (e.g. t h e branch p o i n t s zk f o r w2 = Ifg+‘,zk) a r e W p o i n t s s i n c e f k = l / ( z - z k )

1 Pk,

The s p e c i a l d i v i s o r s D =

i s meromorphic w i t h a p o l e o f o r d e r 2 ) .

N = deg(D)

A

n,

(Ai))

i:S N + J(S):

(P

,,..., PN)

c1

N

+

+ 1,

g, w i t h dimL(D) > deg(D) - g

a r e p r e c i s e l y t h e c r i t i c a l p o i n t s o f t h e Abel map (SN A(Pj)

(i.e.

n,

symmetric product;

rank d i < 9).

-

( 1 / 2 ~ i ) r ~w + s~( pl ) l p w., j = l,...,g (Riemann Now l e t K . = 4 ( 2 a i + B ) J jj Po J c o n s t a n t s ) and K = ( K l,...,Kg). Let 5 = ( q 5 ) be a v e c t o r such t h a t 9 0 and f i x a p o i n t po E S. Then one knows t h a t F ( p ) = o(A(p) -5 K)

,...,

-

A

F has g zeros pl,...,p

1;

Aj(pk)

=

1;

J2wj

9 =

on S s o l v i n g t h e Jacobi i n v e r s i o n problem A.((pi))

cj;

further D =

m u t a t i o n ) t h e pk a r e u n i q u e l y determined. i s h i n g theorem.

1;

Thus f o r D =

has e x a c t l y g zeros pl,...,p

J

1 pk

=

i s nonspecial and ( u p t o a p e r T h i s i s c a l l e d t h e Riemann van-

pk nonspecial, F(p) = @ ( A ( p ) - A(D)

-

K)

o r i t vanishes i d e n t i c a l l y (see here 521 f o r

9 comments on t h i s and c f . [MU1 I f o r d e t a i l s ) .

Note a l s o 2K = -A(KS);

t a t i o n i s clumsy here b u t s o r t o f s t a n d a r d ( o f t e n

2

.\r

A,

(pi)

%

Next c o n s i d e r meromorphic d i f f e r e n t i a l s ( s i n g u l a r i t i e s a r e p o l e s ) .

t h e no-

pi). First

one c o n s i d e r s holomorphic d i f f e r e n t i a l s w (Abel i a n d i f f e r e n t i a l s o r t h e f i r s t k i n d o r DFK) normalized so t h a t l w dk

j

f o r general mero= 2 n i 1 S ~ ~ Now .

morphic d i f f e r e n t i a l s one can add s u i t a b l e holomorphic d i f f e r e n t i a l s so t h a t t h e a p e r i o d s a r e 0.

B a s i c meromorphic d i f f e r e n t i a l s a r e t h e n (A) A b e l i a n

d i f f e r e n t i a l s o f t h e second k i n d wn w i t h a p o l e o f o r d e r n+l a t Q and p r i n -

+

c i p a l p a r t wn = dz/zn+’ kind w

Q

... a t

Q

Q (6) Abelian d i f f e r e n t i a l s o f the t h i r d

have s i m p l e poles a t P,Q w i t h r e s i d u e s +1 r e s p e c t i v e l y .

PQ d i f f e r e n t i a l s one f i n d s

wi

n,

fi(z)dz

For such

= JpQ wi where

l. wn = (1/n!)D:-’fi(Q) and l w bc Q b; PQ a r e b a s i c holomorphic d i f f e r e n t i a l s as above. One notes a l s o

t h a t f o r any meromorphic d i f f e r e n t i a l

1 Respw

REARK 5 - 3 (LINE BUNDLES, PRImE F0Rm, ECC,). o f S ([ FY1 ;GU1,2,4;HW1

;MU1,3;V1]

= 0

( c over

poles).

We go now t o t h e prime form

a r e p a r t i c u l a r l y good r e f e r e n c e s here).

L e t KS be t h e canonical d i v i s o r c l a s s w i t h

L a l i n e bundle such t h a t c,(L)

PRIME FORM

53

-

1 ( c f . Remark 86 e.g. L Q, T*S). Recall H (a*) Q, isomorphy c l a s s e s of l i n e bundles. Pick a l i n e bundle X E H1(O*) such t h a t A 2 = L. This i s done = KS

s y s t e m a t i c a l l y in [ G U l 1 i n terms o f p r o j e c t i v e s t r u c t u r e s e t c . Thus proj e c t i v e s t r u c t u r e s subordinate t o a g i v e n complex s t r u c t u r e a r e uniquely de1 termined by t h e i r coordinate cohomology c l a s s e s $ E H (S,PL(Z,C)) (PL(2,C) = projective l i n e a r group on C ) . Take such a s t r u c t u r e based on T E H 1 (S, SL(2,C)) (note f o r T = (: :) E SL(2,C), = 4rT E PL(2,C) f o r g ( z ) = (az+b)/

+

( c z t d ) ) . Let (Uayza) be a coordinate c h a r t f o r this projective s t r u c t u r e 1 and Tab E Z (SL(2,C)) a r e p r e s e n t a t i v e f o r T. Then t h e za must s a t i s f y za ( p ) = (a,,z,(p) + baB ) / ( c a8 z B ( p ) + da8 ) while the canonical bundle i s de-1 2 f i n e d by t r a n s i t i o n functions k,,(p) = (dza/dzg) = ( c z , ( p ) t d ) (exrepresents X E "9 H (a*) with % X = L. e r c i s e ) . Hence AaB(p) = c z ( p ) + d a8 B

Now a s a model l e t S = P and l e t i : X -+L be t h e corresponding isomorphism; choose a s e c t i o n Jdz E r ( P l - - , X ) such that (Jdz) 2 = dz. This i s r a t h e r too b r i e f a discussion o f such X such t h a t X 2 = L b u t i t shows one p o i n t of view; the matter will be developed more extensively l a t e r . Now h e u r i s t i c a l l y consider the prime form E(z,r;) = (z-s)/JdzJds ( f o r P1 ). If z ' = l / z w i t h d z ' = - ( l / z ) d z one defines ddz' = i J d z / J z and E + ( z ' - r ; ' ) / J d z ' Jdr;' so E i s f i n i t e on P1 X P1 , E has c e r t a i n p r o p e r t i e s (e.g. vanishing f o r z = r ; ) which a r e useful and we now give a more general d e f i n i t i o n . For d e t a i l s see here [FYl;HWl;MU1,3] and we simply follow IDUS]. The presence o f h a l f d i f f e r e n t i a l s Jdz s h o u l d now be no mystery and i s expanded upon below and i n o t h e r s e c t i o n s . (5.1 )

The prime form will have t h e form

E ( P , Q ) = @(v)(A(P)- A ( Q ) ) / ( ~ w ~ ( P ) @ ~ ( ~ ) ( ~ ) ) ~ ( ~ w ~ ( Q ) @ ~ ( v ) ( O ) ) %

2

where v n, ( a , ~ E ) 4ZZg is any odd, nondegenerate, half period ( i . e . 4(a,B) $ 0 mod 2, gradO(v)(O) P 0 ) , o i n, a / a z i , and w . basic holomorphic d i f f e r e n J

Q,

t i a l s (cf.[AG3,4;MUl;SSl;Vl;FYl I and §17,19 f o r much more about a l l t h i s ) . Then E ( P , Q ) i s s i n g l e valued on S c u t along a i , b i and vanishes only f o r P = In going around a i , b i i t i s multiplied respectiveTy by 1 and exp(-%Bii P - fQ w i ) . D i f f e r e n t i a l s of t h i r d k i n d w PQ a r e given by w PQ( R ) = d(1og E ( R , P)/E(R,Q)) (R E S). The wn can be obtained via d i f f e r e n t i a t i n g w ( P , Q ) w i t h Q 1 respect t o Q, where w ( P , Q ) = w ( Q , P ) = (dZdwlogE(P,Q))dzdw; t h u s e.g. w Q ( P )

Q.

54

ROBERT CARROLL

I t w i l l be i n s t r u c t i v e l a t e r t o connect w i t h t h e n o t a t i o n o f 2ai I where ~ , E 6 Rg and T i s a sym[ FY1 I. Thus one w r i t e s e E Cg as ( € , a ) ( T 6 m e t r i c g X g m a t r i x w i t h Rer < 0. Then O T ( E ) ( ~ Q ) O(a,B)(z) w i t h 6 = a, T = w(P,Q)/dw.

= B, and

E

For

= 6.

t i f i e d w i t h Pic’s

r

n,

(Eai1,T)

one w r i t e s J ( S ) =

Cg/r and

= D i v o S / C ( f ) l as i n Remark B4, and v i a c1 w i t h t h e group

o f isomorphy c l a s s e s o f l i n e bundles on S as i n Remark B6. S, n,

L

n,

t h i s i s idenThen f o r a g i v e n 1

T*S can r e p r e s e n t t h e c a n o n i c a l l i n e bundle, l i n e bundles 5

d i v i s o r classes

morphic s e c t i o n

n,

%

E

H (a*)

p o i n t s o f J ( S ) ( e v e r y l i n e bundle has a n o n t r i v i a l mero-

d i v i s o r class).

We w i l l n o t develop t h e prime form E(P,

Q) f u r t h e r a t t h i s p o i n t and r e f e r t o [ FY;HWl;MU1,3]

and l a t e r s e c t i o n s i n

t h i s book f o r more m a t e r i a l i n t h i s d i r e c t i o n . L e t us r e c a l l t h a t f o r s u i t a b l e p l a n a r domains t h e Szego k e r n e l KS i s c h a r a c t e r i z e d b y t h e p r o p e r t y Z, K s ( z , f ) f ( c ) d s

= f ( z ) and has t h e p r o p e r t y under

conformal mapping KS(z,s)dz4dc’ = ?(w&)dw’dW’ ( c f J HW1 I ) . The Bergmann b k e r n e l K (z,;) n, @,(z)Tv(c) ( f o r a complete orthonormal s e t o f a n a l y t i c

1

has t h e p r o p e r t y !ID Kb(z,f)f(s)dxdy = f ( z ) and under conb “b Thus KS corresponds t o a % d i f f e r formal maps K (z,f)dzdg = K (w,G)dwdti. f u n c t i o n s @,)

e n t i a l which i n f a c t l o o k s l i k e 1/E l o c a l l y ( c f . [ FY1;HWl;NIl;Vl a r e e x p l i c i t formulas and we w i l l d i s c u s s t h i s l a t e r i n 519).

1 - there We do n o t

pursue t h i s f u r t h e r here.

(HALF ORDER DZFFERENCXACS7 SPIN SiCRUCellRU, ECC,).

RmARK 6.4

There a r e

v a r i o u s o t h e r approaches t o Jdz and we i n d i c a t e a p o p u l a r p h y s i c s p o i n t o f Thus l e t S be a RS o f genus

v i e w (see e.g. [AG3,4] and c f . a l s o [ A A l ; V l ] ) .

g w i t h Riemannian m e t r i c s a t i s f y i n g ( l o c a l l y ) ds2 = exp(Z@)dwdK = exp(2$) 2 w = x -ixl). One w r i t e s T*S = TfoS kJ (dx2 + dx ) = 26 dwdi (w = xo+ixl. 0 0 1 ww T*O,S f o r forms f(w,w)dw o r f(w,i)dG and takes a g a i n g (1,O) forms wi =

-

fi(w)dw

as a canonical homology b a s i s

Js e

1:

-

(4.&w

= 6ij

j

and

Ibt

w

j

= 0.. = 0 i j ji

-

1-n ) ) . Thus f o r e = ;i w i t h n holomorb, aL p h i c one o b t a i n s ( 1 / 2 i ) J e‘ A e = I m l 1 5 J e > 0 so ImSl > 0 and R E H as note

d e s i r e d (H

A n =

9

2,

( 4 - 0 Jbiq

1.8

a; bi Siega7 h a l f space as i n Appendix B).

g Consider now e.g. L a

c a n o n i c a l l i n e bundle as i n Remark 5.3 d e s c r i b e d b y g

aB

E H1(S,O*).

Then

o n Ua n U may n o t produce a c o c y c l e b u t i t can aB B 1 be a d j u s t e d by a p p r o p r i a t e c h o i c e o f faB= +1 ( f E C (S,Z2) w i t h h i B =

a n obvious c h o i c e haB = +Jg

55

S P I N STRUCTURES

( c f . Remark 87 f o r c o n s t r u c t i o n s and n o t e t h a t t h e c o c y c l e c o n d i t i o n ha8fa!3 1 d e t e r m i n i n g a l i n e bundle i s ga8gBY = c f . [ FT1;GUl 1). Then h ’ E H (S, gw 0 * ) i s s a i d t o d e s c r i b e a s p i n bundle and d i f f e r e n t s p i n bundles can be gen1 e r a t e d b y H (S,Z2) 2 ( Z 2 ) 2 g ( t h u s t h e r e a r e ZZg i n e q u i v a l e n t s p i n structures).

-

More e x p l i c i t l y one can d e s c r i b e s p i n differentials.

s t r u c t u r e s i n terms o f h a l f o r d e r

Indeed choose frames e x p ( z ) = e0 t i e ’ = exp($)dw f o r TTo

+

-

and exp(5) = eo i e l = exp(+)dG f o r Tt1 w i t h as above. Then a c r o s s Uei n U8 exp(2+a)ldwa/2 = exp(2$8)[dwg[ 2 so 2$a = 2$8 + 1ogldwg/dw,12. Hence e: = e

je

e

eei

= (dwa/dw )Idw /dwal and l e f t and r i g h t s p i n o r s J/+ E S’

8 6 form as J/+ = exp(fie/2)na8J/,8 where n

ture.

8’

HeFe one t h i n k s o f s p i n o r s

E

-

1 H (S,Z2)

trans-

determines t h e s p i n s t r u c -

J/, i n frames

(exp(z))’,

(exp(7))’

v i a U(1)

t r a n s i t i o n f u n c t i o n s o r v i a holomorphic l i n e bundles S’ w i t h t r a n s i t i o n f u n c t i o n s naB(dw,/dwg)*’ ( n o t e $+(e z )L , -- J/e(dw)’ cf. here a l s o [ HW1 I f o r

-

Thus s p i n o r s a r e l i k e

some d e t a i l s ) .

4 d i f f e r e n t i a l s dw”.

We n o t e a l s o

t h a t t h e r e i s a correspondence between s p i n s t r u c t u r e s and c e r t a i n t h e t a function indices

(z,n)

§19.

E

E

I).

Thus i n

0~ ( r eEs p . ~:1 ) mod 2, and t h e s e t i n J ( S ) where 0(€‘) €1

= 0 i s a symmetric t r a n s l a t e o f 0 = s e t where @ ( z , f i ) = 0 (0 c h a r a c t -

eristic). Zg-’(Zg

~

( c f . Remarks 84-85 and [ V1 :SS1

h a l f p o i n t s ( E ~ , E ~( )2 ~ ~ E , 2Zg).~ ~c a l l e d even

J(S) = JacS t h e r e a r e 2” ( r e s p . odd) i f

(a,b)

( E ,~ E ~ ) =

-

Each s p i n s t r u c t u r e

one o f t h e s e ZZg h a l f p o i n t s ; t h e r e a r e

%

1 ) odd and Zg-’(Zg t 1 ) even ones.

We w i l l say more on t h i s i n

L e t us remark t h a t t h e i n t e r a c t i o n between RS, t h e t a f u n c t i o n s , KdV

and KP, s p i n s t r u c t u r e s , s t r i n g s , conformal f i e l d t h e o r y ( C F T ) , e t c . i s made even more p r o f o u n d by t h e c l a s s i c a l connections o f t h e t a f u n c t i o n s w i t h numb e r t h e o r y ( c f . [MU1 ] f o r a t a s t e

-

o t h e r r e f e r e n c e s c o u l d become excessive).

R€iTkRK 5.5 (%A FUWCiTL0Wi).

We go n e x t t o t h e BA f u n c t i o n i n t h e s p i r i t o f

K r i t e v e r f o l l o w i n g [ DU1 ,81.

Take an a r b i t r a r y R S S o f genus g. P i c k a L e t D = P1 + + P m.

...

p o i n t Q and l o c a l v a r i a b l e l / k near Q so k ( Q ) =

g

Then t h e r e i s a ( u n i q u e up t o

a nonspecial p o s i t i v e d i v i s o r o f degree g.

c o n s t a n t m u l t i p l e s ) BA f u n c t i o n J/ c h a r a c t e r i z e d by t h e p r o p e r t i e s :

(1)

$(P) i s meromorphic on S e x c e p t a t Q where J / ( P ) e x p ( - q ( k ) ) i s a n a l y t i c ( 2 ) On S / Q J/ has p o l e s o n l y on D. (5.2)

J/(P) = e’:l

Ie(A(P)

I n f a c t $ has t h e form

-

A(D) + U

-

K)/@(A(P)

-

A(D)

-

K)l

be

56

ROBERT CARROLL

where Po C Q i s f i x e d and R i s a meromorphic d i f f e r e n t i a l o f second k i n d

(Q d q ( k ) a t Q) normalized v i a I R = 0, w i t h p e r i o d v e c t o r Uk = J n ( i , k = a, bk Note here t h a t U and t h e e x p o n e n t i a l t e r m a r e l i n k e d v i a R and 1,. ,g). 2 3 2 f o r say q ( k ) = k! + k y + k t one can n o r m a l i z e so t h a t J/ 'L exp(kx + k y +

..

k3t)(l +

1;

near Q ( c f . a l s o [ KR151); t h i s w i l l be used below.

ci/k')

Note

J/ i s unique up t o c o n s t a n t s s i n c e f o r general q ( k ) t h e n u l l d i v i s o r D o f $ w i l l be nonspecial o f degree g (use t h e Riemann v a n i s h i n g theorem) and i f r ( P ) i s another BA f u n c t i o n t h e n r ( P ) / J / ( P ) i s meromorphic w i t h poles i n DJI, hence c o n s t a n t .

To see t h a t

3

P P were chosen t h e n fp,

=

1;

A(P)

nkak t

+

1;

n

-f

J,

n and A ( P ) -+ A(P) + (6y wi). If P + 1 m j U j = lp R +e (M,U) and A(P)

lp 52 -t B

IP

m b we g e t

i s w e l l d e f i n e d n o t e t h a t i f a n o t h e r p a t h Po

s2

-t

1;;

j j PO 2niN t BM. But O(z+2niN + BM) = o ( z ) e x p ( - Y BM,M)

ti

Now t h e exp(q)(l

+

so t h e 0

which i s balanced by a m u l t i -

q u o t i e n t i n (5.2) i s m u l t i p l i e d by exp(-(M,U)) p l i e r exp(( M,U))

- (M,z))

y +

from t h e e x p o n e n t i a l term. above can be computed ( f o r m a l l y ) from (5.2)

= ti(x,!,t)

(and JI =

1;

Ci/k')) and t h i s a l s o l e a d s t o Lax o p e r a t o r s , e t c . Indeed, 2 s e t t i n g u = -2DxS1 w i t h w = 3C1DxS1 3DxS1 3D 5 one o b t a i n s 3x 2 e.g. f o r m a l l y ( - D + Dx + u)$ = O ( l / k ) e x p ( q ) and (-Dt + Dx + (3/2)uDx + w)J/ Y = O(l/k)exp(q). T h i s c h o i c e o f u,w makes t h e c o e f f i c i e n t s o f knexp(q) van3 i s h f o r n = 0,1,2,3. Then d e f i n i n g L and A v i a L = :D + u and A = Dx + (3/ given

2)uDx (-Dt

-

si,

+ w one f i n d s t h a t D J/ = LJI Y + A)$ s a t i s f y the c o n d i t i o n s

and Dt$

= x = 0 by uniqueness f o r B A ) .

AJ/ ( n o t e

9 = (-D

Y

+ L ) $ and

x

=

f o r a BA f u n c t i o n w i t h t h e same e s s e n t i a l

-

s i n g u l a r i t y and p o l e d i v i s o r s as J/ 9

=

-

b u t q e x p ( - q ) and x e x p ( - q )

0 a t Q so

I f one now w r i t e s o u t t h e c o m p a t i b i l i t y

-

c o n d i t i o n f o r D J/ = LJ/ and DtJ/ = A$ as Lt A = [A,L]one o b t a i n s t h e KP Y Y e q u a t i o n (3/4)u = Dx(ut 4(6uux + u x x x ) ) . T h i s s i m p l e d e r i v a t i o n o f KP YY v i a a RS and a BA f u n c t i o n i s s t r i k i n g and g i v e s a c l u e perhaps t o t h e deep

-

mathematical meaning o f such e q u a t i o n s . a r i s e from t h e BA f u n c t i o n .

The o p e r a t o r s L and A f o r example

A c t u a l l y t h e r e a r e many meanings a r i s i n g i n

v a r i o u s c o n t e x t s and v i a v a r i o u s d e r i v a t i o n s ; one would be premature and presumptious a t t h i s p o i n t t o s p e c i f y o n l y one meaning. Going f u r t h e r i n t h i s c o n t e x t c o n s i d e r (5.2) i n t h e form (5.3)

J/(x,y,t,P)

=

P 1 2 ePo (xn +yn

3 + "

)@(A(P)+xU+yV+tW+zo)/e(A(P)+zo)

BA FUNCTIONS where R1 dk 2 l b i R , W. = J Q

..., R 2

+

b; R

1

3

v a r i a b l e s x,y,t.

.... Q 3

%

..., Ui

= J R 1 , V. = b; 1 K, and a i s a n o r m a l i z i n g fac-

d(k3) +

zo = - A ( D ) It i s i m p o r t a n t t o n o t i c e e x p l i c i t l y t h a t t h e " f l o w "

( i = l,...,g),

t o r ( c f . Remark 5.5). of

d(k 2) +

%

57

... a r i s e

from R and i t s e x p o n e n t i a l a t Q.

The "meaning"

9 i n i t s dependence on t h e d i v i s o r D i s discussed i n [ KR151 and we w i l l

r e t u r n t o t h i s i n 112.

One sees t h a t i n t h e ( g e n e r i c ) f i n i t e zone s i t u a t i o n

o f genus g, e i g e n f u n c t i o n s

o f L f o r example must have g p o l e s on t h e as2 = 2DxlogO(xU + yV F u r t h e r u(x,y,t)

s o c i a t e d RS (which do n o t depend on x ) . t tW t

f i n d El

zo)

t c.

For t h i s one uses (5.3) and 9 = e x p ( q ) ( l +

1;

ci/ki)

to

v i a an expansion i n l / k ( c f . IDU1,81 and Remark 5.2).

9 goes as f o l l o w s ( c f . [ DU1,BI). L e t S 1 w i t h a s i n g l e p o l e a t Q so t h a t S +. CP i s a two-

Another i n t e r e s t i n g c a l c u l a t i o n from have a meromorphic X ( P )

sheeted c o v e r i n g w i t h S h y p e r e l l i p t i c and Q a branch p o i n t . near Q so t h a t k-'(P)

k-'(P)

= X(P)-'and

w r i t e $(x,y,t,P)

where cp i s a BA f u n c t i o n w i t h t h e same p o l e s as 3 l a r i t y cp * e x p ( k x + k t ) a t Q. Then one can r e w r i t e t,P)

9t = A9 becomes

9t =

Choose k - l = = exp(yA(P))cp(x,

9 and e s s e n t i a l singu9 = L9 as Lcp = A 9 and

Y Thus a s p e c t r a l parameter A m a g i c a l l y appears

&.

(based on k o f course), i s o s p e c t r a l i t y and a Lax p a i r emerge, Lt = IA,Ll from t h e c o m p a t a b i l i t y c o n d i t i o n , and one has t h e KdV e q u a t i o n ut = %(6uux

Ye mention n e x t t h e dual BA f u n c t i o n ( f o l l o w i n g [ DU81).

Given a ( n o n s p e c i a l )

d i v i s o r D o f degree g and a p o i n t Q E S w i t h l o c a l parameter k-'

near Q one

says a d i v i s o r D* o f degree g i s dual t o D ( r e l a t i v e t o Q) i f D + D* i s t h e 2 n u l l d i v i s o r o f a meromorphic d i f f e r e n t i a l dk + w d k / k + =n(with a

...

double p o l e a t Q). S.

Thus D

+ D*

-29

Q

KS where KS i s t h e c a n o n i c a l c l a s s o f

Given A(KS) = -2K ( c f . Remark 5.2) d u a l i t y o f d i v i s o r s i s e q u i v a l e n t t o

A(D*)

-

A(Q)

+

K = -(A(D)

t h e p o i n t s o f D and

9

Q

-

A ( Q ) + K ) . Given a BA f u n c t i o n 9 w i t h p o l e s a t 2 3 2 exp(kx + k y + k t ) ( l + E1/k + E2/k + ...) near Q

one d e f i n e s 9* w i t h poles a t t h e p o i n t s o f D* v i a (5.4)

9*(x,y,t,P)

where 51" = -El,

= e

2 3 -kx-k y - k t

2 E; = - E 2 + El

- Si

-

( 1 + E i l k + E;/k2 a,

...

Note here

+ @

...) =

99*n i s meromor-

p h i c w i t h a double p o l e a t Q ( t h e zeros o f SZ cancel t h e p o l e s o f

W*).The

58

ROBERT CARROLL

r e s i d u e s a t Q must t h e n be 0 g i v i n g 57 = -El. 9

1

= $

$*a,

x

(DY + L*)$*

(5.5)

= -A

+ 2w

+ A*)$*

= 0

A*

e t c . and (L* = L ,

= 0 ; (DT

To see t h i s n o t e t h e p r o p e r t i e s o f ( D

Y

-

One can proceed f u r t h e r w i t h (3/2)ux)

+ L*)$* and (Dt t A*)$*

as b e f o r e

and check t h e c o m p a t a b i l i t y t o g e t L* = D2 t u*, u* = 25* = -2+, e t c . I n 2 t ..., 1and w o r k i n g f r o m a p a r t i c u l a r one f i n d s t h a t 9 % dk + Jiu(x,y,t)dk/k h y p e r e l l i p t i c s u r f a c e t h i s p r o p e r t y i s u s e f u l i n s t u d y i n g t h e NLS and KP equations ( c f . [CRZy4;DU8l).

6- HMtZCC0NZAN CM0Ry FOR KdU-

The f a c t t h a t Hamil t o n i a n methods a p p l y t o

i n t e g r a b l e systems i s o f course i n t r i n s i c and i m p o r t a n t .

There i s an ex-

t e n s i v e H a m i l t o n i a n t r e a t m e n t o f NLS i n [ F21 and r e f e r e n c e s t h e r e supply a l s o many r e f e r e n c e s f o r KdV ( c f . 510 f o r some o f t h i s ) .

Since t h e r e a r e so

many papers i n t h i s d i r e c t i o n we w i l l o n l y r e f e r t o s p e c i f i c papers when t h e y come up i n t h e d i s c u s s i o n ( b u t see e s p e c i a l l y [ BGl;BV2;F4;G2,3;GDZ;LX4; OLl;Z5]). i n [ DS1

For KdV a n i c e s k e t c h o f t h e H a m i l t o n i a n f o r m u l a t i o n a l s o appears

1, which we f o l l o w a t t i m e s , a s i n d i c a t e d . There i s a d e t a i l e d s k e t c h o f

SEIURK 6.1 (REUZE3Il O F CtA%XCAC EtEtXANZC$).

d i f f e r e n t i a l geometry techniques and c l a s s i c a l Hamil t o n i a n mechanics i n Appendix A.

We s i m p l y r e c a l l here t h e e s s e n t i a l ideas i n a condensed form f o r 1 1 Thus f o r a m a n i f o l d M l e t C ( 9 ) be C f u n c t i o n s i n a NBH o f

i n s t a n t review. q (q

E

M ) and say f E g when ( a / a x i ) ( f - g )

= 0 a t q.

The e q u i v a l e n c e c l a s s e s

1

f' = d f and f d f i s a map C ( 9 ) T*M d e f i n i n g the cotangent 9 space T*M o r T*. (algebraic dual) T* has a b a s i s dxi ( o r dqi) and T = (T:)* 9 q q 9 1 has a b a s i s vi = a/axi ( o r a/%+) For f E C ( q ) w i t h ( d x a / a x . ) = 6ij. jy J

a r e denoted by

(6.1) For f :

(6.2)

-f

-f

v ( f ) = (v,df) = (1p.v

J j'

M

-f

N l e t f,:

(f,v)(g)

f o r g E C'(f(q)). v ( g f ) = f,v(g)

T M 9

-f

(af/axi)dx.)

1

Tf(q)N and f*:

= v(gof);

f*(dg)

=

T;(q)

= ( p

N

af/axj) jy

-f

J

J

T*M be d e f i n e d by q

d(gof)

Then o u r n o t a t i o n p r o v i d e s (*) ( f * ( d g ) , v ) = (dg,f,v)

= (Cp.a/ax.)(f)

so f, and f* a r e a d j o i n t maps.

=

(d(gef),v)

I f J C R i s now

=

CLASSICAL MECHANICS

59 A

M a c u r v e i n M y t h e n a t t E J, T;(J) h a s a g e n e r a t o r w i t h (A) D t h ( g ( t ) ) = C ( a h / a x i ) D t x i = g ’ ( t ^ ) d / d t and g*(d/dt) = go(;) E T

an i n t e r v a l w i t h g: J

-t

g(^t)

(h) = ( l(dxi/dt)a/axi)(h). 2 Now A T*M is g e n e r a t e d by dxi A d x . (i< j ) a n d f o r u = 1 a .id x i o n e d e f i n e s q J ( 0 ) du = 1 1 ( a a i / a x . ) d x . A dxi. F o r vi = a / a x i o n e h a s J J (dxiyvk) (dx.,v ) (6.3) (dxi A dx.)(vk.vm) = J k J ( d x v ) (dx.,v ) i’ m J m C o n s i d e r w2 = 1 d p i A d q i on T*M = U T*M. Here c o o r d i n a t e s are ( p , q ) , q E 1 9 2 M , a n d p % 1 p i d q i E T*M. Let w = p 1. d q i so dw’ = w Let f : T*M + M b e

.

1

q

q.

Let 5

E

T(T*M) so f&

T(M).

Thus l o c a l l y 5

t h e projection (p,q)

-+

1a j a / a p j

+

(note p

T* h e r e ) .

Then w ( 5 ) = 1 piBi = p(f,C) a n d f,S = 1 B j a / a q j . G i v e n 5 as a b o v e d e f i n e I : w1 + 5 w h e r e w’ i s d e t e r -

E

1 BJ. a / a q .J

E

=

1

? 2 5 5 mined v i a w ( n ) = w ( q y 5 ) . Thus f o r IT z : 1 y i a / a p i -+ s i a / a q .1 t o 5 2 = 1 a . d p t b . d q . o n e c o m p u t e s w (q,s) u s i n g ( 6 . 3 ) a n d c o m p a r e s J j J J t o o b t a i n a = B. and b Now l e t H b e a f u n c t i o n o n T*M 1 ( a H / a p i ) d pji t J( a H / a q i )jd q-i . - a jT*h i n k o f dH a s w 1 w i t h a i = aH/aqi

d e t e r m i n e w1 5 coefficients

so t h a t dH

=

a n d bi =

5

aH/api. (6.4)

I t follows t h a t I ( d H ) = I(W15 ) = 5 =

1 (-aH/aqj)3/apj

2 Thus d H ( 5 ) = w ( 5 , I d H ) a l s o d e t e r m i n e s I .

erates a flow g t : T*M t

(aH/apj)a/aqj

+

The v e c t o r f i e l d IdH o n T*M gen-

T*M via t h e s o l u t i o n o f O D E i . ( t ) = C i ( t ) ( 5 % I d H ) . 2’ t 2 Thus Dtg q l t = O = I d H ( q ) . One knows t h a t (gt)*w2 = w w h e r e ( g )*w ( 5 , n ) = t 2 t F i n a l l y we d e f i n e P o i s s o n b r a c k e t s v i a w (g*t,g,q). (6.5)

-+

t {F,H}(x) = DtF(gH(x)ltZo

= dF(1dH) =

2

w (IdH’IdF)

= -{H,F)(x)

T h i s y i e l d s i n p a r t i c u l a r { q k y H l = aH/ap k a n d { p k HI = -aH/aqk so H a m i l t o n ’ s equations are

REnARK 6-2 ($mE mT?C&IQmAt. nECHOD$), We w i l l f o l l o w t h e n o t a t i o n o f [ DS1 I i and thus i f we denote t h e ( q , p ) c o o r d i n a t e s by Y’ ( Y = qi f o r 1 5 i 5 N and yNti = p i ) t h e n {y’,yv) = E” where E” % ( - I0 o1 ) ( n o t e { q i , p j l = Sij w i t h

ROBERT CARROLL

60

t q q . l = Ipi,p.l = 0). Hence f o r a a/ay' and A,B i' J J u one has (summing on repeated i n d i c e s ) I A ( y ) , B ( y ) ? avB.

Thus ( 6 . 6 ) becomes

'i

= ty',HI

=

EpVaVH

two f u n c t i o n s on T*M

a A{y',ywlayB 1J.

= {y',y"l.

a A€'" u

and t h i s form g e n e r a l i z e s t o

systems w i t h noncanonical c o o r d i n a t e s i n t h e form ( 6 ) f"

=

y'

= f u v ( y ) a v H where

T h i s would correspond t o a Poisson b r a c k e t ( + ) { A ( y ) , B ( y ) ? =

a A(y)f'"(y)avB(y)

u

and t h i s i s a p p a r e n t l y a n a t u r a l s i t u a t i o n i n systems Note t h a t i n o r d e r t o d e s c r i b e Poisson brac-

with constraints. (cf. [ D S l I ) .

k e t s one must have fuv(y) = - f V u ( y ) , an i n v e r s e f fAv= 6" = fvxfAu, and ( m ) a f 1J.

u

v i d e s a Jacobi i d e n t i t y Iy',{y

(y) + ,y

A

a f

1J.V

should e x i s t w i t h f

Pi

( y ) +"aVfAp(y) = 0 (which pro-

xv,{y',y

l l + A{ y

11 +

{y",{yh,yul? = 0 ) .

It

i s i n t h i s s p i r i t t h a t one w r i t e s f o r a c o n t i n u o u s system

(6.7)

{u(x),u(y)l

= f(x,y);

=

Iu(x),Hl

=

dyf(x,y)6H/6u(y);

a,

{A(u),B(u)I

=

1 1 dxdy(aA/su(x))f(x,y)(6B/6u(y)) -02

where 6H/Su a r e v a r i a t i o n a l ( o r f u n c t i o n a l ) d e r i v a t i v e s d e f i n e d below ( c f . a l s o Appendix A ) .

rf

3 2 Thus one t h i n k s o f f u n c t i o n a l s H(u) = dx(u /3! +(ux) ) f o r example w i t h 2 6H/6u(x) = 4u + uxx. Frechet d e r i v a t i v e s g i v e one approach t o t h i s o f

-

Thus t h i n k o f H as a map o f some v e c t o r space

course.

E

-f

C ( l e t E be a

Banach space f o r convenience i n d e f i n i t i o n h e r e ) and say t h a t H i s Frechet

( F ) d i f f e r e n t i a b l e a t uo i f t h e r e e x i s t s a l i n e a r map T E E' such t h a t JH(uo+U)

(6.8) as Uull

-f

-

-

H(uo)

0 (u,uo E E ) .

TuI/IIuII

+

0

We r e f e r t o e.g. [ C l ] f o r d i s c u s s i o n o f t h i s concept

and i t s r e l a t i o n s t o Gateaux d i f f e r e n t i a b i l i t y ; here we s i m p l y n o t e t h a t one i s basically looking a t

i n v a r i o u s n o t a t i o n s , where

(

,

)

denotes l i n e a r d u a l i t y .

I n general o f

course norms a r e n o t expected and we s i m p l y make formal c a l c u l a t i o n s o f t h e type ( u

?r

(u,ul ,...,u(n))

say)

FUNCTIONAL METHODS

61

where $ p l a y s t h e r o l e o f a t e s t f u n c t i o n , v a n i s h i n g s u i t a b l y , a l o n g w i t h i t s derivatives, a t

km.

c e p t we n o t e that$,u

€

To be c o n s i s t e n t w i t h t h e Frechet d e r i v a t i v e conE and,of

c o u r s e , d e r i v a t i v e s o f ah/au(k) p l a y e d o f f

a g a i n s t v a r i o u s d e r i v a t i v e s o f $,must v a n i s h a t

km.

I n p r a c t i c e t h e nota-

t i o n is abused i n v a r i o u s ways and as l o n g as such abuse i s c o n s i s t e n t and l e a d s t o c o r r e c t formulas we w i l l use i t when expedient. writes f o r u(y) = (6.11

LI G(x-y)u(x)dx,

Thus e.g.

one

(**) 6 u ( y ) / 6 u ( x ) = 6 ( x - y ) and ( c f . [ D S l D

= )DEH(u(x) + E ~ ( x - Y ) ) I ~ = ~ ~H(U(X))/~U(Y

To see how (6.11) works l o o k a t H(u) = (b'(X-y),f(x))

if

3

dx(u /3!

- +uf)

above ( r e c a l l

= -f'(y)).

Now w r i t e t h e KdV e q u a t i o n wt = wwx t

* Dx(%w

REmARK 6.3 (APPLXAtSZ0W &0 Kh4). wxxx i n t h e form wt =

wxx ) = Dx(6H/6w(x)). we want t w ( x ) , w ( y ) l = ax6(x-y).

For t h i s t o have t h e

+

form wt = tw(x),Hl (6.7) f o r f ( x , y )

= ax6(x-y) one o b t a i n s ( a x 6 ( x - y ) =

To see t h i s n o t e t h a t i n

- a 6(x-y)) Y

m

lI dx(6A/6w(x))ax(6B/6w(x))

=

-1;

dx(6B/6w(x))ax(6A/6w(x))

%lf d y € ( ~ A / ~ W ()Yay() 6B/ 6 w h ) Hence f o r tw(x),Hl we g e t ( f r o m (*)

-

=

( 6B/6w(Y) ay( &A/ 6w(y) )f

rl

above) {w(x),HI = dy(bw(x)/W(y))a Y 6(x-y)a (GH/Sw(y))dy = ax(6H/6w(x)) as d e s i r e d . C l e a r l y t h e Y n o t a t i o n c o u l d l e a d t o problems ( w i t h p r o d u c t s o f 6 f u n c t i o n s e t c . ) b u t i t (SH/Gw(y)) =

if

i s p r o d u c t i v e i f one i s c a r e f u l .

We l e a v e as an e x e r c i s e ( c f . [ DS1 I ) t h e

c o n f i r m a t i o n t h a t t h e b r a c k e t o f (6.12) s a t i s f i e s t h e Jacobi i d e n t i t y . I n f a c t t h e n a t u r a l a l g e b r a o f observables A,B t o which (6.12) would be app l i e d does n o t v a n i s h a s x -+ + b u t r a t h e r has l i m i t i n g values (*A) F, = 1i m X - t ~ m (6F/6u(x)). To a d j u s t t , I t o t h i s one d e f i n e s i n [ F41 a niore genera l bracket

62

ROBERT CARROLL

(6.13)

(A(w),B(w))

(sA/sw(Y))ay(sB/sw(y))

+ %(A-G+

( c f . [ F4;BVZI f o r d i s c u s s i o n and d e f i n i t i o n s here).

-

dY ( s ~ / s w ( Y ) ) a y ( s A / s w ( Y I )

=

-

-

A+G-)

we w i l l n o t d w e l l on t h i s

One can use t h e p r e v i o u s machinery, a t l e a s t h e u r i s t i c a l l y , t o de-

v e l o p a g r e a t deal o f i n f o r m a t i o n and s t r u c t u r e . We n o t e f i r s t what seems t o be p e c u l i a r , b u t t u r n s o u t t o be q u i t e general, namely t h a t KdV has a second H a m i l t o n i a n s t r u c t u r e . 2 w dx, and choose a Poisson b r a c k e t

%[I

(6.14)

{w(x),w(Y)>,

=

(a

3

Thus c o n s i d e r H2(w) =

+ (1/3)(axw(x) + w(x)ax)s(x-y)

(check t h a t t h i s i s a n t i s y m m e t r i c ) .

Then one has

We w i l l d i s c u s s b i h a m i l t o n i a n s t r u c t u r e s a l i t t l e more i n t h i s s e c t i o n b u t r e f e r f o r t h i s and t o p i c s such as r e c u r s i o n o p e r a t o r s , symmetries, e t c . t o

AN1,2;BT1,2;CC1,2;0C2;DS1-3;OVl ;FO2,11 ,12,5;FC1 ;GE4;GC1 ;OEl ;OV1 ;KH1,2;KU5; KN1-4;LIl ,2;MHl,Z;SN2-4;SPPl I ( o t h e r r e f e r e n c e s a r e g i v e n a t t h e end o f 16).

[

R?3tARK 6.4

Now go back t o Remark 2.4 f o r example

(HMZL:&B)NZAN Ei:&RI.ICEl.lRE).

where conserved q u a n t i t i e s c

~ ( o~r

if+ @,dx) ~

and

iI vZndx a r i s e .

L e t us

work here w i t h t h e vn ( f o l l o w i n g [ DS1 I)and s e t p n = ( - l ) n 3 v 2 n w i t h (*@) Hn =

m

H2 =

if

-

(exercise). gree

2

4. DS1 I.

...

( a 3 + ( a w + ~ a ) / 3 ) ( s H ~ _ ~ / s w ( x =) ) a(sHn/sw(x))

(6.16)

in

m

Thus from Remark 2.4 we can w r i t e Ho = 3 i z wdx, H1 = % w dx,lm 3 2 Then i t i s e a s i l y checked t h a t f o r n = O , l , . . dx(w /6 %(wx) ),

Lm pndx.

Note t h a t t h e Hn a r e homogeneous o f degree n + l where a x has de-

To see t h a t (6.16) h o l d s , f o r a l l n i n d u c t i o n i s used h e u r i s t i c a l l y Thus assume (6.16) i s t r u e f o r 0 5 n 5 m.

served, atHm = 0 = IHm,H211(where { A l t e r n a t i v e l y atHm = {Hm,H112

with

Then s i n c e Hm i s con-

, I, r e f e r s t o axs(x-y) = ~w(x),w(y)l,). I , ) 2 as i n (6.14).

Using t h e l a t t e r

HAMILTONI A N STRUCTURE formula one o b t a i n s atHm = We r e f e r t o [DSl

3 w(y)Fm(y)dy where Fm = (ay + ( l / 3 ) ( a y w ( y )

+

Some h e u r i s t i c argument t h e n l e a d s t o (6.16) f o r a l l

w(y)ay))(6Hm/bw(y)).

n.

m

-i,

63

1 for details.

A n e a t e r c o n s t r u c t i o n would produce a l l

t h e Hn u s i n g t h e r e c u r s i o n o p e r a t o r ( c f [ OL21). I n any case g i v e n (6.16)

f o r a l l n one can produce many i n t e r e s t i n g conse-

quences. We c o n t i n u e t o f o l l o w [ DS1 ,2 I here. Thus f i r s t one computes ( s e t 3 E = a + (1/3)(aw + w a ) so t h a t E(6Hn-1/6w) = a ( 6 H n / 6 w ) { H n ,H m 1 l =

(6.17)

( 6Hn-1 /6w

1

=

1:

-1:

dx( 6Hn-1 / 6 w ) E( 6Hm/6w) =

and t h i s equals {Hn-l,Hm+l}l. {Hm,HnIl

=

dx(6Hm/6w(x))ax(6Hn/6w(x))

1:

-1:

dx(6Hm/6w)E

dx( 6Hn-1 / 6 w ax( 6Hmt1 / S w )

By r e p e a t i n g t h i s one f i n d s t h a t {HnyHmIl

= 0 ( t h e Hn a r e s a i d t o be i n i n v o l u t i o n when IHn,HmI

=

=

0 for all

n,m).

S i m i l a r l y {Hn,Hm12

H112.

Thus one has an i n f i n i t e number o f conserved q u a n t i t i e s Hn which i s

= 0 and one o b t a i n s atHn

sometimes equated w i t h t h e idea o f i n t e g r a b i l i t y .

=

{HnyH211 = 0 = { H n y

There a r e v a r i o u s con-

c e p t u a l problems w i t h i n t e g r a b i T i t y o r complete i n t e g r a b i l i t y i n a n i n f i n i t e number o f dimensions which we o m i t f o r t h e moment.

The presence o f such

q u a n t i t i e s however does p r o v i d e an i n f i n i t e h i e r a r c h y o f f l o w s (each a s s o c i a t e d w i t h a d i f f e r e n t " t i m e " v a r i a b l e tn, where an (6.18)

a / a t n below).

anw = {w(x),Hn12 = E(6Hn/6w(x)) = ~w(x),Hnt1Il

Thus

= a(SHntl/6w(x))

G e n e r a l l y t h e r e seems t o be no " p h y s i c a l " i n t e r p r e t a t i o n f o r t h e h i g h e r tn ( c f . however §16,20).

REWRK 6.5 (ACCIBN ANGLE VARIABLE$),

We w i l l show t h a t c e r t a i n s p e c t r a l

data determine a c t i o n a n g l e v a r i a b l e s ( c f . [AB5;DS1 ;N02;21,6,71).

The r e -

l a t e d c a l c u l a t i o n s a r e a l s o o f g r e a t use i n many o t h e r a s p e c t s o f t h e t h e o r y ( c f . [C1,3,5;TB2]).

Thus, c o n t i n u i n g w i t h t h e w form f o r KdV, L = a 2 + w/6 2 + k J/

( c f . remarks b e f o r e (2.14) f o r c o n v e r s i o n s ) we c o n s i d e r IL" + (1/6)wIL

= 0 and t h i n k o f f,, T,R, e t c . as f u n c t i o n s o f w. Consider (see [AB5;DS1; N021, a l l o f whom do t h i s a l i t t l e d i f f e r e n t l y ) Gf+(k,S)/Gw(x) = F, which -

evidently w i l l satisfy

64

ROBERT CARROLL

One can imagine c o n d i t i o n s on 6w(x) f o r example t o j u s t i f y s e t t i n g

W(-m)

=

0 ( c f . [ DS1 I ) and t o handle t h e h a l f l i n e i n t e g r a t i o n one has r e c o u r s e t o h a l f l i n e 6 f u n c t i o n s ( c f . [Cly24;DSl;F41).

Thus n o t e t h a t F o u r i e r Cosine

transform t h e o r y o n [ 0 , ~ ) g i v e s S+(x) = ( 2 / ~ ) 1 ; Coskxdk where 6, i s a h a l f

On t h e o t h e r hand, on t h e f u l l l i n e 6 ( x ) = (1/2~)[:

l i n e 6 function. (ikx)dk =

IT)$

$(5)6,(g)d<

=

exp

Coskxdk = %6+(x) i s n a t u r a l , w i t h (*6) 1; $ ( c ) 6 ( c ) d g = S i m i l a r arguments a p p l y t o

+$(O).

w i t h 6 ( x ) = %6-(x) and %g(O)

=

i:

y i e l d i n g 6-(x)

(-w,O],

0

$ ( n ) 6 ( n ) d n = %J', $(n)6-(n)dn.

Now a p p l y

0

(*6) f o r 6 - ( n ) t o $(TI) = f ( x + n ) ( x f i x e d ) so t h a t + f ( x ) = j m G(n)f(n+x)dn = 1 -w ' 6(y-x)f(y)dy.

(1/12)f+f-.

c12 = 1 / T = W(f+,f (6.21 )

W i t h these conventions t h e n (6.20) g i v e s W ( f + , 6 f - / 6 w )

An analogous procedure g i v e s W ( 6 f + / 6 w , f ) / 2 i k we o b t a i n ( n o t e a l s o cZ2 =

6c 1 2/ 6 w = ( 1 / 1 2 i k ) f + f - ;

) = (1/12)f+f

-W(f+(k,x),f-(-k,x))/Zik)

6cZ2/6w = - ( 1 / 1 2 i k ) f + ( k , x ) f - ( - k , x )

( t h e l a t t e r formula i n v o l v i n g s i m i l a r c a l c u l a t i o n s ) . Now a c e r t a i n amount o f c a l c u l a t i o n , u s i n g f o r m u l a s such as i.rrs(k) = l i m e x p ( i k x ) / k as x

-+ m y

a l l o w s us t o c a l c u l a t e Poisson b r a c k e t s as i n (6.12)

f o r a = c12 and b = cZ2.

Such c a l c u l a t i o n s a r e t e d i o u s ( b u t i n s t r u c t i v e )

and s i n c e t h e y appear i n several e a s i l y a c c e s s i b l e places (e.g. AB5;NE41) we w i l l o m i t them here. -(144k/s)log a(k)

[ C1;DSl;

Then, t h e outcome i s , s e t t i n g P ( k ) =

and Q ( k ) = a r g b ( k ) ,

F u r t h e r f o r B . and mR one can w r i t e ( n o t e cll = 1/c22 when c12 = 0 ) b ( i a j ) -1 = b . = - i m R /d and some c a l c u l a t i o n shows t h a t J (6.23)

Pn = 1448,; 2

Qn = %loglb,l

a l s o form p a r t o f a c a n o n i c a l s e t f o r KdV, i . e .

=

and from

IQn,Qm) = 0 = {Pn,Pm) and

ACTION ANGLE VARIABLES

65

fjmn (see

IQn,PmJ =

l o g a ( k ' ) l = 0.

[ DS1;NOPl f o r c a l c u l a t i o n s ) . Note a l s o t h a t { l o g a ( k ) , These v a r i a b l e s (P,Pk) and (Q,Qk) w i l l be a c t i o n a n g l e v a r i -

a b l e s as d i s c u s s e d i n Remark A31. Now go back t o (2.12)-(2.13)

where t h e n o t a t i o n must be s l i g h t l y a t e r e d f o r w/6). Note T = l / a and a mom-

t h e p r e s e n t c o n t e x t ( s i n c e we go from - u

-+

e n t s w i l l g i v e (2.13) a s b e f o r e w i t h (6.24)

czn+l

-

= - ( i / 7 2 ) j m P(k)kZn-'dk

c1

( 2 / 2 n + l ) ( i / 1 2 ) 2n+l N p'i 2n+l)

0

For (2.12) we have t o t h i n k o f a R i c c a t i e q u a t i o n t

2 ($I)

+ w/6 = 0 and (2.11)

6

A

(i replacing

4)

$"

+ 2ik;'

i n t h e form

A

2

...

- w /36, ). One sees t h a t t h e ( $0 = 0, $, = -w/6, $2 = wx/6, $3 = -wx,/6 n +1 same p o l y n o m i a l s a r i s e and i n f a c t one can w r i t e $n = (-i) f o r vn a s i n (2.17) ( r e c a l l t h e vn a r e d e r i v e d i n a d i f f e r e n t c o n t e x t b u t t h e p o l y A

nomials i n v o l v e d a r e t h e same).

Then, m o t i v a t e d b y t h e e a r l i e r d i s c u s s i o n

o f Hn r e l a t e d t o vnr s e t

and comparison o f terms i n t h e a s y m p t o t i c s e r i e s (2.12)-(2.13) placed by (6.27)

in) gives c

" *$*n+ldx and

~ = ~-(1/2i)2n+1 + ~

Hn = 18(2i)2n+1C2n+,

(1/2n+1)(1/6)

( w i t h $n r e -

Lm

= (-1) 2

2n-'/c

+

p(k)kZn-'dk

2n-1 N p++4

1,

I n p a r t i c u l a r t h e KdV H a m i l t o n i a n H = H2 has t h e form (6.28)

H = 81" P(k)k3dk + ( 1 / 5 ) ( 1 / 6 ) 0

ll Pm5 / 2

3 N

I t f o l l o w s t h a t t h e t i m e e v o l u t i o n o f t h e canonical v a r i a b l e s i n v o l v e s

(6.29)

P = {P,H)

( 1 , 1 as i n (6.12)).

= 0;

Pn = {Pn,HI

= 0; Q

= {Q,H)

3

= 8k ; Qn = 46,

Note here t h a t c Z 2 = b and i n t h e w t h e o r y

3

3 = 8ik b

66

ROBERT CARROLL

3

( n o t - 8 i k b as i n ( 2 . 2 ) . = -u,

t

-+

-t, and t h e n q

To see t h i s s i m p l y n o t e b e f o r e (2.14) t h a t u -+

w/6.

+

q

The change i n t i m e d i r e c t i o n accounts f o r

t h e t s i g n s i n (6.29), which a r e seen t h e n t o be e q u i v a l e n t t o (2.2). F i n a l l y l e t us w r i t e down a s y m p l e c t i c form a p p r o p r i a t e t o t h e a c t i o n - a n g l e One expects something l i k e n = L21jz Sw(x) A Sw(y) dxdy m d u l o

variables.

adjustments a s i n [ F4;BV2]

( c f . (6.13)).

Thus i n canonical c o o r d i n a t e s one

expects (6.30)

=

1"0

S Q A SPdk +

c1N

SQk

A 6Pk

b u t we w i l l n o t do a n y t h i n g w i t h t h i s f o r t h e moment.

R m R K 6-6 ($gmPL€cCZc ICRLICCLIRES). We f o l l o w [ CIS1 I h e r e m a i n l y ( c f . also 1 AF1 ;CP1 ;DS2-4;ANl ;OK1,2 I) i n o r d e r t o i n t r o d u c e r e c u r s i o n o p e r a t o r s as w e l l ( c f . a l s o [ F2;MHl;OL2,3;N02;SHAl;ST1,41). Thus t h i n k i n g momentarily o f f i n i t e dimensional m a n i f o l d s and u s i n g t h e n o t a t i o n o f Remark 6.2 we t h i n k o f = V'f

f(V,W)

iff(V,W)

Ww and f nondegenerate

'V

= 0 f o r a l l V,

%

f

1-1"

n o n s i n g u l a r ( f i s nondegenerate

W fixed, implies W = 0).

t h e i n v e r s e m a t r i x and s e t { p ( y ) , q ( y ) ) construction).

Set

and a two form f = 4fvv(y)dy'l A dy".

v e c t o r f i e l d s V = V'a/ayV =

We w r i t e a g a i n f''

= f'"(y)al-lpavq

for

(= -{q(y),p(y)l

by

H a m i l t o n i a n v e c t o r f i e l d s i n t h e s p i r i t o f Remark 6.1 can be

formed from a s c a l a r f u n c t i o n p v i a X = f'"aPpav so t h a t I p ( y ) , q ( y ) l = P f ( X ,X ) ( c f . (6.5) where I d p X etc.). Note as an e x e r c i s e here t h a t i f P q P 2 df = 0 t h e n t h e Jacobi i d e n t i t y f o r I , 1 i s s a t i s f i e d (and s i n c e dw = 2 1 We n o t e d w = 0 t h i s a u t o m a t i c a l l y h o l d s f o r t h e s i t u a t i o n o f Remark 6.1). 0 -I one a l s o t h a t f'"(y) = iy",y") and when fVv= ( I o ) w i t h f'" =

(-!

speaks o f c a n o n i c a l c o o r d i n a t e s .

Recall a l s o t h a t the flow corresponding t o

i s iP= {y',p(y)) = f""ayp(y). P Now we have seen t h a t KdV i n v o l v e s two H a m i l t o n i a n s t r u c t u r e s and t h i s p r o -

x

duces a number o f i n t e r e s t i n g developments. Li = o ' ( y ) y p 1-1

Remark A27).

-

Hi(y)

(i= 0 , l )

Thus suppose two Lagrangians

d e s c r i b e t h e same Hamilton e q u a t i o n s ( c f .

The corresponding Lagrange equations a r e t h e n f v v ( y ) y

V

=

all

a oo(y) - avoo(y) ( e x e r c i s e ) . 1-11 fllv 1-1 v T h i s g i v e s r i s e t o two s y m p l e c t i c forms f = +fv,dy' A dy tt and F = +FUvdy 1-1

Ho(y) and F,,"y"

dyV.

=

a H ( y ) where e.g.

They a r e c l o s e d ( i . e .

df

'W

= 0 = dF

=

'V

) s i n c e e.g.

A

t h e Bianchi i d e n t i t y

SYMPL ECTIC STRUCTURE

ahfvu

t

67

t a V f x u = 0 and nondegenerate ( e x e r c i s e ) .

allfvA

Poisson b r a c k e t s as above (e.g.

{p(y),q(y)lo

Thus one has two

= fP"(y)avp(y)avq(y))

and t h e

e q u i v a l e n t Hamil t o n e q u a t i o n s become

>'

(6.31)

= {y',Holo

j'

= fv"(y)aVHo(y);

= {y',H1I1

= F'"(y)aVH1(y)

Since t h e e q u a t i o n s i n (6.31) d e s c r i b e t h e same dynamical system we must have (*+) f"(y)a,Ho(y) say (6.16).

and t h i s i s an a b s t r a c t v e r s i o n o f

= F'"(y)avH,(y)

Now c o n s i d e r

(6.32)

= F,,,(y)f"~y);

S:(Y)

U;(Y)

= al.l(F"AaAHl(~))

a!'l

=

Now s i n c e a H 1.10 j" we must have c o m p a t a b i l i t y i n t h e f o r m a a H = a a H which when V" l l v o V l J O

(fpxFhvcould a l s o have been used b u t (6.32) i s s u f f i c i e n t ) . = f

w r i t t e n o u t becomes ( u s i n g t h e B i a n c h i i d e n t i t y ) (6.33)

f,,a"iA

.A

0 = al(f,,y = -aAfll"Y

.A

Note here Uv(y) = !J

-

-

ally

a y" !J

.A

.A

av(fPxY

1

=

f A v + avY f A P

(a,fl

+

--

avfA,)iA

-DtflJv

-

= au(fVAaa,Ho) = av(FVXaAH1).

+

fvAa,,y

+

U"fAP

A

UVfA"

.A

-

h

Thus, w i t h a s i m i l a r

c a l c u l a t i o n f o r Fvv ( u s i n g c o m p a t a b i l i t y ) one has

x

(6.34)

Dtfllv

x

x

+ U,,fAl;

= -Uufxv

DtFM,,

A

= -UPFA" + UvFhll

One can a l s o d e r i v e e q u a t i o n s o f t h e form D t f P w = fuxU; n o t need t h i s .

-

f v A U y b u t we do

F u r t h e r one can e a s i l y c a l c u l a t e now, u s i n g i n f o r m a t i o n a l -

ready given, t h a t

DtS;

(6.35) (exercise

-

= ;)S'

-

UAS"; V A

DtS = [ S , U l

c f . here Remark 2.4 b u t n o t e t h i s i s a m a t r i x , n o t an o p e r a t o r ;

we come back t o t h i s a t t h e end o f Remark 6.6). I t w i l l be o f i n t e r e s t l a t e r ( c f . a9,lO) Kn = ( l / n ) T r S n w i t h KO = 1 o g l d e t S l . (S),[

S,U])

D K = 0. t n

= 0 (exercise

-

t o c o n s i d e r conserved q u a n t i t i e s

From t h e r e l a t i o n Tr(P(S)DtS) = T r ( P

P(S) denotes a polynomial i n S ) one f i n d s t h a t

Note t h i s i s a f i n i t e dimensional system ( S i s 2N X 2N) so n o t

ROBERT CARROLL

68

a l l K, a r e independent. Assume one has N f u n c t i o n a l l y independent K, and consider t h e Nijenhuis ( t o r s i o n ) t e n s o r (6.36)

= - N L ~=

N$:

A s'aa h s'$ - sia,s: - sy(aast - aBsa)

Using (6.35) one f i n d s ( e x e r c i s e ) DtN:B

(6.37)

-

=

In p a r t i c u l a r i f N'

a$

+ U'N6'

U:NYB

+ UYNtB

Aa

( y ) = 0 a t t = 0 i t remains zero.

Now 'N

a$

i s related t o

t h e Kn via (6.38)

la

N&(S n-'

-

= S:(a,S~)(S"')~

=

A sp)(s B n-l

?(a

)B

-

(sn))"a

(S")% A a Sh B = (l/n)StaxTrSn

x = SaaxKn

-

-

x s" - (sn);(aa$ - aBsa)

(l/n+l)aaTrSntl

aaKn+l

We w r i t e this out ( a s i n [ DS1 1) s i n c e t r a c e (and determinant) c a l c u l a t i o n s will be of i n t e r e s t a t many places i n s o l i t o n theory ( s e e e.g. 1 7 ) . Hence i f Nu

z 0 we o b t a i n

(6.39)

S:axKn

aB

=

aaKn+l;

f'"avKn

= F'"aVKntl

and involution follows imnediately from (6.40)

{

K

~ = ~ fuva,,K,,,avKn K ~ ~ =~ F'va~KmavKn+l

=

f'"apKm-lavKn+l

which equals { K m - l y K n + l l o ( c f . ( 6 . 1 7 ) ) . This leads t o { K m y K n I O = 0 and s i milar c a l c u l a t i o n s imply {Km,Kn)l = 0 for a l l m,n. Thus N g g = 0 ensures i n t e g r a b i l i t y and one g e t s a hierarchy o f flows a, 'L a / a t n (6.41)

11

anyp = { Y ~ , K , , I =~ fPvav Kn = {y , K ~ +l1~ = F1l"avKn+l

w i t h Lax equations (6.42)

(un);

=

a;ls

=

1 S,Un] where

au(fVxaxKn)= a u ( F V X a x K n + l )

RECURSION OPERATOR

lW). Now

go t o KdV w i t h F’” ‘L a and f’” ’L M = I n a standard p h y s i c s n o t a t i o n one can w r i t e h e u r i s -

%EElARK 6.7 (APPL1CAC10M CO 3

a + (1/3)(aw + wa). t i c a l l y F(x,y)

69

= ( y l a l x ) = ax6(x-y) and f ( x , y )

We w i l l

= ( y l M l x ) = Mx6(x-y).

want t o deal now w i t h a - l which i n some sense r e p r e s e n t s i n t e g r a t i o n (e.g. sometimes 3 - l

‘L

-c

or

l E ).

Here we w i l l w r i t e f o l l o w i n g

= (y1a-l I x ) = E(X-y) where E ( Z ) =

+ for

I

DS1 1, F-l (x,y)

z > 0 and = -4 for z < 0.

Then a x €

( x - y ) = 6 ( x - y ) = - a ~ ( x - y )and o b v i o u s l y Y (6.43)

1:

dzF(x,z)F-l(z,y)

’

For S” one t a k e s now ( x (6.44)

S(x,y)

-t

=

lf dzF-’(x,z)F(z,y)

y; o p e r a t o r a c t i o n as i n

fE d z F - l ( x , z ) f ( z , y )

=

= Ma-’

= 6(x-y)

(XI‘ ly))

a 2 + (2/3)w + (1/3) a w 1 a - l

=

T h i s w i l l be t h e famous r e c u r s i o n o p e r a t o r t o which r e f e r e n c e s a r e g ven a t t h e end o f t h e s e c t i o n (one must be c a r e f u l n o t t o equate aw w i t h wx i n t h e Note t h a t S i s n o n l o c a l because o f t h e 2 - l term. Now 3 w r i t e KdV as w = K ( w ) = a w + w ( a w ) so t h a t o u r second o p e r a t o r U ‘L U” w i l l operator calculus).

a3 + aw ( c f . ( 6 . 3 2 ) ) .

be U = Sw/sw = 6K/6w =

(6.35) corresponds t o DtS = we compute SU = ( a 2

+

(2/3)waw

dz(S(x,z)U(z,y)

+ (2/3)w + (1/3)(aw)aq1)(a

3

-

U(x,z)S(z,y)) 5

t

aw)

=

(x

-+

y ) and

3

a + a w + (2/3)wa

+ (1/3)awa2 + (1/3)waw w i t h s i m i l a r c a l c u l a t i o n s f o r US.

lows t h a t DtS = [ U , S ]

REXIARK 6.8

if

1-I

Then t h e Lax t y p e e q u a t i o n 3

It f o l -

i n o p e r a t o r form.

(REFERENCE$).

We had i n t e n d e d o r i g i n a l l y t o add a separate sec-

t i o n on symmetries, b i h a m i l t o n i a n s t r u c t u r e , R m a t r i c e s , r e c u r s i o n o p e r a t o r s , Back1 und t r a n s f o r m a t i o n s , f a c t o r i z a t i o n , e t c b u t o t h e r m a t e r i a l was w r i t t e n f i r s t and we r a n o u t o f space. Hence we l i s t here a number o f i m p o r t a n t r e ferences i n t h e s e d i r e c t i o n s , w i t h a p o l o g i e s f o r omissions. c i t e [ AD1 ;AJ1 ;AN1-4;AKl ;AF1 ;A11 ;DDS;BE8;BOLl ;BULZ;BV2;CL1-3;CPl CC1 ,2; DA1,2; DE1-3; DV1-6; DN1 ,2 ; OZ1; DS1-4; FC1-5; FJ1-4; F02,5

Thus l e t

US

;BT2,4-6;CGl;

,7,10-12,21;

F2,4;

-

FD1-5; GEl -4; GC1; GI1 5; LI1-9; LZ1; LV1; KU5; KD1-3; KN1-15;MH1-4 ;MOK1-3;MORl ;MV1 ,2; OK1,Z;GDl ,2;GP1 ;I01 ;KH1 iOL1-3;ORl ,2;RQ1 ;ROG1 ;MAR1 ;SG5;SHA1 iSPP1-3;SAMl ;STAl; TU1-4 ;SN1- 5; ST1,4 ;VS1; WA3 ;WC2 ;W4,8

,9; WN1- 3 1.

ROBERT CARROLL

70

7. VECERFIILIUANC IREUHHBDS F0R KdV AND KP; C N fLlNCCZ0W. We embark now on an a n a l y t i c a l procedure to c o n s t r u c t tau functions by determinant methods. Tau functions a r e a principal theme we wish to develop and they have many o r i gins and meanings. In the f i n i t e zone s i t u a t i o n of 53-5 we encountered thet a functions a n d these a r e e s s e n t i a l l y the tau functions f o r such s i t u a t i o n s . We w i l l s e e how such functions a r i s e a n a l y t i c a l l y i n 57, a l g e b r a i c a l l y in §8, and geometrically i n 511,13 ( a l s o via physics i n §14-22). We use the tau function theme a s a unifying d i r e c t i v e i n much of what follows. For b i l i n ear and elementary determinant methods we l i s t a few references here most relevant t o this s e c t i o n ; thus see e.g. [CF1;C13,17-19;D1,8;DY1,2;FNl;HI1-3; KQ1- 3; KW1; HB1 ;MY1 ;MT1; NM1;Ol ,2;OHl; P1-5;WBll ( o t h e r references a r e 1 i s t e d l a t e r ) . We will begin w i t h a medley o f themes on KP t o set the s t a g e and then go t o determinant methods. A t various stages here and i n 58,12,13,11 f o r example we will s e e many f a s c i n a t i n g i n t e r p l a y s between determinant methods, Wronskian techniques, t h e general Sat0 theory involving i n f i n i t e Grassmannians, Lie algebra techniques, e t c . This i s a l s o e s p e c i a l l y pronounced i n Chapter 3 o n physics.

RB'IARK 7.1 (HEURL$CLC$ ON KP). The basic s c a l a r hierarchy i s t h e KP (Kadomt s e v - P e t v i a s v i l i ) s i t u a t i o n , and KdV i s i n f a c t a reduction o f t h i s (explained l a t e r ) . We give here some formal background based on [ D1 ;MS1,21 f o r t h e hierarchy and will t r e a t other aspects l a t e r . A more rigorous treatment of some o f t h i s following [MSl-71 appears i n 612. Thus we consider ( c f . Remark 2.4) a Lax operator of t h e form ( a % a x = a l ) (7.1

We s e e t h a t one can take a. = 0 s i n c e e'Le-' = a t (ao - a l $ ) t ... a n d equations Lw = Aw, anw = Bnw can be transformed into s i m i l a r equations via w = vexp(-$) w i t h a. = ale. One takes now x = ( x l , x z y . . . ) (sometimes x 'L x l ) and 8-l 'L or f o r example. For P = 1 p.8' we w r i t e P, = 1 pja j J ( j 2 0 ) and P- = P - P,. We w r i t e as i n Remark 2.4 (*) Bn = Ly a n d Bn will evidently have t h e form Bn = a n t 1:-*b .a'. We take a n = a / a x n and connJ s i d e r the wave functions w s a t i s f y i n g

Lt

(7.2)

Lw

=

kw;

-4

W

a nw

=

B w

n

KP HIERARCHY

71

Compatabil i t y o f t h e s e equations r e q u i r e s anL = [ B n y L l ; anBm

(7.3) and (7.3),

-

amBn = [ B n ,Bm ]

g i v e s t h e KP e q u a t i o n s .

s i n c e (7.3),

implies

[ B n y B m l = (anLm

-

We n o t e t h a t (7.3)b f o l l o w s from (7.3),

(A)

anLm = [ Bn,L

-

[BnsBml)+ =

amLn

m I f o r any m and hence anBm - amBn m ([ B ,L 1 - 1 BmsLnl - [ B ,B I ) + = ([Bn -

nn m 2 LnyB - L m l ) + = 0 ( s i n c e e.g. Ln Bn = L - ) . Note i f one assumes L = B2 i s a m 5 2 2 2 d i f f e r e n t i a l o p e r a t o r ( L 2 = L+) t h e n w i t h L2 = a + b and L w = k w we have

-

t h e KdV s i t u a t i o n ( f o r k

ik).

-+

For wave f u n c t i o n s i t i s n a t u r a l t o c o n s i d e r c(k,x)

=

1;

( 0 )

w

i(w(k,x)exp(C(k,x))

where

T h i s i s formal here; l a t e r we w i l l i n t r o d u c e convergence -2

xnkn.

We l o o k f o r a "gauge" o p e r a t o r P = 1

i d e a s as needed.

+ wla-'

+ w2a

+

...

a

-t

such t h a t

L P =Pa ; anp = -L;P

(7.4)

To see t h a t t h i s e x i s t s f o r m a l l y s i m p l y w r i t e o u t LP =Pa w i t h say L =

1,

m

~ - ~ a -and j P = 1 +

1"1 p-J. a - j

t o get

For f i x e d v t h e r i g h t s i d e i n v o l v e s o n l y p ~ 1 , p ~ 2 y . . . y p w + 1

( v = -1,-Z,...).

and one can s o l v e r e c u r s i v e l y f o r pv by t a k i n g i n d e f i n i t e i n t e g r a l s o n l y once a t each s t e p ( c f . [MSl with [C,a] c-l p-l

I).

There i s o f course some a m b i g u i t y s i n c e PC

= 0 i s a l s o a gauge o p e r a t o r when P i s ( n o t e (PC)a(PC)-'

= Pap-')

b u t (7.4),

below i n Lemma 7.2.

i s confirmed.

The a m b i g u i t y w i l l be discussed

Now one t h i n k s o f a-'exp(kx)

= ( l / k ) e x p ( k x ) which usu-

a l l y d o e s n ' t make any sense a t a l l b u t i s amazingly p r o d u c t i v e . -fm

X

If a-l

Q

f o r example and Rek < 0 t h e n -fxm exp(kx)dx = e x p ( k x ) / k and we w i l l deal

w i t h such c o n t e x t s i n p r a c t i c e .

Thus t h e t h e o r y i s n o t vacuous a t l e a s t .

Note now t h a t f o r w = Pexp(c) ( P = 1 e x p ( c ) where ^w(k,x) = 1

(7.6)

= PCa

anw

=

+

1-1 wik-i.

+

1"1 wia-i)

one has f o r m a l l y w = $(k,x)

Hence g i v e n (7.4) we have

a Pe' + PkneeS= -L-Pe n c + paneeS= -L!PeF n

+

LnPeeS = BnPe'

72

ROBERT CARROLL

s i n c e Pap-’

+

= L i m p l i e s Ln = P8’P-l

and Bn = Ln.

To show t h a t P can be

found s a t i s f y i n g (7.4)b we f o l l o w [!IS21 ( f o t - m a l l y here, t h e a l g e b r a w i l l be

,.

a m p l i f i e d i n 512).

1” Lndt + n’

Z =

(7.7)

Thus w r i t e (tn

Zc =

*

-1”1

xn)

Lndtn -

-

Then t h e Lax e q u a t i o n becomes dL = [ Z,Ll ( e x e r c i s e

n o t e dL =

1”1 anLdtn)

and

t h e i n t e g r a b i l i t y c o n d i t i o n (7.4)b i s c a l l e d t h e Zakharov-Shabat ( Z S ) equat i o n defined v i a

dZ = Z

(7.8)

A Z

One w r i t e s t h i s o u t t o g e t (7.4)b

(exercise).

Here we can t h i n k o f Z as a

c o n n e c t i o n f o r m o n a n a p p r o p r i a t e L i e a l g e b r a bundle and t h e Lax e q u a t i o n says L i s h o r i z o n t a l w h i l e (7.8) says Z i s f l a t ( c f . Appendix A ) .

Then,

f o l 1owing [ MS2 I

LEtKtA 7.2.

L s a t i s f i e s dL = [ Z,Ll i f and o n l y i f t h e r e e x i s t s a gauge opera-

t o r P s a t i s f y i n g dP = ZcP (which i s (7.4)b). Proof:

L e t dL = [ Z,L] and Po s a t i s f y L = PoaPil

above).

-’

D e f i n e Z i = PilZcPo

-

PildPo as a gauge t r a n s f o r m a t i o n o f Zc by

(gauge t r a n s f o r m a t i o n s g a r e d e f i n e d by U =

).

P i ’ dPoP,’

dL = [ Zc,L];

(exercise

-

C

f i c i e n t s (i.e.

[ Zo,a

[ Z oC, a l [ Zc,L]

-

Note L = PoaPi’

Now Zc f l a t i m p l i e s ,Z;

-

I = 0).

(L;

= -Ln) and

o b t a i n e d v i a gauge t r a n s -

Next one checks t h a t Zz has c o n s t a n t c o e f -

Indeed

= P i 1[PoZ~Pil,PoaPi 1 ]Po = pil[Zc

[ dPoPi’ ,L] )Po = Pil(dL

i m p l i e s dL = dPoaPil

-

-

-

dP P-l,L1P0 0 0

=

dL)Po = 0

poaPi’dPoPi’).

t h e r e e x i s t s a gauge o p e r a t o r C w i t h [ C , a l

-

+ gUg-l and ( P i ’ ) ’

n o t e i n t h i s d i r e c t i o n t h a t Zn = Ly = Ln + L i

formations, i s f l a t (exercise).

cise

g’g-’

dZc = Zc A Zc

Ln commutes w i t h L, e t c . ) .

Pi’(

+

Now one checks t h a t

(7.9)

(7.10)

(such a Po i s c o n s t r u c t e d

Then one can say

= 0 such t h a t Z i = dCC-l

(exer-

t h i s i s c l e a r l y c o m p a t i b l e upon l o o k i n g a t dC = ZEC b u t one must show

KP HIERARCHY

-

73

necessity e x e r c i s e ) . Now s e t P = P C and gauge t r a n s f o r m Zc by P t o g e t (PoC)-ld(P0C) = C- 1Po- 1 OZC PoC C-’Pi1(dP0C + PodC) = C-’(PO’ (PoC)-’ZcpoC

-

ZcPo

-

PO’dP )C

-

-

= PoCa(PoC)-l

e v i d e n t l y Pa:-’

-

C-ldC = C-’Z;C

C- 1dC = 0.

= PoCaC-lPil

o b v i o u s (see a l s o remarks below).

Hence dP = ZcP as d e s i r e d and

= PoaPi’

= L.

The converse i s

QED

I t w i l l be i n s t r u c t i v e t o w r i t e down some c o n s t r u c t i o n s from

MS2 I i n a more

o r l e s s formal manner here and t h e r i g o r o u s framework w i l l be d i s p l a y e d more c o m p l e t e l y i n 112. above).

Z = PQP-’

+ dPP-’

Indeed, t o see t h i s s i m p l y w r i t e PQP-’

-

P (P as

Then

(7.11)

= Z

lrn a”dt, b y 1

Consider a gauge t r a n s f o r m o f Q =

=

1;

Lndtn =

1 (Ly

+ L_”)dtn = Z

-

Zc

Now l e t E = 11 auauy U E Z) where t h e au a r e say f o r m a l power

dPP-’.

s e r i e s i n t = (tlyt2,...)

w i t h r e s t r i c t i o n s on o r d e r o f t h e form:

There

e x i s t s W such t h a t ord(au) > u-N, f o r a l l u > > O ( t h i s w i l l be discussed i n more d e t a i l i n 112

-

c f . a l s o 113).

same o r d e r r e s t r i c t i o n s . gauge o p e r a t o r s 1 for U

E

+

L e t EX = { A

1”1 aUa-”,

EX t h e r e e x i s t s P

Similarly l e t

E

E

E; A(0)

E

D =

G ; A - l E E l , where G

and l e t DX = { A E D;A(O) G and Y

E

avaul with the = 1 ;A-’

E

Dl.

Q,

Then

Dx ( u n i q u e l y ) such t h a t (6) U = P-’Y

( B i r k h o f f decomposition EX = GDX, G n DX = 1

-

t h i s i s proved i n S12).

One

can use t h i s i n s o l v i n g t h e KP system i n t h e form dP = ZcP ( c f . Lemma 7.2). Note here f o r t h e converse p r o o f i n Lemma 7.2, g i v e n dP = ZcP w i t h L = Pap-’ one has dL = dPaP-l + Pa(-P-’dPP-l) and [ ZC,Ll = [dPP-’,PaP- 1 I = dPaP-l

-

F u r t h e r [ Z,Ll = [ Zc,Ll

(PaP-ldPP-l).

s i n c e [ L n y L l = [ Z-Zc,Ll

g i v e n t h e B i r k h o f f decomposition ( 6 ) one can prove

CHEOREII 7.3.

The Lax e q u a t i o n dL = [ Z,L]

EX w i t h a =

1“1 andt n .

= 0.

Thus

( w i t h L = Pap-’).

s o l v i n g dL = [ Z,Ll i s e q u i v a l e n t t o s o l v i n g dP = Zc

cf.

MS2 I )

i s equiva e n t t o DU = nU f o r U

This e q u a t i o n i s c o m p l e t e l y

Now

E

n t e g r a b l e i n t h e sense

o f Frobenius. Proof:

Suppose dL = [ Z,L].

Zc A Zc a l o n g w i t h P can f i n d Y

E

E

Then one has Z and Zc w i t h dZ = Z A Z and dZc =

G as above s a t i s f y i n g dP = ZcP.

DX such t h a t Z = dY Y - l

i n terms o f power s e r i e s ) .

Since Z i s f l a t one

( t h i s i s done b y w r i t i n g o u t anY = BnY

D e f i n e t h e n U = P - l Y ( Y and P known).

It f o l -

74

ROBERT CARROLL

-

-

lows t h a t dU = P-ldY P-ldPP-lY = P-ldY P-ldPU w h i l e from (7.11) 1 1 1 = dYYHence PnU + dPU = dY and nU = P- d Y - P- dPU = dU.

Conversely l e t dU = nU and use ( 6 ) t o w r i t e U = P - l Y . dYY-’,

pap-’

t

.

dPP-’

Since PnP-’ = PdUU-lP-l

and Zc = dPP-’.

= dYU-lP-l

-

dPP-lYU-lP-l

= dYY-’

t i o n o f Zc i s a p p r o p r i a t e .

-

= PEP-ldY

-

= Z

-

1

,Z =

P-ldPP-lYIU-lP-l

Zc we see t h a t t h i s d e f i n i -

Then by Lemma 7.2 L s a t i s f i e s t h e Lax e q u a t i o n

9 ED

as d e s i r e d .

R m R K 7.4, (7.12)

dPP-’

Set L = Pap-

For U = dU =

-

1;

nu

=

1 uuaV lm {I,, (auv/atn)av 1

I,, {(auu/atn)

- 1,n

lV 1:

-

(y)(aiuv)a v t n - i } d t n

=

n i (i)a u,,-.,+~ l a v d t n

Thus dU = nU i s a system o f e q u a t i o n s w i t h c o n s t a n t c o e f f i c i e n t s ( + ) auv/ n n i The ( u n i q u e ) s o l u t i o n t o dU = nU w i t h U(0) E P; g i v e n atn ( j ) a Uv++i’ i s f o r m a l l y ( m ) U ( t ) = exp(lm tnan)u(o).

lo

1

Now look a t t h e i n i t i a l v a l u e problem f o r dL = [ Z , L l o r (+*) dP = ZcP =

-1;

(PanP-l)-dtn

P.

F i r s t f o r e x i s t e n c e o f a s o l u t i o n one s i m p l y w r i t e s

U ( t ) = e x p ( n ( t ) ) P(O)-’

= exp(1;

tnan)P(0)-l

Y(0) = 1 i s s t i p u l a t e d i n DX so U(0) be t h e unique f a c t o r i z a t i o n i n C.DX; above.

(P(0) being given).

P-l(O).

where L ( 0 ) = (PJP-l)(O) i s

For uniqueness i n t h e L e q u a t i o n

g i v e n ( L ( 0 ) now g i v e n i n s t e a d o f P ( 0 ) ) . E

Let then U ( t ) = P-’(t)Y(t)

i t f o l l o w s t h a t P ( t ) s a t i s f i e s (**) as

The c o r r e s p o n d i n g L s a t i s f i e s dL = [ Z,L]

t a k e any P1,P2(0)

Qs g i v i n g t h e same L ( 0 ) and s e t U i ( t )

Since P1 = P2C ( t = 0 ) w i t h [ a , C ] w i t h exp(n(t)) the Ui(t)

Note t h a t

= exp(a(t))P;l(O).

= 0 from remarks above and s i n c e C commutes

s a t i s f y U1 = C

-1

U2 and b y unique f a c t o r i z a t i o n U. =

so one has P ( t ) = P 2 ( t ) C ( n o t e CU1 = C P ( t ) - ’ Y 1 = (P1(t)C-’\-’ P Y1 = U2 r e q u i r e s P 2 ( t ) = P , ( t ) C - l ) . But L ( t ) = P,aP;’ = P23P’;’ i n this siPi(t)-’Yi(t)

t u a t i o n so L i s u n i q u e l y d e f i n e d ( t h e a m b i g u i t y i n C does n o t a f f e c t L ) . L e t us s t a t e t h i s as ( c f . [ M S Z I )

EHHE0RBJ 7.5.

The e q u a t i o n dL = [

Z,L I, L(0) given, has a unique s o l u t i o n

given v i a U ( t ) = exp(n(t))U(O), U = P-lY,

and L = Pap-’.

REEVIRK 7-6 (0N CHE HZR0CA BZLZNEAAR ZDENCZQJ).

We go n e x t t o a b r i e f and

75

HIROTA BILINEAR IDENTITY h e u r i s t i c discussion (following

I Dl I )

o f residue calculations leading t o the

Hirota b i l i n e a r i d e n t i t y

IC w(x,k)w*(y,k)dk

(7.13)

= 0

where C i s a c o n t o u r around k =

w = Pexp(c), and w* = (P*)-'exp(-c).

my

This i s an enormously i m p o r t a n t formula c o n t a i n i n g a gread deal o f geometric and c o m b i n a t o r i a l i n f o r m a t i o n .

The r e s i d u e c a l c u l a t i o n around

-

i s however

n o t v a l i d f o r i n v e r s e s c a t t e r i n g s o l u t i o n s and t o remedy t h a t we show i n 511 how a c o r r e s p o n d i n g formula can be o b t a i n e d ( c f . [ C13,16-191). o t h e r a l g e b r a i c and g e o m e t r i c a l ways t o d e r i v e (7.13),

f i n e P* = 1 +

1;

w?a-i

1;

Note here f o r P = 1 t

dy = wJxm f d y + =

(1 +

1:

/," w'Jm fdcdy, Y

we de-

w* = ( 1 +

- a ) so L*P*-lexp(-c)

1;

wtk-i)exp(-c)

= -P*-'aexp(-E)

1

+ f)'

Thus we w r i t e , a l o n g w i t h w = P e x p ( t )

etc.).

= w*(k,x)exp(-c).

a l s o from Lw = kw we have L*w* = kw* s i n c e L = Pap-' (a* =

wi(x)a-i

-q wfdy = -c w(/"Y

t h a t under s u i t a b l e hypotheses

wik-i)exp(c),

1;

( n o t Gi) and t h i s can be r e w r i t t e n a s P* = 1

(-a)-'wi

( n o t e e.g.

i n various specified

f o r example, so one s h o u l d r e g a r d t h e

contexts, which a r e covered i n §8,9,13 p r e s e n t development as h e u r i s t i c .

There a r e

Note

i m p l i e s L* = -P*-'aP*

= kw*.

1

F o r P = a . a j and Q = b . ( - a ) j one has f ( t , t ' ) = I P(t,at) J J exp( t k ) Q( t ' ,a )exp( -t ' k )dk/ hi where f ( t ,t ' ) Y ( t-t ' ) = ( PQ* ) 6 ( t - t ' ) ( Y ( t ) =

TElTtA 7,7,

a-'a(t)

i s t h e Heavyside f u n c t i o n ) .

Proof:

Pexp(tk) =

1aj(t)kjdep(tk)

1aj(t)bj(t')IC

(7.14)

1i+j-< -1

-

and Q e x p ( - t k ) =

k i+je(t-t')kdk/2ni aj ( t ) b j ( t ' ) ( t - t ' )

1 bJ. k j e x p ( - t k )

= 1 =

-1-i-j

/ ( -1 - ij)!

represents the i n t e g r a l (simple residue c a l c u l a t i o n ) . Q =

so

On t h e o t h e r hand f o r

1 b.3') J

(7.15)

PtQ;6(t-t')

= Ptrt,6(t-t')

( t ) b i (t )ati +6j( t-t' ) + li+j<-la

li+j,Oa

-

-

=

1 a J. ( t ) bJ. ( t ' ) a ~ + j s ( t - t ' ) j ( t ) b i ( t' ) ( t-t ' )-l-i-j

=

Y/ (-1 -i - j )!

76

ROBERT CARROLL

s t a t e d ( ( )! = ( - 1 - i - j ) ! ) . The lemma proves (7.13) f o r x

QED

x'

2 2 . Indeed f o r Q = P*-' one has PQ* = 1 so (PQ*) = 0. Moreover one has J C ( a / a x l r w ( x , k ) w * ( x , k ) d k = 0 f o r any a 1. 0 (formally d i f f e r e n t i a t e (7.13) in x, f o r x i 4 x1 and then s e t x i = x l ; a l l o t h e r xi = x i ) . B u t s i n c e a k w = Bk(x,a)w we know ( a l ) f f i ( a 2 ) a z . . . ( a n ) % w = 1 c j ( x ) ( a l ) j w and hence Jc (a,)"' (a2Iq2 ... ( a n ) " " w ( x , k ) w * ( x , k ) d k This implies J C w ( x ' , k ) w * ( x , k ) d k = 0 f o r a n a l y t i c w, a t l e a s t when = 0. only a f i n i t e number of hierarchy v a r i a b l e s a r e involved, by Taylor's theorem. W e d e f e r consideration of a n a l y t i c functions f n i n f i n i t e l y many variables a t this point. We see from t h e proof o f Lemma 7.7 how c e r t a i n algej

=

jy

j

b r a i c and combinatorial c a l c u l a t i o n s a r i s e in determining (PQ*)-6 ( t - t ' ) and these a r e then augmented by i n j e c t i o n of B k r e l a t i o n s via d i f f e r e n t i a t i o n a s above.

Then ( c f . [ D1

I)

Let w and w* have t h e form w = wexp(5) and w* = w*exp(-C) PR0P09ZEZ0N 7.8, a s before and assume (7.13). Then w i s a KP wave function and w* i s i t s a d j o i n t wave function.

Then by Lemma 7.7 (PQ*)- = Proof: Define P and Q by w = Pe 5 and w* = Qe-'. 0 or PQ* = 1 (note P a n d Q have only negative powers of a by v i r t u e of t h e form of ;,$*).

Hence (7.13) implies Jc

(an -

Bn)w(xyk)w*(x',k)dk = 0 (sim-

ply d i f f e r e n t i a t e ) and we note t h a t

(an

(7.16)

-

B,)W = ( a n p + k n P

(anP

t

-

B,P)~' = ( a n p t Pan

L ~ -P BnP)eS = ( a n p

t

-

BnP)eS =

(Ln)-p)e'

Since anP t L n P has order e 0 we have a n P + L_"P = 0 by Lemma 7.7 again. Hence

(an -

Bn)w = 0 and w i s a wave function f o r KP.

from PQ* = 1 Rl3MRK 7.9

.

Q ED AND EAU FUNEZ0W).

(DAVE FUNCCZ0N$, V€RCEX 0PERAC0%,

defines ( D k i s t o be distinguished from t h e

Here Res f ( k ) d k = ( 1 / 2 n i ) J C f ( k ) d k .

The r e s t follows

a j

Next one no confusion should a r i s e )

One shows t h a t d w ( x , d x ) = 0 ( s e e below)

VERTEX OPERATORS: TAU FUNCTION and

T

i s defined by

(i =

-

$(x,k)

- d l o g T ( x ) = w(x,dx);

(7.18)

cf. [ D l

-3,log.r

77

I)

= Res k

n

k=co

(Ijlk-j-l l aj -

Dk)logidk

T h i s l e a d s t o (see below) t h e v e r t e x o p e r a t o r e q u a t i o n (VOE) (7.19)

w(x,k)

= {r(xl-l/k,x2-1/2k2,..

w*(x,k)

= {T(xl+l/k,x2+1/2k

.)/T(x))

e S ( x y k ) Y.

',...)/T(x)~

e -S(x,k)

V e r t e x o p e r a t o r s w i l l a r i s e f r e q u e n t l y i n t h e remainder o f t h e book i n b o t h p h y s i c a l and mathematical c o n t e x t s .

L e t us p o i n t o u t here an o b s e r v a t i o n o f

J. S z m i g i e l s k i , namely, v e r t e x o p e r a t o r s a t a p r i m i t i v e l e v e l a r e o p e r a t o r s V s a t i s f y i n g [ x , V I = aV and [ a x , V l = BV. Thus e.g. i f V f = e x p ( i k x ) f ( x - l / i k ) = e x p ( i k x ) G - f , w i t h G - f = f = e x p ( - a x / i k ) f , t h e n (xV - V x ) f = x e x p ( i k x ) f

-

exp(ikx)(x-l/ik)ff' = i k e x p ( i k x ) f -

= ( l / i k ) V f while (axV t

exp(ikx)fl

-

-

Vax)f = dx(exp(ikx)f-)

exp(ikx)f' = ikVf.

-

exp(ikx)

To prove (7.19) e t c .

d e f i n e f i r s t ( c f . [ Dl;FK41)

at)

(7.20)

-1 = ( l / l - t )+ t - k - t )

=

( t h i s i s l i k e a D i r i c h l e t k e r n e l l_"eikx). f(t,t)$(t/k)

and

f o r m a l l y go t o (7.13) w i t h x ' = x . - l / j t ; j J

lc i(x,k)G(t,)G

(7.21) where G

G

%

l/(l-k/tl

Then e v i d e n t l y f ( t , k ) i ( t / k )

-

l/jt; t o g e t

t,)^w*(x,k)dk/(l-k/t,

)(l-k/t2) = ((tl/k)/(l-tl/t2))(~(k/tl)

tl/k)(l-t2/k). jt;)exp(ly

-

t

note

E ( k / t 2 ) ) + (tlt2/kz)/(1= ~ ( x , k ) ~ * ( x . - l / j t ~ l/

( k / t 2 ) j / j ) and l o g ( 1 - x ) =

-1;

J Now t h e l a s t

xj/j.

t e r m i n (**) makes no c o n t r i b u t i o n t o t h e i n t e g r a l (7.21)

(no power o f l / k )

and t h e i n t e g r a t i o n g i v e s t h e n (*&) $(xytl )G(tl )G(t2)G*(xytl) G(t2)i*(x,t2). have ;(x,k)-'

= ~(x,t2)G(tl )

S e t t i n g now tl = k ( a f t e r t h e i n t e g r a t i o n ) and ti' = 0 we = G(k)G*(x,k)

t a i n s (*+) i?(x,tl)/G(t2)$(x.tl $-l(x,tl)

(**I

-

Observe here t h a t w(x,k)w*(x',k) (k/tl)j/j

= 0

)(l-k/t,)

G t ) f ( x ) = f(xl-l/t,x2-1/2t2,...))and

above ( i . e .

=

Now t o show dw = 0

f ( t , t ) = Ic f(t,k);(t/k)dk/Zaik.

(*A)

= (G(t2);(x,tl))-').

(note G(k)l = 1).

) = G(x,t2)/G(tl Set now f ( x , k )

Putting t h i s i n );(x,t,)

(*&I one

( n o t e here e.g.

= logG(x,k)

obG(t2)

and (*+) y i e l d s

78

ROBERT CARROLL f(x,k)

(*MI

G(k)f(x,t)

( n o t e e.g.

G(t)logi(x,k)

=

Consequently

(1 k+'a

(7.22)

-

+ f(x,t)

= G(t)f(x,k)

logG(t)G(x, k ) .

-

V

Dk)f(x,k)

+ lk-"-'a"f(x,t)

= G(t)(Zk-"-laV-Dk)f(x,k)

This implies (7.23)

w(x,dx)

-

= G(t)w(x,dx)

df(x,t)

e x t e r i o r d e r i v a t i v e r e l a t i v e t o x ) . Then d i f f e r e n t i a t i n g a g a i n one has (d dw(x,dx) = G(t)dw(x,dx) so t h a t t h e c o e f f i c i e n t s o f dw(x,dx) must be cons t a n t functions, i.e. w(x,dx)

1J 1

one o b t a i n s d f ( x , t ) f(x,t)

= G(t)h(x)

i n t o (*.) aij

J1

= ia..dxi

A d x . w i t h aij J

1J

-

= d(G(t)h(x)

= -a..

€

J1

- 1 ( l / i t i)aijxj).

h(x)

i ( l / i t )aijxj

-1 1 a 1. .J ( l / i k i j t j

-

h(x)

gives 0 =

= -a..).

dw(x,dx)

Hence

C.

+ d h ( x ) f o r some f u n c t i o n h and p u t t i n g t h i s i n (7.23)

=la..x.dxi

This i m p l i e s aij

s e q u e n t l y up t o a c o n s t a n t

-

There r e s u l t s

+ b ( t ) f o r some 9.

l/itijkj)

Putting t h i s

= 21a..(l/ikijtj)

(since

1J = 0 and hence dw(x,dx) = 0 as s t a t e d .

Con-

i s w e l l d e f i n e d by (7.18).

T

Next we n o t e t h a t t h e d e f i n i t i o n o f w and w* v i a

as i n (7.19) i s s i m p l y a

T

F i r s t n o t e t h a t , u s i n g t h e v e r t e x o p e r a t o r X(k) de-

" r e p h r a s i n g " o f (7.18). f i n e d by

n

m

X ( k ) f ( x ) = e5 ( x y k ) e - l l ( a n ' n k

(7.24)

e '(x,k)e-'(a,k

-

-1 )f(x);

7=

) f ( x ) = e 5 f(x,-l/k,x2-1/2k

(X*(k) = e-'exp(l: T.

an/"')).

I)

Also n o t e t h a t e.g.

D

Now a s f o r r e p h r a s i n g we observe t h a t f o r

DG(k)f(x) = (&)T

(7.26)

+

1 k - j - l a j f - 1 a.f(l/k'+') J

k-j-'a.T J

= 0.

0 = (1k-j-la.i J

Dk$)r

(7.25) i m p l i e s =

1k-j-la j -

Hence i f

= 0.

Therefore

-

=

,...)

(al ,4a2,a3/3

one can w r i t e (7.19) i n t h e form ( c f . [ D l

2 ,...)

+ clk-j-'a.r

J

GT

=

G(k).r/

Dk one has

= G(k)T one o b t a i n s

HIROTA EQUATIONS T h e r e f o r e d i v i d i n g by

and t a k i n g r e s i d u e s v i a a kn m u l t i p l i e r g i v e s

Res k n ( D i / 6 ) + anlogT

0

(7.27)

GT

79

which i s (7.18).

(iUR0CA EQMCZ0W). We c o n t i n u e w i t h t h e development i n [ D l ]

R?ZlMRK 7.10

and w i l l d e r i v e t h e H i r o t a equations.

Thus f i r s t from (7.19) and (7.13)

IC ?(xl-l/k,x2-1/2k

c(x-x',k)dk

(7.28)

2y...)~(x,'+l/k,...)e

One defines elementary Shur polynomials v i a exp(S(x,k))

= 0

= ZjLokJsj

( x ) (these

w i l l be discussed more t h o r o u g h l y i n 58) and changing v a r i a b l e s in (7.28) 0 =

(7.29) % .,

/c

-1

e'(ay'k

(T-

Ic ~ ( x ~ - y ~ - l / k2 -y, x2 -1/2k2,...)~(x1+yl+l/k,...)e~2~~yyk)dk

)(T(x-y)r(x+y))e -25(yyk)dk =

=

Ic ( ~ k - j S j ( ~ ~ X T ~ T + ) C k m S m ( - 2 y ) d k

~ ( x - y ) , 'I+= ~ ( x + y ) ) . Hence t h e H i r o t a b i l i n e a r i d e n t i t y (7.13) and

=

t h e r e p r e s e n t a t i o n (7.19) g i v e

Now d e f i n e t h e H i r o t a b i l i n e a r e x p r e s s i o n (ai = a/ayi) (7.31

1

Pf-g = P(al

,...) f ( x l

..)g(x, +yl,. . . I ly=o

-yl,.

An e q u a t i o n Pf.g = 0 i s c a l l e d a H i r o t a e q u a t i o n and we can r e w r i t e (7.30) M

as f o l l o w s . =

sj+l

Note f i r s t Sj+l ( a y ) T ( x - y ) ~ ( x + y ) = Sj+,

(~u)exp(~ml!Y,(a/aum))T(x-u)T(x+u)

T ( X ) where (7.32)

Iuz0

=

($)T(x-~-u)T(x+~+u)

luz0

s ~ ( Q+ w ~( I m 2 1 y m x m ) T ( x )

= ( X ~ ~ ~ X ~ ~ X ~ / Then ~ , . .t h. e) .H i r o t a e q u a t i o n s a r e

1;

sj(-2y)sj+l

( ~ ) e x p ( ~ m ~ l y s x S ) T ( x ) . T ( x )= 0

and ( e q u a t i n g c o e f f i c i e n t s o f powers o f y t o 0 ) t h i s i s t h e KP h i e r a r c h y . We n o t e t h a t among t h e f i r s t equations w i l l be (P(D) (7.33)

(al 4 + 3a22

-

4a 1 a 3 ) T . T

=

%

P(x))

o

which w i l l y i e l d t h e c l a s s i c a l KP e q u a t i o n s utx + 3uYY + 3(u2) xx

+

uxxx =

O

80 (xl

ROBERT CARROLL

n,

x, x2

%

y, x3

n,

2

t) w i t h u = 2axlogr.

To check t h i s n o t e t h a t t h e

f i r s t Shur p o l y n o m i a l s a r e

S1 = xl;

(7.35) (cf. [ Kl]

-

2 3 S2 = x1/2 + x2; S3 = x1/6 + x x

+ x3;

...

t h i s w i l l be developed i n 58 from a n o t h e r p o i n t o f v i e w ) .

c l a s s i c a l KP e q u a t i o n s a r i s e e.g.

The

i n t h e t h e o r y o f w a t e r waves.

Rm\ARK 7,If-(DECERF;IIIW4C RECE0l)S FOR Icats),

We r e f e r h e r e e s p e c i a l l y t o 101 ,

One c o u l d b e g i n w i t h KP b u t i t i s m r e i n s t r u c t i v e t o

2;P1-5;C13,17-191.

work w i t h KdV s i n c e more i n f o r m a t i o n i s a l r e a d y a v a i l a b l e i n 52-6 f o r background and comparison.

L e t us f o l l o w [ O l 1 and begin w i t h t h e b a s i c b i l i n e a r

H i r o t a equations f o r KdV i n t h e form qt + 6qqx + qxxx = 0, namely (7.36)

ax(at

+

3

ax]T.T

=

0

2

w i t h q = 2 a x 1 0 g ~ . Never mind t h a t t h e d e r i v a t i o n o f t a u f u n c t i o n s and H i r o t a equations i n Remarks 7.7-7.9

does

not a p p l y

t o inverse scattering solu-

t i o n s f o r example, t h e H i r o t a b i l i n e a r formula does have a v e r s i o n i n such cases ( c f . 511 and CC17-191).

Even more g e n e r a l l y e x p l i c i t c a l c u l a t i o n s i n

t h e d e t e r m i n a n t c o n t e x t w i l l g i v e v a r i o u s H i r o t a equations d i r e c t l y ( i n f a c t one can c o n s t r u c t t h e d e t e r m i n a n t s based o n (7.41)

f o r example) b u t a b i -

l i n e a r i d e n t i t y has n o t y e t been d e r i v e d f o r t h e general d e t e r m i n a n t construction.

T h i s would correspond t o d e r i v i n g t h e b i l i n e a r formula from one

H i r o t a e q u a t i o n such as (7.36)

by j u d i c i o u s placement o f t h e h i e r a r c h y v a r -

i a b l e s and seems e m i n e n t l y p r o v a b l e ( s i n c e from t h e development below and i n

511 we know how t o i n s e r t t h e h i e r a r c h y v a r i a b l e s ) .

However we do n o t p u r -

sue t h i s here. Note t h a t (7.31) can be w r i t t e n f o r monomials

and f o l l o w i n g [ 01 ] one t r i e s t o f i n d a s o l u t i o n o f (7.36) i n t h e form

( o n l y x1 = x and x 2 = t a r e needed h e r e ) .

It f o l l o w s e a s i l y t h a t

DETERMINANTS FOR KdV

1:

(7.39)

w i t h fo = 1.

ax(at + a:)fn-i-fi

81

o

=

T h i s g i v e s a l i n e a r e q u a t i o n f o r fn when fiy 0 5 i 5 n-1, a r e 3 + ax)fl-l = 0 has a general s o l u t i o n

I n p a r t i c u l a r ax(at

known.

fl =

(7.40)

1,

e kx- k3td; ( k )

i s a n a p p r o p r i a t e c o n t o u r i n t e g r a l and d$ i s a s u i t a b l e measure.

where /,

r:

F o r convenience one t h i n k s o f change v a r i a b l e s k

-c-i-

-+

-c+i-

(c > 0); we w i l l e v e n t u a l l y

i k f o r KdV and work i n t h e upper h a l f plane.

-+

39)-(7.40)

there results

(7.41)

fn = ( l / n ! ) j r

3 eI 1n (kix-kit)

... I r W ( k i - k j ) / ( k i + k j ) l

Now ifAn i s a n n X n m a t r i x w i t h e n t r i e s 2ki/(ki+k.) then

T

i n (7.38) can be w r i t t e n

(7.42)

= 1 +

T

= 1 +

where o

n

(7.43)

1;

1;

...I,

(l/n!)I,

(1/n!)lxm

..

.c

=

-Ir

nd;(ki)

i n t h e (i,j) p o s i t i o n

J

3 det(An)e 1 (kix-kit) ndi(ki) =

In

d e t ( a n ) ndsi

i s a n n X n m a t r i x w i t h (i,j)e n t r y F(t,xi+s.) F(t,s)

From (7.

3 L,kS - k t e du(k)

J

where ( f o r d $ = kdp)

Equation (7.42) has t h e form o f a Fredholm d e t e r m i n a n t and m o t i v a t e d b y t h i s (and o f course t h e GLM e q u a t i o n s i n t h e background) one d e f i n e s (7.44)

D(t,x,z)

-

= -F(t,x+z)

1”1

(l/n!)q

...$ d e t ( d n )

where fin i s a n ( n + l ) X (n+l) m a t r i x w i t h f i r s t row F(t,x+z), F(t,x+sn); rest.

f i r s t column F(t,x+z),

Note t h a t

(A*)

D(t,x,x)

ndsi F(t,x+sl)

F(t,sl+z),...,

r e l a t i v e t o t h e f i r s t column and n o t e t h a t (7.45)

(l/n!

-

)r

...Jxm

(l/(n-l)!)C

,...

F(t,sn+z); and On f o r t h e = ~ ~ ( t , x( e ) x e r c i s e ) . Now expand d e t ( f i n ) r

det(nn)$dsi

= -fn(t,x)F(t,x+z)

-

(q ...Ix det(fi~-l)F(t,s+z)n~‘ldsi)ds m

82

ROBERT CARROLL

where

i s o b t a i n e d from

(7.46)

D(t,x,z)

Assuming .(t,x) (7.47)

by r e p l a c i n g z w i t h s .

-

= -F(t,Z+X)T(t,X)

Lm

It f o l l o w s t h a t

D(t,x,s)F(t,s+z)ds

# 0 one has t h e n a M e q u a t i o n f o r K(t,x,z) + F(t,x+z) +

K(t,x,z)

= K(~,x,z)/T(~,x)

= 0

K(t,x,s)F(t,s+z)ds

Under s u i t a b l e c o n d i t i o n s o n F t h e s o l u t i o n s o f GLM equations a r e unique ( c f .

1) and we s i m p l y assume such c o n d i t i o n s a r e met. Then, s t i l l f o l l o w i n g [ 01 1, one can show d i r e c t l y t h a t a s o l u t i o n K o f (7.47) ne-

[ C1 ;CD1 ;LT1 ;MR1 ;TN1

cessarily involves (7.48)

= D(t,x,X)/~(t,X)

K(t,x,x)

= axlOg.r

To see t h i s m u l t i p l y (7.47) b y D(t,x,z)

Jxm D(t,x,z)K(t,x,z)dz

(7.49)

K( t ,x,s)F( t ,s+z)dsdz =

,,”

-r

=

and i n t e g r a t e i n z t o g e t ( c f . (7.46))

+ Jxm D(t,x,z)f(t,x+z)dz

K( t ,x,s)($

K(t,x,s)(D(t,x,s)

=

D ( t ,x,z)f (t,s+z)dz)ds

-h

D(t,x,z)

=

t F(t,x+s)r(t,x))dS

c

T h i s i m p l i e s (u) D(t,x,s)F(t,x+s)ds

=

I,”

K(t,x,s)r(t,x)F(t,xts)ds

u s i n g (7.47) a g a i n t h e r i g h t s i d e o f (u) i s -T(t,x)(K(t,x,x)

t

and

F(t,2x))

w h i l e from (7.46) t h e l e f t s i d e o f (a)i s - F ( t , E x ) r ( t , x ) - D(t,x,x). T h i s 2 i m p l i e s (7.48) and q ( t , x ) = 2axK(t,x,x) = 2axlogT w i l l s a t i s f y KdV ( n o t e we a r e d e a l i n g w i t h t h e q e q u a t i o n here, q (0)

o f 51 a f t e r (1.24)).

(7.50)

-

KxX(t,x,z)

R,

-u, and t h i s changes t h e s i g n i n

F u r t h e r formal c a l c u l a t i o n y i e l d s KZ,(t,x,z)

+ U(t,X)K(t,X,z)

To see t h i s d i f f e r e n t i a t e (7.47) t o g e t (Ae) KXH

-

K(t,X,X)Fx(t,X+Z)

and ( A d ) KZz + Fzz t tegrate by p a r t s i n

-

KX(t,X,S)

S,xF(t,x+~)

/xm K(t,x,s)FZZ(t,s+z)ds

(Ad),

-

4

= 0.

assuming e v a l u a t i o n a t

j u s t i f i e d i n classical situations KZz + Fzz

+

-

= 0 t

Fxx

-

(K(t,x,x)xF(t,x+z)

KxX(t,x,s)F(t,S+z)dS

-

c f . [Cl;LTl;MRl]

= 0

Now use Fzz = Fss and i n vanishes ( t h i s can be ).

K ( t , x y x ) F x ( t s x + z ) t Ks(t,X,S)]S=XF(tyX+Z) +

One g e t s

c

(A+)

0 =

Kss(t,x,s)F(t,

MARCENKO EQUATION Note a l s o t h a t axK(t,x,x)

s+z)ds = 0. Subtract

(A+)

q(t,x)K)ds

+ asK(t,x,s))ls,x.

= (axK(t,x,s)

from (A@) t o g e t now Kxx

-

-

KZz

so a d d i n g q t i m e s (7.47) t o t h i s g i v e s Kxx

+

83

-

KZz

+

m

/x

(Kxx - K )FdS = 0 ass q ( t , x ) K + Jx ( K x x - Kss

qF +

T h i s i s a homogeneous form o f M e q u a t i o n and by unique-

= 0.

ness o f s o l u t i o n s t o (7.47) one concludes t h a t (7.50) i s t r u e . I t w i l l be c o n v e n i e n t t o change v a r i a b l e s now ( k

(7.51)

F(t,z)

e

=

-+

i k ) and w r i t e

i k z + 8 i k 3 tdA

where t h e c l a s s i c a l i n v e r s e s c a t t e r i n g s i t u a t i o n corresponds t o dA = R(k,O)dk/Z*

a s i n (1.25).

= C and

E v i d e n t l y (7.51) i s much more general and

w i l l l e a d t o p o t e n t i a l s q o f many types.

The s i t u a t i o n i s analogous t o t h e

c o n s t r u c t i o n o f p o t e n t i a l s by Newton-Sabatier (NS) methods i n t h e c l a s s i c a l

I).

t h e o r y ( c f . [ C2,24;CD1

The o b v i o u s r e s t r i c t i o n s i n v o l v e t h e e x i s t e n c e o f

( o r (7.43)) and s u i t a b l e growth and r e g u l a r i t y o f F

t h e i n t e g r a l i n (7.51)

and K t o enable us t o d i f f e r e n t i a t e and i n t e g r a t e by p a r t s ( p l u s uniqueness f o r the M equation).

T h i s package o f requirements deserves f u r t h e r s t u d y

b u t we do n o t d w e l l on i t i n t h i s book.

We w i l l assume whatever p r o p e r t i e s

a r e needed f o r c a l c u l a t i o n s . i n n o t i n g here t h a t such p r o p e r t i e s do h o l d f o r many i n t e r e s t i n g s i t u a t i o n s .

RETIARK 7-12 (W9mPk0kZC$). Go back t o (2.9) and c o n s i d e r now f+ = e x p ( i k x 2 + g(k,x)) (g(k,m) = 0 ) which l e a d s t o a R i c c a t i e q u a t i o n gxx + g, t 2 i k g x t g = 0 and an expansion from which (7.52)

gx

'L

1;

Rn/(2ik)n;

g =

1;

(l/(Zik)'))i_x

Rndy

One has R1 = -q, R 2 = qx, R3 = - q xx - q2, e t c . as bef o r e and ( s i n c e f + e x p ( - i k x ) + a = c12 as x -t - m y Imk > 0 ) as i n (2.12) one f o r Imk > 0,

Ikl

f i n d s ( c f . [ NE1 I (7.53) (a = l/T).

-+

a.

-

n o t e g(k,m)

= 0)

= loga(k)

-Irn1 ( l / ( 2 i k ) n ) l z

g(k,-m)

'L

Rn(y)dy

Now t h e v e r t e x o p e r a t o r e q u a t i o n (7.19)

( o r (7.25))

v e r s i o n i n t h e form (7.54)

-

W = X_(k).r/.r

Y

= e S ( x ' k ) r-(x,k)/.r

n.

= e'<(k)r(x)/r(x)

has a KdV

84

ROBERT CARROLL

icr

= x ~ k2n+l ~ Y+ where x = ( X ~ ~ X ~ ~ . x. .n.,) ,x 1 sometimes, x3 = t3 n., 4 t , and g - ( k ) T ( x ) = ~ ( x ~ - l / i k , x ~ - 1 / 3 i k ~ ,). We w i l l see how v e r t e x o p e r a t o r

...

e q u a t i o n s a r e proved i n t h e d e t e r m i n a n t c o n t e x t below ( f o l l o w i n g [ P1-31). Thus'(7.59) a p p l i e s t o t h e t a u f u n c t i o n (7.42) where we w r i t e i n t h e h i e r archy s i t u a t i o n n,

F ( x t y ) = ($o(k,x),$o(k,y))A

(7.55) for x =

=

( X ~ ~ X ~ ~ . Y. .= ) (~Y ~ ~ Y ~ , . . . ) , and

17-19]).

1:;

e'(kyx)eS(kyY)dA = FA(x+y)

x

~ = ~Y

+~ f ~o~ r n+ 2 1~ ( c f . [ C13,

For t h e p r e s e n t d i s c u s s i o n assume a c l a s s i c a l s c a t t e r i n g s i t u a t i o n

w i t h dn = R(k,O)dk/En, i n (1.25)).

R

E

r

S, and

=

(-my-)

( o r more g e n e r a l l y use (R,C)

as

W i t h t h e h i e r a r c h y v a r i a b l e s i n c l u d e d (7.53) s t i l l a p p l i e s and

we can use (7.54)

t o e s t a b l i s h a d i r e c t l i n k between a and

T.

Thus g

Q,

lOg(T-)/T and (7.56)

a(k) =

(T-/T)-

1i m

= X-t-m

(T-(x,~)/T(x))

..). C o n c e p t u a l l y t h i s i s i n t e r e s t i n g because,

( n o t e a ( k ) depends on x3,x5,.

e.g. when a has no zeros, i t shows t h a t t h e i s o s p e c t r a l m a n i f o l d ( d e f i n e d by a o r T = sll)

i s c o m p l e t e l y determined v i a t h e t a u f u n c t i o n ( c f . a l s o 911).

Moreover i f we w r i t e o u t f o r m a l l y (7.57)

-rx/ik

T- = T ( X )

-

~

~

(Tx/T)-/ik

-

~

- / r 3 2/ 3 i kk3

~+ ~ ~ ~ ~3 +/ ... 6 i k

t h e n i n t h e n o t a t i o n o f (7.56) (7.59)

a(k) -1

-

2 ( ~ ~ ~ / ~ ) - +/ 2... k

+ ... and t h i s would a l l o w us a l s o t o determine t h e Lz Rndy by means o f t h e t a u f u n c t i o n . As an example c o n s i d e r t h e 1 s o l i t o n t a u f u n c t i o n 3 5 ) 2(-llx + ll x3 - ll x5 + (7.59) T = 1 t e Now f o r m a l l y l o g ( 1 - x ) =

-1;

xn/n so ( A m ) l o g ( a )

Q,

(Tx/T)-/ik

...

( n > 0 ) . T h i s can be i n c l u d e d i n t h e framework o f (7.51) o f (7.35) b y simp l y h a v i n g a s u i t a b l e 6 measure a t one p o i n t . Then T = 1 + exp(2(-nx t 3 3 rl x 3 - ... ) ) e x p ( 2 ( o / i k - I-I / 3 i k 3 + ) ) and as x - i t h e exponents domi-

...

-f

~

NEWTON SABATIER POTENTIALS nate w i t h

og(.r-/.r)

2(n/ik

-

TI

3 /3ik3 t

can be eva u a t e d v i a l o g ( ( k - i n ) / ( k t i n ) ) %

(2/i)zi

...) .

85

On t h e o t h e r hand

/f

Rndy

( c f . (2.13)) and (7.53) g i v e s l o g a

so (7.56) i s c o n f i r m e d f o r t h i s example.

-l)n((s/k)2nt1/(2n+l),

We w i l l see i n § 9 t h a t s i m i l a r c o n n e c t i o n o f s p e c t r a l data w i t h t a u f u n c t i o n s occurs i n AKNS s i t u a t i o n s . L e t us say t h a t , g i v e n FA i n

R€IXARK 7-13 (MEMBN SABACZER elJPE PoCENC%A#)* (7.55),

KA i s t h e s o l u t i o n o f

(7.60)

KA(x,z) t F,,(xtz)

+

/xm KA(x,s)FA(s+z)ds /x I

m

x Q ( t a k e K A ( x y z ) = 0 f o r z1 < x1 and n o t e I =

+

KA(xys)$o(k,s)ds where $,(k,x)

(7.61)

, ds

= dsly

e t c . w i t h x~~~~

W r i t e a l s o i n t h e same n o t a t i o n ( a * ) $,(k.x)

z2n+l f o r n 2 1 ) .

I,"

m

= 0

= exp(y(k,x)).

= G0(k,x)

Put (7.55) i n (7.60) so

+ ~ $ o ( k y ~ ) y $ o ~ k+y ~ ) K~ AA( X , S ) ( J l o ( k y s ) , J l o ( k y z ) ) , d s

0 = K,,(x,z) = K,,(x,z)

+ (~,,(k,x),~o(kyz))A

Hence ( c f . (1.22)) (7.62)

K,,(x,Y)

= -(JIA(k,x),Jlo(k,y))A

T h i s formula i s u s e f u l i n p r o v i d i n g examples ( c f . [ C2,13;CD1 I). Thus l e t dA(k)

%

cs(k-m) (Im(m) > 0 ) so K,(x,y)

= -c$,,(m,x)Jlo(m,y)

and Jl,,(k,x) = T h i s i m p l i e s Jl,(m,x) = $o(myx)/

$ o ( k y x ) + cJIA(m,x)$o(m,x)$o(k,x)/i(ktm). (1 - c$o(m,x)/2im) 2 and KA(x,y) = -c$o(m,x)$o(m,y)/(l

(7.63)

-

2 c$o(m,x)/2im)

From (7.63) one can make computations on K A ( x y x ) t o determine t h e NS t y p e p o t e n t i a l q,(x)

etc.

REEII\RK 7.14 (F0RmLW Z t W U 3 Z N C CAU FLINCCZQW, D R S S Z N G KERNELS, AND SPECCRAC DAM). We w i l l use l a t e r a d i f f e r e n t formula f o r t h e VOE i n v e r i f y i n g i t f o r KP

as follows. exp(F)

+

l a [ P2]. From (-)

I n t h e KdV c o n t e x t t h i s has a c o n n e c t i o n w i t h (7.56) *

t h e VOE $,

KA(x.s)exp(?(k,s))ds.

- A

= X - T ~ / T ~ i s e q u i v a l e n t t o exp(S)T-/T,,

=

We t h i n k o f K a s t r i a n g u l a r (K,(x,s)

=

86

ROBERT CARROLL

= 0 f o r sl

< xl)

and r e c a l l t h a t x

(7.64)

T!/T~

= 1 +

~

- ~’2n+1+

jxm KA(x,s)e ik(s-x)ds

f~o r n 2 1.

Hence t h e VOE i s

= 1 + e - i k x *KA(x,k)

A

where K A ( x y k ) = f K A ( x , s )

(7.65)

(Fourier transform).

Consequently from (7.56)

a ( k ) = lim X-t-m (1 + e-ikxtA(x,k))

T h i s a l l o w s one i n p r i n c i p l e t o determine a i n terms o f K., i s more i n t e r e s t i n g s i n c e one t h i n k s o f K

A

T~

A c t u a l l y (7.56)

a s t h e fundamental o b j e c t ( n o t

such m a t t e r s w i l l be discussed a g a i n i n v a r i o u s p l a c e s .

or

a case can be made f o r KA as a fundamental o b j e c t .

Perhaps

We n o t e f u r t h e r t h a t

under s u i t a b l e c o n d i t i o n s o f growth and r e g u l a r i t y on KA ( m e r c i f u l l y unspecif i e d h e r e ) t h e f o l l o w i n g formal c a l c u l a t i o n s can be made. We t h i n k o f K A Y 2 2 T ~ qA , b u t w r i t e K, T , q and r e c a l l t h a t a x K + q ( x , t ) K = a K and ( ( a x + a ) 2 Y Y K(x,y,t))lx=y = D,K(x,x,t) = axlog.r = q/2. L e t us compute (0.) K (x,y, 2* YY t ) e x p ( i k y ) d y = - k K(x,k,t) t ( i k K ( x , x , t ) K (x,x,t))exp(ikx). On t h e o t h e r A Y hand ( 0 0 ) Kxx = K(x,y,t)exp(iky)dy = (-DxK(x,x,t) - ikK(x,x,t) KX(x,

I,”

-

air

/,”

x,t))exp(ikx) + 2^ -k K qexp(ikx).

-

-

A

Kxx(x,y,t)exp(iky)dy.

(T-/TIXx

T h i s l e a d s t o ( 0 4 ) Kxx t qK =

Consequently i f (7.64) holds, o r e q u i v a l e n t l y t h e V O E i s

i n f o r c e , then, w r i t i n g q = (7.66)

A

~ ( T ~ / Twe) o ~btain

an a p p a r e n t l y new o b s e r v a t i o n

+ 2 i k ( ~ - / ~= )-2(Tx/T)x(T-/T) ~

We n o t e a l s o t h a t f o r JI = $A = ( ~ - / ~ ) e x p ( C t)h e e q u a t i o n (**) $ “ t kL@ = -q$ i s t h e same as (7.66).

T h i s however i s b a s i c a l l y t h e same d e r i v a t i o n

s i n c e (*+) f o l l o w s from $ = JI0 +

/,”

Some comg i v e n Kxx + qK = K YY. p u t a t i o n ( e x e r c i s e ) shows t h a t (7.66) can be c o n f i r m e d from t h e H i r o t a equa-

tions.

K$,ds,

A c t u a l l y t h i s must be t r u e s i n c e t h e H i r o t a equations a r e t h e o n l y

c o n c e i v a b l e method o f comparing t h e two s i d e s o f (7.66) d i r e c t l y .

-

Also

s i n c e we w i l l l a t e r prove i n Ill t h e H i r o t a b i l i n e a r i d e n t i t y f o r s c a t t e r i n g s i t u a t i o n s ( A = Rodk/2s, T =

(-my-),

w i l l be v a l i d f o r some A a t l e a s t .

no bound s t a t e s ) , t h e H i r o t a e q u a t i o n s On t h e o t h e r hand (7.64) f o r KdV w i l l

f o l l o w from t h e VOE f o r KP, b u t a p r o o f o f t h e H i r o t a b i l i n e a r i d e n t i t y f o r general A has n o t y e t been w r i t t e n down.

It would be o f i n t e r e s t t o f i n d

such a p r o o f o r t o f i n d a p r o o f o f a l l t h e H i r o t a equations i n t h e d e t e r -

DRESSING KERNELS

87

minant context ( s e e remarks a t t h e beginning of Remark 7 . 2 ) . Note here (under our usual hypotheses of growth and r e g u l a r i t y f o r K ) t h a t i f one can e s t a b l i s h a l l t h e Hirota equations by separate arguments, thus confirming (7.66), and i f e.g. T - / T -+ 1 and ( T - / T ) ~ 0 s u i t a b l y a s x -+ m (plus unique -+

s o l u t i o n s of the M equation), then (7.66) will imply t h e V D E , which would give another proof of VOE. The proof o f V O E given below (from [ P21) i s more s a t i s f a c t o r y however. Further e x p l o i t a t i o n of t h e methods of [ P1-51 should provide a n e x p l i c i t proof o f t h e Hirota b i l i n e a r i d e n t i t y f o r determinant constructions. Let us emphasize t h a t t h e determinant constructions a r e very important in t h a t they provide a way of constructing s o l u t i o n s a n a l y t i c a l l y , including t h e inverse s c a t t e r i n g s o l u t i o n s , i n which the tau function appears immediat e l y w i t h a c e r t a i n amount of a l g e b r a i c s t r u c t u r e i m p l i c i t via the determina n t constructions.

The program of determining a l g e b r a i c s t r u c t u r e a l s o pro-

ceeds via asymptotic expansions as i n § 1 , 2 where one a r r i v e s a t an i n f i n i t e number o f conserved q u a n t i t i e s Hn and the higher KdV flows. We emphasize again t h a t t h e Hirota b i l i n e a r i d e n t i t y is primarily geometrical , a l g e b r a i c , and combinatorial.

I t will a r i s e i n a group t h e o r e t i c context i n 18 a n d i n

a Grassmannian context i n §11,13, a n d i t contains within i t s e l f a l l t h e Hiro t a equations. The need f o r showing how this information is encoded i n anal y t i c s i t u a t i o n s i s t h e r e f o r e of some importance; one a l s o wants to connect

this information d i r e c t l y t o t h e flow h i e r a r c h i e s . Let us p o i n t o u t another use of (7.64) (assumed t r u e f o r t h e A in question) n i n determining an asymptotic expansion of K A ( t , x , k ) in k . T h u s f i r s t we use the expansion f o r (7.67)

T /T

-(l/ik)Tx/T t

based on say (7.57) to w r i t e from (7.64)

-

(1/2k 2)

(1/24k 4 ) a x4T / T -

-

T ~ ~ / T

(1/3ik 3 ) T ~ / T t (1/6ik 3 ) T ~ ~ t ~ / T = emikx[(x,k,t)

KA). Now assuming everything makes sense we can write ( w i t h our usual assumptions of s u i t a b l e growth and r e g u l a r i t y f o r K and thinking o f a+’ + - J ~ ~ (T

2,

T ~ ,K

(7.68)

^K(x,k,t) =

eikXlm ((-l)ntl/(jk)ntl)anKl = 0 y y=x 1 ;

anKl (-a)-n-leiky y y=x

ROBERT CARROLL

88

e x p ( i k x ) / i k and a n K l

where a-’exp(iky) could w r i t e

(7.69)

Y=x

Y

i(x,k,t)

= K(x,y,t)oeiky

=

1;

means a n K(x,y,t)ly=x. Y

Then one

a~Kly,x(-a)-n-loeiky

The i d e n t i f i c a t i o n now ( i n a s p e c t r a l s i t u a t i o n f o r example) w i t h (7.67) g i v e s an e x p r e s s i o n f o r 1 + K c o e f f i c i e n t s a r e made up o f

T

Q ,

P as a p s e u d o d i f f e r e n t i a l o p e r a t o r whose

f u n c t i o n expressions.

Thus g i v e n (7.64) (e.g.

i n a s u i t a b l e s p e c t r a l s i t u a t i o n ) one has a formal expression, based o n i d n e n t i f i c a t i o n o f a KI w i t h t h e T f u n c t i o n c o e f f i c i e n t s i n (7.67) Y Y=x -3 1 + K % P plr 1 - ( T X / T ) a - ’ + ’i(Txx/T)a-2 + ((T3/3T) - (Txxx/6T)a (7.70)

+

... -T(x2n+7 -

I n particular

(-l)na-(2n+1)/(2n+l))/T(x)

T ~ / T=

K(x,x)

X-(-ia)T/T

‘L

K ( X , Y ) ~ ~= =b ~ xX/~. Y f u n c t i o n s c o n s t r u c t e d as above t o

(which we know) and e.g.

I n t h e KP s i t u a t i o n c o n d i t i o n s on FA f o r

be Cm a r e i n d i c a t e d i n [ P2].

K,

=

T

Conditions p e r m i t t i n g the c a l c u l a t i o n s w i t h

i n Remark 7.15 here have n o t been s t u d i e d .

Our purpose i n w r i t i n g such

formulas i s p a r t l y t o show what i s f o r m a l l y going on i n u s i n g p s e u d o d i f f e r e n t i a l o p e r a t o r expansions f o r P and t o i n d i c a t e t h e need f o r some i n v e s t i g a t i o n o f hypotheses e t c . Note t h a t t h e

+ FX K b u t F,

M e q u a t i o n (7.60) can be w r i t t e n as

= 0 where Kx %

I,”

%

/xm K corresponds

F a r b i t r a r i l y specifies 1

F = -(1+K )- K o f course. X

g i v e n F,,K,,T,,

and q,

i f q,

to stipulating the triangularity o f K

/,” .

so K = -(l+Fx ) - l F and $ = Q0 + K$o =

K + F + KxF = 0 o r K + F

-

We assume unique s o l u t i o n s f o r K (l+Fx)-lF$o = ( l + F x ) -1 Q0. Also

$0

I n t h e s p i r i t o f NS methods ( c f . [ C2,13;CD1 1) i s a potential giving r i s e t o a spectral situa-

t i o n as i n 51,2 t h e n t h e wave f u n c t i o n s $, w i l l s a t i s f y (me) $; + q$, = 2 -k $*. I f i n a d d i t i o n $, -+ $o and + $’ a s x -+ m (which can be assured t h e n qA = J1 where q~ i s t h e wave f u n c t i o n v i a c o n d i t i o n s on T-/T o r on K),

$i

f o r t h e s p e c t r a l s i t u a t i o n based o n F,K w i t h (A,?) ‘L (R,C) a s i n (1.25) 2 ( s i n c e $” + q$ = - k $ w i l l have unique s o l u t i o n s w i t h p r e s c r i b e d a s y m p t o t i c c o n d i t i o n s on

$,$I).

T h i s means K = K, and hence F = FA ( t h e a c t u a l con-

DRESSING FOR KP

89

t o u r 'i: might be movable here due t o a n a l y t i c i t y of c o u r s e ) . However t h i s says i n a general way t h a t i f a A c o n s t r u c t i o n ' l e a d s t o a s u i t a b l e s p e c t r a l s i t u a t i o n (not q E S b u t more i n t h e s p i r i t lf ( l + x 2 ) [ q [ d x < a ) then i n f a c t (A,?) E ( R , C ) . This is not too surprising perhaps b u t i n t h e event t h a t one has bound s t a t e s this should s p e c i f y t h e p o s i t i o n s and normalizing constants. Thus b a s i c a l l y only c e r t a i n (A,?) can give r i s e t o spectral s i t u a t i o n s and one can tell i n advance whether o r not this will happen. P r a c t i c a l l y this is of some value ( c f . [ C2;CDl I ) .

REmARK 7.15 (DRESdLIQG n€CH0D$ FOR KP). Go back now t o [ 01 I and e s s e n t i a l l y r e p e a t the constructions (7.38)-(7.45) f o r KP. Thus t a k e t h e KP basic equa2 t i o n a s (0.) utx + 3uyy + 3(u ) x x + uxxxx = 0 which i n b i l i n e a r form involves

(axat + 3a 2 +

u

2ax10gT 2 4 Again work from (7.38) w i t h 2 ( a t a x + 33; + a x ) f n - i (7.71)

Y

8;)T.T

= 0;

=

n-1

=

-11 ( a t a x

+

2 3ay

+

a4x )

f n - i ~ f i . Then one takes f l a s an i n t e g r a l of a s u i t a b l e exponential a s bef o r e and c o n s t r u c t s f n , T , F, D, K e t c . . W e p r e f e r t o follow [ P21 here where a l l t h e hierarchy v a r i a b l e s a r e i n s e r t e d from t h e beginning and t h e presentation is connected to dressing ideas. First make a change o f v a r i + a b l e s t -+ - 4 t so t h a t the basic KP equation is now (&*) (3/4)u = (ut 3 YY %faxu t ~ U U ~ ) ) ~ .Let x 5 (xl,x2,.,.) (or sometimes x 'L. x l ) and f o r any two hierarchy v a r i a b l e s x,y we will always s t i p u l a t e t h a t xn = yn f o r n 2 2 . Set a n = a/axn and a = a, w i t h c ( k , x ) = 1; x n k n . We have a l r e a d y developed t h e hierarchy p i c t u r e i n Remark 7.1 and we sketch t h e "dressing" procedure of Zakharov-Shabat now (this i s b a s i c a l l y just a g l o r i f i e d transmutation idea b u t t h e context has been extended i n many ways - c f . [ P2;Zl ;C22-24,1, 2,5,6,13]). We t h i n k of dressing a c t i o n i n x1 a n d o p e r a t o r s K,- w i t h kernels a l s o denoted by K operate on functions $(XI i n t h e x1 v a r i a b l e . Thus e.g. m

$ K+(xl

f

,~~,...;y~,y~,...)q~(y~,y~,...)dy~( w i t h xi = yi f o r i 2 2 ) . T h i s is written a s (K+$)(x) = K+(x,y)$(y)dy. 2 The dressing idea now i s t o take some "bare" operator Mo, e.g. M = a x o r 0 1 a J.aJ and "dress" i t t o a more complicated operator M via kernels K,.- Thus one wants (4A) M(l + K -+ ) = (1 + K+)M Let t h e operators work on nice func- 0 t i o n s $ 6 C i f o r example (eliminating boundary conditions) and we will want Kt$ =

/,"

.

90

ROBERT CARROLL

M t o be a d i f f e r e n t i a l o p e r a t o r . 1

+

F such t h a t ( 6 . )

f a c t o r i z a t i o n holds.

Guided by KdV one can ask f o r an o p e r a t o r

(1 + K+)(1 + F) = 1 + K- and we assume t h i s canonical One shows now t h a t t h e d r e s s i n g o f Mo t o M i s t h e same

f o r K,- i f and o n l y i f 1 + F commutes w i t h Mo. same w i t h ( c f . P2 I )

To see t h i s assume M i s t h e

M(l+K-) = M(l+K+)(l+F) = (l+K+)Mo(l+F);

(7.72)

M(l+K ) = (l+K-)Mo Assuming (l+K*)-’

= (l+K+)(l+F)Mo

makes sense we have [Mo,Fl = 0.

Conversely i f [ M o y F ] = 0

t h e n ( 6 6 ) (1+K_)Mo(l+K-)-l = ( l + K + ) ( l + F ) M o ( l + K - ) - l = (l+K+)Mo(l+F)(l+K-)-l = (1 + K + ) M(1 ~ +K+) - 1 i s t h e same. We remark i n passing t h a t (l+K+)-’ can e x i s t b u t n o t (l+K-)-’

P21) so t h e demonstation above i s n o t a l l i n -

(cf.

c l usive. The c o n d i t i o n IMo,FI = 0 i s f r e q u e n t l y o u t as

(7.73)

Mo(ax)F

-

+

FMo(aZ) = 0;

1 a J. ( x ) a i F ( x , z )

=

1 (-az)J(F(x,z)aj(~))

To see t h i s s i m p l y w r i t e o u t t h e a c t i o n on t e s t f u n c t i o n s

I$

E C i ( n o t e gen-

e r a l l y t h e a . c o u l d be m a t r i c e s ) . I f one dresses Mo + aa t o M + aa t h e n J + Y Y F(x,y,z) w i l l s a t i s f y ( W ) aa F + Mo(ax)F FMo(az) = 0 ( e x e r c i s e ) . The Y c o n d i t i o n [ Mo,F] = 0, M a d i f f e r e n t i a l o p e r a t o r , a l s o i m p l i e s t h a t M w i l l

-

0

be a d i f f e r e n t i a l o p e r a t o r .

T h i s can be v e r i f i e d by d i r e c t computation a s

i n [ Z1 ] or by t h e f o l l o w i n g argument from [ P21 (assume (1+K )-’ e x i s t s ) . d e f i n i t i o n K- i s l o w e r t r i a n g u l a r ( o r l o w e r V o l t e r r a ) and K+

M = (l+K-)Mo(l+K-)-’

= (l+K+)Mo(l+K+)-l.

i s upper.

By

Now

The f i r s t e x p r e s s i o n i s d i f f e r e n -

t i a l + l o w e r t r i a n g u l a r and t h e second i s d i f f e r e n t i a l + upper t r i a n g u l a r w h i l e d i f f e r e n c e i s zero. and ’V

Hence 0 = P(x,d) + V+ + V

upper ( l o w e r ) V o l t e r r a .

= ti(x-a)

with P differential

Apply T = P + V+ + V- t o a 6 f u n c t i o n 6a

t o g e t 0 = Ttia = V+(x,a)

( x < a ) o r V-(x,a)

(x > a).

V+ = 0 and hence P = 0, so i n p a r t i c u l a r M i s d i f f e r e n t i a l . Now f o r KP, f o l l o w i n g [ P 2 1 we want K

such t h a t

This implies

DETERMINANTS FOR KP

91

n p l u s a c a n o n i c a l f a c t o r i z a t i o n ( 6 0 ) . T h i s l e a d s t o [Mo,FI = 0 = I an-a , F l n n n i n t h e form (bm) anF axF t ( - 1 ) a F = 0 f o r F(X,Y) ( a x - a/axl, a ,I, a/ayl, Y Y an % a/axn, and r e c a l l xn = yn f o r n 2 ) . An easy formal g i v e s f o r example

-

(7.75)

-

anKt(x,y)

B~K,(x,Y)

+ (-l)'a;K,(x,y)

o

=

and t h e wave f u n c t i o n s a r e determined v i a (+*) w = Pe' t

= ( 1 t Kt )eE = e E ( x ' k )

For F now i t i s n a t u r a l t o t a k e

Jxm K,(x,z)exp(S(k,z))dz.

where A i s a c u r v e o r r e g i o n i n C

2

and dp i s s u i t a b l e (examples below).

As

an exercisecheck t h a t F s a t i s f i e s ( b m ) . W r i t e now (7.77 where

Sj

%

(Sj,x2,x3,...)

and

nj

%

n1

(7.78)

T(X)

=

1;

...

...lXm F(nl ... ")nn

(l/n!

Define

(yj,x2,x3,...).

n dyi

;

( c f . (7.42) and (7.44) b u t n o t e t h e r e a r e some n o t a t i o n a l changes i n v o l v i n g exp(5)).

W r i t i n g o u t 1 + K- = ( l + K t ) ( l + F )

i n terms o f t r i a n g u l a r i t y we have

f o r x1 < z1 , a GLM e q u a t i o n (7.79)

K,(x,z)

+ F(x,z)

+

h K,(x,y)F(y,z)dy

= 0

and t h e t a u f u n c t i o n i s e s s e n t i a l l y t h e Fredholm d e t e r m i n a n t f o r t h e t r u n c a t e d o p e r a t o r Fx ( = F on [ x , m ) ) .

I n o r d e r f o r t h i s t o make sense assumptions

o f t h e t y p e (+A) s u p ~ F ( s , t ) ~ ~ l t s ~ " ~< l t tf ~o r" a ( s , t 03

v i s i o n e d i n [ P21, based on c a l c u l a t i o n s i n [ P51. c l a s s and i t s Fredholm d e t e r m i n a n t . i s d e f i n e d . t a i l s on convergence here however. mention [ S I M l

<

m,

any a, a r e en-

Then Fa w i l l be o f t r a c e We w i l l n o t go i n t o any de-

For Fredholm determinants e t c . l e t us

I.

As before we now have (+a) K+(x,z)

= D(x,z)/T(x),

D(x,x)

= a,T(x),

and u =

92

ROBERT CARROLL

2 2axlogr(x).

That K+ so d e f i n e d s a t i s f i e s (7.79) i s a more o r l e s s standard

c a l c u l a t i o n (Cramer's r u l e ) , as above f o r KdV, o r c f . [ RZ1 I. Note a l s o from

-

an

Bn = P ( a n

-

an)P-'

we g e t Can-Bnyam-Bml

and Bn w i l l be

= 0 ( o r (7.3)b)

a d i f f e r e n t i a l o p e r a t o r v i a (bm) and remarks above (P = 1+K+ h e r e ) . n e c t i o n t o u comes from B2 = a

The con-

2

+ u and w r i t i n g o u t [ a y -B 2 ,a t -B 3 I, w i t h B3 3 a + (3/4)(ua t au) + w, g i v e s wx = ( 3 / 4 ) u and ut-4(uXXX + 6uux) = wyy from which (am) f o l l o w s .

One can f o r m a l l y determine t h e Bn d i r e c t l y from t h e

d r e s s i n g equations (7.74) b y d i r e c t c a l c u l a t i o n ( c f . [ Z1,ZI). e.g.

u ( x ) = 2axK+(x,x),

2

w(x) = (3/2)((ax

-

One o b t a i n s

2

a Z + u ( x ) ) K ( x , ~ ) ) l ~ = ~and , this

i s analogous t o r e p r e s e n t i n g P = 1+K+ as a p s e u d o d i f f e r e n t i a l o p e r a t o r v i a 2 d e r i v a t i v e s o f K+ as i n (7.70). To check t h a t u = 2a 1og.r i n (+@I i t s u f f i c e s t o show t h a t D(x,x)

= axT(x) and t h i s can be achieved by d i f f e r e n t i a -

t i n g T i n (7.72) and making a few elementary d e t e r m i n a n t c a l c u l a t i o n s (exercise

-

c f . [ P21). i m a g i n a r y ) and KP2 real, x 2j We r e f e r f o r t h i s t o [ C13;AB2-4;F01,

It t u r n s o u t t h a t t h e r e a r e 2 cases KP1 (x2j+l

( x j r e a l ) which a r e q u i t e d i f f e r e n t .

2,6;WC1 I. For KP1 a u s e f u l F was d e s c r i b e d i n [ MN1 I o f t h e form (7.80) G e n e r a l l y f o r purposes o f a n a l y s i s i n [ P2] one works w i t h F(x,z)

i n (7.70)

i n a r e g i o n A where Rep(s) 5 - 6 < 0 < 6 5 Req(s) i n which case F i s a n a l y t i c i n x,z and decays e x p o n e n t i a l l y t o zero as z1 sumed t o s a t i s f y l i m s u p l x n l "n as i n d i c a t e d one w i l l have x

E

T

H one shows i n [ P2] t h a t

= 0 (one says x

+

a n a l y t i c (exercise). T E

Cm.

The v a r i a b l e x i s a s -

03.

E

H then). For f

E .$

€

H andA

i n (7.80) and

F u r t h e r i f A i n (7.76) i s a r e g i o n A s

w i t h p = p ( s ) , q = q ( s ) , and dp = d p ( s ) ( s = (s1,s2))

-q ( s )

For x

satisfying p(s) =

-

0, t h e n T > 0 f o r KP ( c f . [ P21). T h i s s i t u a t i o n p r e v e n t s s i n and dp 2 g u l a r i t i e s i n u = 2a 10gT and i s s a t i s f i e d e.g. when p,q = i k w i t h k r e a l . There a r e many examples i n [ P2] o f s o l i t o n s o l u t i o n s f o r KP and we w i l l n o t

d w e l l on t h i s here ( c f . a l s o § 9 ) . Next c o n s i d e r t h e v e r t e x o p e r a t o r e q u a t i o n ( V O E ) as i n (7.19). the proof, f o l l o w i n g

We s k e t c h

P2 1 o f

CHE0REm 7-16, L e t F be as i n (7.76) w i t h A a r e g i o n As as above w i t h Rep(s)

VERTEX OPERATOR EQUATION

93

< - 6 < 0 < 6 < Req(s). Assume dv i s suitable so t h a t T ( X ) i s well defined a s a Fredholm determinant. Then f o r Rek < 0, X ( k ) . r ( X ) = w ( k , x ) T ( x ) .

Proof: First note t h a t regions w i t h p a n d q on the imaginary axis can be envisioned via analytic continuation o r via suitable hypotheses on d u , preserving algebraic relations; the theorem here i s semiformal since du i s n o t made precise, e t c . In any event as in (7.64) we write w = (ltK,)exp(S(k,x)) and the VOE i s equivalent t o (7.81)

( G ( k ) - 1)T(X) =

D(x,z)ek('l

-

'l)dzl

(recall xn = zn f o r n 2 2 and G ( k ) . r = T - = ~ ( x ~ - l / k , x ~ - l ',...)). /Zk The proof here (from [ PZ]) uses a l o t of determinant calculation and i s rather long; however i t reveals a great deal of important structure which we will need t o refer t o l a t e r . There i s another proof i n [ P1 I which does not use the e x p l i c i t form of F. Thus from (7.78) the general term in the s e r i e s f o r " T i s (l/n!)& I" det ((F(nj,nk)))ndyi where nj = (yjyx*,...). Applying 1 XI the vertex operator t o t h i s general term we find t h a t G(k)/,m det ( ( F ( n j , n k ) ) ) ndyi = Ixy d e t ( ( G ( k ) F ( q j . n k ) ) ) ndyi. T h i s ' i s veiy cute. The e f f e c t of G ( k ) i s t o s h i f t the lower l i m i t s of integration in the intehen change gral t o xl-l/k and from xn t o xn-l/nkn inside the integrand.

...

. .,,"

...c

the variables of integration y

j

-f

y.+l/k. J

Now

where p = p(sj) a n d q j = q[s.) and Sn is the permutation g r o u p . B u t one j J has G(k)exp(S(x,p)) = exp(S(x,p))(l-p/k) (exercise) and applying t h i s t o

(7.82)we get G(k)det((F(nj,nk))) = le5n(-1 1.n; I exP(S(vjyPj) - S ( n r ( j ) ~ q j ) ) ((k-pj)/(k-qj))du(sj). Integrating with respect t o y l , ...yn and interchangi n g the order of integration gives

ROBERT CARROLL

94 where dp = d u ( s l )

...d p ( s n )

a n d d y = dyl

...dy,.

The i n t e g r a t i o n s w i t h res-

p e c t t o t h e yi v a r i a b l e s a r e e a s i l y c a r r i e d o u t p r o v i d e d Re(p.-q ) < 0 f o r J k a l l j , k and one g e t s

=

I ...I E n ( x , s ) h n D n d p

-

...

w h e r e E,(x,s) = e x p ( S ( x , p , ) + + t;(x,Pn) - S(x.q,) ....- S ( x , q n ) ) , A n n IT1 ( k - p j ) / ( k - q j ) , a n d Dn = d e t ( ( l / ( q -p 1)). One i s a b l e t o c o n c l u d e t h a t j

k

=

.

t h e g e n e r a l term o f ( G ( k ) - l ) . r ( x ) is ( l / n ! ) f ..f E n ( h n - l ) D n d p . T u r n i n g t o m m t h e r i g h t s i d e o f ( 7 . 8 1 ) we n e e d t o c a l c u l a t e t h e i n t e g r a l ( l / n ! ) J x ...I

...

;:(F

)

RD)exp(k(zl-xl)dzldy l...dyn nh

no = ( z l , x *,...I (7.83)

where

TI

j

*,...) .

= (yjyx

X

Let us p u t

a n d s ' u b s t i t u t i n g F from ( 7 . 7 6 ) we g e t

(l/n!)l dp

1

(-1)ITjm

msnti

...Ix, e k(Z1-X1

e - ~ ( ~ I T ( 0 ) Y q o- ) I n t h i s i n t e g r a t i o n du = dp(so)

.*.

-

c(nIT(n)Yqn) dy

...d v ( s n )

a n d t h e p i n t e g r a t i o n is a n n+l f o l d The y i n t e g r a t i o n r e d u c e s t o a p r o d u c t

i n t e g r a l w h i l e d y = dzldyl.. .dyn. e ~ ( x ~ ~ o ) ...I., - ~ ~ el k kz 1~-S(z,qIT-i

(o))n;

Carrying out t h e integrations gives Summing o v e r

) e S ( ~ y ~ o ) .+.+S(X,P,) .

x,

e S ( Q j s P j ) - c ( l l j s q I T - i ( j )1d y

(q

e x p ( S ( x , p .J) - S ( x , q J.))/(q ~ - ' ( 0 ) - ~ ) i n Sn+l i n ( 7 . 8 3 ) we g e t EoEnDn+l,O w h e r e

(qIT-j(l)-pl).... Eo = e x p ( ~ ( x , p o ) - S ( x , q o ) ) , En i s g i v e n a b o v e , a n d Dn+l,O i s t h e d e t e r m i n a n t o b t a i n e d from Dn+l by r e p l a c i n g po by k. Now by r e l a b e l i n g t h e v a r i a b l e s i n a n o b v i o u s way t h e i d e n t i t y ( 7 . 8 1 ) i n v o l v e s (7.84)

(l/n!)j

IT

...I EnDn(An-l)dp

= (l/(n-l)!)J

...I EnDn,ldp

...,s n a n d

is t h e d e t e r D n,l m i n a n t o b t a i n e d by r e p l a c i n g p1 by k i n D n . Now t h e r e i s a c e r t a i n s y m m e t r y On t h e o t h e r hand En is symmetric i n ( 7 . 8 4 ) , namely p1 i s m i s s i n g i n D n , l . i n s1 .,s If we i n t e r c h a n g e s1 a n d s . i n t h e r i g h t hand s i d e o f ( 7 . 8 4 ) n' J

w h e r e t h e v a r i a b l e s o f i n t e g r a t i o n r u n o v e r sl,

,. .

FOR KERNELS

SPECTRAL FORMS

95

i s transformed i n t o Dn

t h e n Dn,l

Therefore t h e r i g h t s i d e o f (7.84) can ,j* be w r i t t e n as a sum o f n terms and (7.81) reduces t o showing (+&) Dn(An-l) =

Dn,l

.-. +

+

where D i s o b t a i n e d by r e p l a c i n g p . by k i n t h e d e t e r n ,j J The determinants Dn and Dn,, can be e v a l u a t e d by u s i n g t h e

DnynS

m i n a n t f o r Dn.

i d e n t i t y ( c f . [ PY1 I)

w i t h a s i m i l a r e x p r e s s i o n f o r Dn,k. t h a t Dn,,,/Dn

C a n c e l l i n g o u t common f a c t o r s we f i n d

= (An/(p,-k))Q(pm)/P‘(pm)

where P(k) =

ny

(p.-k) J

and Q ( k ) =

Thus (+&) can be w r i t t e n as An-1 = A n I Y Q(pm)/(pm-k)P’(pm).

IIc (qj-k).

D i v i d i n g t h i s e q u a t i o n by An i t s v a l i d i t y f o l l o w s from t h e p r i n c i p a l p a r t s expansion f o r t h e meromorphic f u n c t i o n A-1 ( k ) = Q ( k ) / P ( k ) .

QED

REEIARK 7.17 (C0mPLECENE.SS AND SPECCRAL F0wnd FOR KERNELS). L e t us w r i t e w T -1 (l+K+)exp(S) and w* = (l+K+) exp(-S). One w i l l have w and w* a n a l y t i c i n some l e f t h a l f p l a n e w i t h a s y m p t o t i c b e h a v i o r wexp(-S) exp(S)

Q

Q

=

1 + O ( l / k ) and w*

1 + O ( l / k ) ( f o r p r o p e r t i e s o f KP wave f u n c t i o n s see e.g. [ AB4;MNl

I).

Assume we can work i n Imk 5 0 and we w i l l show t h a t ( c f . (1.13)) (7.86)

(l/Zni)Li:

w(x,k)w*(y,k)dk

( r e c a l l xn = yn f o r n 2 2).

= 6(x-y)

W r i t e (l+K;)-’

= 1

+ L- where L- i s l o w e r t r i -

a n g u l a r ( t a k i n g t h e transpose reverses t h e t r i a n g u l a r i t y ) .

Writing t h i s out

one has(1 + K + ) e x p ( € ) - ( l + L-)exp(-S) = e x p ( k ( x - y ) + e x p ( k x ) / z L-(y,n)exp (-kn)dn + exp(-ky)lxm K+(x,s)exp(ks)ds (-kn)dn.

+

I,”

K + ( x , s ) e x p ( k s ) d s i t L-(y,n)exp

Now r e c a l l t h a t Laplace t r a n s f o r m a t i o n t h e o r y g i v e s (1/2ni)[;:

+ I,”K+(x,s) K+(x,s)L_(y,~)s(s-n)dsd~ = ~ ( x - Y )+

exp(kz)dk = 6 ( z ) so one o b t a i n s o n t h e l e f t i n (7.86) 6 ( x - y ) G(s-y)ds +

/z

+ I,” $ K + ( ~ , ~ ) L - ( y , s ) d s (upon i n t e g r a t i o n

L-(y,n)G(x-n)dn

+ L-(y,x) +

But T f o r y < x t h e l a s t 3 terms a r e 0 w h i l e f o r y > x we w r i t e o u t ( l + K + ) o ( l + L - ) K+(x,y)

= 1 t o g e t K+(x,y)

+ L-(y,x)

t i o n s h o l d by c o n t i n u i t y as y

+ -f

K + ( x , ~ ) L - ( y , s ) d s = 0. x and t h i s proves (7.86).

i n (7.86)).

The k e r n e l rela; Now u s i n g (7.86)

we can f i n d f o r m a l l y s p e c t r a l expressions f o r v a r i o u s k e r n e l s as i n (1.22). Thus ( c f . [C6,13]) (7.87)

P(x,y)

f o r P = 1 + K+ = kerP(y

-f

x) = (1/2ni)jl~w(k,x)e-~(kyy)dk

ROBERT CARROLL

96

(under the k i n d s o f hypotheses i n d i c a t e d ) . To s e e this t h i n k of P a s chara c t e r i z e d by i t s a c t i o n on e x p ( S ) and w r i t e formally ( P ( x , y ) , e x p ( c ( m , y ) ) ) =

/Iw(kyx)(l/2ni)[:

exp((m-k)x)dxdk = /f w ( k Y x ) 6 ( m - k ) d k = w ( m , x ) . We a r e thinking here o f s i t u a t i o n s like (7.80) where Fourier theory i s natural and use here a two sided Laplace transform. Similarly l e t Q(x,y) = ker(P*)-’ ( y -+ x ) and one shows (7.88)

Q(x,y) = ( l / 2 ~ i ) f i zw*(k,x)e 5 ( k t Y ) d k

Again P*-’ i s characterized by i t s a c t i o n on exp(-6) so a computation a s T above gives (7.88). We s e e a l s o t h a t P ( x , y ) ’L P ( y , x ) % kerP*: y -+ x so 1 t h a t formally one has P*-’(l/2ni)JC w(y,k)exp(-S(k,x))dk = ( 1 / 2 n i )

PiitPs

JC w(y,k)w*(t,k)dk = 6 ( y - t ) by (7.86). Regarding t r i a n g u l a r i t y we r e c a l l w(x,k)exp(-E(x,k)) ‘L 1 + O(l/k) f o r Rek < 0 and I k l w i t h w(x,k) analytic -+

Consider t h e n i n P(x,y), w(x,k)exp(-E(k,y)) = w(x, For x > y one can c l o s e C = (-i-,i-) t o t h e k)exp(-S(k,x))exp(S(k(x-y)). l e f t t o get P ( x , y ) = 0. S i m i l a r l y i n Q(x,y) one has w*(x,k)exp(f(k,y)) = w*(x,k)exp(E(k,x))exp(S(k,y-x)) and f o r x < y one c l o s e s t o t h e l e f t t o get

i n some l e f t half plane.

NXSY) = 0. Now consider 1 + K- = (1 + K+)(1 + F) w i t h kerF = F a s i n (7.76). We can use the s p e c t r a l formula now f o r P = ker(l+K+) t o obtain (ker(l+F) = 6 + F , E % ker(1 + K-)) (7.89)

J*

-5(q*t)

E(x,t) = (P(x,y),G(y-t) + F ( y , t ) ) = P(x,t) + (1/2ni)jc w ( x , k ) { e-S(k’Y),eE(PyY))d~dk=

P ( x , t ) + f, w(xYp)e-‘(qyt)dp

( s i n c e (exp(-S(k,y)),exp(E(p,y))) = Pai6(k-p), e t c . ) . Note a l s o ( u s i n g (7. Thus i f one has 8 6 ) ) (Q(x,y),w(x,m)) = exp(E(m,y)) so Q(x,y) ‘L P-’(y,x). the canonical f a c t o r i z a t i o n 7 + K- = (l+K+)(l+F) w i t h K- ( x , t ) = 0 f o r t > x t h e n i n analogy t o (1.22)-(1.23) we obtain formally PROPO9)SIZIO)N 7-18. Given a canonical f a c t o r i z a t i o n 1 + K- = ( L + K + ) ( 1 + F) w i t h F determined by (7.76) and P ‘L 1 + K, g i v e n i n (7.87) one has formally

SPECTRAL FORMS FOR KERNELS Observe here from (7.90)

-1,

axlog.r =

(7.91)

97

t h a t one can w r i t e f o r m a l l y w(x,p)e -5 (PY )dp

which has a c e r t a i n charm ( c f . [C6,13]), L e t us emphasize however t h a t f o r KP i n general l+k- i (l+K:)-’ For KP we have (C

i s t r u e f o r KdV). (-S(k,y))dk

(y

+

%

1+K+

(-im,i=))

%

(recall this

( 1 / 2 n i ) J C w(x,k)exp

( 1 / 2 n i ) l C w*(x,k)exp(S(k,y))dk

x ) and (l+K:)-’

(y

+

x).

Thus f o r m a l l y (l+K:)-’(w(x,m))

(7.92)

=

lc e S ( k/2ai yy )f )w*(x,k)w(x,m)dxdk (l

and one wants here t h e o t h e r h a l f o f t h e completeness r e l a t i o n (7.86), namWe n o t e t h a t o u r c o n s t r u c t i o n e l y ( W ) ( l / Z d ) J w*(x,k)w(x,m)dx = 6(k-m). was based o n P*-’

o f (l+K;)-’

w h i l e by d e f i n i t i o n s (l+K+)-’:

w

-t

exp(S).

T h i s says t h a t f o r m a l l y ( W )

One is perhaps “ d r e s s i n g ” a completeness r e l a t i o n here.

must be v a l i d .

f o r KdV one has ( c f . Remark 1 . 2 ) f o r K- d e f i n e d v i a ? o r formula ( c f . (7.93)

(*+),I. The f a c t o r i z a t i o n

(l+K+)-’(l+K-)

= (1/2n)‘)’

(1/2a)’(f

f

= w*

b e i n g c h a r a c t e r i z e d b y P*-’(exp(-S))

Q ,

(f+(x,-k)

Thus 1+K- = ( l + K I ) - ’

%

T(m)f-(x,m)eimSdmdkdx

T(m)f-(x,m)eimS+ikydmdkdx

f+(x,k)T(m)f-(x,m)eimS-ikydmdkdx

= 6(5-y) + F(S,Y)

1

same

1+K- = ( l + K + ) ( l + F ) a l s o h o l d s s i n c e

( 1 / 2 ~ ) ~ff T(k)eikYf-(x,k)j

+ R(k)f+(x,k))j

v i a (1.18),the

Now

+

=

f f R(k)f+(x,k)T(m)

+

F

i n Remark 1.2 agrees w i t h t h e f a c t o r i z a t i o n 1+K- o f

and i n v o l v e s T f - = f; + For KP one would need some formal e x p r e s s i o n (+.) w*(k,x) = w(-k,x)

( U ) . The reason f o r t h i s i s c l e a r from (7.93),

Rf+.

+ 2 a i I A G(k+q)w(x,p)dp (7.94)

(l+K;)-’

f o r example.

Then f o r m a l l y ( C

= (1/2ni),fC eS(k’y)w(x,-k)dk

( 1 / 2 n i ) f C e S(-kyY)w(x,k)dk

+

%

(-im,im))

f,w ( x y p ) e E ( - q y x ) d p =

+ I, w ( x y p ) e - E ( q y y ) d p

98

ROBERT CARROLL

( c f . (7.90)). I n o r d e r t o have C(-k,y) = -S(k,y) we want yi = 0 f o r i even ( t h i s i n v o l v e s e.g. t h e BKP h i e r a r c h y - c f D1,31). Then f o r m a l l y (l+K;)-’ = (ltK,)

t

p e c t (*.)

J h w(x,p)exp(-C(q,y))dv.

I n general however one s h o u l d n o t ex-

o r any v e r y n i c e r e l a t i o n between w and w*.

Indeed Lw = kw and

L*w* = kw* and i n general one does n o t have s o l u t i o n s o f t h e same d i f f e r e n t i a l e q u a t i o n as w i t h T f - and f+. We n o t e f o r BKP, L* = -aLa-’ and L(a-’w*) 1 = - k ( a - w*) = - k ( a - l w * ) so perhaps a - l w * ,I, w ( - k ) i n some sense.

RrmARK

7-19 (UARZACZBW

ON DECERmZNANC tALtULAEZ0W).

I n [ P1 ] t h e idea i s

t o g e t as much m i l e a g e as p o s s i b l e o u t o f r e l a t i o n s l i k e

(7.95)

atdet(l+F) = det(1tF) Tr(Ft(l+F)-’ )

One d e f i n e s F x @ ( x ) = Jm F(x,y)$(y)dyl x2 ca

XI

,...;o+xl,y2 ,... ) @ ( o ; x 2,... )do

Jo F(s+xl,x2

,... ;s+xl,y2 ,... )ds

change o f v a r i a b l e s y1 = s+xl.

(as b e f o r e ) and F x @ ( s ) = 1; F(s+xl,

( r e c a l l xn = y, =

TrF,

/x

m

=

for n

F(y ,x

2).

Then TrFX =

,...;y1,y2 ,... )dy A

S i m i l a r c o n s i d e r a t i o n s a p p l y t o TrF

the Plemelj-Lidskij results (cf. [ SIMl

I)

via a

so by

one has t h e n d e t ( l + F X ) = d e t ( l + F x ) .

We w i l l use these f a c t s a l s o i n 511 f o r t h e H i r o t a formula. i n t e g r a l o p e r a t o r s A$(s) = J r a(s,u)$(o)do

Now d e f i n e f o r

(.*) AL$(s) = aLA@(s) = J;

asa(s,

It f o l l o w s t h a t e.g. o)$(o)do; AR@(s) = aRA$(s) = 1; (a,a(s,u))@(o)do. (AB)L = ALBL b u t (AB)R = ABR and an o p e r a t o r [ A 1 i s d e f i n e d as [ A 1 = -a(O,O)

( c f . a l s o [P31), so t h a t e.g.

[ A ] = Tr(AL + AR).

[ A ] can be determined and used i n p r o v i n g e.g. 7.14) by e n t i r e l y d i f f e r e n t methods.

Many o t h e r p r o p e r t i e s o f

t h e VOE f o r KP ( c f . Theorem

The p r o o f t h u s appears more e l e g a n t

b u t i t i s j u s t a s l o n g ; no a t t e m p t i s made t o compare hypotheses s i n c e t h e format i n v o l v e s d i f f e r e n t k i n d s o f p r o p e r t i e s .

99

CHAPTER 2 SYSTEMS AND ALGEBRAIC METHODS

8, ORBZCS 0F CHE UACLlm.

from C K 1 , Z I

The m a t e r i a l i n t h i s s e c t i o n i s l a r g e l y e x t r a c t e d

( c f . a l s o EMLzJ).

There i s l i t t l e one can do t o improve on

CKIJ

as an i n t r o d u c t i o n t o h i g h e s t w e i g h t r e p r e s e n t a t i o n s o f i n f i n i t e dimensional L i e a l g e b r a s b u t we w i l l add and s u b t r a c t and m o d i f y a b i t i n hopes o f u s i n g t h i s approach a l s o as an i n t r o d u c t i o n t o t h e whole area o f L i e a l g e b r a s and For some b a s i c ideas see Appendix A.

representation theory.

t o p i c s i n a way t o m i n i m i z e t h e p a t h t o t h e KP equations.

We w i l l p i c k

Some p r o o f s o f

p u r e l y a1 g e b r a i c m a t t e r s w i l l be o m i t t e d w i t h a p p r o p r i a t e r e f e r e n c e s p r o v i d e d o f course.

There i s more on Kac-Moody a l g e b r a s i n 610 and more r e -

l a t e d a l g e b r a i n Chapter 3.

REIURK 8.1 (ZNFZNZCE DZFIIENSL0NAt U E ACPjWW AND ECEtXR0W). L e t V = aZ C v . be an i n f i n i t e dimensional complex v e c t o r space w i t h b a s i s I v . 1 . One J J t h i n k s e.g. o f v . as a column v e c t o r w i t h 1 i n t h e jthp o s i t i o n and 0 e l s e J where.

An element i n V has o n l y a f i n i t e number o f nonzero components a

j' z; a l l b u t a f i n i t e number o f aij a r e 01. L e t Define g l m = {((a..))iyj 1J E.. have 1 as t h e (i,j) e n t r y and 0 elsewhere so t h e Eij form a b a s i s o f gl, TJ E v i d e n t l y E..v = 6 . v . and E. .E = GjmEin w i t h (*) [EijyEmnl = 6 j. m E i n 1J k Jk 1 I J mn gl, can be viewed as t h e L i e a l g e b r a o f GLm = { A = ( ( a . . ) ) , i,j E 'ni E m j * 1J Z; a l l b u t a f i n i t e number o f aij 6ij a r e 0; A - l e x i s t s ) . Define also

-

-

= {((aij)),

i , j E Z; aij

number o f nonzero d i a g o n a l s , and (8.1)

Akvj

= v ~ - ~A k ; =

and e v i d e n t l y [ A

A,) jy

li-jl >>01.

= 0 for

Bm i s

Then i n

a L i e algebra.

xm one

zm

has a f i n i t e

D e f i n e now

1 Ei,i+k

= 0 w i t h AkE

,A

Now one goes t o t h e D i r a c t h e o r y o f e l e c t r o n s ( c f . [ K1 I f o r p h i l o s o p h y ) .

....

L e t Fo = A l V be Consider AmV w i t h vacuum v e c t o r $o = vo A v - ~A v - ~A where ( 1 ) io > i-l> t h e " s t a t e space" w i t h elements )I = v. A v . A 10

1-1

...

-.-

ROBERT CARROLL

100

(2) i k = k f o r k << 0. Thus J, has an equal number of electrons (positive index o r energy) a n d holes ( a n unoccupied negative index or energy). One defines the energy or degree of J, as (8.2)

deg(q) =

1;

(i-s

+

1 (is

s) =

> 0 present)

For k a positive integer l e t k = ko t kl + This partition defines a unique kn-l. (8.3)

JI = v . A v . A Jo

J-I

... A v .

A

... t

kn-l

v - ~A v - ~ - A~

J-,+,

- 1 i s 5 0 absent) with ko ?, kl

2

...

...

with j-i = ki-i (i= O,...yn-l). For example take k = 6 = 3 t 2 t 1 so jo = j-, = 3, j-, = 2-1 = 1 , and j-, = 1-2 = -1. Thus J, = v3 A v1 A v - ~A v - A~ (n=3). Then deg(q) = 3t1-(0-2) = 6 = k. Clearly i f i s the l i n e a r span of vectors of degree k and p ( k ) i s the number of partitions of k then

$

...

(8.4)

$;

Fo = 8 k€Z

F: = C q o ; dim(Fi) = p ( k )

...

W e can also deal w i t h Fm = linear span of J, = v . A v A where ( A ) i m > Im im-, im-l > ( B ) i k = k t m for k < < O (the reference vacuum i s qm= V m A Vm-l A 1. Then

...

(8.5)

...

deg(q) =

1;

1( i s

(im-s+s-m) =

>

m present)

-

( i s 5 m absent)

and one writes F = 8Fm = AmV ( F = space spanned by semi i n f i n i t e monomial s ) . In physics m i s called charge. As for Fo) one has Fm = 1 FF with dim(FL) = p ( k ) ( c f . ( 8 . 4 ) for Fo).

REmAaK 8.2 (CR0W MD AtC€3BG REPRECIENCACZ6W).

Next one defines represen-

tations R of GLm a n d r o f glm in F by

(8.6)

R(A)(vi

A I

v . A ...) 12

=

Av. A Avi A 'I

...

1

...; r ( a ) ( v i I

A vi A

... )

=

2

(note r(E. . ) ( v i A v i A ) = 0 i f j 4 { i l y i2,...} and = vi n . . . h v i A 1J I z &-I vi A v . A if j = ik). W e recall ( c f . [MLZ;ST8;VR1,21) t h a t for f i n i t e 1k+r dimensional F a representation of a Lie group G (resp. l i e algebra 5) on F

...

REPRESENTATION3

101

i s a homomorphism s: G -+ G L ( F ) (resp. d s o r P: + L(F) = g l ( F ) ) ; r e c a l l GL(F) r e f e r s to i n v e r t i b l e o p e r a t o r s , which a r e represented by matrices f o r dimF < m ( c f . Appendix A ) , and s is a l s o assumed here t o be Cm o r a n a l y t i c . For i n f i n i t e dimensional r e p r e s e n t a t i o n s one needs t o t h i n k o f various matt e r s such a s the existence o f a dense subspace o f a n a l y t i c vectors i n domain d s ( a ) f o r a l l a , b u t he simply do not need t o go i n t o a l l t h a t . T h i s will be c l e a r from the development which follows. Now from (8.6) one has the rel a t i o n (). e x p ( r ( a ) ) = R(exp(a)) and we observe t h a t r: Fm + Fm so r = @ rm where rm '~rr i n ( e x e r c i s e - note J, = v . A A vim-; v ~ - ~ - , 5 ltn N o w define a p o s i t i v e d e f i n i t e Hermitian form r(Eim,m). .r(Eim-k, m- k )$,,,). ( 1 ) on F by taking semi i n f i n i t e monomials J, a s above t o be an orthonormal b a s i s . Let w(a) = aT = at and i t i s immediate t h a t ( 0 ) ( r ( a ) $ l $ ' ) = (J,lr(a+) $ I ) which implies t h e r e p r e s e n t a t i o n r : g l m + L ( F ) is u n i t a r y (and ( , ) i s c o n t r a v a r i a n t ) . The decomposition F = @? is a l s o orthogonal and one can show t h a t the rm a r e i r r e d u c i b l e ( c f . [ K1,4] f o r proof - t h i s means i n part i c u l a r t h a t ? does not have a proper nonzero i n v a r i a n t subspace under rm

...

...

.

(a)).

Now elements of

Am

have t h e form a k = X i E i , i t k and i n representing a k i n L ( F ) via r an adjustment must be made f o r k = 0 ( s i n c e r(a0)$,,, = ( A m t Am-l t ...)I),,, formally and the s e r i e s could diverge). Hence define (8.7)

m

(E..) = r (E..), i # j o r i m

1~

1~

= j >

0;

r^m ( E 1. .J)

=

r(E..)-I (i ( 0 ) 1J

mtl

Then :(ao)$,, = (17 hi)@, i f m 2 1 ; = hi)$, i f m 5 -1; and = 0 i f m = 0 . Also one checks e a s i l y t h a t ( 4 ) [; ( E . .),; ( E . . ) ] = Fm(Eii) r*m ( EJ.J. ) + m ij m ji a ( E i j Y E j i ) I where a(Eij.Eji) = - a(Eji,E. .) = 1 f o r i 5 0 , j 2 1 and u ( E i j y

-(lo

Emn ) = 0 otherwise. one has

zm

-

1J

The o t h e r brackets behave "normally" and summarizing

Now extend Gm t o by l i n e a r i t y t o get a p r o j e c t i v e representation (due t o t h e a term). T h i s can be made i n t o a l i n e a r representation of t h e c e n t r a l extension Am = Am @ Cc of Am w i t h c e n t e r Cc and bracket (+) [ a , b l = ab - ba t a ( a , b ) c by s e t t i n g F m ( c ) = 1 ( c f . [K1,2,4;02] f o r background). Here a ( a ,

102

ROBERT CARROLL

b ) i s a 2 cocycle, l i n e a r i n each v a r i a b l e , and d e f i n e d above on t h e Eij. One extends w t o Am b y w(c) = 1 and t h e r e p r e s e n t a t i o n s

qm a r e

then u n i t a r y

I n p a r t i c u l a r we l o o k a t t h e commutative subalgebra A C

as w e l l .

t e d by t h e s h i f t o p e r a t o r s Ak o f (8.1).

-

t i v e representation -central

genera-

One has ( e x e r c i s e )

-

Fm o f

T h i s w i l l correspond t o a f e r m i o n i c u n i t a r y r e p r e s e n t a t i o n l e d o s c i l l a t o r a l g e b r a ( c f . below

A,

t h e Fm spaces

-

"fermions"

t h e so c a l

-

the projec-

e x t e n s i o n i s c r u c i a l i n d e t e r m i n i n g t a u below).

Now t h e o s c i l l a t o r a l g e b r a A i s a complex L i e a l g e b r a w i t h b a s i s {an, n

E

Z,

A) s a t i s f y i n g (8.10)

[%,an]

Thus [ao,an] C[ x1.x2,

=

0; [a,a,J

= 0 so a.

= m6m,-n

(m,n

E

Z)

One speaks o f a Fock space B =

i s a c e n t r a l element.

. . . I o f polynomials i n i n f i n i t e l y many v a r i a b l e s .

Then t h e f o l l o w -

i n g r e p r e s e n t a t i o n o f A i s f a m i l i a r from elementary quantum mechanics. (8.11)

an =

Enan;

-1 amn = T ~ nx, E ; ~ a.

= PI; $ =

Take

51

One checks e a s i l y t h a t i f % # 0 t h i s r e p r e s e n t a t i o n o f A o n B i s i r r e d u c i b l e (exercise

-

n o t e any polynomial

+.

1

N

vacuum by successive a p p l i c a t i o n s o f

t h e an, n > 0 ( a n n i h i l a t i o n o p e r a t o r s ) , w h i l e a_,, generate a r b i t r a r y p o l y n o m i a l s from 1 ).

n > 0 (creation operators)

This r e p r e s e n t a t i o n o f A would be

c a l l e d a bosonic r e p r e s e n t a t i o n and one checks t h a t A 2 A .

Further there i s

d e f i n e d an a n t i l i n e a r a n t i - i n v o l u t i o n w on A by w(an) = a_,, = a: = h.

with

w(n)

Then g i v e n a r e p r e s e n t a t i o n o f A on a space V h a v i n g a (vacuum) v e c t o r

v w i t h % ( v ) = Rv ( R

4 0) t h e monomials a!k)v

=

... a_k;(:v)

are linearly

independent; i f t h e y span V t h e r e p r e s e n t a t i o n i s i r r e d u c i b l e and e q u i v a l e n t D e f i n e now on V a form

t o t h e r e p r e s e n t a t i o n on B above. ( v l v ) = 1 and s t i p u l a t e t h a t a;!

... a > ( v )

w i t h norm (8.12)

( a -kv a k- v ) = :I

kj!(h/jIkj

(

I

)

such t h a t

should form an o r t h o g o n a l b a s i s

BOSON FERMION CORRESPONDENCE These p r o p e r t i e s d e f i n e c f . [ K1 I ) .

(

103

-

( u n i q u e l y ) as a c o n t r a v a r i a n t f o r m ( e x e r c i s e

)

Think now o f V = B and d e f i n e n e x t t h e vacuum e x p e c t a t i o n v a l u e

f o r a n a r b i t r a r y polynomial i n B as ( m ) ( P ) = c o n s t a n t t e r m i n P. that (w(P)) = P,Q

E

(P)

where w ( P ) = F ( a k / k ) ( t a k e TI =

E,

One sees = 1 h e r e ) and f o r any

B define

T h i s w i l l be a c o n t r a v a r i a n t H e r m i t i a n form and hence e q u i v a l e n t t o (8.12) above by uniqueness.

R m R K 8.3 ( W E B0S0N FElKIZBIQ CORRUP0NDENCE). boson f e r m i o n correspondence.

Thus we have a f e r m i o n i c r e p r e s e n t a t i o n

o f t h e o s c i l l a t o r algebra A 2 A

)$m ( 0 < kl 5

...rm(Aik,

m

F

= $Fk.

One can now d e s c r i b e t h e

A

Pm

C Am on Fm such t h a t elements (**) r.,(A-

... 5 k;,

k =

1 ki)

form a b a s i s o f

kS

)

and r e c a l l

This r e p r e s e n t a t i o n i s isomorphic t o t h e bosonic r e p r e s e n t a t i o n

,...

m

C[xl I ( i . e . B i s a copy AB A L e t r = u!rmuil be t h e t r a n s p o r t e d

o f A on B v i a (8.11) and we s e t om: Fm -+ Bm

We n o r m a l i z e b y mapping $, 1. AB A m r e p r e s e n t a t i o n o f A on Bm ( i . e . r ( A . ) = u r (A.10:). Corresponding t o m~ m m J ? f3F: we have Bm = $6; ( k E Z+) where deg(x.1 = j (exercise see [ K1 I J f o r d e t a i l s h e r e ) . One a l s o has a t r a n s p o r t e d c o n t r a v a r i a n t form ( ) *B s a t i s f y i n g ( 1 11 ) = 1 and ;:(Ak)' = r m ( A - k ) . By uniqueness o f such forms o f 6).

-+

-

I

(

I

agrees w i t h (8.13).

)

There a r e now two q u e s t i o n s about urn ( w h i c h r e -

presents t h e boson f e r m i o n correspondence). (vim

A v.

lm- 1

A

... )

F i r s t f i n d t h e polynomials om

and second, extend t h e r e p r e s e n t a t i o n o f A t o Am (see

114 f o r a n o t h e r r e a l i z a t i o n o f t h i s v i a f r e e f e r m i o n o p e r a t o r s ) . For t h e second q u e s t i o n i t i s e a s i e r t o work w i t h u = @am: F = @Fm -+ B =

@Em, where i n B one i n t r o d u c e s an i n d e x i n g parameter and r e p l a c e s Bm by zm -1 Bm so t h a t B becomes C[xl,x ...A;z,z-1 -1 I = C[x j y *z,z I ( x = (xlYx2 ,... ). -1 Define now l i n e a r f u n c t i o n a l s v*: E V* W r i t e t h e n rB= uru and r = uru

,%Iy

= a l g e b r a i c dual o f V by 6ij

.

= v*(v.)

J

I

w i t h V * = $Cv*.

J

t i o n o p e r a t o r s a r e d e f i n e d as f o l l o w s . (8.14)

^v(vi,

A

V

il

= f(Vi)Vi, I

A

...) A

V i

= v A vi 3

A

..,.-

For v

I

A

Vi

f(V.

'2

2

J Wedging and c o n t r a c -

E

V and f

A

...; i ( v i

)Vi I

A Vi 3

V* w r i t e

E

A

I

A vi

... +

a

A ...)

.-.

1 04

ROBERT CARROLL

y?J

A

Note t h a t vi and

a r e adjoint r e l a t i v e t o

These operators

jy

( [ a , b l + = a b + ba)

I

)

and G ( E . .) =.;:Gi

Hence

1J

i , j E Z l generate a Clifford algebra determined by

One checks e a s i l y from (8.15)-(8.16) t h a t

;*Ik

(

=

(*A) [ ; ( A . ) , c k ]

J

ck-j and

[

P ( AJ . ) ~

= -;;+j.

v.v? i n Fm. Since the transforms o f ti Now one wants r B ( E . .) where E i j 1J 1 J and ?; are complicated i t i s e a s i e r t o work w i t h generating functions, so J one considers A

(8.17)

X(u)

=

1

jEZ

u j c j ; X*(u) =

9

1

j€Z

u-j;?

J

where u E C , u 4 0. Since X ( u ) i s defined by an i n f i n i t e s e r i e s i t maps F” into the formal completion ^$“‘l in which i n f i n i t e sums of semi i n f i n i t e Write ? = @? so monomials are permitted. Similarly X*(u) maps Fm A t h a t ~ X ( U ) I J ” and uX*(u)u-’ map B 4 where B denotes formal power s e r i e s From the formulas above i n (x;z,z-’) which a r e polynomial i n z and z-’. -+

6

A

*

These equations hold in F a n d under u : F (8.19)

AB

r (A.) J

= $(A.)U-’

J

=

a/ax

j

=

^Fm-’.

A

A

-+

a j ’-

B they will hold in B.

P B (-J~.)

= jx

Thus

j

in particular. Define now vertex operators r ( u ) = uX(u)u-l a n d r*(u) = uX*(u)o-’ a n d one has immediately (*.) [ a r ( u ) ] = ujr(u) with [ x r ( u ) l = jy jy (u-J/j)r(u). Similar formulas hold for r*(u) via (8.18). Such relations determine r ( u ) and r*(u) u p t o constants a n d one has

PR0P0SICZ0N 8.4. (8.20)

r(u)

On

^Bm

= u m+l ze

r*(u)

1” u j x j e-l; ( u - j / j ) a j . , = u -m z -1 e -1“ u J x j exp(1;

(u-j/j)aj)

VERTEX OPERATORS Proof:

(1;

L e t ( T U f ) = f(xl+l/u,x2+1/2u

(u-j/j)aj).

-(u-j/j)Tu.

(1; with

L

,... )

105

so by Taylor’s formula TU = exp

One checks t h a t I x r ( u ) T U I = 0 from (**), and [x.,T ] = j’ J U This means r(u)Tu has no d i f f e r e n t i a l p a r t so r ( u ) = zg(x)exp

-

( u - j / j ) a . ) w i t h g t o be determined (exercise c f . [ K21). From (**)a J [ a exp(-Z; u j x j ) l = - u j e x p ( - l y u j x . ) one gets [ a j’ exp(-I; u j x j ) r ( u ) l jy

= 0 and consequently r ( u ) = cm(u)zexp(ll

mJ

uJxj)exp(-ll

.

(u-J/j)aj).

c (u) i s obtained from observing t h a t t h e c o e f f i c i e n t o f $m+l m m+l expansion o f X(u)$, i s u S i m i l a r l y one gets (8.20)b.

in

The term

Pt’

i n the

.

QEO

BEIIAltK 8-5 (IIORE 014 OERCEX OPERAC0lU AbID SlUfR fllwCFXW).

Define now R(u)

f(x,z)

= uzf(x,uz)

Then (8.20) (8.21)

so f o r f(x,z)

= zmg(x) one has R(u)f(x,z)

has t h e general form ( i n any r ( u ) = R(u)ely

uJx e xj p ( - l y ( u - j / j ) a j ;

r * ( u ) = R(U)-1 e

-1”

u j xj exp(1;

Consider the generating f u n c t i o n (*&) t i o n o f t h i s generating f u n c t i o n i n

F?

(u-j/j)aj)

1 uiv-jEij under

( i , j E Z). i s X(u)X*(v)

r e l a t i o n exp(aax)exp(bx) = exp(ab)exp(bx)exp(aax),

i n Bm (assuming say I v / u l v/u)-’ (8.23)

i s needed i n (8.22),

cl).

For

^rB

The representaand, using the

w i t h (8.21 ), one gets

an adjustment o f

1;

( u / v ) ~ = (1

(i/(i-v/u))((u/v)mr(u,v)

F i n a l l y one goes t o t h e determination of

-

- A vA m ( v 1m lm-1

(I

1)

...)

i n B.

d e f i n e the elementary Shur polynomials ( c f . remarks a f t e r (7.28)) (8.24) Thus Sk(x) = 0 f o r k < 0, So = 1, and f o r k > 0 (8.25)

-

and one o b t a i n s

i -jAB u v r (E. .I = i,j e Z m 1J

1

= um+’zrn+’ d x ) .

im)

via

First

ROBERT CARROLL

106 We n o t e a l s o A = {A1

S,

... 2 A k

2 A2

3 + x Z y S3 = x1/6 + x 1 x 2 + x 3'""

2

S2

xlY

=

= 4xl

Now t o each

> 01 one d e f i n e s a Shur polynomial

SA(x) =

(8.26)

...

S A3

SA,-l

Sh3-2

. ... .. . . (SA(x) i s a k X k determinant).

4

x3, S2,2 = x1/12

Then e.g.

2

-

i-l>

ao(v.

... and

10

A v.

1-1

A

2

= x1/2

One checks t h a t S,

x1x3 + x2¶....

....

nomial o f degree / A 1 = X1 + A2 +

CHE0RER 8.6.

S1,l

...)

3

-

-

x2, s2,1 = x1/3 i s a homogeneous p o l y -

Then one can prove

...( x ) ,

= Sio,i-l+l,i-2t2,

where io >

i-k = -k f o r k s u f f i c i e n t l y large.

The s t r a t e g y here f o l l o w i n g CK1 1 i s t o compute (**) uo(Ro(explyiAi) B ) ) = Ro(exp( CyiAi)P(x) f o r P(x) = uo(vio A vi-, A .) (vm i A vi-, A ( n o t e here e x p r ( a ) = R(exp?) and (*m) Rm(A)(viT A v. A ) = xdet 1W-1 (i 1 has i n d i c e s jm> J,-~ >-..with (i) % (im,im-ly A ( j ) Vj, A ,,V,j A * a . 9 A") ( j1 .), and denotes t h e m a t r i x l o c a t e d on t h e i n t e r s e c t i o n o f t h e rows j m y Proof:

.. .

..

...

..

jm-l '..

. and

.

columns im, im-l '.. o f A

E

GL,).

comparing t h e c o e f f i c i e n t s o f t h e vacuum ( r e c a l l problem a r i s e s here s i n c e exp( CyiAi) group t h e aij

Ern= { A = ( ( a .1J. ) ) ; - 6ij w i t h i 2 j

{((aij)); CL,

and

i s not i n

i , j E Z; A - l

a r e 01.

V

=

I lcivi;

1).

A technical

One uses a l a r g e r

e x i s t s ; a l l b u t a f i n i t e number o f

n

on

U ~ ( I ) ~ )=

GL,.

The c o r r e s p o n d i n g L i e a l g e b r a i s

i,j E Z; a l l b u t a f i n i t e number o f aij

3, act

The r e s u l t w i l l f o l l o w by

w i t h i 2 j a r e 01.

=

Then

3.

I t i s easy t o see t h a t

3 , and ELm o n

F ( c o n s t r u c t e d from V )

ci = 0 f o r j >> 0

r and R extend t o r e p r e s e n t a t i o n s o f

3,

and R(expa) = e x p ( r ( a ) ) w i t h f o r m u l a s l i k e (*.)

preserved.

Now r o ( A k )

'L

ak

f o r k > 0 so RBo ( e x p l y j A j )

(8.27)

= e x p ( l m y.a.1 1 J J

L e t F ( y ) be t h e c o e f f i c i e n t o f 1 when t h e o p e r a t o r i n (8.27) i s a p p l i e d t o P(x).

Thus

EQUATIONS

107

AkSk(y) which can be regarded as a m a t r i x

A w i t h Amn = Sn-,,(y)

(m,n

(*+) reduces t o uo(vi,

Z).

E

A vi-,

A

... ) ) .

A vi-, A uo(R(A)(vio f o r (i) = (ioyi-ly...)

Since Sk = 0 f o r k < 0 one has A E

...)

Kmand

= c o e f f i c i e n t o f $o i n t h e expansion o f

t o be d e t A (i1 ( j1 and t h u s equals Si,i-,+l

T h i s can be read o f f from ( * m )

,...

and ( j ) = (0,-lY-2’...),

( f r o m (8.26) and t h e d e f i n i t i o n o f A).,

Hence F ( y ) = Sioyi-,+,,

...(y)

= P(y).

As a c o r o l l a r y one shows a l s o t h a t (8.29)

..

and notes t h a t

, ,,,-, -m+l,

A v im-, A .) = ,S-,i

Um(Vi,

(

S,ISp)

= 6

with

h,v

< I > defined

...( x ) as i n (8.13).

REmARK 8.7 (EHE KP EQlAtz0rU). L e t R = GLm.l be t h e o r b i t o f t h e vacuum i n B and we w i l l see t h a t p o l y n o m i a l s T E B a r e c h a r a c t e r i z e d by b e i n g s o l u t i o n s c o r r e s p o n d i n g uo: Fo -+ B y 1 A

... where

0 5 n 5 k-1

uum o r b i t i n T E

j

$,

j E GL,-$o

B (C

9 ) then

j EZ and c o n v e r s e l y i f

) = 0.

5

P,

T

# 0, and

T

O

Now any

f o r g = (AT)-’f.

be aij

and aij

=

1 a..:.(T) J1 1

then

=

j

R.

To see

A0

= w f o r w = Av and Ro(A)FRo(A)-’

L e t t h e m a t r i x elements o f A and A - l

(A*)

T E

) = 0 f o r j > 0 so (A*) I;.($ J O (A)$, f o r A E GLm. From t h e

1a..v

J I i’ gives 0 =

(AT)-lv? =

1 ikjvc,

?, 1 Ro(A)vjRo(A)-

i n t h e b a s i s vi and

(T)

1 (1

Ia .:*(.I) = $kjaji)?i(T) I;;(T) kJ k The converse i s more o r l e s s s t r a i g h t f o r w a r d ( e x e r c i s e (T)

Hence f o r

O has t h e form T = R

T E

so t h a t Av

Applying R(A) t o

‘ki

1-n ’

= R(Ah0.

s a t i s f i e s (8.30), J

O

n

A v,i = v.

= 0

J

d e f i n i t i o n (8.14) we see t h a t Ro(A)CRo(A)-l =

= vo i

$,

?r

is a s o l u t i o n o f

T

I;?(T)

r E

J

J

S,

GLm d e f i n e d by Av-,

which i m p l i e s S, E R (we u s e n f o r t h e vac-

t h i s n o t e t h a t $.($ ) = 0 f o r j 5 0 and ;?($

I:Z($

E

B o r Fo).

1 Cj(=)

(8.30)

... and

i-., = -n f o r n 2 some k. Av = v f o r a l l o t h e r j, one has $,

each A a s i n (8.26),

Now i f

$o = vo A v - ~A

n,

For A

, and

Indeed i n t h e

F i r s t one sees t h a t t h e S, E 0 .

o f t h e H i r o t a equations.

1 akjaji A

IRo(A)?3Ro(A)

which i s (8.30).

-

cf

LKW.

=

-1

ROBERT CARROLL

108 Next c o n s i d e r (")

Fo B Fo and C r x i ,xi,.

1 ui-jvi(7)

IX* u)r =

X(U)T

if t h e c o n s t a n t t e r m vanishes 1, x"

..;XI'

... I

r*(u)

( u s i n g x"); ue

(8.31)

a'

where

j

'L

1;

-f

r(u) ( u s i n g

see (8.20) f o r t h e formulas. u j ( X jl - X I !j)

a/ax' j-

exp(-l;

R i f and o n l y

T E

. .I

2 C[xl ,x2,.

( = polynornia? r i n g i n x ' , x " ) .

t o t h e bosonic r e p r e s e n t a t i o n v i a X(u)

(AA)

@ v J ( T ) so

The isomorphism Fo

(u-j/j)(ag

-

Then

(AA)

extends t o Transform

and X*(u)

XI)

-f

becomes

ai))T(X')T(X'')

I n t r o d u c e new v a r i a b l e s x ' = x-y and

XI'

= x+y so x l - x " =

I t f o l l o w s t h a t T E C[xl,x 2 y . . . ] , T = 0, i s i n fi Y' ifand o n l y i f t h e c o e f f i c i e n t o f uo vanishes i n t h e e x p r e s s i o n

-2y and a '

- a"

=

-a

Now use t h e H i r o t a n o t a t i o n o f (7.31) and w r i t e

7

= .(1/3)a yay...) Y (aY\ Yz Expand t h e terms i n (8.32) i n terms o f Shur polynomials i n t h e form

Put t h e t e r m independent o f u equal t o zero t o g e t

T h i s can be r e w r i t t e n v i a A .,

(8.35)

Sj+l (a,)T(x-y)~(x+y)

N

s ~ ( a+u ) ~e x p ( l

s,l

where

?=

Y

a

)T(x-u)T(x+u) us

(xl ,+x2,x3/3,.

EHEBRER 8.8.

.. ) .

luZ0

Sj +1 ( ? u ) ~ ( ~ - ~ - ~ ) ~ ( ~ + y= + ~ )

=

lu=o

=

s ~ (?)exp(I: + ~

Y,X~~(X)-T(X) S

l

Consequently one has proved

A nonzero polynomial

T

belongs t o R i f and o n l y i f t h e f o l l o w -

i n g H i r o t a equations a r e s a t i s f i e d (8.36)

1:

Sj(-2y)Sj+l(F)e17

'sXs

T(x)-T(x)

= 0

Thus we have another p r o o f o f (7.32) and a c o n s t r u c t i o n o f t h e KP h i e r a r c h y ; equations such as (7.33)-(7.34) powers o f t h e yi t o zero.

a r i s e upon e q u a t i n g c o e f f i c i e n t s o f t h e

109

AKNS SYSTEMS

9, A K G Sl@Jb&Eills, We begin with a sketch o f ideas f o r NLS (nonlinear Schrodinger equation) following [ F2;C131 and then develop some AKNS theory follow-

i n g [ NE1;FLly4;C181 ( c f . a l s o [ BG2,3;TK2,31).

For c l a s s i c a l references to AKNS systems we mention [AB5;Cl;NE41. There i s a l o t of work, some very rec e n t , on inverse s c a t t e r i n g techniques f o r mu1 tidimensional systems and f o r n X n systems. We had intended t o develop t h i s b u t came u p s h o r t of space (some of this appears already i n [BE41 and some of i t will be i n a f o r t h coming book [ KN21 i n any e v e n t ) . Thus l e t us supply here a l i s t of r e f e r ences, with apologies f o r omissions, namely [ AB2-4,12,13;BTl-14;AK2,3;BE1-7; CA2; CZ1; DI6; DN3; F01,4,14-20; HT1,2; G X 1 ; DF5; KT1 ;J1,2 ;KNl-lO;MN1-3; NZ1; SCLl ; ST1 -7; SUN1 ;N1,2; SN5; LP1 ;WC1,2 ;ZH1-3; 23 1.

RUlARK 9.1

(RZEGIANN-HZL%ERJb& = RH ZDEAB AND DRE$BZNC).

Following [ F21 r a t h e r

extensively a t f i r s t l e t us look a t some evol u t i o n - s c a t t e r i n g problems on (--,a) in a matrix form. A typical model here i s the NLS equation where u = u + XU v = v + AV + u0 = J E ( Oq 60 ) ( E = * I ) , u1 = ( 1 / 2 i ) u 3 = O 1 ol’ O z1 i c l q l u3 - i J E ( - q * : x ) y V, = -Uo, and V 2 = -V1 (u3, u1 = ( 01 /12 i ) ( o - 1 ) , Vo ( =0 -i) ( ), and u2 = a r e Pauli m a t r i c e s ) . One considers U , V a s connection 1 0 c o e f f i c i e n t s i n a t r i v i a l bundle R 2 X C 2 over R 2 w i t h

-

(9.1)

Fx = U(x,t,A)F; Ft = V(x,t,h)F

where F i s e.g. a 2-vector o r a 2 X 2 matrix. The compatibility condition f o r s o l v a b i l i t y o f (9.1) i s Fxt = Ftx and this can be regarded a s a zero curvature equation (*) U t - V x + [ U , V l = 0, equivalent t o the NLS equation 2 (A) i q t = - q x x + 2 E l q l q . Gauge transformations F G(x,t,A)F a r e associated w i t h maps U -+ GXG-l + GUG-’ a n d V -+ GtGml + GVG-’ leaving (*) i n v a r i a n t so t h a t gauge equivalent connections involve the same NLS equation (A). One determines t r a n s i t i o n matrices T(x,y,A) f o r t h e spectral problem -+

(9.2)

D x T ( x , ~ , A ) = U(x,A)T(x,y,A); T(x,x,A)

=

I

where t = to i s fixed (and suppressed). One has T(x,y,A) = T(x,z,A)T(z,y,A), T(x,y,A) = T-l (y,x,X), a n d DyT(x,y,A) = -T(x,y,A)U(y,X) by known theorems ( c f . [ C1,201). Now l e t E(x-y,A) = exp{(x/2i)(x-y)031 be t h e s o l u t i o n of Ex = U E w i t h

-

110

ROBERT CARROLL

E l x Z y = I when Uo = 0.

= y lim ~ T~( ~ ,"~ , A ) E ( ~ , A )e x i s t s f o r A r e a l

Then T+(x,A) -

and one can w r i t e T-(x,A)

(9.3)

= E(x,A)

+

1:

r-(x,z)E(z,A)dz;

T+(x,A)

= E(x,A)

+ jxm r+(x,z)E(z,A)dz

1 2 i 1 L e t T+- = (T, - T+), f o r s u i t a b l e P,.T column vectors, and then e.g. T- and 2 T+ can be a n a l y t i c a l l y extended t o Im > 0. On d e f i n e s a l s o ( 0 ) T(A) - 1 i m a Eb E(-x,A)T(X,y,X)E(y,X) as x + m and y - 0 and one can w r i t e T(A) = (b --) = d e t S+(x,A) where T (x,A) = T+(x,A)T(A) w i t h a ( A ) = det(T-(x,A) 1 T+(x,A)) 2 7 1 and b(A) = det(T+(x,A) T (x,A)) ( c f . [ F21 and n o t e t h a t T+ i s unimodular). -f

Thus a(A) extends a n a l y t i c a l l y t o I m A > 0.

+ 0 for

c r e t e spectrum ( i . e . a(A)

We w i l l assume t h e r e i s no d i s ImA > 0 ) i n o r d e r t o s i m p l i f y t h e formulas.

The s c a t t e r i n g m a t r i x S(A) has t h e form ( d e f i n i t i o n )

and l / a (resp. b/a) is r e f e r r e d t o as a t r a n s m i s s i o n ( r e s p . r e f l e c t i o n ) coefficient.

One s e t s now S-(x,A)

= (T+(x,A) 1

2 T-(x,A))

and S+(x,A)

= ( T1 (x,A)

2

( S - i s a n a l y t i c f o r ImA < 0 ) so t h a t S,- s a t i s f y Sx = US and extend r e s p e c t i v e l y t o t h e upper o r l o w e r 2 h a l f planes a n a l y t i c a l l y . Then i n f a c t

T+(x,A)) S (x,A)

( c f . [ F21) and s i n c e d e t S+ = a ( A ) one d e f i n e s G (x,A)

= S+(x,A)S(A)

= S-(x,x)E-'

( x , ~ ) and G+(x,A)

Riemann-Hilbert (RH) problem (G(x,A) G+(x,A)G-(x,A)

(9.5)

1 where G(A) = ( - b

so t h a t Gk s a t i s f y t h e

= a(A)E(x,A)S;l(x,A)

= G(x,A)

or

= E(x,A)G(A)E-~(x,A)) -1

G- = G+ G

E i

1 ) i s a p r i o r i d e f i n e d o n l y o n t h e r e a l l i n e and G, - a r e t o

extend a n a l y t i c a l l y t o t h e upper o r l o w e r h a l f p l a n e r e s p e c t i v e l y ( a l s o G, and G

Q

I + o ( 1 ) as

1x1

.+

m).

Next i t can be shown t h a t

(9.6)

G&(x,A)

= I +

G;l(x,A)

1" @+(x,s)ekiAsds; 0 = I + 1" A+(x,s)e 0

G(A)

iAsds

=

I +

lz @(s)eiAsds;

-

RIEMANN HILBERT PROBLEM

111

Then t h e a n a l y s i s o f t h e RH problem above ( v i a F o u r i e r t r a n s f o r m o f (9.5)) reduces t o t h e Wiener-Hopf (UH) e q u a t i o n ( s At(x,S)

(9.7)

where @(x,s) = (-

+ @(x,s) +

0)

1"0 ~,(XyS)@(x,S-S)d5

E'(-S-X)) 0

B(S-X)

f o r B(s)

=

0

( 1 / 2 1 ~ ) l z b(A)exp(-iAs)dh.

Once A + i s determined from (9.7) one can express @ - v i a @-(x,s) = @(x,-s) + The GLM equations (9.8) below f o r t h i s problem can

A+(x,S)@(x,-s-S)dS.

be d e r i v e d d i r e c t l y i n t h e standard manner (see below) o r can be determined They have t h e form

from t h e WH e q u a t i o n . r+(x,y)

(9.8)

+ A(x+y) +

c

r,(x,s)A(sty)ds

= 0;

( t h e f i r s t e q u a t i o n f o r y > x and t h e second f o r y < x ) .

r(A)

-

= b/a,

7= -

+

Here one w r i t e s

b/a, w(x) = (1/41r)jz rexp(iAx/2)dA, F ( x ) = ( 1 / 4 n ) L I Yexp 0 1 0 0 u+ = ( o o), u- = (l o), A ( x ) = w(x)u- + E;(x)u+, and X ( x ) =

(-iAx/Z)dA, E;(X)O-

-

;(X)IS+

(we c o n t i n u e t o assume no d i s c r e t e spectrum).

The " s t a n -

dard" d e r i v a t i o n o f t h e GLM equations (9.8) goes as f o l l o w s ( c f . a l s o 51). 1 1 2 One w r i t e s e.g. ( l / a ) T (x,X) = T+(x,h) + rT,(x,A) where r = b/a ( f r o m T- = T+T) and i n s e r t s (9.3); 1 0 rt(xyy)(o) t w(xty)(,)

rt.

t h e n t a k i n g F o u r i e r t r a n s f o r m s t h e r e r e s u l t s (y > x ) 0 + J," rt(x,s)(l)w(s+y)ds = 0 which l e a d s t o (9.8) f o r 2 2 S i m i l a r l y one uses Tt/a = ?T! + T- ( v i a T, = T - T - l ) and t h e r- formula

i n (9.3) t o g e t t h e

r-

e q u a t i o n i n (9.8) v i a F o u r i e r t r a n s f o r m s .

Further

r e l a t i o n s between WH and GLM e q u a t i o n s a r e s p e l l e d o u t i n [ F21 and v a r i o u s advantages o f one o r t h e o t h e r approach a r e discussed (see a l s o below). F i n a l l y t o deal w i t h t h e t i m e e v o l u t i o n o f s p e c t r a l data and s o l v e i q t = 2 t 2 ~ l q lq w i t h q(x,O) given, one f i r s t determines s p e c t r a l data (e.g. 2 -qxx b,a)at t = 0 from d i r e c t s c a t t e r i n g a t t = 0. Then (6) OtT(t,A) = ( i h / 2 ) 2 [ u 3 , T ( t , h ) ] y i e l d s at = 0 and bt = - i h b. The e q u a t i o n (6) f o l l o w s by d i f f e r e n t i a t i n g Tx = UT i n t and u s i n g Ut = V x V(x)T(x,y)

-

T(x,y)V(y).

Then as 1x1

+

my

-

[U,V]

V(x,h)

-f

t o g e t (+) Tt(x,y) = 2 ( i h / 2 ) u 3 and (6) r e -

s u l t s upon m u l t i p l y i n g (+) b y E(y,A) o n t h e l e f t and by E(-x,A) and t a k i n g l i m i t s x

-f

my

y +

--

(cf. (a)).

on t h e r i g h t

Then one uses i n v e r s e s c a t t e r i n g

112

ROBERT CARROLL

w i t h data a(A,t),

-

= u 0 ( x ) = k ( u3r k (x,x)03

t ) v i a say ( m ) U:(X)

= -Uo(xyt) =

t o determine t h e WH o r GLM k e r n e l s and thence q(x,

b(A,t)

r +- ( x , x ) )

o r ~i[Q-(x,o,t),u3]

L2[Qt(XyOYt)y~31.

We r e c a l l now ( c f . 5 7 ) t h a t t h e r e i s a n i c e i n t e r p e r t a t i o n and expansion o f RH methods i n terms o f t h e d r e s s i n g techniques o f [ Z l - 3 1 ( c f . a l s o [ C1,6-20, 22-25;F2;N02;P2]).

I n a sense t h i s i s s i m p l y a r e p h r a s i n g o f t h e r o l e p l a y -

ed by c l a s s i c a l t r a n s f o r m a t i o n = t r a n s m u t a t i o n o p e r a t o r s which i s p a r t i c u l a r l y w e l l adapted t o s o l i t o n problems. Thus one dresses bare o p e r a t o r s Mo i = mia t o M v i a t r i a n g u l a r V o l t e r r a o p e r a t o r s K,- (based on k m ) i n t h e form (**) M(l+K*) = (ltK,)Mo. We assume M i s a d i f f e r e n t i a l o p e r a t o r and gener-

1

0

a l l y w i l l deal i n s i t u a t i o n s where t h e r e i s a canonical f a c t o r i z a t i o n (1+K+) (1+F)

=1 +

K

(1

n,

I).

G e n e r a l l y ( c f . 87) (**) w i l l i n v o l v e M a l s o being

a d i f f e r e n t i a l o p e r a t o r and r e q u i r e s t h a t 1+F commute w i t h Mo. t i o n FMo = MoF i n k e r n e l form i s w r i t t e n as

1 mi(x)axF(x,z) i

means t h a t 0 =

-

(*A)

Mo(Ox)F

1 (-aZ)i(F(x,z)mi(z))

-

The c o n d i -

FMo(Dz) = 0 which

( t h e mi

can be mat-

r i c e s ) . I f one dresses Mo + aD t o M + aD t h e n F(x,y,z) w i l l s a t i s f y (**) Y Y crF + Mo(Dx)F FMo(Dz) = 0. Y Now f o l 1owi ng [ F2 ] ( c f a1 so [ AH1 ;FD1 ;GEl -4; NE1; N02; P2 ;ST3 ,5 ,7; SM1-3 ;21 -3 I)

-

.

we s t a r t w i t h c o m p a t i b l e Uo and Vo where DxFo = U°Fo and DtFo = V°Fo and dress them t o g e t s o l u t i o n s o f Ut VF, U(x,O,A)

= U0(x,A),

-

and V(x,O,X)

V x + [ U,V 1 = 0 where (*4) = Vo(x,X)

Fx = UF, Ft =

( t i s o c c a s i o n a l l y suppressed

and e.g. Uo = U o f NLS). Thus one can t a k e Uo,Vo and Fo as known ( i n [ F21 wir(xyt)/(A-ki)r + kw k ( x 3 t ) ) one c o n s i d e r s u*, VO, u, v o f t h e form

1

11

and p i c k s a s u i t a b l e m a t r i x f u n c t i o n G ( A ) on a s u i t a b l e c u r v e [ F21 f o r d e t a i l s i n p a r t i c u l a r s i t u a t i o n s

-

r

i n C (see

t h e r e i s no general r e c i p e ) .

This g i v e s r i s e t o t h e RH problem

(9.9)

G(x,t,A)

Here G,

= G'

ior of

r (r

= G+(x,t,A)G-(x,t,A);

G(x,t,X)

= Fo(x,t,~)G(~)F~l(x,t,~)

a r e t o have a n a l y t i c c o n t i n u a t i o n s i n t o t h e i n t e r i o r o r e x t e r and G g i v e n

-

see below f o r a n example).

One assumes t h i s RH

problem has a s o l u t i o n ( n o r m a l i z e as i n [ F21 t o e l i m i n a t e gauge e q u i v a l e n t s o l u t i o n s etc.).

D i f f e r e n t i a t i n g i n (9.9) one can d e f i n e

NONLINEAR SCHRODINGER EQUATION

A l s o F+ = G;’Fo

and F- = G-Fo s a t i s f y

(*+I DxF

113

= UF; DtF = VF.

Thus t h e

i s determined v i a G+ and Uo and e v e r y t h i n g reduces t o

e v o l u t i o n o f U(x,t,A)

r

s o l v i n g t h e f a c t o r i z a t i o n problem ( a f t e r d e t e r m i n i n g

and G ) .

Recall t h a t

t h i s i s e x a c t l y what t h e WH t e c h n i q u e does and e s s e n t i a l l y what t h e GLM met h o d accomplishes i n a d i f f e r e n t way.

r+ or

t h e kernels

The r e c o v e r y o f p o t e n t i a l s through

i s s i m p l y a t e c h n i c a l s t e p f r o m t h i s p o i n t o f view.

@+

It

i s now p o s s i b l e however t o rephrase t h e f a c t o r i z a t i o n p o i n t o f view i n a L i e We w i l l g i v e here

t h e o r e t i c c o n t e x t which i s v e r y e l e g a n t and meaningful. o n l y a b r i e f sketch o f t h i s following

F21 ( c f . a l s o [BG2,3;FLl;TK23]

and

remarks l a t e r i n § 9 on AKNS systems)

REl’tARK 9.2

(N0NI;ZNEAR BCHR0DZNGER =

NU E I A Q 3 0 N ) .

We use t h e NLS model as

a v e h i c l e t o i l l u s t r a t e t h e t h e o r y ( c f . [ F21 f o r f u r t h e r d e t a i l s ) .

Thus

assume no bound s t a t e s and suppose b(A) = bo(A) i s determined from i n i t i a l data q(x,O)

Then (9.5) a p p l i e s (where t i s suppressed) and we

= q,(x).

w r i t e now G(x,t,A)

= G+(x,t,A)G-(x,t,A)

2 (9.11)

G(x,t,A)

= e”’

(where b = b,exp(-iA

2 ta3G(x,A)e-’iX

z t ) and

2

0 ta3 =

(

iix-il - bne -

2 t

boe-iAx+ix 0

1

One takes t h e c o n t o u r t o be t h e r e a l a x i s and G+(x,t, - -1 We r e c a l l a l s o (9.10) i n t h e form (f.) U(x,t,X) = -G+ A) -+ I a s +. D G + G; 1 (Xu3/2i)G+ = DxG G - l + G-(Ao3/2i)G11. R e c a l l U = Uo + xul, U1 = x + 0 3 / 2 i , and we use t h e s i t u a t i o n q = 0 ( i . e . Uo = 0 ) as t h e base problem (i. (G(x,A)

= G(x,O,A)).

I A ~ -.

--

e.

Uo

= Aa3/2i,

U

(9.12)

=

T h i s can be w r i t t e n i n t h e form

etc.).

E*G>( x o 3 / 2 i )

.Lc

= Ad*G-fxo3/2i

)

N

where Ad*gU = 9 ’ g - l

+

gUg-l i s a c e r t a i n c o a d j o i n t a c t i o n (we o m i t here t h e

a l g e b r a i c s t r u c t u r e f o r t h e moment a t l e a s t Now s e t (A*) h+(x,t,A) = G~(x,O,A)G+(x,t,A)

-

c f . Appendix A f o r d e t a i l s ) . and h-(x,t,A) = G-(x,t,A)G- -1 (x,

0 , A ) so t h a t h ( t ) = h+h- has t h e form

( 9.1 3)

h = G i l ( x ,0, x)eL”

2 3‘t

G+( x, 0 ,x ) G- ( x, 0 ,A )e

-$iA

2

to

3G11

(x,O,A)

114

ROBERT CARROLL

and i s expressed d i r e c t l y through t h e s o l u t i o n o f t h e RH problem f o r t = 0. ( 0 ) ( A o 3 / 2 i ) = ~ * G - ( 0 ) ( X o 3 / 2 i ) so

Then U(0) = fi*G;'

U ( t ) = :*h;'(t)U(O)

(9.14)

= z*h-(t)U(O)

N

( s i n c e Ad*Gil

= z*h;lE*G;l(0)

etc.)

Now J = F(x,A)C(X)F-'(X,X)

f o r F' =

U(0)F and C ( h ) a n a r b i t r a r y i n v e r t i b l e m a t r i x l i e s i n t h e c e n t r a l i z e r o f U

(0) w i t h r e s p e c t t o (h;'J)U(O) g,

= h,

G* ( i . e .

= G*h;'U(O)

and E * ( h - J ) U ( O )

= z*h-U(O).

2

= h-(x,t,h)G-(x,O,h)exp(%ih

and g-(x,t,A)

G*

= U(0)) so one can w r i t e e.g.

fi*JU(O)

Hence we can use

(Am)

ta3)G11 ( x , O , h ) t o o b t a i n

g ( t ) = g+9- w i t h 2 g(x,t,X)

(9.15)

= G i l (x,0,h)et2ix

G+(x,O,X)

t"3

T h i s i s now i n a " c a n o n i c a l " form where g ( t ) i s a one parameter group which can be r e p r e s e n t e d f o r m a l l y as e x p ( - t v H ( U ( 0 ) ) ) G-(x,O,X))

we have a l s o U ( t ) = G*g;'(t)U(O)

( c f . [ F21).

Note ( u s i n g F

and U ( t ) = f i * g - ( t ) U ( O ) .

2,

The

r e p r e s e n t a t i o n i n terms o f VH i s connected f o r m a l l y t o H a m i l t o n i a n equations The a l g e b r a

based on H i n a s u i t a b l e L i e t h e o r e t i c c o n t e x t (DtU = CH,UIo).

u s u a l l y has t o be a d j u s t e d t o each p a r t i c u l a r model so we do n o t deal w i t h t h i s h e r e ( c f . Appendix A ) .

We remark i n passing t h a t " g e n e r i c a l l y " under

s u i t a b l e " c e n t r a l e x t e n s i o n " o f t h e L i e framework one a r r i v e s a t an indent i f i c a t i o n o f H a m i l t o n ' s e q u a t i o n s o n a reduced phase space w i t h a zero c u r v a t u r e equation.

For e l e g a n t L i e t h e o r e t i c and a l g e b r a i c t r e a t m e n t s o f NLS

and AKNS systems see a l s o [ BG2,3;DRl;IM1,2;GE1-3;NEl;AC1-3;PEl

] and 510.

For f u t u r e a p p l i c a t i o n l e t us a l s o o r g a n i z e a l i t t l e d i f f e r e n t l y some o f t h e Also we want t o make c o n t a c t w i t h t h e 1 0 Thus f i r s t E = e x p ( h x u 3 / 2 i ) w i t h a 3 = ( o -1 ) so

o b j e c t s which a r i s e i n t h i s s e c t i o n . AKNS framework o f [ AB5;Cl (9.16)

E =

[

I.

exp(-%ihx)

0 exp(4ixx)

0

1

6) ( h )T+1

A l i t t l e comparison g i v e s now (I$,$,@,$ a r e AKNS v e c t o r s and q 2 1 2 ( x ) 2, $(%XI; T+(x) % $(%x); T - ( x ) % $J(%X); T - ( x ) % -$(%x). Thus S-f

(x/2),

S,

'L

n,

($:?I

1 Eb) and w r i t i n g o u t S- = S+S w i t h (A*) S = ( l / ~ ) ( - ~

(I$ +)(x/Z),

g i v e s i d e n t i f i c a t i o n o f a and b w i t h t h e c o r r e s p o n d i n g o b j e c t s i n t h e AKNS theory.

We n o t e t h a t

E

= 1 corresponds t o

^b = -6, a^ = a

( f o r r e a l A ) and

MISCELLANEOUS CONNECTIONS

+ 21q12G (which becomes i q t =Aqxx iqt = -qxx c 2 one has e.g. r,1 = K and w r i t i n g r, = (r+1 r,)

-

115

2 ) q l 2 q f o r q + ii). F u r t h e r = K (K,; i n AKNS t h e o r y ) .

r,2

For completeness we w r i t e a l s o

and ( A 0 ’G; = ( l / a ) ( $ e % i A x Jle-%iAx) = ( r e c a l l W($,$) = a = $1$2 - I$ $ (l/a)S+E-’; G- = ($exp(%iAx) 2-$exp(-fiAx)) 1 = S-E- 1 . Next we n o t e t h a t t h e WH k e r n e l s a r i s i n g v i a F o u r i e r t r a n s f o r m f r o m

(which l e a d s a l s o t o t h e d r e s s i n g formulas and c o a d j o i n t a c t i o n ) must be r e l a t e d t o t h e GLM k e r n e l s a r i s i n g v i a F o u r i e r t r a n s f o r m from ( r = b/a, F = (l/a)S, = S, t (l/a)St(b 0 R e c a l l here S, = (T-1 T+), 2 S- = Ez/a) (4)

-ib).

1

(T,

2 T-),

o r S = (l/a)StG

G- = Gi1(EGE-’)

= S+S

(G = aS as i n (9.18)),

one notes i n general t h a t , f o r u n s p e c i f i e d g e n e r i c R+,IJI,U,

R,

-

The l a t t e r o b s e r v a t i o n shows-the e q u i v a l e n c e o f (A+)

R- = R+$(l-U)$-’.

and (9.18);

n o t e here t h a t (A+) can be w r i t t e n a s

ft

Now e.g. l o o k a t S. = EX t @-EXe-iAsds 1 2 f; (@-exp(-+iAx) @-exp(%iAx))exp(-iAs)ds. (9.20)

and

R- = R+IJIU$-~ E

0

+

2 T-) = E +

= T,1 = (,)e 1

+Ax

,

-

1 r,(x,s)e

-+i As ds

1 1 Consequently (Am) $0 (x,k.(E;-x)) = r+(x,E). Similarly 2 2 fr @-exp(-iAs++iAx)ds from which (a*) %@-(x,+(C+x)) =

/xm

1

Hence e.g.

+ jm@1(x,s)e - iAs -4i Ax ds

(,)e

so (T,

( f r o m (9.6))

Sx

2 r-exp(+iAs)ds =

r t (x,-E).

Next one

o b t a i n s TE = E2 + r,(x,s)exp(+iAs)ds 2 = aE2 + %Gm A,(x,+(E-x))exp(+iAE) 20 2 2 0 dS where E2 = (l)exp(%iAx) so we w r i t e (0.) %aAt(xy+(~-x)) = r+(x,E) t (1) x 1 1 ( l - a ) & ( x - c ) and f i n a l l y T1 = E, + lm r (x,s)exp(-+iAs)ds = aE1 t a:/ A,(x,s) 1 1 1 exp(-%Ax t i A s ) d s so ( 0 . ) %A+(x,f(C+x)) = r-(x,-c) + (o)(l-a)s(x+E). L e t us connect t h i s now w i t h t h e framework o f [ BE1-7;LEl;NOl;STl-7].

Thus

116

ROBERT CARROLL

9a “:I.

one uses a general c o n t e x t ( z i n s t e a d o f A here,

+

( -4i

Q$ so t h a t f o r NLS J =

-

SO ( 0 4 ) mx

(l = (

O ) and

4i

JECl

m a t r i x ) ( 0 6 ) qX = ZJJ, One s e t s J, = mexp(xzJ)

= DZm = Qm and from $t = b$ w i t h b ‘~rbozm a s 1x1 -+ m 1i m -1 = 0 ) one o b t a i n s v i a S = x~ J,(x,z,t)J,(-x,z,t) , St = [ z mbo,S].

z[J,ml

([ bo,J] The s o l u t i o n t o ( 0 4 ) w i t h m bounded and m m = 1

(9.21)

+

s i n c e Dz? -Famu r t hmve r where Dv,

= %D,

m = 1

Q:

+

-m

is

(5

a/aT, ! - f a )

=

h ( d c A dS) = Qzm and Dz(m-’%n)

one has D% ,

+

(1/2si)j

d:

( d e f i n i n g C and T, Tm = mv). a solution.

(c-z)-’

1 as x

This implies

= 0.

= 0 so v = exp(xzadJ)w(z) = exp(xzJ)w(z)exp(-xzJ)

=

(9.22)

(1/2si)/

-f

(c-z)-lmv(dc

+ CTm

A d s ) = 1 + C(mv) = 1

Some c a l c u l a t i o n shows t h a t (9.22)

For comparison purposes l e t us n o t e t h a t f o r NLS,

and

i s i n fact

E

= exp(xAJ)

so $ = mE w h i c h i m p l i e s ( d e f i n i n g m,- by a n a l o g y )

G-

(9.23)

= S-E-’

R e c a l l here D T, ++,

and D S

%

m-;

G+-1

= (l/a)S+E-’

%

m,

+ QT, S- = (T+1 T-) 2 % $-, ( l / a ) S + = ( l / a ) ( T -1 T+) 2 % + Q j S ( n o t e here e.g. ax(T+1 T-) 2 = (axT+ 1 axT-) 2 and z(T+1

= AJT,

x -

= (AJ

k

2 1” +2 T ) = (sT+:T-)).

-

-

-

- zrK t ( F . below), J E C = Cartan subalgebra o f a - Br(z,Q), Br(z,Q) = J J L i e algebra (diagonal m a t r i x J), Q E ( o f f diagonal m a t r i x Q), and d e f i n e Now go t o [ ST5,7] and w r i t e DZ =

ax

zadJ, Dx = a x

ZJ

Q,

Dt = a

1;

F = rnKm-’

( K E C).

Under s u i t a b l e hypotheses (e.g.

Q E 3 = Schwartz space)

1;

F.z-j, Fo = K, and i n any event (from ( 0 4 ) ) (om) axF - z[ J , F l - [ Q , F l J = 0. T h i s l e a d s t o t h e analogues o f Lenard r e c u r s i o n s ( b * ) [ J,Fj+l ] = a F x j [ Q , F . ] and t h e F. a r e polynomials i n Q (and x d e r i v a t i v e s o f Q ) o f o r d e r J J j - 1 (cf. [ ST1-7;SY1,2]). I J,Fr+l I F u r t h e r t h e s t i p u l a t i o n Dx,Dt I = atQ F =

-

-

= 0 y i e l d s t h e n o n l i n e a r e v o l u t i o n e q u a t i o n f o r Q ( n o t e i f one denotes t by

tn we have anQ = [ J,Fn+l

I

i n a hierarchy format).

o f s e c t o r s where m i s meromorphic e t c .

We o m i t here a d i s c u s s i o n

To g e t t h e t i m e e v o l u t i o n o f spec-

t r a l data i n t h i s c o n t e x t one checks f i r s t t h a t J, = mexp(xzJ + z r K t ) s a t i s f i e s Dx+ = Dt$

= 0.

It f o l l o w s t h a t

117

COADJOINT ORBITS

axm = z[J,ml + Qm; atm = zr[K,ml

(9.24)

Set now VXYt = e x p ( x z J + t z r K ) V ( z ) e x p ( - x z J t i o n i n d i c a t e d , i f e.g. Cv

(m+ = m-Vv)

on c

-

tzrK) and t h e n under t h e e v o l u -

i s t h e jump o f m across a s e c t o r boundary

Vv(x,S,t)

i t f o l l o w s t h a t VV(x,S,t)

(see [ST5,71

+ B,(z,Q)m

= Vv(E)X’t

where Vv(S) i s d e f i n e d

for further details).

-1 Now we connect a l l t h i s w i t h t h e framework o f [ F21. Thus G % m.-, G+ ‘L m+, -1 2 2 E % exp(xzJ), _G+G = m+ m- = exp(L,ix tu3)Goexp(-1-,iX t a 3 ) = TGoT-l where Go =

-

E G E - ~ = E ( - ~ Eb)E’l m+-m-

(9.25)

( c f . (9.11)),

h- = G - ( t ) G - -1 ( 0 ) = m-(t)m:’(O),

(O), gtg-

m-(t)Tm:l since

,

= rn+(TE)(l-G)(TE)-l

= g =

(t)m_(t)Tm:’(O)

= m,(O)Tm+ -1 ( 0 )

m- = m+TEG(E-1 T-1 ) = m,EGE -1 a t t = 0 we have mI’(0)

One w r i t e s t h e n g ( t ) = m,(O)Tm;l(O) here.

= exp(-tvH(U(0)))

R e c a l l now U ( t ) = G*g:(t)U(O)

= G*m-(t)Tm:l

-1 -1 -1 E m+ ( 0 ) ) .

= EG

but H i s not displayed

= $*g-(t)U(O)

U ( t ) = &*m+(t)m;’(O)U(O)

(9.26)

(t),

= m+(O)TEG(TE) -1 m--1 (01, g+ = h+, g- =

h,h-

m,(O)m;

h+ = G i l (O)G+(t) = m,(O)m;’

so t h a t

(O)U(O)

2 ( T = exp(-x t a 3 / 2 i ) = exp One s h o u l d p r o b a b l y j u s t t a k e U(0) a s g i v e n here. R e c a l l

N

3/ 2 i ) = i i * G - ( 0 ) ( X a 3 / 2 i )

where U(0) = Ad*G;’(O)(ho (-ht(Xo3/2i))).

N

a l s o G * g U = g ’ g - l + gUg-’ m;’(t) m;(t)rn;

so e.g.

Ad*m+(t)m:(0)Uo

+

m+(t)m;’ (O)U(O)m+(O)m;’(t). 1 ( 0 ) - m+(t)m;’ (O)m;(O)m;’(O)

(9.27)

(Dxg:

+ Qtm+(t))m;’

)gil

(t)

= ml(t)m:

- m+(t)m;

Now (U = AJ and hence (9,

( t ) - m+(t)m;

(o){z[J,m+(o)]

On t h e o t h e r hand m+(t)m;l(0)U(O)m+(O)m+

m;’(t).

+ Q)(Dx(m+(t)m;’

(0)) =

= m+(O)m;’(t))

1 (O)m;(O)m;’(t)

= {Z[J,m+(t)]

Q0m+(O)}m;’(t)

-1 ( t ) = m+(t)m:

(O)(zJ

+

Qo)m+(o)

It f o l l o w s t h a t (as d e s i r e d )

(9.28) m;’(t)

+

= Dx(m+(t)m;’(0))m+(o)

rd*g;’(t)U(O) =

ZJ

-

= z[J,m+(t)]m:

zm+(t)Jm;’(t)

( t ) + Q ( t ) + zm+(t)m+-1 (O)m+(o)J

+ Q ( t )+ zm+(t)Jm;’(t)

= ZJ

+

Q = U(t)

118

ROBERT CARROLL

This theme will be picked u p again i n 110 ( c f . also Appendix A ) .

(em HZERARCfQ FRAIIE30RK

We want t o draw t o gether and display some connections between inverse scattering and sol iton hierarchies for AKNS situations. The main idea i s t o indicate how the continuous spectrum is related t o various algebraic and geometric points o f view (cf. 911 a n d [ C6,13;17-191). Thus we connect various p o i n t s o f view and relate various canonical asymptotic expressions f o r wave matrices i n vol v i n g tau functions t o the appropriate "dressing" gauge transformations. We show how connection o f wave matrices to the hierarchy picture requires certain natural choices of dressing based on R H factorizations, etc. The continuous spectrum i s emphasized throughout and serves a s a guide i n selecting the correct wave matrices. Determinant constructions o f kernels a n d t a u functions a r e related t o AKNS kernels a n d some structure f o r kernels based a t *- i s established. Completeness relations a n d Marzenko equations a r e developed i n various contexts. Let us comment briefly on one point o f special i n t e r e s t , Thus one knows that t a u functions a r i s e naturally i n various a1 gebraic and g r o u p theoretic constructions related to "sol i t o n mathematics" (as indicated a t many places i n t h i s book). In many such developments a grading or indexing parameter A o r k i s subsequently identified w i t h a spectral variable related to some Lax operator and t h i s has various ramifications i n terms o f algebraic curves, t a u functions, Grassmannians, e t c . The constructions frequently involve loop groups and current algebras 1 based on S however, and when one attempts t o relate the spectral variable t o situations involving classical inverse scattering on the l i n e there are conceptual problems (some o f which are discussed i n §7,11 a n d i n [ Cl7-191). In particular S' i s not R a n d one cannot simply make a linear fractional transformation o r a simple deformation. Now the use o f S1 does not a f f e c t the solitons b u t some adjustments in the algebraic theory are needed i n order t o accomodate the continuous spectrum ( c f . I l l and [C6,17-19]). Thus e.g. f o r KdV one can work w i t h Hardy spaces H2' in the upper a n d lower half planes and develop the geometry, vertex operators, t a u functions, Grassmann i a n s , e t c . directly from continuous spectrum i n p u t . The development involves t a u functions ( o r "singular" theta functions following [ MC1 ,10;E31) obtained via determinant constructions equivalent to those o f [ 01,2;P1-51. I n REiRARK 9.3

FOR AKW SgSEnk).

AKNS SYSTEMS

11 9

p a r t i c u l a r t h e t a u f u n c t i o n s a r e o b t a i n e d d i r e c t l y from d e t e r m i n a n t c o n s t r u c t i o n s as a r e t h e r e l a t e d d r e s s i n g k e r n e l s (corresponding t o Marzenko k e r n e l s i n inverse scattering theory). t h e general s p i r i t o f §7,11

,

Moreover t h e d e t e r m i n a n t c o n s t r u c t i o n s , i n

include automatically a possible contribution

from a c o n t i n u o u s spectrum, and t h i s makes i t p o s s i b l e f o r t h e "meaning" o f a s p e c t r a l presence o r s p e c t r a l component from t h e r e a l l i n e t o emerge i n t h e r e s u l t i n g t a u f u n c t i o n s and d r e s s i n g k e r n e l s . We w i l l want t o develop some d e t e r m i n a n t themes f o r AKNS systems b u t f i r s t l e t us r e w r i t e a l i t t l e some

of t h e NLS development i n Remarks 9.1-9.2.

We

recall therefore

F

X

= UF; Ft = VF; Ut

-

V

+[U,V] = 0

X

I n o r d e r t o compare n o t a t i o n s w i t h [ NEl;FL1,41 one makes a change o f v a r i a b l e s t (9.30)

- t and q

-+

2 2 2 Ft = Q F, Q = QoX + QIX

+ Q;,

= kl)

(E

l a t e r (up t o f a c t o r s o f $) which l e a d s t o

-+

(E

= 1)

Qo = ( 1 / 2 i ) 0 3 ;

I n [ F 2 ] ( g o i n g back now t o t h e n o t a t i o n o f (9.29)) m a t r i c e s T(x,y,h)

and "wave f u n c t i o n s " T,(x,A)

a l y t i c a l l y extendable t o I m X > 0. f o r convenience

-

one c o n s t r u c t s t r a n s i t i o n 2 1 2 1 = (Ti. Tt) w i t h T- and T, an-

We t a k e T(X) = (: ):

w i t h T- = TtT

(E

= 1

s i n c e t h e r e a r e t h e n no s o l i t o n s t h i s c o n c e n t r a t e s a t t e n F u r t h e r i n a d d i t i o n t o a l l t h e formulas 2 we have e x p l i c i t l y b ( X , t ) = b(X,O)exp(-iA t), a ( X , t ) =

t i o n on t h e c o n t i n u o u s spectrum). i n Remarks 9.1-9.2 a(X,O),

and (6.)

+ lbI2)Xn-'dh

=

loga(h) m

Q

lrnPn(q,q)dx

icy

€In/An

where ( f o r

E

= 1 ) In = (l/h)L:

(Pn = polynomial i n q,q x,...).

gous t o KdV a s y m p t o t i c expansions as i n 51,2,6,7.

log(1

T h i s i s analo-

Such formulas g i v e a d i r -

e c t c o n n e c t i o n between c o n t i n u o u s spectrum and eventual h i e r a r c h y o b j e c t s In. We go now t o [ BGZY3;AB5;FL1 ,3;IM1,2;NEl NE1 ] f o r t h e moment.

;P1,2;TKl

,2;W1,5]

b u t m a i n l y t o [ FL1;

We w i l l examine AKNS i n an a l g e b r a i c framework b u t

120

ROBERT CARROLL

o m i t t i n g much o f t h e a l g e b r a .

Thus one c o n s i d e r s AKNS w i t h sl(2,C) 0 m a t r i c e s ) i n t h e t y p e I 1 p i c t u r e o f I NE1 1 where

(trace

h Here Qn = ( f n -Fn) and tl = x w i l l be " s p e c i a l " (see [ NE1 ] f o r o t h e r c h o i c e s o f special variable). 29) f o r NLS w i t h

E

Note a q a i n t h a t t o connect w i t h t h e n o t a t i o n o f ( 9 .

= 1 one takes r = q and t h e n t

b o t h e r t o a d j u s t f a c t o r s o f % here). formally related t o

(-;

y)

-f

-t w i t h q

The wave f u n c t i o n s J,

Q v i a a hierarchy connection

(6.)

%

loP hnr;-ny

etc. with h

T h i s is c o n f u s i n g i n

0

= f

-

= h

0

w i t h s u i t a b l e F i n Imr; > 0 o r I m c < 0.

The o n l y case where (6.)

on t h e r e a l l i n e i s when t h e s p e c t r a l t e r m b ( c ) = 0 (see below). t h e h i e r a r c h y c o n n e c t i o n ( 6 0 ) and e = -2ie2,

axfl

= 2 i f 2y...y

one o b t a i n s e.g.

2 ( n o t e 2Q (% )

1;

e

cmn, f n

and a f t e r s p e c i f y i n g h2,

( c f . (9.30) f o r r =

2

%

Qo =

= 0, e q, fl = r i n o u r formu1 1 = -ia w i l l r e q u i r e F diagonal as x -+ ?m. 0 3 b u t we c l a r i f y t h e s i t u a t i o n below by w o r k i n g

= -i,e

NE1 1

F here can be

FNn where [ Nn, h e Since we w i l l have Q = ( f - h ) , h =

= 0).

-10 l a t i o n , t h e formula Q = FQoF , Q 0

ij (we do n o t

Q = FQ F - ' ;

f o r example w i t h anF = QnF (more g e n e r a l l y anF = QnF

Q I = 0 b u t we w i l l deal w i t h N

-+

Q ( A ) i n (9.30)).

Ti -

can h o l d Thus w i t h

1; fnc.-ny one has h3, ... as i n d i c a t e d =

axel below

modulo f a c t o r s o f L,)

The c a l c u l a t i o n o f t h e hn f o l l o w s from

some L i e a l g e b r a i c machinery i n v o l v i n g c o a d j o i n t o r b i t s , t h e Adler-SymesKostant lemma, L a x - K i r i l l o v brackets, e t c . (see Appendix A f o r background and some d e t a i l s ) . We w i l l o n l y show here t h e c a l c u l a t i o n s f o r m a l l y based k on S : Xjcj Xjr;jtk where X = f_X.ci E = : sl(2,C,S) (M < m v a r i e s ,

1

-+

1

R

Xi E s l ( 2 , C ) ) . One w r i t e s $,(X) = -YS X,X) ((X,Y) = lisjSnTrX.Y., n = 0 here f o r convenience), DE$(Xt~Y)I, = t v $ ( X ) , Y ) so v $ ~ ( X )= - S k X,J HJ k ( X ) =

r=

[ n n SkX , X l w h e r e [ X , Y l = rk&tj=kc[Xi,Y.l, k r = T + ? , {l-,X. -1 cJ1,7;= M J J X j c j , M < -1, and nn: i-+ i s t h e c a n o n i c a l p r o j e c t i o n . For X = Q one k k k k has S Q = c Q, ~~e Q = Qk, and H k ( Q ) = Q ,Q1. Hence t h e Hamilton equak k hihk-i t t i o n s a r e Q % akQ = Hk(Q) = [ Q , Q ] . One f i n d s t h a t (bk(Q) =

{Io

-lo

=

AKNS SYSTEMS t

eifk-i

c - ~i n

= coefficient o f

121

the series f o r - ( h

2

t

e f ) and on phase

space t h e +k a r e i n i n v o l u t i o n ( { $ k y $ m l = 0 f o r {$,$)(X)

v $ ( X ) ] ) ) so we l o o k a t $,(a)

= ck.

This l e a d s t o c 2 =

= -(X,[n,V$(X),nn

-YS2Q,Q)

= 2elfl

-

Taking c 2 = 0 one o b t a i n s

4 i h 2 f o r example and g i v e s h2 i n terms o f c2.

(9.32) and t h e procedure g i v e s e v e n t u a l l y a l l hn ( c f . [ FL1;NEl I). F i n a l l y t o g e t conserved q u a n t i t i e s i n [ FL1;NEl ] one f i n d s t h a t akejtl

akfj+l

= ajfktl

a J.e k t l

-

= alFkj akhj+l a r e t h e corresponding f l u x e s ) . ved q u a n t i t i e s and t h e F jk

(9.33)

=

= Fjk w i t h and akhjtl = a j h k t l . Then one determines Fk j and aiFkj = ajFik (a, % a x e v e n t u a l l y t h e hjtl a r e conser-

F k j = Tr(kQoQktj

(k-l)QIQktj-l

t

..- t

It f o l l o w s t h a t

Qk-lQjtl)

k@ktj

2 = a (log-r)/at a t ( c f . also and a t a u f u n c t i o n can be d e f i n e d v i a (66) F kj k j [ 862 ,3; DK3 I).

-

The f i r s t few h . a r e e.g. h2 = - i q r / 2 and h3 = + ( r q X qrx) ( f o r c2 = c3 = J 0); a l s o r e c a l l ho = -iand hl = 0 i n o u r f o r m u l a t i o n . Up t o c o n s t a n t f a c t o r s these correspond t o t h e d e n s i t i e s one o b t a i n s from a s y m p t o t i c expansion o f l o g ( a ) , l o g ( $ ) i n [AB5;C1] ( c f . below). Thus e.g. from [ A B 5 ] we can 1 2 w r i t e axF = Q F and atF = Q F i n o u r p r e s e n t n o t a t i o n as i n (9.32) which 2 corresponds t o c l a s s i c a l AKNS n o t a t i o n w i t h A = -i(L,qr t 5 ), B = -i(-L,qx + i q c ) , and C = -i(L,r t i c r ) . X 1 axF = Q F determined v i a (9.34)

$

1 -i3x (,)e and

%

$

’L

(y)eiLx

Then one has wave f u n c t i o n s $,;,$,$

%

and

(-y)eisx %

(;)e-jsx

as x

-+

as x

--; -+

-

A#%

It f o l l o w s t h a t $ = a$ t b$ and

-

_ -lx-+i m $2e x p ( - i s x ) . -

Then e.g.

C1 = -f q r dx, e t c .

= -$$ t b$ w i t h a ( c ) =

log(a)

%

satisfying

1”0 Cn/(2ic)”’

and s(5)

w i t h Co = -f qrdx,

X

REltARK 9.4

(SPECCRAL AsrlJl’IPCOClCS ARD CALI FLIrWEltXI$).

from [ NE1 ] i s r e l e v a n t here. for

F i n t h e form

A further construction Thus upon p o s i t i n g a n a s y m p t o t i c e x p r e s s i o n

ROBERT CARROLL

122

one p u t s t h i s i n t h e formulas a k F = Q k F and equates t h e c o e f f i c i e n t s o f powers o f <. u s i n g t h e equations f o r e,f,h a r i s i n g from a k Q = [ Q k ,QJ,Q = T h i s l e a d s e.g. t o 2 i e = b ( i - h ) , 2 i f = c ( i - h ) , h = -i- k i ( e c + f b ) , ( f - he ) . Now d e f i n e T by (U) and s e t u = Tel w i t h p = rfl. Then and ec = fb. (9.36)

b(tks<) = (1/<)el(tk+i/2k< k

(1/5)fl(tk-i/2k<

k

= (l/<)fl-;

= (l/<)el+;

c(tksc) =

$+$ = lOg(T-/.C);

$-JI

lOg(T+/T)

=

+ic c k t k

D e f i n e now v e r t e x o p e r a t o r s (&+) X + ( c ) = ee x p ( k i 1 ak/2kc ) and t h e r e r e s u l t s

where T+ = T ( t k t i / 2 k g k ). -

-(i/2c)X+o

X T

(9.37)

F

Q

(l/-r) (;/2<)x-P

One notes t h a t exp(2$) =

’

“3

= (

x+T

= (i-h)/Zi

T+T-/T’

= Q with

- *F ( ck iu3)?-’

ak??-’

k

a completeness formula (6.) a r i s e s from [ AB1

ticular J

C

S e t t i n g now ( s i n c e

I.

= Qk, so Q

k

is a

A

gauge t r a n s f o r m a t i o n o f ( - i u 3 c ) under F.

m)

= (l-%bc)-’.

l o g ( 0T - / d lOg(T+/T) 0 ) from (9.36))

we a l s o o b t a i n ? ( - i o 3 ) ? ’

at

i;

I n [ NE1 ] i t i s a l s o claimed t h a t

( 1 / 2 n i ) J C F(x+y,c)F-l(x-y,c)d<

=

I (C a c i r c l e

T h i s i s somewhat m y s t e r i o u s however s i n c e i n par-

would have t o be a formal e x p r e s s i o n as i n o u r development i n

Ill ( o r a pure s o l i t o n s i t u a t i o n would be r e q u i r e d ) .

However u s i n g ( 6 . )

w i t h (9.37) does l e a d f o r m a l l y t o t h e H i r o t a e q u a t i o n s so i t deserves f u r t h e r i n t e r p e r t a t i o n ( p r o b a b l y as i n § 1 1 ) .

L e t us a l s o s p e l l o u t here some

f u r t h e r connections t o c l a s s i c a l AKNS and s p e c t r a l data. $, e t c . as i n (9.34) -+ - m

Then

and Y

-f

@

0

= YA and

as x

-

o = ( 4 -$) and -f

( j -$)

(note

Go

Q

9 = ( j JI)

so o

E i n say (9.18),

= ( $ $ ) S where

-f

oo

Thus w r i t e f o r = e x p ( - i c o 3 x ) as x

with 5

Q

+A o r x

Q

4z).

TAU FUNCTIONS

123

1 Note here @ f o r example s a t i s f i e s a x @ = Q @ b u t t i m e e v o l u t i o n f o r tk, k 2 (**) aka = Q k @ - @(oilQ:eo) where Qk = l i m Q k as x + 2, must be a d j u s t e d v i a 0

km.

Then Q,“

= -igku

mula @ = 0 A as x 0

3

+ m

akA = [ - i c ku3,Al = 2 i r ; k (,,O og )

(9.40)

f r k 2 = 0,

a kb

whence a a = a

(1;

mi’Qke and one o b t a i n s , u s i n g t h e a s y m p t o t i c f o r 0 0

=

k

ak;

= 2 i c b, and

( - i c u 3 t k ) ) s a t i s f i e s f o r k 2 2, akF

kA

= - 2 i c b.

k

= Q F.

We n o t e t h a t F = @exp

Next we observe i n (9.37)-

(e ,f ) = ( q , r ) 0 (and i n f a c t ( e , f ) -+ 0 ) which i m 1 1 p l i e s b,c -+ 0 and hence any F r e p r e s e n t e d i n t h i s form must be diagonal as (9.38) t h a t as x

x

-+

?I-

-f

( n o t e a l s o f o r b = c = 0,

= T’ o r T + T - / T ~ = 1 ) . Hence i n part t i c u l a r such a formula cannot r e p r e s e n t e o r \Y and we must deal w i t h F o f -+ t m

t h e form (*A) S, (not Imc = 0).

=

T T-

( $ $); S- = ( $ -$) f o r example w i t h Imc > 0 o r I m c < 0

This i s discussed l a t e r .

On t h e o t h e r hand i f t h e s p e c t r a l

h

data b,b a r e zero t h e n from

a,;

=

I$

$ = : we I have )

@,Y diagonal a t

Lm.

Now w r i t e (9.41)

T =

(‘;IT

Tt/T

lim x++w

); T

and l e t f = Fao f o r Q0 = e x p ( - i c x o 3 )

Q

E.

Consider a s i t u a t i o n where (9.28)

applies asymptotically i n 5 f o r a r e g i o n c o n t a i n i n g r e a l r;. Then as x -1 -1 = Since @ = Y A we o b t a i n +my f T k @ which i m p l i e s ET-’ = ‘Y and sT

-+

-+

@.

0

A = TtTI1

(9.42)

Q

[

(T-/T

0 t

0

(T+/T)

Thus n e c e s s a r i l y b

( f ’ means l i m ( f ) as x + + ) . If (9.37)-(9.38)

CHEOREl 9.6,

I+( T t / T 1-

(T-/T)-

=

*b

1

= 0 and f o r m a l l y

hold asymptotically i n 5 i n a region includA

i n g r e a l r; t h e n A has t h e form (9.42) w i t h t h e s p e c t r a l data b = b = 0. Hence a

(T-/T)’(T+/T)-

Q

EXAEIPI;E (9.6). A

= 1,

el

(9.43)

-

=

el

and

Q‘

+

Q

( T ~ / T ) (T-/T)-.

-

Consider t h e example o f [ NE1 1, p. 173. Thus fl = r = -q, k ” -k and = t - i u (cl = E t i q ) , H1 = 2 i c cltk, H1 = - 2 i l cltk,

T = 1 + (1/16n 2 ) e (-4nx t

2iq t k ( $ - E ; ) )

k R e c a l l a l s o T + ( t ) = (tk ?I i / 2 k r ; ) ( 5 #

-

c1 h e r e ) so one has

(Ek =

k

c1

-

$)

N

124

ROBERT CARROLL

I t follows t h a t a s x (9.45)

( T +-/ T ) -

-

-f

m,

T

and

T+

-

-f

1 , while a s x

3 -m

e x p ( T 3 i ~ d cT 21; E k / 2 k s k )

m k Thus e.g. l o g a % log(Tt/T)- % - 2 i d s - 212 Ek/2ks and we can compare with k (6.) when E = -1 and b = 0 i n which c a s e ( c f . [AB5;NE1 1) I k = - ( l / i k ) ( t l k k e l ) and l o g a % - i l y I k / c . T h u s I k = ( l / i k ) E k f o r k 2 and I1 = 2 i n which a g r e e s w i t h (9.45). A r e l a t e d example f o r KdV i s given i n 17.

We r e c a l l some r e s u l t s o f [ AI13;KQ1-4] s p e c i a l i z e d t o AKNS. T h u s i n t h e n o t a t i o n o f [KQl-31 s e t Lv = i ~ - l ( -a ~ Q)v = A V and vt = A(Q,A)v where v ( ' I ) , T = u3 = -:), Q =

RElURK 9.7

(DECZEIZNANC ?'IECH@D$ FOR AKW).

o q

( r o), AZ2

=

(9.46)

A =

t

- A l l y and

1,3

n An(Q)X = a o +~ al(TA

a 3 ( T ~ 3t iqx2

(A

va

t +T(Q

2

-

Q ~ ) At

t

iQ)

t

+q 3 -+qXx

a ( T A ~t i q 2

t

4-r(Q2- Q x ) )

- Li "&Qx I )

Our i n t e r e s t i s i n a 3 = 0 + ..., and we f i r s t connect t h i s t o the preceeding n o t a t i o n . S e t ( a x - Q ) v = -i.rXv = - i o Av o r a v = (-iAo3 t Q)v which i s the 3 X same a s ( 9 . 3 1 ) ( A % 5, Q' % -iAo3 + Q = XQo + Q l ) . Now Q, i n ( 9 . 3 2 ) and Q2 2 2 = A Qo t AQ, t Q, = - i h u3 t A Q t Q, w i l l provide i d e n t i f i c a t i o n s w i t h (9.46) v i a a 3 = 0 and a 2 = - i (so the a 2 term i s -iu3X2 t X Q - + i a 3 ( q 2 - 4,) and then - + i o 3 ( Q 2 - Q,) i s e x a c t l y Q 2 ) ; hence t a k e a. = a l = a 3 = 0 i n (9.46) 2 w i t h v t = Q v . Now we write

e

(9.47)

so I#J+

%

iAx0

3

($+I -

X I+,

= I + -

dy e

i Xya

3 (Q$+) -

0 a s x + ?m. Thus i n 0 $- = $t$ which corresponds

exp(-iAxa3) =

We write assuming no d i s c r e t e spectrum we s e t 1 r ( 9 48) M = r+l = s12/s22 = r,2 $- ~ - 0 .

[

r -1

= -s

/s 12 11 = :/a;

:' 1; r-2

= -s

21

/s

22

=

-b/:;

previous n o t a t i o n , t o Q = YA so $ = A

-^b/e';

$t =

yAand (a - b ) and b a '~r

rt2 = sZ1/sll = b/a;

F -+ ( w ) = ( 1 / 2 ~ ) 1 1( M +-( A )

-

1)

DETERMINANT METHODS FOR AKNS

M equations a r e ( +

Then t h e

%

cm

T B,(x,Y)

%

y < x)

B+(x,z)F+(z+y)dz -

B+(x,y) + Ff(x+3/) ?

(9.49)

-

y > x;

&-

T + F+(x+y) t

125

= 0;

T T F+(z+y)B+(x,z)dz -

fm

= 0

t r a n s p o s e ) and (BN % o f f diagonal, BD % d i a g o n a l ) BD (x,x) = ?"./- +m$2 N 2 (u)du; B (x,x) = t%Q(x) ( $ means QQ). Comparing w i t h [ A B l ; C l I we have (B. T

%

J, = (y)eisX

(9.50)

+

F ( x ) = ( l / Z i ~ ) l f(b/a)eicxdc;

0 (,)F(x+y) = 0;

+

I,.

K =

= 0;

); F

K(x,Y)

= ( F0 oA); -F

K(x,y) f

-icx +

= (,)e 1

A

(1/2n)lI

p(x) =

K(x,s)F(s+y)ds

K1 K1

IJJ

K(x,s)eicsds;

c

(t/i)e-icxdJ;

t?(x,y)

o n - (,)F(x+y) -

F(x+y) +

t(x,s)e-issds; t

?(x,s)?(s+y)ds

4-K(x,s)F(s+y)ds

= 0

K2 K2 Thus $+

/xm ( k

%

Y

=

( j J,)

(recall

K)exp(-icsa3)ds o r I

means K = B+.

$-%

-

-$I) and

($

exp(-icxu3) =

Therefore F+ =

we have e.g.

( j J,) -

/xm K(x,s)exp(-icsa3)ds

e

-i <xu 3 =

which

F and hence ( n o t e 0 ,= FD ) r+2= b/a

= sZ1/sl1 a -b T h i s means f o r Sr = ( b a ) , s , ~= a, sZ1 = b, s Z 2

-$/$ =*s 1 2/ s 22' = a, and s 1 2 = -b. Kato i n [ K Q l ] uses I$+d e f i n e d on (x,+m) w i t h and r+l = A

IJJ+(nyC,~) = v+(o,C,~)

(9.51

l p d c F +-( o , s h + ( s,Sy1-I)

+

T T T T T -1 T From (9.49) we have now (1 + F+)B+ F., - - = -F+- and B+- = - ( l + F+) where s % ( x y j ) , o % ( y , k ) , j,k,m = 1,2, $ + ( n , ~ , - l ) % (B,)jk(XsY)

":1

and 1 dc ,I, Thus K

dz; a l s o v+(n,C,u)

B+ w h i l e J,+

BI.

Y), o = ( y Y 2 ) , 6 + ( X , l ) % a

1

1

F+(y,z)$~+(z,x )dz. (9.53)

2

one can w r i t e - e . g .

(+*I

a

IJJ+(Y,x)

t

a

for i

= v+(Y,~)

1-1 =

- I,"

s

%

(z,m),

= Fimk(z+y).

z l , B12(x,y) $

F,($,;)IJJ+(z,x)dz

= K1(x,

1

-

But n

1

v+(Y,x)

= PF+jk(x+y) and F+(n,c)

Note t h a t

= -Fl2(xy)

a 1

= F(x+Y), F+(Y,z)

1 1

F+(Y,z) = F + 2 2 ( ~ + z )= 0; J,+($,;) = Kl(x.Y) A

= K1(x,z);

...

+

= F + 1 2 ( ~ + z ) = -?(Y+z);

= B12(X,Y);

l

l

$+(z,x)

= Bll(x,z)

5

126

ROBERT CARROLL

SO (+*) i s K1(x,y)

H(s,y)

=

/,”

+

c

;(y+z)?l(xyz)dz

as i n (9.50).

W r i t i n g now

from (9.50) we o b t a i n

-

K1(X3Y) = ;(X+Y)

(9.54)

c

= ;(x+Y)

F(s+z);(z+y)dz

Kl(x,S)H(s,y)ds;

-

~ 2 ( ~ y ~ ) H ( ~ y sK)2d( xsY; y ) = -H(X,Y) A

K1 (x,Y)

-

= -H(Y,x)

i 2 ( x Y y ) = -F(x+y)

-

K2(xys)H(s,y)ds;

&

m

*

K1 (x,z)H(y,z)dz

which w i l l be e q u i v a l e n t t o t h e f o r m u l a t i o n o f [ K Q l ] .

Such equations can be

as i n §7 d i r e c t l y .

s o l v e d by t h e determinant methods o f [ 01,2;P1-41

One

wants t o do t h i s from t h e t h e o r y o f [ KQ11 d i r e c t l y however s i n c e i t generali z e s n i c e l y ( c f . a l s o [AS1-3;CFl;KQ2-41). (9.55)

-

Jl+(u) = (1

uF+)-’v?(u)

-

(1 + v R + ) ( l or R

-

F

-

-

RF = 0 = R

“m (9.56)

K,

f o r m 2 1 (K.

0

[nl51

1

-

FR.

Here 1

= det

I K Q ~one J writes

= (1 + wR+)v+(u); - -

wF+) - = 1 = (1

[ F,(nlYcl)

’L

-

v F-+ ) ( l + w R +)

p).

6(y-z)(;

)

F,(r)my
5,

- * .

= 1 ).

F

Thus i n

... . ..

W r i t e now

F+(rll.cm) F*(nmycm)

Here t h e p r e s e n t a t i o n i n [ KQ1 1 i s u n c l e a r b u t t h e de-

t a i l s can be determined v i a [ C F l 1.

F i r s t we w r i t e t h e formulas.

i s a m a t r i x d e f i n e d by

, [ “ “1“1

Km+l

(9.57)

5

“m

...

]

;I

P

means rl

P

i s missing.

...

[“I n1

= K;

r),

p,(“,nP)K: where

I

[

... nm

“

r)

5

“1

Then ( +

’L

1-

llm

F+(Q,c)

-

... rl ... rlm ... n p ... nm

: I

y > x;

-

‘L

y < x)

Thus KY” -

DETERMINANT METHODS FOR AKNS

where J d n

j

=

'4-

+m

127

d q . and

J

We provide now t h e following i n t e r p e r t a t i o n f o r (9.56)-(9.57), e t c . followi n g [ CF1 ]. (9.60)

T h u s one will need i n (9.58) KE

[ "n1 ... "] . .. urn

=

1KT

(Y, y j l )

... ( y m y j m )

(yl , j , )

...

1

bmYjm)

(sum f o r j, , . . . j , = 1 t o 2 ) . This will then provide t h e appropriate indices In a term i n t h e determinant of (9.57) f o r terms F+(nj,nk). A

index appearing i n a P s e p a r a t e sum and n having i t s own index 1 o r 2 (along w i t h 5). Note n = P ( y p y l ) and n = ( y p y 2 ) will b o t h be included i n (9.57). P REI;1ARK 9.8 (S0NE 3ACKGR0LIND ZNF0RlMCZ8)N). We record now some miscellaneous f a c t s from [ KQ1 I. One wants the above t o hold f o r p = -1. Kato shows this is O K i f e.g. Fk2, ( z t y ) = cF+12(y+z) when c < 0 i s real (such a s i t u a t i o n p r e v a i l s f o r NLS f o r example). In t h a t case t h e M equation (9.51) has f o r example one sums again over ( y i , j i ) w i t h t h e n

unique s o l u t i o n s f o r u <

E

R and g + ( p , x ) = 0.

In general one wants ( O b ) 1p1

IIF i 1I-l f o r convergence of the Fesolvant s e r i e s . t

Here

o ) so the F; terms have bounds (+*) IFi2(u)1 5 ( 1 / 2 a ~l h l~I ^ ~b / a " l d h and l F i l ( u ) l 5 (1/2a) Now F ' ( u ) = (1/2a)/ (M+-l)(ihT)exp(iX?w)dh w i t h M,-1

=

(Or

fa,( A ( Ib/aldA. One can work e.g. i n a context where r k l y rk2 E S (Schwartz space). Hence the i n t e g r a l s i n ( 0 0 ) will e x i s t i n reasonable circumstances.

-m

More t o t h e p o i n t we consider Fi2(u) = ( 1 / 2 ~ l ) j I(g/:)Aexp(-ihw)dh

and Fil(u)

128

ROBERT CARROLL

Now given r l , r 2 E S one has Arl, A r2 E b and the transforms F i 2 ( w ) , F i l ( w ) E S. Hence t h e r e will be a c l a s s C o f s i t u a t i o n s where IIF+II+l < 1 i n which case ( + b ) holds f o r w = -1. Note t h a t t h e requirement f o r F+ e n t r i e s t o be in S in Kato is t h a t ( W ) Rea(A) 2 - 3 = 0 f o r A E R where a ( X ) = I. anAn which means f o r us a ( X ) = a x2 = - i x 2 must be imaginary, which holds f o r h real as desired. Generally we will a s sume c l a s s C s i t u a t i o n s a r e present. Then one gets (=*) Q-+ j k ( x , t , u ) = T2u G+(S',S,FI,x,t)/g+(u,x,t) where 5 = ( X , j ) , 5' = ( x , k ) . Note Q+jk(x,w) = r2 N $ + ( 5 ' , 5 , w ) ( B + ( x , x ) = %Q(x) f o r w = -1 1. Also time dependence e n t e r s here via ( a = -ix2, -2at = 2ix 2 t ) = ( 1 / 2 r ) i z A(b/a)exp(iAw)dA d i r e c t l y .

(9.63)

F,12 ( w , t ) = ( 1 / 2 ~ ) J r k l e

i (i h w - z a t )

dA; F+12 = ( 1 / 2 a )

The f a c t t h a t Q-+ j k ( X , ! J ) = T ~ ? J R * ( S ' , S , ~ J , X ) with Q-+ j k ( X , P ) Q +kJ. ( x , ~ ) = F ~ P R -+ ( S . ( , F I , X ) is c a l l e d redundancy. Thus F+R, -+ Q, w i t h conditions on Q i j . Now f o r t h e tau function aspects -f

REPIARK 9.9 (C@lPARIS0N O f KERNELS AND SPECCRAL AsgllPC0EZCS). We now w r i t e down t h e wave functions from B+- a n d look a t t h e dressing version based o n

Remarks 9.1-9.4 f o r NLS.

Thus r e c a l l B+ % (n,c,-l,x)/g+(-l,x), a n d g+ corresponds t o ~- + ( =t )T ( t k-+ i / 2 k s k ) c a r l i e r and T-+ lim X m ,' ( x , ~ ) ) =) ~$ +~( ( y , k ) , ( x , j ) , - l ) and s i n c e 'Y (-ihsog)ds from Remark 9.8 we obtain $) Q

(2

(-iAsu3)ds/

+T ( x ) .

(9.66)

JI

=

T K , $+ B+ , $ + ( n , E , - l ) = -G+ + a tau function T ( X ) say ( r e c a l l ~ ( x ) ) . Also B+(x,y) means ( ( B + - exp(-ixxa3) = jm B+(x,s)exp = exp(-ixxu3) - ' /x m G+(s,x)exp T Q

We w r i t e t h i s i n component form a s

1 -iXx (,)e

- /xm

G+lT (S,X)e-iAsdS)/+T(X);

COMPARISON OF KERNELS

4-G12(s,x)eiXsds/+r(x)

-

$ = (1 )e

129

We a l s o want t o s p e l l o u t t h e t h e o r y based a t

( c f . a l s o [ CF1 I ) .

-m

us w r i t e @ = ( 4 -;)

i n AKNS form w i t h say ( l / a = T, l/a^ =

(9.67)

-isx +

T+ = (,)e 1

If K - ( x , ~ ) e - ~ ~ ~ dT+= s;

Thus l e t

7)

( - ]0) e i g x

+

+ 1,x nK-(x.s)eigsds A

( t h e use o f T,T here is i m p o r t a n t and i s discussed l a t e r ( c f . a l s o 57). On t h e o t h e r hand i n K a t o ' s f o r m u l a t i o n we have from (9.48), B (x,y) = ( 1 / 2 1 ~ )

-

lI ($-

- exp(-iXxo3))exp(ihyo3)dh 4-

(9.68)

= e-ixxO

l:

3 +

or

(0

a)

B - ( ~ , y ) e - ~ dy ~~~3

1 ( n o t e i n [ CF1 I, K1 (x,Y) = ( 1 / 2 a ) I m ( + e x p ( - i c x ) (,)exp(-icx)~xp(iEy)dt;, -0 K 2 ( x y Y ) = (1/2~)[: ( 4 2 e x p ( i 5 x ) (l ) e x p ( i g x ) ) e x p ( - i g y ) d 5 so (@lexp(-iEx) 1 $ 2 e x p ( i c x ) ) % 4- w i t h B1 (K1 K2) t h u s e.g. $1 = (,) + exp(i5(x-y))K1

-

-

-

Q

(x,y)dy e t c . ) . (y,k),

5

= (Xyj),

[f

K ) (P (K- ?i )!); f o r n = 'T $- % B- ( v = -1 i n a l l o f t h i s

Consequently we w r i t e B- = ( K

4-(nyC)

%

((B-))jk(xyy);

now) and ( m A ) $ _ ( n y 5 ) = - G ~ ~ ~ , 5 , - 1 , x ~ / g ~ ~ - 1 W , ex ~w. r i t e g - ( - l y x ) = - T ( x ) T now and from (9.68) and ( m A ) one has 0 = ( 4 - $ ) = 4 = e x p ( - i h x a ) G-

-

(S,X)eXp(-ihSu,)dS/-,(x).

3

-

[t

Thus a g a i n i n components

Now p u t AKNS i n d r e s s i n g form f o l l o w i n g Remark ( 9 . 5 ) . Thus r e c a l l some no1 2 1 2 1 . 1 2 t a t i o n : (T+ T+) % Y ; (T- T-) % Q; T, % $; T+ % $; T! 4; Tf 'L -$; w h i l e 1 2 1 2 R e c a l l a l s o YA = @, (mo) S, (T- T+) %*(I$J I ) and S- = (T, T-) % ( $ -;). a -b a $ A = ( b $), A i l = (-., $1$2 02q1 = a, s'; = (l/a)('2 - ' I ) , SS+S, -$? 41 1 -b S 'L ( l / a ) ( - b l)y G- = S-/E, and G+ = aESil ( E ( x ) % exp(-%iXxo3) 'L e x p ( - i c x Q

-

a3)

-

we i g n o r e t h e f a c t o r o f % i n o u r AKNS development).

e t c . w i t h t h e AKNS terms $,$,

...

We use now S,

(which d e f i n e s T l Y 2 f o r AKNS).

I n dealing

w i t h a s y m p t o t i c s f o r Y o r 0 t h e o n l y common r e g i o n o f v a l i d i t y f o r 5 is t h e real l i n e

(a'

priori).

Then one has f o r example from 4 = a? + b$,

= : +I)

130

ROBERT CARROLL n

AA

t b$,

$ = b4

4

-

A

A

-icx (9.70)

'Y

A

a$, J, = a4 + b$ as x (e

-f

O

e

iex) ( x +. -1;

t h e formulas

+m

-+

I

'Y +.

(x

-+

--);

Given 5 r e a l t h e r e i s no.way t o simply d r o p t h e o f f diagonal terms i n (9.70)

-

4

u n l e s s b = b = 0. I n terms o f h i e r a r c h i e s e t c . i f one a l l o w s a formula Q = -1 w i t h F +. Bo = e x p ( - i c x u 3 ) as x -+ m o r x -+ -m then s i n c e Q -iu3 a t FQoF n . +- we g e t - i o&3 = @05 0B -0l and hence ( c f . Remark 9.5) Qo = - i u 3 . Thus Q i s on an o r b i t o f Po. I f now F QoB a t t h e o t h e r e n d p o i n t (say F % Q, F a. a t -+

+.

-f

-a,

and F

-+

BOA a t

a)

t h e n one needs - i u g = moB$oB-l~~l

which i s O K i f B

v

commutes w i t h Qo = - i o 3 .

But B = A f o r example does n o t commute w i t h - i u 3

A

u n l e s s b = b = 0.

Hence i n t h e n a t u r a l domain o f

-

i n g f u l f o r p o s s i b l e a s y m p t o t i c expansion i n -1 ,.. (Qo = - i u 3 ) .

<

h i e r a r c h y c o n n e c t i o n Q = FQoF

(9.38) w i t h F = B o r Y as i n Remark 9.6.

c where

B o r 'Y a r e mean-

say we have a v i o l a t i o n o f t h e Hence one cannot use (9.37)-

Gle w i l l see below however t h a t t h e

m a t t e r i s more i n t e r e s t i n g w i t h G,

Indeed g o i n g back t o (ma) we con-

s i d e r G,

A l s o we have t h e r e l a t i o n s ( e

etc. f o r I m c > 0 o r G- f o r I m y < 0.

e x p ( i < x ) , e-

%

G-

= S-E-'

Therefore s i n c e e.g. as x

+.

--

'L

exp(-i<x))

Consequently we can w r i t e (9.72)

t

-

= ( T j T2)E-l

exp(2icx)

-+

%

f

0 as x

f o r Imc < 0 e t c . one has

o

-+

.I-

-be

$et

m

n -

]

E-l; m

G-

%

[ ae ] be'

e'

E-l; -m

when I m c > 0 and e x p ( - Z i < x )

-+

0

ASYMPTOTICS Thus G- and G i l

131

b o t h t a k e diagonal form as x

mains o f d e f i n i t i o n I m g < 0 o r Img > 0. use o f (9.37)-(9.38)

+

i n t h e i r r e s p e c t i v e do-

f-

T h i s a l l o w s one t o e n v i s i o n now t h e

e t c . as reasonable a s y m p t o t i c expansions i n g i n these

regions. L e t us examine t h e a s y m p t o t i c form now i n c o n n e c t i o n w i t h (9.37)-(9.38).

We

do n o t expect m a t t e r s t o reduce t o t h e s i t u a t i o n o f Theorem 9.6 ( u n l e s s perA

haps b = b = 0 ) s i n c e we a r e w o r k i n g i n d i f f e r e n t r e g i o n s .

The t a u func-

t i o n s i n v o l v e d h e r e a r e n o t g e n e r a l l y t h e same a s those i n Theorem 9.6 and Example 9.7 (where a

%

and $

(T+/T)-

Q

I n t h e p r e s e n t s i t u a t i o n we

(T-/T)-.

l o o k a t (9.37)-(9.38)

and observe t h a t a conponentwise v e r s i o n o f t h e diagon

a1 terms g i v e s ( f r o m

(9.69))

(9.74)

- ?ol

=

+

-

= +r e- i c x

- -re- i c x - E l

GIlle-iseds.

,

GTlle-issds*

, ra2 = +r e i c x -

GIZ2ei “ds ;

+

TO^ = - re i c x - 1,x

T igsds G-22e

;’o (Fo = e x p ( - l y Now as i n Remark 9.5 we expect t h a t F- = G-Fo and F+ = GF k i s t k u 3 ) ) w i l l be wave f u n c t i o n s F i n (9.37)-(9.38) so t h a t F % ( l / ~ ) X( T %

) f o r s u i t a b l e T. We s h o u l d have t h e n f o r I m c > 0 say, w i t h F g i v e n by Gi1F0 = ( l / a ) ( T ! Tf)E-’Fo % ( l / a ) ( T ! Te)exp(-il; t k c k o 3 ) = ( l / a ) ( T -1 T+) 2

F ’ = ( l / a ) ( $ $)F;

f o r some

0

(9.75)

T-/T

%

T

(l/a)+eiCx;

T+/T

%

(l/a)$2e - i c x

- i c x = -T Now i n (9.74) we w r i t e ( A g ) ( ~ 6 )T+e +T$2exp(-icx) = +T - Jxw G+22exp(-ig(x-s))ds T + = t. Q

-

eic(x-s)ds = -t; G-11 I f t h i s i s t o be compatLw

i b l e w i t h (9.75) we must have (m+) aT-/T = - C / - T and aT+/T =

+t/+T.

from r e s u l t s f o r t h e KdV e q u a t i o n s and KP equations ( c f . §2,3,7)

Now

we have,

t a k i n g KdV as an example (9.76)

$+ = e i gx

K(X,Y)

and 1+K- = (l+K;)-’ o t h e r d e t a i l s here.

+ +

K(x,s)eiesds;

F(x,Y)

+

;/

T$- = e -igx

K(x,z)F(z,y)dz

w i t h 1+K- = ( l + K + ) ( l + F ) .

+

lf K-(x,s)e-iesds;

= 0

We o m i t t h e form o f F and 0 t h

Then one can c o n s t r u c t f+v i a t a u f u n c t i o n s and d e t e r -

ROBERT CARROLL

132

CI

minant methods a s above and w = $+ = X-T/T = ( T - / T ) e x p ( i Furt h e r T$- represents the a d j o i n t wave function w* which i s expressed a s ~ + T / T = (T,/T)exp(-i l ~ ~ ~ ~ The~ point r ; i s ~t h a~t i t~ is ~t o t)a l l y, natural to

have wave functions w,w* defined via kernels K = K, a n d K- based on It- nevert h e l e s s represented through t h e same tau function. T h u s i n (e+) i t is not a t a l l unseemly t o expect T' and C' t o be expressed via a single T , even though T' a r e constructed v i a d i f f e r e n t kernels, e t c . Finally l e t us check t t h e asymptotics. From ( m b ) , (9.58), e t c . one has ( m m ) T 1 as x m with t V 1 ; -T 1 a s x .+ -a w i t h -C5+ 1 while one knows a s x m y $ l e x p ( i c x ) a and a s x - m y $2exp(-ir;x) a . Thus F ( l / a ) ( $ $)E-lFo = ?Fo = Gy1F0 A * 1 0 'Ia a s i n (9.73) .+

.+

-f

-+

-+

-f

-+

Q

-+

will have asymptotic form (***) F 'L ( o l / a ) m ; F % ( +&/ +T 1 ( x -+ m ) and + a ( x wh-ley from ( m + ) , (*A)

-m

- m ) with-C/-r a 1 ( x + -m). Similarly f o r F = G-Fo we would obtain tau funct i o n s compatible w i t h t h e asymptotics i n (9.73) f o r G- (see below). Hence -+

(x

-f

m)

and

.+

-+

.+

mfQ)~m 9-10.

In t h e regions Imr; > 0 o r Imr; < 0 one can determine tau funct i o n s and vertex operator equations based on ( m b ) , ( m + ) , providing t h e asym-

p t o t i c formulas (9.37)-(9.38). In p a r t c u l a r f o r G;'Fo one has (T-/T)- = t + + lim l i m (l/a)-C/-T = l / a and ( T + / T ) = xG-Fo we ( l / a ) E/ T = l / a . For T X"= lim t + + lim have ( T - / T ) - = X-f-m &/ T = 2 and (T+/T -e/-T = 2. X-+m 1 2 Proof: For G- = ( T + T-) = ( I $ -;)E-' we have (9.73) w i t h T / T 'L &exp(ir;x) A and T + / T -$2exp(-ir;x) ( T is a d i f f e r e n t tau function than in the siQ

Gil

Q,

T h u s a s x -, (T-/T) -* 1 , (T+/T) ^a while as x .+ - m , (T-/T) :, ( T + / T ) .+ 1 . Also from (9.741, - r$2exp(-icx) = -& = - T - jmG- 22eXP(i5 + A t t m T ( s - x ) ) d s a n d r q l e x p ( i c x ) = C = T - /x G + l l e x p ( i c ( x - s ) ) d s . Hence T - / T +c/+T and T+/T -c/-T e t c . Q ED tuation).

.+

-f

-f

-

A

Q

4

Note i f b = b = 0 so a: = 1 one has t h e Sam: asymptotic f o r a 0 mulas in (9.73) b u t these do not correspond t o t h e form ( o a ) o f Examfle 9 . a 0 $)E-' f o r example gives the c o r r e c t F-m = ( 7. In Example 9.7 F Q ). A 04 Now f o r b = = 0 ( w i t h a$ 1 ) one has $ = a$, $ = -a$, $ = -a;, $ = a @ so ? Q -a$) w i t h G- 'L ( j-;) which a l l f i t s together. T h u s ( j-a;) = ( j 1 0 A n 1 0 k ( o a ) and ( j-a$)Fo = ( $ - $ ) F o ( o a ) s i n c e Fo = e x p ( - i l y tkr;u 3 ) commutes 1 0 with ( o a ) . RElMRK 9-11.

r\

(4

A

(i

A

-i)

COMPLETENESS AND SPECTRAL FORMS with our use o f dressing kernels K*etc.

133

from the r i g h t and l e f t we will ex-

amine various El equations and conpleteness r e l a t i o n s . In p a r t i c u l a r some F i r s t (assumi n t e r e s t ng f a c t s emerge about t h e asymptotic behavior of K-. ing no d s c r e t e spectrum) one knows f o r AKNS t h a t completeness is expressed c f . [ ABlY5;C1;NElJ and remrks below)

through (9.76)

( ( ) x means t h a t the f u n c t i o n s have argument x ) .

(again (

This i m p l i e s i n p a r t i c u l a r

argument x ) .

More g e n e r a l l y f o r C,? c o n t o u r s above and below the z e r o s o f a and $ r e s p e c t i v e l y one uses fX

1 0 ( o ) G ( ~ - Y ) = ( l / 2 n ) l C (l/a)(:l

(9.78)

;

%

A

Ix($* J , Iydr; ~ -

4

- ( l / Z n ) J ? (l/i)?;$:dg

( & ) x ( @ 2 $1 )ydc; ~ ( x - Y ) = ( 1 / 2 n ) I c (l/a)$;$dr, (1/2r).fC (l/a)$;$:dc

We r e c a l l a l s o W(u,v) A

= u1v2

-

u2v1; W($,$)

(1/2n)l[ (l/:)

-

*x.y

(1/2n)!z A

=

(l/a)$2$1ds

A

r\

= a ; W($,$) = a ; -W($,$)

=

b;

A

W($,J,) = b and from (9.50), (9.67) one has (9.79)

e i gs ds;

T$ = (,)e 1 -icx ( 0, ) e i g x +

$ =

t

lt

K-(x,s)e-igsds;

,,” K(x,s)eigsds;

Recall a l ssoo from 1I C1,4,7,8,10 J f o r y (9.80) (9.80)

;;(X,Y) (X,Y)

>>

j

?$ =

= (;)e-jcx

( - ,0) e i g x t x AK-(x,s) t

,,” K(x,s)e-igsds A

xx

$(c.,x)e’cYdc; -i SYd = ( 1 / 2 a ) Jl C ~ ( c , x ) e ’ c Y d c ;K(x,y) == ((1l / Z2 n ))..fft? $ ( c , x ) e -i cyd A A

Let us derive equations e q u a t i o n s f o r KK - and KK- i n tthe h e same manner o p e r a t e on on $ aand i n (9.79) by by ( 11 // 22 ~~ )) //exp(icy)dc e~~x p ( i g y ) d c (( yy
exp(-igy)dg (y << xx )) r e s p e c t i v e l y t o obtain obtain for y exp(-icy)dc i s no continuous spectrum) t u a l l y i f tthere here

<<

(C xx (C

f o r AKNS. AKNS. Thus xx )) and ((l/Zn)/; l/Zn)/; ffii

== CC =

(-m,m) (-my,)

even-

134

ROBERT CARROLL

We n o t e t h a t t h e r e seems t o be a c h o i c e p o s s i b l e w i t h t h e c o n t o u r s i n t h e e q u a t i o n s o f (9.81 ); however one sees i n

$,$

states

- and

below

-

1 C81 when t h e r e a r e no bound

t h a t t h e c o n t o u r s a r e w e l l chosen here.

t i o n here (no d i s c r e t e spectrum) we w i l l use C =

?/:

(b/a)$;

*

=

Now $/a =

(-my-).

A

= -$ + (b/a)$ and we observe t h a t 0 = Ic ($/a)(XJexp(icy)dc

(~/^/a")(x,c)exp(-icy)de f o r y

0 = /;

Ic $(x,c)exp(-icy)dc A

*

For o u r s i t u a -

f o r y < x.

n

r

x.

/?

Also

8=

Then p u t

$(x,c)exp(ir,y)dg

$/a

-

j + and

0 = *

A

n

(b/a)$ and $ = (b/a)$

-

This leads t o

$/a i n (9.80)-(9.81).

Under s t a n d a r d hypotheses one has f o r y > x

EHE0REl3 9-13, 4

(9.82)

K(x,y)

K(x,y)

= (l/Zn)l;

= ( l / Z i ~ ) j ' ~~(c,x)eiSYdc

$(c,x)e-ieYdc

= - ( 1 / 2 ~ ) j ' ~(b/a)$(c,x)eiCYdc;

= ( 1 / 2 1 ~ ) k (b/a)$(s,x)e A A ~

-'cYdr,

while f o r y < x

R€X~~IRK 9-14 (IMRCENK0 EQUAEl0Wg)- R e o r g a n i z i n g i n Theorem 9.13 we can w r i t e n

(9.84)

( 1 / 2 a ) l C (b/a)$(c,x)eieYdc

$(c,x)eiSY

dr, = K-(x,y)

(Y <

(Y < x);

XI;( 1 / 2 ~ ) l ;

A

4

= -K(x,y)

(Y

>

x);

( 1 / 2 a ) l c (&:)$(c,x)e A

(bla)$(c,x)e-iSYdc

= K(x,y)

(1/2a)k

-ieydc

=

(b/a) (x,y)

(Y > x )

A

I n t h e event C = C =

-?

(-m,m)

one has a s i t u a t i o n l i k e KdV ( p a i r i n g K- w i t h

and K w i t h ?-) w h i l e i n s i t u a t i o n s l i k e NLS where a,;

t h e r e w i l l be f u r t h e r l i n k a g e .

and b y $ a r e r e l a t e d

Thus f o r KdV r e c a l l from

I

C131 f o r example,

w i t h no d i s c r e t e spectrum ( c f . a l s o §2,3,7) (9.85)

( 1 / 2 1 ~ ) l zReikYf+(k,x)dk

{

K_(XYY)

- K(,

x ,Y

1

(Y < x ) (Y > x )

MARCENKO EQUATIONS

135

L e t us a l s o w r i t e down t h e corresponding M e q u a t i o n s ( c f . (9.50)).

Thus

for y > x (9.86)

+

F ( x ) = ( 1 / 2 r ) f C (b/a)eiCxdg;

+

(Y)F(x+y)

K(x,s)F(s+y)ds

= 0;

? ( x ) = ( 1 / 2 v ) l c (h$)e-igxdC;

?(x,y)

-lxm ;(x,s)F(s+y)ds

K(x,y)-(0)F(x+y) 1 "

= 0

Now we can d e r i v e by s t a n d a r d methods s i m i l a r M e q u a t i o n s f o r k e r n e l s o f t h e T form B o r B ( o r f o r t h e k e r n e l s r- o f [ C13;F2] f o r T- i n Remark 9.5) b u t

-

A

-

n o t f o r K-,

On t h e o t h e r hand we w i l l see t h a t f o r completeness r e l a -

K-. 6

t i o n s t h e K- K- form i s e s s e n t i a l . A

and K

-

Thus f i r s t we i n t r o d u c e new k e r n e l s Kn

B- o r r- t y p e k e r n e l s ) where T,T a r e d e l e t e d i n (9.79).

(analogous t o

Thus w r i t e (9.87)

$ = A

A

Then w r i t e 4 = $/a (9.88) e igs ds);

-

lt K-(x,s)e-icsds; = (-l)e 0 igx + 1 : i-(x,s)eigsds +

$ = (i)e-icx

(b/$)$ and

$/a = (;/a)((b)e

;/a*

= ((,)e 1

-igx

-i gx

+i:K-

$

= (;/a)$

t

1:

-

$/a and i n s e r t (9.87) t o g e t

K-(x,s)e-i'sds)

x,s)e-i'sds)

+

-

((-Y)eigx

(b/a^)( (-y)eigx

+

l:

i-(x,s)

t~~~-(x,s)eZds)

Now m u l t i p l y t h e f i r s t e q u a t i o n by ( l / Z n ) r c exp - i s y ) d T ,

(e% = exp(ir;s)).

t h e second e q u a t i o n by (1/27r)I? exp(igy)dg, t o o b t a i n (y < x ) Y)

+

4

where (**el G(x) n = (1/2a)JC ( c / a ) e x p ( - i g x ) d g ; dg.

For t h e K- K

-

G(x) = (l/Zn)J$ (b/:)exp(icx)

k e r n e l s o f (9.79) t h i s procedure o f d e r i v i n g a M e q u a t i o n

w i l l break down (as a l i t t l e experiment shows).

We summarize i n

A

M e q u a t i o n f o r K - , K- i s g i v e n by (9.89), w i t h T t h i s w i l l be e q u i v a l e n t t o (9.49) f o r B- o r B - .

C€t€@R)RM 9-15, The

REIMRK 9.16

(PR0P€fWU OF KERNELS).

6 ( x - y ) = ( 1 / 2 n ) i l T$:$zdk

(we),

F o r KdV t h e completeness r e l a t i o n

can be w r i t t e n o u t i n terms o f k e r n e l s ($+ =

and

136

ROBERT CARROLL ‘v

(ltK)exp(ikx),

T$- = ( l + r - ) e x p ( - i k x )

as (+*&)

0

K-(x,y)

+

K+(y,x)

u

K+(y,c)~-(x,S)dc

Note e v e r y t h i n g i s 0 f o r x < y. u

K-)(l+K+) w r i t t e n o u t .

t

1’ Y

1 t K - i s p e c u l i a r t o KdV.

( x > y) where 1tK- = (l+K:)-’,=

This i s i n f a c t j u s t t h e r e l a t i o n 1 = ( 1 +

Now 1 + B = (l+K:)-’

=

example so AKNS c o u l d have v a r i o u s f e a t u r e s .

i s n o t t r u e f o r KP f o r

l+K-

We n o t e t h a t GLM equations u

correspond t o 1 +K

= (1 + K t ) ( l +F) w h i l e orthogonal it y corresponds t o (1 +K-)

= (l+K+) -l;f o r KdV t h e s e a r e “ e q u i v a l e n t ” ( c f .

+

(a;

b$),qTo;

I

C131 and §2,3,7

f o r Parse-

Now i n (9.76) i f one w r i t e s f o r example $1) = 0 1 T $2 (l o ) y $ = (JI1 J12) = row v e c t o r ) and expands i n terms

Val formulas e t c . ) . (alA=

A

o f K and K, F and F, i t f o l l o w s t h a t t h e completeness r e l a t i o n i s e q u i v a l e n t

M equations (9.86). ThePe should be an e q u i v a l e n t v e r s i o n i n terms A T T AT o f K-,K-,G, and ^G i f we w r i t e e.g. $ = b$ - a? and $$ o1 = $(b$ - a $ )ul. to the

n

A

We o m i t t h e c a l c u l a t i o n s here ( c f . remarks below).

L e t us a l s o w r i t e o u t

h

I n KdV cases some r e l a t i o n s between 1+K- = ( l t K + ) ( l + F ) and 1+K- = (ltK:)-’. rc. T Thus (**+) ( 1 + F = F and one can ask whether t h i s enough f o r K- = K-.

-’

-

( l + K T- ) ( l + K- - ) = 1 + = (l+F)(ltK-)-’; (l+K:)-’ = (l+KT)-’(l+FT) = l + K - ; K* FT = l + F = ( l t K + ) - l ( l + K - ) = ( l t K“T- ) ( l t K - ) . This says (l+K:)(l+t-) = ((l+KT)

-

k

(lt?-))T = (lt ??)(ltK-),

f o r which a s o l u t i o n i s K- = K-,

b u t t h i s does n o t

seem t o be necessary.

A p o i n t which needs some c l a r i f i c a t i o n here i n v o l v e s t h e K- n o t a t i o n i n r e l a t i o n t o ( l + K * ) e t c . Thus i n KdV one has ( c f . Remark 9.11) T$- = e x p ( - i c x ) t

it

K-(x,s)exp(-igs)ds,

which makes i t appear as i f T$-

--, which i s n o t t r u e ( u n l e s s

I g I i s large).

e x p ( - i c x ) as x

-+

T h i s i s however t h e c o r r e c t

formula f o r T$- based on s p e c t r a l c o n s t r u c t i o n s and on d e t e r m i n a n t arguments ( c f . §2,7). (ikx)

To c l a r i f y t h i s r e c a l l here t h a t examination o f c ( x , y ) V

-+

f, and y: f+ + e x p ( i k x ) ) as i n ( m ) l - ( + * ) l

h e u r i s t i c a l l y t o (**.)

K-(x,y)

gous argument c l a r i f i e s t h e T

Q

( T - l ) G ( x - y ) as x,y l / a and

f

(?: exp

(before (1.27)) leads -+

-m(y < x ) and a n a l o -

= l/$ f a c t o r s i n t h e AKNS formulas

such as (9.67) and (9.84). L e t us i n d i c a t e some k e r n e l r e l a t i o n s i n v o l v e d i n completeness f o r AKNS. Set

0 1 where al = (l o ) .

This can be w r i t t e n o u t v i a k e r n e l s as

COMPLETENESS

S)U~~(X-S)

- 1:

~-(x,s)(O

137

I)b(y-s)ds

Combining we o b t a i n 1 T 0 = ( 0 ) K (yyx)al

(9.92)

+

K-(X,Y

Hence f o r y > x e v e r y t h i n g i s 0 ( n o t e one can always i n t e r c h a n g e x and y ) and f o r y < x we g e t a general r e l a t i o n between t h e k e r n e l s (analogous t o f o r KdV).

(*6)

We can w r i t e h e r e A

b u t w i l l r e f r a i n from f u r t h e r d e t a i l .

Note t h e r e w i l l be some s i m p l i f i c a A

t i o n when t h e r e a r e no bound s t a t e s ; i n t h a t case from (9.84)

K- and - K a r e

A

paired, a l o n g w i t h K and K-.

REmARK 9.17 ($0nE CRAs$RMUlIAN ID=).

One would 1 i k e t o have a v e r s i o n o f

t h e Grassmann f o r m u l a t i o n o f 511 f o r t h e AKNS s i t u a t i o n . The n a t u r a l f o r H = L 2 ( S1 ,C 2 ) i s used. I n

mat i s i n d i c a t e d i n [ PO21 where a H i l b e r t space

2

2

o u r f o r m u l a t i o n we would want L (R,C ) w i t h tik c o r r e s p o n d i n g t o t h e n a t u r a l Hardy spaces based o n upper and l o w e r h a l f p l a n e a n a l y t i c f u n c t i o n s . f =

(1 alj:j

and e.g. H (*A*)

1 a 2 j ej) E involves a

e = $exp(-ikx),

example. notation)

j , ' e

Thus from T$ =

L

2

for e

= a =

b a s i s v e c t o r s i n H'

j = 0 f o r j < 0.

2j $exp(ikx),

$

+ RJ, and

e

One c o u l d d e f i n e BA f u n c t i o n s

= @ e x p ( i k x ) , and

?$ = R^$ -

Thus

as i n d i c a t e d i n 511,

J,

g-

= ;exp(-ikx)

for

we g e t ( w i t h some abuse o f

ROBERT CARROLL

138 Te- = eA t R,e;

(9.94)

-?:

=

s-t-e; R, = Re2ikx;

2- = Re- 2 i k x 4

(4;-)

e Thus = R ( 6 ) for R = (R+ ) and one should be a b l e t o c o n s t r u c t Gras-1 Grassman; o b j e c t s H ( R + , A - ) e t c . a s i n 111. We will n o t pursue t h e matt e r f u r t h e r here. 10. bQMtE t Z E CHZ0RECZC BECHHODS.. We will sketch here some work i n [ 862-41 which develops t h e Toda-AKNS theory i n a Lie t h e o r e t i c context. In p a r t i c u l a r one obtains many o f t h e r e s u l t s o f [ FL1 ;NE1 ] ( c f . §9) i n an elegant man-

ner plus o t h e r r e s u l t s o f various types (some r e l a t e d t o [W2,51). Extensions and v a r i a t i o n s of this appear i n [ IM1,2] based on [ DR1 I . We will s t a t e some f a c t s o r r e s u l t s f o r general simple Lie algebras

b u t will be primariAs usual t h e r e is a v a r i e t y of ( c o n f l i c t i n g ) notation i n various papers and books and we will s h i f t notat i o n i f i t seems d e s i r a b l e ; however i n any s e c t i o n t h e terminology should be c l e a r and we will provide bridges where needed. For i n f i n i t e dimensional Lie a1 gebras one has [ K1,2 I and a sketch i n [ ML2 I f o r example; we will not t r y t o give a bibliography f o r Kac-Moody (KM) algebras b u t mention e s p e c i a l l y [ FK1-5;FPl ;LK1-3;K1-5;TKZy4,51 f o r background. In f a c t t h e presentation here can serve a s an introduction t o KM algebras. For those who are n o t a l g e b r a i s t s (such as the a u t h o r ) one should be a l e r t e d t o t h e enormous amount o f beautiful s t r u c t u r e l y i n g i n t h e algebra; i t i s e s p e c i a l l y pleasi n g t o read about this i n t h e books [ K2;FK41 f o r example. All we can do here is t o pick o u t l i t t l e bits a s needed.

l y i n t e r e s t e d in

= gl(2,C) o r s l ( 2 , C ) .

(RElMRlG 0H A t ) . Since we will be concerned primarily w i t h s l 1 (2,C) and A1 l e t us give some d e t a i l s here ( c f . a l s o Appendix A ) . Thus l e t g = sl(2,C) w i t h basis h = e = ( 0o 1o ) y and f = ( O O ) so [ h , e l = 2e,

REmARK 10.1

(0l-P).

[ h , f ] = -2f, and [ e , f l = h.

Set

’ Y O

= g

I C[t,t-’ I w i t h b a s i s h I tm,e I tm,

and f I tms a t i s f y i n g (10.1)

[ h 5 t m , e 5 t n l = 2e I tm+n; [ h It m , f I t P

[ e I t n y f I t P l = h I tntp (note C[t,t-’ I r e f e r s t o polynomials i n t,t-’).

NOW

i s defined a s

Cc

LIE THEORY

139

extension) w i t h (note f o r d = t D t y ( d tm- tn )o =

Q @ ,t J g @ Cc ( c e n t r a l

m6m,-n)

+ m$,,,-nTr(xy)c [x I t m , y I t n l = [x,yl Itm+n

(10.2)

One o f t e n uses t h e Killing form TradXadY t o represent a b i l i n e a r form ( X , Y ) ( o r ( X l Y ) ) and f o r sl(2,C) this reduces t o TrXY. A l t e r n a t i v e l y one can describe

via generators eo = f It , el = e B 1 , fo =

e I t - l , f l = f I1 , h 0 = - h I 1 + c , and hl = h I 1 w i t h (10.3)

[h , h ] = 0; [ e i , f . ] = 6 i j h i ; 0 1 J

[ h i y e . ] = A..e J

*

]J j'

[ h i , f j ] = -A i j f j'*

[ e i y [ e i , [ e i , e j l l l = 0 = [ f i , [ f i , [ f i y f J. l l l 2 -2 ( t h e l a t t e r f o r i # j and A = ( - 2 2 ) = Cartan m a t r i x ) . Note c 'L ho + hl 1 and [c,^gl = 0. This algebra $ is sometimes c a l l e d ( A 1 ) ' (derived a l g e b r a ) 1 *e fie and A1 = = @ Cd where = [g , g 3 w i t h ( d tDt)

te

[X B t" + uc + vd,y I t n + c';

(10.4)

+ v^d] = [ x , y ] BI tm+n +

vny Itn - Cmx It m + mtim,-nc We r e c a l l now a l i t t l e Lie t h e o r e t i c notation a s follows. One w r i t e s (cor o o t s = gi 'L Ha; and we will use A f o r t now, omitting sometimes I) V

(10.5)

=

"I

-1

=

"1

e; eo = Eao; f o

Af;

f0 =

h,;

[eo,fo] = [f,e] + 1 6

A

OYO

V

= h = hl; a =

E-ao;

0

=

-h+c = h o,* el = e; f l = f ; e 0 =

el = E" 1 ; f l = E-",;

[ e , f l = [e1 , f 1 I =

( f l e ) c = - h + c; ( f l e ) = Trfe = 1

By abuse of notation one o f t e n writes h

'L

a but

the context will c l a r i f y the

meaning e a s i l y . In standard notation c1 'L a1 w i t h (*) g" = { X E g ; [ H , x ] = a ( H ) x f o r H E h ) , go = Ch, " ( H a ) = 2 where [HayYcr] = -2Ya, [Ha,XaI = 2Xay and [ X,,Ya] = Ha with [ X , Y l = ( X I Y ) h a f o r Xa€ g", Ya€ g-a (we will w r i t e a l s o a(H) = ( a , H.r) ) . Thus Xu 'L e , XVa 'L Ya 'L f , Ha = h" = h. One w r i t e s (A) s12 = n _ 8 h 8 n+, n+ = s t r i c t l y upper t r i a n g u l a r , n- = s t r i c t l y lower t r i a n g u l a r , and

te =

-

n+ +

(10.6)

h

n+

Ch 8 Cc 8 Cd w i t h 1; lks12 = Ce + 1; A k s12; An- = Cf +

1;

A - ~ ~ I ~

ROBERT CARROLL

140

A The Weyl group W of i s generated by conjugations via f = (-10 1o ) a n d T = A 0 v (o ,/A) ( T k , k E Z i s t h e kth power of T a n d r = ra , al = h a ). T h u s r ( h ) = - h (note r-’ = -:) and rhr-’ = - h ) , w i t h ( 0 ) r ( c ) = c , r ( d ) = d , T ( h ) = h + 2kc, T k ( c ) = c , T k ( d ) = d - k h - k 2c , r 2 = 1 , T r = rT-k, a n d k ,k 1 W = CTk. Tkr, k E 2 3 . Further f a c t s about A, amd i t s representations appear i n t h e t e x t a s needed; i n p a r t i c u l a r see Remark 10.3 f o r more d e t a i l s on

(p

weights and representations. NOW go to [ 8G31.

(ZNCRBDLICCZ0N CO KAC-R0ODg = Km ALGEIRM).

RBlARK 10.2

c =C[A,X-’ 1 I

Take g to be a f i n i t e dimensional simple Lie algebra over C , g. a n d l e t

9” = @;

(10.7)

q ~ :

Cc be defined by t h e 2 cocycle X

-f

C; for P,Q

$J(P

1

(here (

E

C[X,X-’],

I x,Q I Y )

x,y E g ,

= Res(aAP.Q)(xlu)

)

Killing form). The degree derivation d : -+ d = A a A with d ( c ) = 0 . Then t h e untwisted a f f i n e KM algebra *e Ae Cd and = [ g .g I w i t h r e l a t i o n s

s*

(10.8)

[ Ak

Ix + vc + vd,A P Iy + cc + Gd] = AktP t vpAp

PD y

-

~k

vkx

PD x

+

k6

4

i s defined by i s $e = $ (3

te

I[ x , y ] + k+p,OC

In w h a t follows we will occasionally use terminology o r f a c t s about Lie a l gebras without much background o r motivation. Since this is f a i r l y standard we simply r e f e r t o e.g. [SER2;K2] f o r d e t a i l s . Now f i x a Cartan subalgebra h am) t h e simple roots and e

C

g and l e t A E h* be t h e roots with

1 aiai 2

the highest r o o t .

(al,...,

Choose root vectors

Ea,E - J form an s l ( 2 , C ) t r i p l e The vectors e i = E , f i = E-ac generate g (Chevalley g e n e r a t o r s ) ; t h e = a; H a r e simple coroots and h = @lCgi. The Killing form remains nondegenerate ai r e s t r i c t e d t o h and induces an isomorphism v : h h*. Define a b i l i n e a r For a E A define r e f l e c t i o n s rcl: h -+ h* form o n h* by (ale) = (v-’aIv-’B). by ( 0 ) ra(X) = X - ( 2 ( X l a ) / ( a ( a ) ) a , A E h*. The Weyl group W is generated by rcc-= r i . Define f u r t h e r Ahe = h B Cc @ Cd. An element a E h* extends t o a l i n i a r map a: -+ C via ( a , c ) = ( a , d ) = 0 ( ( 4 ~ )a ( c ) , e t c . ) . Then ;e Ea such t h a t ( E a I E e a )

=

2/ja/

so EayE-c,yHa

= [

zi

-f

^he

KAC MOODY ALGEBRAS has r o o t space decomposition

^Se

fi

=

8 8;;

141

(y E

2) where 2 = Zre

U

$im

g i v e n by ( 4 ) The i m a g i n a r y I j S + a ; j E Z,a E A } ; $im = C j 6 ; j E Z/{Ol}. r o o t 6 i s d e f i n e d by &Ih = 0, ( 6 , ~ ) = 0, (6,d) = 1 and one has ( m ) *e g j6ta =

i:6 = Chj

C A j I;E,

I h.

9“ w r-1i t e

For C h e v a l l e y generators o f again.

(e

I e-ey

L e t eo = h

fo = A

Ea-

first 1 1

I

i s a c e r t a i n sum o f r o o t s ai and we r e f e r t o [ K 2 1

i s s p e l l e d o u t when needed). and c =

m The c o r o o t s a r e

n

v v

1 a.aI,’i s

Hence (**) h

=

gi

t h e canonical c e n t r a l element w i t h

$CZi I Cd.

Let

st (resp.

5e

%

=

l o we need o n l y A1 which

-

t-) c

(0 5 i 5 rn)

= [eiyfil

iip o s i t i v e

integers.

be generated by e ,e I\

~

A

A

(resp. foy ..., fm);t h e n (*A) = n- B) he @ g = n- 8 h B) n+. em t e n d t h e K i l l i n g form t o i n t h e s t a n d a r d way w i t h r e s t r i c t i o n t o 4

A

G,;

ie

A

( 2 ( A l a ) / ( a ] a ) ) a ;A E

(te)*. Then W,

^he

-

aty

i = 0,

...,m

still

is

It c o n t a i n s an a b e l i a n normal ( t r a n s l a t i o n ) subgroup

t h e a f f i n e Weyl group.

= r6-ai ri y

T generated by Ti

generated by ri = r

lY..’

Now ex-

*

A

R E ~ A R K 10.3

0

D e f i n e f o r r e a l r o o t s a , ra: h* + h* v i a (*@) r a ( h ) = h

nondegenerate.

d i r e c t product

fi

= 6-8 and -a

a.

) i s a system o f s i m p l e r o o t s

Then (aoYNl,...,a

(0 5 i < m ) generate g.

and eiyfi

and 1 IE-a.

= ei

L

ee be r o o t v e c t o r s

i = 1,.

4

. .,m,

w i t h W/T 2 W and h i = W o( T (semi-

c f . [ FK41).

(50CI.E REPREdENCACZ0N CHE0Rg).

A

$

m d u l e (V,n),

i s a h i g h e s t w e i g h t module w i t h h i g h e s t w e i g h t A

E

TI:

9*

+

End(V)

h* i f t h e r e e x i s t s vh E

V such t h a t r ( h ) v A = (A,h)v,;

(10.9)

(U

‘L

enveloping algebra

{v

E

:*;VA

nilpotent.

For any A

E

= 0; V = U($)V,

ne * one can a l s o phrase t h i s i n terms o f g , he, e t c . )

-

V; n ( h ) v = (A,h)v,h = 01. V i s i n t e g r a b l e if n(e.),

Then V = 8VA, V, = {A E

T(;+)V,,

=

:*

1

E

GI.

n(fi)

The w e i g h t system i s P ( A ) (i= O,...,m)

t h e r e e x i s t s (up t o isomorphism)

are locally

a unique i r r e -

d u c i b l e h i g h e s t w e i g h t module w i t h h i g h e s t w e i g h t A; we c a l l t h i s L(A). ( n ) i s i n t e g r a b l e i f a n d o n l y if CA,;~) l e d dominant i n t e g r a l ) . t e g r a b l e modules

>O

a r e d e f i n e d by (Ai,;.)

L

f o r a l l i (such weights a r e c a l -

One can w r i t e suEh A as A =

t h e fundamental w e i g h t s Ai

-f

E Z

J

lom kiai,

= 6..

1J

ki E Zg0 where

( 0 5 i , j 5 m). The i n -

L(A) can be p r o v i d e d w i t h a H e r m i t i a n form HA: L(A) X L(A)

C u n i q u e l y determined by t h e r u l e HA(aA(x)v,w) = -HA(v,nA(wo(x))w)

(con-

142

ROBERT CARROLL

travariance) f o r a l l x E rc

$ and

v,w E L(A) ( c f . [ F K 2 ; K Z l ) .

Note a l s o t h a t

one w r i t e s ( a a.) = 2 =q (H .). One d e f i n e s here t h e a n t i l i n e a r i n v o l u t i o n i' I at V V wo v i a w (ei) = -fiy w ( f . ) = -eiy w ( a . ) = -a. (i= ~ , . . . ' m ) . 0

0

1

0

1

1

now one c o n s t r u c t s t h e homogeneous r e a l i z a t i o n o f t h e b a s i c r e p r e s e n t a t i o n

).

L(Ao) ( c f . CBG3;FKZ;K2,5;TK51 hR = BRX

L e t hl,...,hm

be an orthonormal b a s i s f o r

i ' The homogeneous Heisenberg subalgebra (HSA) i s d e f i n e d by A

(10.10)

A

A

A

s = s- 8 cc 8 s+;

4

(same a , i ) ;

S+ =

t-

(i> 0; a = lY...,m);

IPCP;

= Bcql

1 1 i = h- /i B ha; p l = ( h a l h a ) - 1 B ha

a b = 6 . .& c. One has [ pi,q.] Now c o n s t r u c t a h i g h J 1.l a h a a e s t w e i g h t module o v e r i n V = C [ x y l w i t h (*6) "(pi) = a/axiy R(qi) = x and a ( c ) = 1. One has (*+) -wo(p;) = ( h a l h a ) -1 A -i191 ha = i ( h a l h a ) - l q ; ( ni' ote (we emphasize i > 0 h e r e ) .

(pa) and a,($)

71

A

a r e t h e r e f o r e c o n j u g a t e o b j e c t s r e l a t i v e t o HA); vA(pq)

1

and

correspond t o a n n i h i l a t i o n and c r e a t i o n o p e r a t o r s .

R,($)

L(no)

B n(Ao) where n(Ao) = I v E L(Ao); IT (;+)v = 01 (vacuum vecA, One can i d e n t i f y n ( A o ) w i t h t h e space spanned by ea, a E Q=8zoli ( i

= C[x!]

tors).

Note i n general < A

= OY...,m).

0

,: 0 )

The w e i g h t s f o r L ( A o ) f o r sl(2,C) (10.11)

= 1 and (Ao,Xi)

are ( a

P ( A o ) = {Ao+B-(%1fi12+k)6;f3

%

a1

%

= (Ao,d)

(10.12)

h)

E

one has a l s o

I BCeka

L(Ao) z C [ x i l

where h =

(01-10)

and i E

(k

E

Z);

a(iih/2)

= a/axi;

( n o t e t h e n i l p o t e n c y o f n(ei), TA 4).

(r;)'

~ ( X - ~ h / i=) xi

Z+.

One w r i t e s i n general r," = exp(v(Ea))exp(-r(E-,))exp(a(Ea))

-f

= 0 (i21).

Q, k E Z,o) = {ho+ma-(m L +k)&; m E Z,k E Z,o) -

For g = sl(2,C)

:W":

Now (*=)

for a

E

ire A

n(fi)

i m p l i e s n i l p o t e n c y o f n(Ea), a E A'

^w"

IT

i s t h e group generated by r; = r'a ; ( i = O,...'m) and r; r i*' W i s a s u r j e c t i v e homomorphism o n t o w i t h k e r n e l t h e group generated by -f

(ri i s g i v e n i n (*.)).

T," = rx-, r:

fl(Ao)

-f

Q(Ao)

For s u i t a b l e g t h e t r a n s l a t i o n o p e r a t o r s a r e

( a E A ) and TR i s generated b y t h e T.;

A l l that

R E PR ES ENTAT I 0 NS concerns us here i s t h e case g = sl(2,C) exp((k+l)a).

143

where T" = {T,T-l}

Ze

Also ( c f . Remark 10.1) we can w r i t e

w i t h T-+1e ka =

= B

Cc B Cd f o r

t h e a s s o c i a t e d a f f i n e KM a l g e b r a o f t y p e A1. The homogeneous HSA i e r a t e d b y pi = %A h, qi = ( l / i ) A - ' h , and c (iE Z+) as i n (10.10). E

pi(Pk

Ie x p ( k a ) ) = aiPk

I

A typiIexp(ka), Pk E C[xil and one has f o r i E Z, Iexp(kw); qi(Pk Ie x p ( k a ) ) = xiPk B exp(ka); c(Pk

L(Ao) has t h e form

cal P

1 Pk

e x p ( k a ) ) = Pk B exp(ka).

The element h o f t h e Cartan subalgebra a c t s o n l y

on t h e second f a c t o r v i a h(Pk B e x p ( k a ) ) = (ka).

$ i s gen-

(

ka,h)Pk

Iexp(ka) = 2kPk Iexp

The element d a c t s on b o t h f a c t o r s and,defining

deg(xi)

= i w i t h deg

(PQ) = degP + degQ, one gets f o r a homogeneous polynomial P, d(P B e x p ( k a ) ) + k 2 ) P 19 exp(ka).

= -(degP

1 A few f u r t h e r f a c t s f o r t h i s A1 s i t u a t i o n can be e x t r a c t e d from [ K l l . Thus ^he = Ca + Cc + Cd ( a % h, a % a 1 ) w h i l e ( a , a ) = 2, (c,d) = 1, and a l l o t h e r and v i a ( I ) and we r e p a i r s vanish. Thus one sometimes i d e n t i f i e s fie ^e, c a l l t h e n o t a t i o n ( , ) f o r ( h ,h ) d u a l i t y ; thus f o r a1 % ;El we w r i t e ( a 1' V = 6i a ) = (al,a1 ) = 2, e t c . The fundamental w e i g h t s Ai s a t i s f y ( Ai,a.) 1 J V V (i,j = 1,2) where a1 % a % h and a. % C - a (ao % C-al i s b e t t e r o f course

Ahe

b u t we i d e n t i f y c and c* say

-

"h*

s i m i l a r l y d = d*).

Then one can w r i t e A.

%

d

and A1

Note t h a t t h e i m a g i n a r y % d+kl checking e a s i l y t h a t (Aiy:.) = tjij. J V r o o t 6 i s d e f i n e d b y ( 6 , c ) = 0, ( 6 , d ) = 1, a n d ( 6 , h ) = ( & , a l ) = 0. Thus

(6,d)

%

(6,d)

v

v

= CxJ

If,

= a o . + a1

'L

= 1 makes p o s s i b l e an i d e n t i f i c a t i o n o f 6 w i t h c and s i n c e c fie he a + a we can w r i t e 6 = a + al. Also gjSta, = ciJ e, gj6-a, 0 1. 0 = C A J I h.

i;6

One remarks here t h a t t h e r e a r e v a r i o u s choices o f HSA p o s s i b l e i n general and t h i s i s discussed i n ['K2,5;BG1-3;TK2,4,51. t e d i n t h e s t r u c t u r e o f t h e vacuum space n(Ao).

The c h o i c e o f HSA i s r e f l e c For example w i t h sl(2,C)

t h e r e a r e e s s e n t i a l l y two i n e q u i v a l e n t HSA, t h e p r i n c i p a l and homogeneous, and t h e r e s p e c t i v e vacuum spaces a r e 1 dimensional and i n f i n i t e dimensional ( c f . [ K5;TK5]

i n particular).

The corresponding i n t e g r a b l e systems o f PUE

a r e d i f f e r e n t a l s o w i t h p r i n c i p a l HSA

RECIARK 10.4

(KAC l!l00D!J GR0UP5).

%

KdV and homogeneous HSA

We c o n t i n u e t o f o l l o w [ BG31.

%

Toda-AKNS.

L e t G be t h e

connected and s i m p l y connected group corresponding t o g. By c o n s i d e r i n g a f a i t h f u l r e p r e s e n t a t i o n G can be r e a l i z e d a s a subgroup o f SLn(C). L e t

5

144

ROBERT CARROLL

denote polynomial maps g: S1 E

SLn(CIA,A-'l);

g(A)

E

T h i s has L i e a l g e b r a

GI.

s c r i b e the central extension 0

Cc

+

+

t h e i n t e g r a b l e r e p r e s e n t a t i o n s nA:

g

-+

-+

To e l i m i n a t e dependence on rn

G = {g

(A*)

I g.

C[A,A-'I

To de-

0 on t h e group l e v e l one uses

EndL(A).

+

c=

D e f i n e $A = group o f i s o -

morphisms o f L ( A ) generated b y exp(tnA(ei )) ,exp(tnA(fi C).

rcI

G w i t h pointwise m u l t i p l i c a t i o n ;

+

. .,m;

) ) ( i = 0,.

t

E

A sum o v e r t h e fundamental

so one can use and l e t G = t h e u n i v e r s a l group generated b y exp(tn(ei)), exp(t

n = @l~i,

-

A

One can show t h a t G i s a c e n t r a l e x t e n s i o n o f G b y Cx,

n(fi)).

A

i . e . we have

;

A

+ G * G + 1. L e t U+- be t h e subgroup o f generated re ( t E C ) . Then one can show t h e r e e x i s t s a B i r k h o f f b y exp(tn(ECY)), CY E A+(Bruhat- Kac-Peterson) decomposition

an e x a c t sequence 1

* Cx

-

A

A

(10.13)

A

U wH U,

U

-

w€wn ( c f . [ PR1

A

A

G =

I - we w i l l say more on t h i s l a t e r ) .

Now c o n s i d e r e.g.

(A0 - A )

h =

E

s12(C,[ A , A - l ]

).

t h e polynomial l o o p group SL2(C[ A , A - l l

) b u t note t h a t exp(h) i s n o t i n I n o r d e r t o deal w i t h such crea-

t u r e s t h e c o m p l e t i o n i s used ( c f . 512, Appendix B y and see [ DF1-4;PRl ;SJ1 ,21 f o r more i n t h i s d i r e c t i o n ) . satisfying

(AA)

Thus c o n s i d e r w e i g h t f u n c t i o n s p: Z

p(k+m) < p(k)p(m),

+

(0,m)

p ( 0 ) = 1 (so p ( k ) 1. 1 ) .

p ( k ) = p(-k),

For v a r i o u s reasons one uses weights o f n o n a n a l y t i c t y p e so l i m p ( k ) ' j k = 1 as k

-+

m.

Examples a r e p ( k ) = 1 o r p ( k ) = (1 +

Wiener a l g e b r a A l y m l a k l p ( k ) (ak

P %

as t h e Banach space o f f u n c t i o n s S1 Fourier coefficients).

comndition on p means t h a t i f f (A@)

x

for x =

+ ac E

j Then

E

A

1 aijAJ

I

x. E

=

1J l a i j l p ( i )

=

P

+ lal.

P i s a Banach a l g e b r a and so i s

i;'

E

Then C [ A , A - l I

0.

Define the

C w i t h norm IIfll

$

P

1 la. .Ip(i) J ',

Let g i f p(k)

P

and f o r

and

Gp

clkl".

=

P i s dense and t h e

has no zeros on S1 t h e n l / f E A

P s e t IIxII

s e t IIxII

P L i e group ( A h )

1 k l la, a >

x^ = 1

P' aijAj

Now

8

be t h e c o m p l e t i o n s . D e f i n e t h e Banach

-+ = Cg E GL ( A ), g(A) E G I . One can show t h a t exp: P n P P P ( i n t o ) c o n t a i n s a NBH o f t h e i d e n t i t y i n ; i f p i s n o n a n a l y t i c as above G"P i s connected and s i m p l y connected. For PAg P choose p from t h e f a m i l y (A+) = exp(t/kl'/'), t > 0, 1 < u < 2. Then t a k e t h e H i l b e r t c o m p l e t i o n P,,t H ( A ) o f L ( A ) r e l a t i v e t o HA and t h e r e e x i s t s a dense subspace o f H ( A ) on

4

which o p e r a t o r s e x p ( n A ( x ) ) , x E g

P'

are well defined.

One d e f i n e s

2'P

as t h e

KAC MOODY GROUPS

145

group generated by these o p e r a t o r s and makes t h i s u n i v e r s a l by u s i n g Now t o g e t t h e B i r k h o f f decomposition use G = SLn(C) as a model.

71

= $vi.

Any A €

GLn(A ) has a f a c t o r i z a t i o n A = A DA+ where P

A,,A+-'

(10.14) and

1;

GL (A'); n p

E

D = diag(hk'.

.. hk")

1 Here i n d = w i n d i n g number o f t h e image o f S under

ki = i n d detA(X).

t h e map i n d i c a t e d and A'

'L subalgebra o f A c o n s i s t i n g o f f u n c t i o n s whose P P F o u r i e r s e r i e s has o n l y nonnegative (resp. n o n p o s i t i v e ) powers o f h ( c f .

[ G J l I ) . F u r t h e r i f A E SL ( A ) t h e n A,,A2 E SL (A*) and 1 ki = 0 ( c f . [ B G n P n P 3 1 ) . Now l e t = subgroup o f SL ( A ) c o n s i s t i n g o f elements ( A m ) A, + hA1

+

;P+

n P upper t r i a n g u l a r and w i t h 1 on t h e d i a g o n a l .

... w i t h A.

by exp(x),

(F+)pi s

x E

+: P (here +;

N+

U

P

i s generated

= proj(n^+) under t h e n a t u r a l p r o j e c t i o n

i t s completion).

-f

:and

L e t H = s t a n d a r d Cartan subgroup o f SLn(C), genw k h "), ki = 0, and W t h e Weyl group o f SLn

.

e r a t e d by m a t r i c e s d i a g ( h k l . .

--

-

1

(G-1

ti

Then (.*) SL ( A ) = U& , Up wH (where i s defined analogously t o ,-s n P R A* A + A+ U wH U (here U (U ) = U'). One can l i f t t h i s t o as (*A) G = U P P *+ P P wad P P P = completion of U i n G etc.). One wants now a s l i g h t m o d i f i c a t i o n o f P' these B i r k h o f f decompositions a s f o l l o w s . For U E ?I- and V E w r i t e (Uo E P P U , Vo E U+; U, c SLn(C) a r e upper o r l o w e r t r i a n g u l a r m a t r i c e s w i t h 1 o n

2

'6

t h e diagonal )

+

U = (1

(10.15)

+ U2h-'

UIA-l

v

+ ...) Uo;

= V0(l

Then one has a f a c t o r i z a t i o n ( F E

-

(10.16)

m a . -

g = g-gog+;

and t h i s extends t o

REl'tAlW 10.5

(vo = v,

2, w E

L ( A ~ ) ) . I:

(10.17)

0 T

=

P

+

g- = 1 + U1h-'

A

A

<

...;

E

2P '

Consider

^tho

4

= g-gog+ f o r

v 2h2

)

u

(ENFUNCCZW).

"A g E G A

$

+ V,h +

= highest weight vector

+ ...)

-

go = UowHVo;

epa

(p

E

Z)

Vlh

+

..

IT'

= Ag -v0;

a l s o w r i t e x-w f o r

II

(x)w, x E

t h e homogeneous r e a l i z a t i o n o f L(Ao) we have

1 T0~ ( X B)

+

=

and t h e o r b i t 0,

-

N

g+ = 1

A0

146

ROBERT CARROLL

w i t h T'(x) = 0 f o r a l m o s t a l l p E Z and nonzero e n t r i e s a r e p o l y n o m i a l s i n P t h e xi. The T o ( x ) s a t i s f y H i r o t a equations e t c . b u t t h e xi a r e formal v a r i P ables t o i n t r o d u c e general complex v a r i a b l e s ti one uses a t e c h n i c a l con1 t r i v a n c e . Working w i t h A1 i n t r o d u c e an isomorphism o f L(Ao) v i a ( 0 0 ) $ ( t )

-

m

= exp(ll

A

tipi)

6

(J, = J,vac).

Now

jvac 4 ^cno

b u t i f one takes a l m o s t a l l ti =

0 t h e n i t belongs t o an a r b i t r a r y c o m p l e t i o n ?iAo f o r p as i n (A+). P as a s h i f t o p e r a t o r

It acts

Now expand . c i ( x t t ) i n formal v a r i a b l e s xi w i t h complex c o e f f i c i e n t s which a r e f u n c t i o n s o f ti.

The zero o r d e r t e r m i s

T i ( t )

and can be i s o l a t e d b y

p r o j e c t i n g (10.18) on exp(ma) u s i n g t h e H e r m i t i a n form H = HA on L ( A o ) v i a

Here t h e

? are

t r a n s l a t i o n o p e r a t o r s on eka terms (vo

%

= 0 in -m a r e vacuum e x p e c t a t i o n values o f ( 0 6 ) J, = T

i s a polynomial one may i n f a c t t a k e a l m o s t a l l t nents T o ( t ) m The elements

1 ) and s i n c e . c i ( t )

'Am

3"

The compo-

(0.).

. ~,,,~i $ E tA0 P ' A

w i l l p r o v i d e t h e c o n n e c t i o n between r e p r e s e n t a t i o n t h e o r y

and zero c u r v a t u r e equations (see below). Now from t h e B i r k h o f f decomposition ( c f . (10.16))

(w

E

so

{lyr~Ao>; k

w jm = $ J - J , ~ J ,where ~ AmAm

= k(m,t);

e = el,

3" E ZAo has a P

$

A

A

= g-gog+

One gets from (10.19)-(10.20)

f = fl). V

(note

decomposition

and exp(bme) s t a b i l i z e vo and alvO

= ( Ao,L1,

vo = 0 w i t h

(

hoyZ0 )vo

= vo w h i l e H i s c o n t r a v a r i a n t )

(thus

T!(t)

(10.22)

= 0 i f and o n l y i f k

T(t) = Nrn.m*m $-q0Qt;

am $o = Bo

# 0). E

P r o j e c t i n g on

SL2(C);

T!

= 1tAlA-'t

P

one has

...; @+

Nm

= 1tC A

1

+

..

~

ZERO CURVATURE EQUATIONS

147

Note t h a t (10.22) does n o t h o l d on t h e zero s e t o f .i(t).

R€IRARK 10.6

(ZERO CURVRCURE EQM&UX3). We s t a y away now f r o m t h e zero s e t

o f r o ( t )= c l o s e d nowhere dense s e t i n Cn where s i n g u l a r i t i e s o f s o l u t i o n s

m

From ( 0 6 ) one has

o f t h e zero c u r v a t u r e e q u a t i o n s c o u l d a r i s e . pi$m which becomes (ai

T h i s i s an e q u a t i o n on t h e a l g e b r a

6,

(10.24)

= $A

i

Ig

I3 CC;

i2 0

n II

A,

(g

A

2 gp.

P

i- = 8i
+P take$,(t)

=

D e f i n e now

Ig

hi

A

A

Then (@+) $p = g-p 8 gtp.

e t c . t h e c l o s u r e s r e l a t i v e t o p.

with

a/atiqm

= a/ati)

Now

# 0 and t h e l e f t s i d e o f (10.23) i s decomposed a c c o r d i n g t o ( a + ) ; "rn "m "m Then (10.23) i s = t o f o r t h e r i g h t s i d e w r i t e R (pi) = R-(pi) t Ro+(pi). (10.25)

nm

(ai

+ ;!(P~))($-)

(i = lY2

-1

= 0;

-

(ai

"m Am R ~ + ( P ~ ) ) += ~0+

D e f i n e c o v a r i a n t d e r i v a t i v e s by

¶...).

(om)

*m Di

=

ai -

im (pi). O+

Then

c o m p a t a b i l i t y c o n d i t i o n s f o r t h e second equations i n (10.25) a r e ( 6 * ) 0 = "m "m 'm "m [ DiyDj]. One checks t h a t [ DiyR ( p . ) ] = 0 so t h e fim(p.) a r e (up t o m u l t i p l i J A J c a t i o n by powers o f A ) r e s o l v a n t s f o r Di 2 l a [ DK1 I. Various d i f f e r e n c e equations a l s o a r i s e here i n v o l v i n g Toda l a t t i c e s t r u c t u r e i n m b u t we r e f e r t o [ BG3] f o r t h i s .

RZmARK 10.7

Now one c a l c u l a t e s t h e c e n t r a l comAm "m L e t c y = c e n t r a l t e r m o f R ~ + ( P so ~ ) (&A) c y = I I , ( R ~ + ( P ~ ) )

(REtiOCVANC DEeOl"PO$ICZ0#).

ponent o f (&*I.

From (10.25) we know

(n* = p r o j e c t i o n on t h e c e n t e r Cc o f $ e ) .

(10.26)

"m

II,(R~+(P~))

Am

Am

= n*((ai$o+)($o+)

-1

)

m d m Am -1 and s i n c e t h e c e n t e r C h one has ( 6 0 ) ci = II*(a.$ ( $ ) ) ( e x e r c i s e 1 0

Now f o r

[BG3]). (10.27) Y

(c = a

0

A $om

+ E,).

T;

0

f 0 one can use k = o i n (10.20) t o o b t a i n

= eamf ,Arc

+ (Ay-A:)gl

P u t t i n g t h i s i n (6.)

ebme implies

- cf.

148

ROBERT CARROLL 0 c$ = (a.hm)c = (ailogrm)c

(10.28)

1 0

$’ t h e c e n t r a l component o f (6*) i s a.cm = a P I J j Hence one can p r o j e c t t h e zero c u r v a t u r e equations (6*) o n t h e l o o p

Since t h e 2 c o c y c l e $ = 0 on c y = 0.

algebra w i t h o u t l o s s o f information (i.e.

t h e c e n t r a l components a r e t r i v i a l ).

Take now t, = x d i r e c t l y i n t h e c o v a r i a n t d e r i v a t i v e ( e ~ ) . One has

-m

ox

(10.29)

(F r e f e r s

=

F,

to

-

ax

t

Ro+ ( p1

etc.).

s i t i o n o f (66) t m ( p l )

1

-m

=

ax

-

A I4h

rn

-

($ qo)

Now qm i s t h e c o e f f i c i e n t o f A’ 4

= ($-)

-1 h B t2hTy

( n ( h B 4h) =

Ie i n t h e decompo-

a/axl;

note

$!

involves

A)

n e g a t i v e powers o f h so t h e r e a r e no t powers o f X i n (10.29) beyond

and t h e r e f o r e qrn i s a l s o t h e c o e f f i c i e n t o f 1-l Ie i n t h e decomposition o f Since ($!)-’J-,h5! (?:)-’+hT!. one can w r i t e ( c f . [ BG31)

H(fo.voy (c~)-’4h$!v0)

= H(+h($!)-’’.

Ie ) .

( h e r e fo = A-1

d i f f e r o n l y b y c e n t r a l terms

Ie . v o y ( $A m- ) -1,5h A$_vo)/H(A-’ m

qm = H(X-’

(10.30)

and (;:)-’ah$!

Ie.vo,X -1 Ie . v o ) =

f o ~ v o ,4rn $ ~ . v o ) ; $ = Hermitian conjugate

Note here a t r a n s f e r o f h

and H ( f v ,f v ) = H(v e f v ) w i t h e o f o

g,

2,

-

i n v o l v e s 2 minus s i g n s

foeo = -h+c so e f v = ( f o e o 0 0 0 0 C J ’ O O O 0 0 0 h+c)vo = ( f o e o t ao)vo = vo ( s i n c e eOvO = 0 by (10.9) and Eovo = ( A ,.’)v

“

0

vo). Ifnow

Thus H(vo,vo)

m -1 ($-I

j u g a t e belongs t o UD and the weight o f a vector. = 6ij;

(10.31)

1). . . e x p ( r A O ( x n ) )

= exp(IrA (x, onto

then, u s i n g c o n t r a v a r i a n c e ( M ) 1

v

a.

+

v

al

(j!)-’t =

1 +

At

($-)

-1 t

.fovo

V

= f v

0 0

u-P i t s conan o p e r a t o r r a i s i n g

Since

where

Now f o + v o has w e i g h t A.

Am

so ( 6 m )

w e i g h t i s A.

= 0

= 1 i s s t i p u l a t e d here.

= e x p ( - r A (w ( ~ n ) ) ...exp(-rA~(wo(xl))). )

tj:)-l+

0

-

A+ i s - ci0

+ w O y1~

E

E

and t h e o n l y h i g h e r Therefore ((Ai,.’.)

C.

= c; a1 = h ) = $l*fo.~o +

4h($m)-1t.fo.vo (A

( r e c a l l nA(h)vo = (A,h)v

0

v 0’

)v = 1 0

= ~ A o - a o , aVl ) f o . ~ o

J

+

fO+VO

and n o t e t h a t ( A -a 0

0

yz1

)

= - ( a ’c-; 0

0

)

=

2 - ( a0 ,c)

RESOLVANT DECOMPOSITION = 2

149

s i n c e the roots a a r e extendable t o *h e w i t h (cr,c) = ( c r , d ) = 0 ) .

f e r r i n g t o ( l O . l l ) , T'exp(ka) -1 [BG31 - f, = A e ) so

Now re-

exp((kf1)a) a n d one has T-v, = -fo-vo ( c f .

=

H e u r i s t i c a l l y , f o r Tvo look a t the weights from ( l O . l l ) , A. + ma - ( m L t k ) s 2 ma - ( m tk)(6+a0). where say vo A. 'L 1 E n(A ) ( w i t h m = k = 0 ) . Then Q

=

0

Tvo 2, exp(a) will have weight A 0 -a 0 ) corresponding to m = 1 , k = 0, which i s the weight f o r f v ( t h e - s i g n must then be checked using t h e formula T," = * rn e t c . );. T : : 0 now t h i s becomes ( c f . (10.20) e t c . ) r6-a a (10.33)

qm = -H(vo,T

-1.m

-A"'

vole o = -H(v,,J,

'm+lv )e-'o

m = -TO

/

0

m+l m '

Similarly rm = Next one considers t h e compl e t a T(p,) w r i t t e n a s (**) "R(p, ) = A I 4h + n) 1; A-i PD R T . We know R: = ($qo) and f o r t h e o t h e r terms one w r i t e s (10.34) (pi

2,

Ai

(10.35)

R T = ResR ( p , ) = Res'jim(pi) = Res((?y)-'ai$y)

B 4 h and one uses ( a i Am I/I= 1

t bm(n

(A*'

t

Am

:?(pi))($-)

-1

= 0).

One can w r i t e now

I e ) ) + am(nA,(A-' I h ) ) + cm(nA, (A-'

I f ) ) ...

_ah):

-1 a ( c f . (10.20)). Hence a f t e r projection (4.) 7; = 1 t A :(, + ... and hence (**) R T = :,"( -aia a i b 2 ) , ai = a / a t i . Using (10.35) one can c a l c u l a t e (10.36)

bm = H ( A - ~

nm I e-v,,$-.v,)

am = H(A-l I h . v o , $ ~ . v o ) / H ( h - l

=

I h-vo,A-'

m q;

nm m., cm = H ( A -1 I f.vO,J,-~VO)=-r

I h-vo)

= H(1-l

rm

I ~h.vo,+--vo)

=

( e x e r c i s e - t h e r e a r e a number o f v e r i f i c a t i o n s needed here b u t most o f t h i s 0 involves k = 0 i n 10.20) and c f . has been sketched before - r e c a l l :T (10.28) f o r c y ) . Hence f i n a l l y

+

150

ROBERT CARROLL

RI= [

(10.37)

0

a . a 10gT

aiq 0 - a .1a xIOgT,

a.r 1

depends on o n l y a f i n i t e number o f ti, R T = 0 f o r say i >

and s i n c e :T

so ii"(p

1

1 N,

) i s i n t h e polynomial l o o p a l g e b r a g.

I n t h e usual ( " c l a s s i c a l " ) t h e o r y o f r e s o l v a n t s ( c f . [ DK1 ,4;G2-9])

t h e mat-

r i x c o e f f i c i e n t s o f r e s o l v a n t s a r e p o l y n o m i a l s i n some fundamental q u a n t i t i e s (e.g.

qm and rm)and t h e i r x d e r i v a t i v e s ( x

ai d e r i v a t i v e s e t c . .

example we have

'L

tl).

-

B u t i n (10.37) f o r

However t h e t a u f u n c t i o n s obey many

e q u a t i o n s and i n [ 8631 one shows how t o reduce t h i s t o t h e " c l a s s i c a l " s i t u a Thus e.g.

tion.

t h e m a t r i x e n t r i e s i n Rm(pl) can be w r i t t e n as p o l y n o m i a l s

i n qm,rm and t h e i r x d e r i v a t i v e s (as i n § 9

-

cf. also

comments on v a r i o u s c o n s t r u c t i o n s o f t a u f u n c t i o n s ) .

DK3;DZy5;P01-31 f o r I n p a r t i c u l a r t h e zero

c u r v a t u r e e q u a t i o n s (&*) become t h e AKNS f a m i l y f o r m u l a t e d now o n a l a t t i c e

m E Z w i t h t h e f i r s t 2 members c o r r e s p o n d i n g t o NLS. -m -m

I=

(10.38)

[ D2,Dx

RBIARK 10.8

( A K W COtUERVAtZON

0

f o r some f i x e d ti (say tl

%

'L

a2qm = acqm

x).

LWS).

-

Thus

m 2 m 2 ( q ) r and a2rm=

2 m

-ax'

-t

m2 2q(r )

L e t f be t. d i f f e r e n t i a l polynomials 1

The t . d e r i v a t i v e o f f can be c a l c u l a t e d

J

from t . d e r i v a t i v e s o f t h e generators o f ti d i f f e r e n t i a l polynomials, i . e . J from t h e c o e f f i c i e n t s o f Rm ( p . ) , which come from zero c u r v a t u r e equations o+ 1 (10.39)

[1?,'2 =l 0; J im

a J. i mO + ( p 1. )

= [Di,Rot(pj)l "m "m

Thus t . d e r i v a t i v e s o f ti d i f f e r e n t i a l polynomials a r e a g a i n ti d i f f e r e n -

J

t i a l polynomials.

One c a l l s f a ti conserved d e n s i t y i f f o r a l l j t h e r e

One can e x i s t s a ti d i f f e r e n t i a l polynomial g . such t h a t (+*) a . f = aigj. J J f i n d such q u a n t i t i e s i n v a r i o u s ways (two a r e g i v e n i n [BG31). One d e r i v a i s (+6) t i o n goes as f o l l o w s . Recall t h a t t h e c e n t r a l component o f ?(p.) m A m -1 .n1 One has from FK1 I f o r $ E G ci = a,(Ad(@-) (pi) = ( a i l o g r i ) c . x E gp, k' Here g i n d i c a t e s a E C ( W ) Ad;(; t a c ) = Ad:(?) t TrRes($-'(aA:)x") t a)c. d

t h e image o f

$

m

2

N

Then s e t (em) Fij = aiajlogTm P P', A m -1 i n t h e form ci = r,(Ad($-) (pi) ( r e c a l l

under t h e p r o j e c t i o n

and t h i n k i n g o f (10.26)-(10.28) *m A m -1 Am m *m R (pi) = (+-I pi@-, ci = r*(Ro+(pi))

-t

Am

= r*(R

(pi)))

one can w r i t e

0

CONSERVATION LAWS FYj = ajTrRes(Sm(a,$-)-m -1 pi)

(10.40)

= TrRes(((a~m)(a,(~m)-l)

( a h a j (JI4- -1 )pi) = TrRes(($_R-(pj)(aX($-) Nm-m Nm -1 - T r R e s ( ~ m a X , ( ~ y ( pyn j ) ) (-1 ~ -pi) )

a x (?(pi ) )

-

= -TrRes(aX(~(pj))"R(pi))

= TrRes

(iim( p

)a,

f o r a l l j a ti conserved d e n s i t y .

-

used t h e r e

(ii!+( pi 1) 1

Jk

b y showing t h e

= TrRes(iim(pj)aX(~+(pi))) i s also

m m Thus g i v e n q ,r one

One can a l s o r e v e r s e t h e procedure.

tAewhich

E

= a.Fm shows t h a t FTj i s

c f . 59).

REnARK 10.9. finds

=

= TrRes($m(pj)

This improves [ FL1;NEll

L i e t h e o r e t i c meaning o f FYj e t c . (Fij

+ Sy

vJyah(gy(pj)($-) -m -1 ))pi)

FYj i s a ti d i f f e r e n t i a l polynomial and akFTj

Thus

rm.

151

$ has

Suppose

determines a p o i n t i n t h e group o r b i t g i v i n g r i s e t o qm, been found;

3" =

then one has (.*)

T-m(exp(I:

p.t.));. 'dm One g e t s f a c t o r s $-, $+ , and equations as b e f o r e (.A) a7 i !+ = Ro+(pi)$o+; -mun m ai$--m = pi$--m $-R o+ ( p i) ( c f . (10.25) r e w r i t t e n w i t h (10.23)). Now Ro+(pi) i s an x d i f f e r e n t i a l polynomial determined by qm,rm so, g i v e n such a s o l u Am

-m

Am

-

t i o n , one s o l v e s ( D A ) ~ , ~ , then d e f i n e s -m The s o l u t i o n o f t h i s i s pi$

Tm = $ ' ;;:+,

aijm

and t h e n d e r i v e s

=

.

In

cn

-m $ = e 1 p it i -m $ ( 0 ) = e 1 tipi $y(O)'j;:+(O)

(10.41)

Thus f o r t h e r e v e r s e problem we s h o u l d t a k e = ?G!(0)Tm

t o be a n a r b i t r a r y l i f t o f

( 0 ) t o ?iAo t o g e t g up t o a m u l t i p l e X

E

Cx.

The r e m a i n i n g

:O

freedom i n g c o n s i s t s o f p o s s i b l e c h o i c e s o f i n i t i a l c o n d i t i o n '3;m(O),

(0).

One shows i n [ BG3] t h a t

f o r a n y T t + ( O ) E SL2(C[,A]).

has a l o c a l s o l u t i o n ;!+(-) On t h e o t h e r hand

( D A ) ~i

E

?+:

SL2(A;)

s only solvable i n

SL ( A - ) f o r s p e c i a l i n i t i a l c o n d i t i o n s T y ( 0 ) ( c f . [ BG31 f o r d e t a i l s ) . 2 P m m The c o n c l u s i o n i s t h a t i f q ,r a r e s o l u t i o n s o f t h e Toda-AKNS system s a t i s m m N f y i n g ( D O ) t h e r e e x i s t N,M such t h a t q ,r # 0 f o r N < m < M; q f 0, rNs M M m m .,tn f o r N 5 m 5 M, 0, q z 0, r $ 0; q ,r a r e r a t i o n a l f u n c t i o n s o f t,, then t h e r e e x i s t s an element T = i - v o o f t h e o r b i t i n GAo t h r u vo h a v i n g

..

components

0

i n t h e homogeneous r e a l i z a t i o n o f L ( h o ) such t h a t ( ~ 6 qm ) = m f o r N 5 m 5 M. - T ~ + , / T ~ and r = TO / m-1 'm 0

0

, T

152

ROBERT CARROLL

11. &HE HZR0CA 3ZI;ZIEAR ZDENtZt3J.

The Sat0 Grassmannian (UGM) i s developed

i n 113 and t h e Segal-Wilson c o n s t r u c t i o n i s discussed a t v a r i o u s p l a c e s ( c f . §19,21 f o r example).

The t h e o r y o f Grassmannians i n t h e c o n t e x t o f Banach

m a n i f o l d s e t c . i s developed i n [ DF1-4;SJl;PRl

I i n d e t a i l and we w i l l o n l y

comment b r i e f l y o n t h i s ( c f . a l s o

We w i l l g i v e an e x p l i c i t con-

§19,21).

s t r u c t i o n o f t h e Segal -Wilson determinants and wave f u n c t i o n s f o l l o w i n g [ H E l l 2 1 f o r H = L ( S ,C) ( c f . a l s o [ PO1,2]). Then we w i l l show how t h i s c o n s t r u c t i o n can be adapted t o t h e Grassmannian o f [ M C l O ] f o r t h e s c a t t e r i n g s i t u a t i o n o f KdV and we w i l l d e r i v e a H i r o t a b i l i n e a r i d e n t i t y f o r i n v e r s e s c a t t e r i n g s o l u t i o n s o f KdV ( c f . [ C18-191).

(&HE tmP GROW APPROACH C0 HZR0CA FOR KP),

L e t us d e r i v e H i r 2 1 o t a formulas i n t h e s p i r i t o f [ H E 1 1 ( c f . a l s o §7,8,13). Thus l e t H = L ( S , i C ) with Fourier series f(h) = aih , ai E C, f E H. L e t (*) Hp = ai I L P hi) E Hyip E Z. The spaces Ho and Ho a r e denoted by H+ and H- (Ho = H- = -1 aih I ) . One w r i t e s H = H, 8 H- and GLres(H) = automorphisms g o f H

REitARK 11.1

{Im

1

{I-,

which can be w r i t t e n r e l a t i v e t o H, b and c H i l b e r t - S c h m i d t = HS.

@ H- a s

(A)

g = (:

,b),

a and d Fredholm,

The connected components o f t h e Grassmannian

Gr(H) a r e g i v e n by (11.1)

a bd ) ( oH+ ) ; i n d ( a ) = p)

GrP(H) =

(GLFes)(H+) = I(:

!)Hp;inda=Ol

One can check t h a t GLOres(H) a c t s t r a n s i t i v e l y o n each G r P ( H ) ( c f . [ PR1 1). L e t now A : H -+ H denote m u l t i p l i c a t i o n by X and s e t

r+

(11.2) For Wp (11.3)

E

i =

reL: ti'

E

Iti 1 ( 1 + E ) i

GLores(H);

<

f o r some €1

GrP(H) l e t W r+p = {y

w

E

r+;orth.proj.:y

-1

Wp

-+

H i s a bijection) P

As i n [ S E l I r + p i s nonempty and one a s s o c i a t e s a wave f u n c t i o n Q W ( y , c ) t o W where y E r + p and 1 < I c l < 1, v i a W = W P'

HIROTA FOR KP

153

where i t i s stipulated t h a t the L 2 boundary value JIw(y,A) E W = W, A E S1 , W P for a l l y E r,?. One r e c a l l s here t h a t an operator A: E F i s Fredholm i f ~1 = dimkerA and 6 = dimcokerA a r e f i n i t e with indA = CI - 6 (ind i s a horntopy invariant). An operator A: E + F ( E , F Hilbert) i s Hilbert Schmidt (HS) -f

-

when 1 IIAeil12 < whenever ei i s a n orthonormal basis of E. The requirernents on J I ~above determine J I ~uniquely ( c f . [ SE11 a n d § 1 9 , 2 2 ) and the flows

r, a r e of course connected t o the KP hierarchy; one should be scrupulous in distinguishing A y A and 5 . For simplicity we will deal only with p = 0 here and refer t o [ HE11 for general p ( c f . also [PRlI). Then for the adjoint wave function one notes f i r s t that i f W E Gro(H) then the orthogonal projection i d L + H- i s Fredholm of index 0. For y E r+ a n d W E Gro(H) one checks easily t h a t ( 0 ) (y- 1 W ) I =y*w where y * i s the a d j o i n t of y . Now interchange t h e roles of H, and H- a n d the Grassmannian Gr*(H) connected t o the new decomposition H = H- 8 H, conI tains a l l of the W . The a d j o i n t flows are given by

As in (11.3) for p = 0 now a n d W A =

rW

(11.6)

For

E

I;

E

W L and l / l k

r-

I

r- ; o r t h . p r o j . <

101

<

E

Gr;(H)

A l A y- W

-+

H- i s biject.)

=

{y*-l; y

E

r,W 3

1 the adjoint wave function J I W l has the form

i s uniquely determined by the requirement t h a t the L I 1 WA E rvalue J I ~ L ( ? , A ) E W for A E S , and a l l

As above

2 boundary

.

The program now i s t o show t h a t various Hirota bilinear i d e n t i t i e s for the ( k , p ) modified KP hierarchies are simply perpendicularity relations between wave functions ( c f . [ HE1;Jll ). Thus for p , k E Z, k 2 p , l e t W E GrP(H) a n d Wk E Gr k ( H ) with W 2 W k (e.g. W = g-H and Wk = g.Hk with g EP G L t e s ( H ) . P P P L Then W ~ WCt a n d since $ k ( y , A ) = J, ( y , ~ )E wk w i t h J I ~ ( ; , A ) = y I ( ~ , A ) E w P P Wk P W P one has

154

ROBERT CARROLL

I

.

( n o t e de = dA/iA o n S

1

and ';i

Here

l/X).

8 ) f o l l o w s immediately by c o n s t r u c t i o n . t i o n , such r e l a t i o n s w i l l h o l d f o r case p = 0 i n d e t a i l . sjAj)

and t h e r e l a t i o n (11.

= (y*)-'

b u t we w i l l t r e a t o n l y t h e

= (y'*)-',

Thus, g o i n g back now t o p = 0, l e t

w r i t e (+) $W(y,A) = ( 1 t bk(;)h-k)exp(c

v*

More g e n e r a l l y l a t e r , by s t i p u l a -

1 ak(y)h-k)exp(5)

( c f . (11 .7)).

(5

=

1 tihi)

W

E

Gr0(H), and

and =: ;

h(1

t

1

Then (11.8) becomes

Now t o express m a t t e r s v i a t a u f u n c t i o n s one f o lows [ S E l ] ( c f . §19,21) and f o r W E Gro(H) l e t

where wt = I d . + Trace c l a s s . form i s e v i d e n t ) . (11.11)

-1

%

a b

dl( c f .

One d e f i n e s a t a u f u n c t i o n v i a ( y =

T,(Y)

( c f . §19,21).

Let y

-+ a

det(w,

Given y-lW

-+

-1

(A)) plr

for y

E

r W+

(the

e x p ( 1 tiAi))

bw-)

H, a b i j e c t i o n one checks immediately t h a t

(w"

-1 a b -1 -1 a b ) ( $+ A )Ht yibw). Note here, f o r y 2 % ( o d)Y TW(YlY2) % y 2 y1 w % ( o d w-1 a B then = awt + 6w- and ^w- = 6w- so ( m ) T , ( Y , ) = det(w, + I f y1 % ( o -1 -1 A A . 1 * a Bw-), T-(Y ) = d e t ( 1 + a bw w; ), Gt = awt + Bw-, w- = 6w-, and T (Y Y 1 A w-12* 1-12 = det(w,+a bw-). Thus (11.12T f o l l o w s from (**) 1 + a-'bsw-(aw, t BW-) a(wt t a-'gw-) = awt t BW- t a -1 b6w- = t a - ' b t - and p r o p e r t i e s o f d e t . Q

it

^wt

Now f o r (11.13)

Is1

> 1 say l e t

qs

%

1

-

(A/<) = exp(-ly

(Aj,/jsj) E

rt

This i s t h e way i n which t h e v e r t e x o p e r a t o r a c t i o n f o r KP i s r e p r e s e n t e d i n

-

[ SE11 and we w i l l g i v e a KdV v e r s i o n below; a general v e r s i o n seems t o be

m i s s i n g a s remarked by J. S z m i g i e l s k i .

Let y2

q5 w i t h decomposition y";'

TAU FUNCTIONS

155

~n

a b ) as above and y 2 T , (a dm ( t < 0 ) and a:’

k).

D i r e c t c a l c u l a t i o n shows t h a t (*A) b: A t t + ct:l:’_ atXt ( C L a 5 ) i s a l-dimension-tl 4-1.. a1 m p . S i m i l a r l y ( * a ) c.1’ -+ - C - ~ / C and a b:l-, c j X j -t - C - ~ / C ~ ; J k ( A / < ) i s a 1 dimensional map. Consequently ( c f . [ S I M 1 1 ) (*r) d y ) = d e t A A-1 W Z (1 t a-’bG ) = 1 t T r ( a - ’ b F ) where K- = w-w+ and one w r i t e s (*+) w-(Xm) =

(

lo

-’

1-m

w

-f

At

-f

$:I:’_

sm

h

S

(m L 0, wsm = ws,(yl)).

(11.14)

$, q c ,)

-1

- = I1 +

( n o t e by (*A) T r ( a - l b F - )

It f o l l o w s t h a t (9‘

)5

Ws,o(~l

= y,)

S

= c o e f f i c i e n t o f Xo

i n (a

-1

#”

On t h e o t h e r

bw-)(Xo)).

hand f o l l o w i n g [ S E l ] ( c f . a l s o §19,21) one can d e f i n e t h e f u n c t i o n GW(y1,X) o f (11.4) as -I

(11.15)

= 1 + z-(l)

$W(ylyh)

1 +

1

-m

w

s,o

(yl )As

-

A

’; corresponding t o 1 ( i . e . qw(yl,X) i s t h g unique e l e v e n t o f yW 1 w 4 1 W = y- W s i n c e W = (Q’_)w; H, = ( i T ) H + = y;’W). T h i s shows t h a t

-

n o t e here

n

The formulas i n [ J1 ] suggest what t o do f o r q W i a n d f o l l o w i n g t h i s one con-

w i t h G a s i n (11.12). Thus we l o o k A-lA As b e f o r e (*.) d e t ( 1 + a bw-) = 1 + and (*+) we o b t a i n (A*) ;-llbG-(Xm) =

side:sAq-’ i n [HE11 and computes $q): a b ‘ a t (o and a p o s s i b l y d i f f e r e n t y,’.

r)

-(‘-I

,m

From (*a)

( c f . (*@)).

Tr(^a-’bc-)

/<)I:

t h e r e f o r e (w- 1 ,m = ‘-1 ,m(yi (11.17)

T$q:)

Now f o r y

W

E

r+,

= (y*)

i n t h e sense t h a t (y;W

-

l y (-“wh_,h,)

t (

-

= 1

1.

,yl

h-,z-ht)

E

-1

For

7

$w~(qlyX)

r-wL , w- -%

A

W ) = ( W ,W)

= 0).

X

-1

bw-) = - ( l / c ) l ;

~ - 1 (Y’ , ~)Cm A-1 w-wt

A

, one

-

-W_*

sees t h a t ( I )H- = yTW

I

u

= 0 ( r e c a l l t h a t W = y;’W

Now one r e q u i r e s

unique element corresponding t o ( l / h ) (11.18)

Tr(a

(AA)

1)

(1/5)1;

-1

w - ~,,,/?? and

A -IAN

( X / C . ) ~ . Consequently

E

$wI(ql,X)

and f o r m a l I

A

E

YTW t o be t h e

H- so t h a t

-

g i v e n by (*+) we h a v e

(A@)

=

-1:

w-1 ,m Am

and consequently i n

ROBERT CARROLL

156

(7 =

an i n t e g r a l l i k e ( 1.8) we would have from (11.17)

ljh)

(11.19)

TWLG1

The o r t h o g o n a l i t y o f (11.8)-(11.9)

i s b u i l t i n t o the construction f o r

+w

and

$ w based ~ on t h e same y1 and w i l l be r e q u i r e d by s t i p u l a t i o n f o r d i f f e r e n t yl.

Note here i f y1

i n (11.7)

( w i t h si

(-1 t J. A J )

where

Ek

from t h e same y, i s automatic.

exp(1 tiA

%

-Ti)

%

w - ~,k-l

i ) then y*-l

we o b t a i n

and f o r n

e x p ( - l iiAei)

Gwdy.,A)

(A&)

from (11.19).

t h e exponeht

%

= h(1

+

1;

-+

A

ck(y)A-k)exp

Thus f o r $w and $ w b ~u i l t up

terms i n (11.9) v a n i s h and t h e o r t h o g o n a l i t y

I n any event t h e r e s u l t i n g H i r o t a formulas (11.8)-(11.9)

can

be w r i t t e n

can i n f a c t w r i t e (11.20) i n t h e form (11.21)

4

(t.-l/jAj):

W J

W

A

where .rw(aj) = -rw e x p ( 1 a j )

RErtARK 11.2

(P;RA5$FzAX!UAW

3

(tl+l/jAJ)e j

( c f . (11.13))

AND t A FOR KaU ON CHE i;I!E).

We go now t o i n v e r s e

s c a t t e r i n g s o l u t i o n s f o r KP o r KdV c o n s t r u c t e d e.g. as i n 57 v i a d e t e r m i n a n t methods o r GLM equations.

The main p o i n t one must emphasize here i s t h a t T -1 wave f u n c t i o n s w = (1 + K+)exp(S) and w* = (1 + K+) exp(-E) as i n Remark 7.15 a r e a n a l y t i c i n l e f t h a l f planes w i t h wexp(-S) and w*exp(S)

Q

1 + 0(1/

one k ) as I k l -+ m (Rek < 0 ) . For KdV wave f u n c t i o n s + ) I - ( o r f -+ ) as i n §;,2,7 i s d e a l i n g w i t h a n a l y t i c i t y i n t h e upper h a l f plane and c o r r e s p o n d i n g asympt o t i c s t h e r e as I k l

-+

m.

Except f o r pure s o l i t o n s o l u t i o n s these wave func-

t i o n s are generally not a n a l y t i c a t

m

i n H i r o t a formulas i s meaningless.

The l i n e s

and c o n t o u r i n t e g r a t i o n a t - i m

+

i-o r

--

-+

m

m

arising

arising i n

completeness r e l a t i o n s such as (7.86) o r (1.1 3) cannot be deformed t o be c i r c l e s around

m

f o r example and i n any event these formulas express some-

t h i n g d i f f e r e n t (namely completeness).

Some s t u d y o f t h e r e l a t i o n s between

GRASSMANNIANS FOR KdV ON THE LINE

157 Thus what we w i l l do

completeness and H i r o t a equations m i g h t be p r o f i t a b l e .

i s e x p l o i t t h e Grassmannian p i c t u r e o f [ M C l O ] f o r KdV s c a t t e r i n g and t r a n s p o r t t h e machinery o f [ HE1 I e x h i b i t e d i n Remark 11.1.

This w i l l a l l o w us t o

g i v e a p r e c i s e geometrical v e r s i o n o f t h e H i r o t a b i l i n e a r i d e n t i t y f o r i n v e r s e s c a t t e r i n g s o l u t i o n s ( p e r p e n d i c u l a r i t y o f wave f u n c t i o n s ) which we express f o r m a l l y i n t h e c o n v e n t i a l c o n t o u r i n t e g r a l f o r m a t by an a l g e b r a i c correspondence which i s n o t based on r e s i d u e c a l c u l a t i o n s .

T h i s corresponds

t o e x p r e s s i n g t h e Hardy space geometry i n a formal way by mapping b a s i s e l n n-m-l ements en -+ w and w r i t i n g (enye,) % ( 1 / 2 v i ) + w dw = 6mn f o r some G =

Ic f o r m a l l y ( t h e r e s u l t s were announced i n [ C16-191). F i r s t we s h i f t t o KdV and r e c a l l some n o t a t i o n ( c f . 51,2,7). We c o n s i d e r 2 2 t h e KdV s i t u a t i o n L$ = ( D + q)$ = - k and $t = B$ = -4QxXx 6qqX 3qx$

+

w i t h qt + 6qqx + qxxx = 0 ( q r e a l ) . e x p ( ? i k x ) as x =

-+

+-

a r e no bound s t a t e s and l e t q

E

= (1/2v)j:

t i o n o f t h e M equation (+m) Then q ( x , t )

f o r y > x.

-

One d e f i n e s J o s t s o l u t i o n s w i t h f,= sZ2, R = sZ1, RL = s12, and r f ( k ) Q

and w r i t i n g T = sll

f - ( k ) = f ( - k ) we have T f - = R f + + f;

i n v o l v e s (A+) F(z,t)

-

Assume t h e r e

3 (Schwartz space).

+

R(k,D)exp(ikz

+ F(x+y,t)

K(x,y,t)

= 2DxK(x,x,t)

c h y v a r i a b l e s x = (xl ,x3,. ..),

+ f:.

w i t h Tf+ = R f

x = x

The c l a s s i c a l p i c t u r e 3 8 i k t ) d k and K i s t h e s o l u t

/xm K(x,s,t)F(s+y)ds

Now i n t r o d u c e h i e r a r -

s a t i s f i e s KdV. sometimes, x3

%

t 3 = 4 t , and u n l e s s

1 o t h e r w i s e s p e c i f i e d , f o r h i e r a r c h y v a r i a b l e s x,y we s t i p u l a t e x for n

(**I

2 1.

Set (**) r ( x , k ) =

F(x,y)

=

= ic;

Ir $o(x,k)$o(y,k)dA

~ = ~y2n+l + ~

xZn+, k2n+' and + o ( x y k ) = e x p ( r ) w i t h where r,A can be i n

= ($o(x,k),$o(y,k))A

general any " s u i t a b l e " c u r v e and measure (i- = sically).

= 0

and dA = Rodk/2n c l a s -

(-my-)

The M e q u a t i o n can be s o l v e d (and d e r i v e d ) v i a d e t e r m i n a n t meth2 = 2Dxlog.r(x) ( t i s

ods as i n 57 and T ( X ) i s d e f i n e d w i t h q ( x ) = 2DxK(x,x) suppressed).

One d e f i n e s wave f u n c t i o n s now as

T$- = e x p ( - y ) +

+,(s,k)ds;

it

(00)

K-(x,s)exp(-r(s,k))ds

+

/x

m

K,(x,s) T -1 ( f o r KdV K- = (ltK,) ). = $,

$,

N

The v e r t e x o p e r a t o r e q u a t i o n i s now t a k e n as (VOE) v

T = $+T;

X + ( ~ ) T = exp(?)G+(k).r

This can be proved as i n

§:

= exp(T&,

X-(k)-r = exp(?)c-(k) 3 = e x p ( r ~ ) . r ( x 1 + l / i k , x 3 + 1 / 3 i k ,...).

f o r KP (Theorem 7.14).

The r e l a t i o n ( l + K T ) ( l +

K+) = 1 l e a d s d i r e c t l y ( v i a F o u r i e r t r a n s f o r m ) t o t h e c l a s s i c a l completeness r e l a t i o n (1.13),

-

m

namely ( C ) (1/2v)Jm T$-(y,k)$+(x,k)dk

= ~ ( x - Y ) . This i s

based e n t i r e l y o n t h e s t r u c t u r e $+ = (l+K+)exp(?) and T$- = (1 +K-)exp(-?)

158

ROBERT CARROLL

T w i t h (1+K-) = (l+K+)-'

I f one works f o r m a l l y w i t h p s e u d o d i f f e r e n -

f o r KdV.

1;

w ( ~ ) a - %~ 1 + K+ as i n 17 w i t h w = Pexp(r) n formal r e s i d u e c a l c u l a t i o n s l e a d (when t h e y make sense) t o

t i a l gauge o p e r a t o r s P = 1 + $+ t h e n e.g.

t h e H i r o t a b i l i n e a r formula (7.13), (7.28)

%

(11.21

1).

namely (H) 9 w(x,k)w*(y,k)dk

= 0

(or

Hence h e u r i s t i c a l l y one can say t h a t t h e conceptual

background f o r (H) and (C) i s e q u i v a l e n t , namely, w = P e x p ( r ) and w* = P*- 1 N

exp(-S),

with P

l+Kt,

%

etc.

a n a l y t i c wave f u n c t i o n s $+, g e n e r a l l y do n o t make sense.

The problem i s o f course t h a t f o r h a l f p l a n e

T$-

as above (Imk > 0 ) t h e r e s i d u e c a l c u l a t i o n s

The H i r o t a formula (H) s h o u l d have general

meaning however s i n c e i t s b a s i c c o n t e n t i s geometrical and c o m b i n a t o r i c and we g i v e below a n a t u r a l development f o r t h e a l g e b r a o f t h e h i e r a r c h y framework i n t h e s c a t t e r i n g s i t u a t i o n . Now go t o t h e Grassmannian p i c t u r e of[MC10],

which i s one o f a s e r i e s o f

papers on t h e geometry o f KdV (namely[MCl , 6 - l l ; E 3 ] ) b r i e f l y i n §4.

which we discussed

The theme o f these papers i n v o l v e s a l g e b r a i c geometry and

d i v i s o r t h e o r y and t h e p r o o f s a r e c o n s t r u c t e d a c c o r d i n g l y . a d i f f e r e n t p o i n t o f view i s p r e f e r r e d however.

For o u r purposes

Thus f o l l o w i n g [ M C l O ] one

l o o k s a t Schwartz c l a s s p o t e n t i a l s q w i t h s c a t t e r i n g data sij bound s t a t e s ) . sZ1(O) = -1,

Thus sll

as above (no

= s Z 2 = T, sZ1 = R, e t c . and i n p a r t i c u l a r s2, E

I s Z 1 ( k ) l < 1 o t h e r w i s e , Zij(k)

-

ti,

s . . ( - k ) f o r k r e a l , s11s21 t 'J 2t s12Z11 = 0, sll i s an o u t e r f u n c t i o n o f c l a s s 1 + H , and sll ( 0 ) c o r r e s 2 ponds t o a s i m p l e r o o t . Here H2+ = FL2[0,m), H2- = FL (-m,O], and t o make =

t h e n o t a t i o n s c o i n c i d e w i t h [MClO], we t a k e F f = ( 1 / 2 ~ l ) j If ( x ) e x p ( i k x ) d x = f ( k ) unless otherwise specified. j e c t i o n L2

+.

HZt

( r e s p L2

+.

L e t p ( r e s p . 1 - p ) be t h e o r t h o g o n a l p r o -

H 2 - ) and t h e n G r = I H C L2; ( 1 - p ) : H

1-1 , onto, w i t h bounded i n v e r s e } i s t h e Grassmanian o f [ M C l O I .

+

H2- i s

One shows

2

G r correspond u n i q u e l y t o sZ1 a s above v i a ( 0 6 ) H = L n {f; 2+ f- + s Z 1 f E H I . W r i t i n g r f ( k ) = f - ( k ) = f ( - k ) some u s e f u l f a c t s h e r e a r e t h a t such H (1) f

E

E

H i f and o n l y i f f + rsZlf

E

H2-

(i.e.

i f and o n l y i f ( 1 - p ) f = (1

t

2-

prsZl)f) (2) I f h E H then (1 + prspl)f = h can be s o l v e d f o r f E H v i a ( 1 - p ) f = h and p f = - ( l + p r s 2 1 ) - 1 p r s 2 1 h ( 3 ) H I = (1 + rsz1)H2+ is orthogonal t o H ( n o t e (rszlp)*

t

(f,g-)

= (f-,g),

( f , g ) = 1 fgdk,

The Baker-Akhiezer (BA) f u n c t i o n f o r H i s d e f i n e d as t h e unique e E 2+ H2+ w i t h e- + sZ1e E 1 + H and we w i l l use t h i s d e f i n i t i o n . I n [ M C l O I

etc.).

1

= prsZ1, r H 2 + = H2-,

BA FUNCTIONS

159

-1 a formula e = 1 - ( l + p r s Z 1 ) prsZ1 i s w r i t t e n down b u t we p r e f e r t o t h i n k o f t h e r i g h t s i d e as an operator below. Our e x p l i c i t e will be defined i n

terms of f,.Now r e c a l l t h a t elements (11 .22)

en = ( - i / J 7 1 ( k + i ) ) ( ( k - i ) / ( k + i ) ) n

form an orthonormal b a s i s o f H

2+

f o r n 2 0 and of H

2-

f o r n 5 -1.

Recall

a l s o t h e notations ( 0 4 ) Fx$(s) = r; F(s+x,cs+x,x . . ) $ ( c s ) d ~ and FxQ.(s) = 2'' F(s,a,x2, ...)$I( 0 ) d a o f 17 a n d note t h a t TrFX = TrF, again. I t follows t h a t d e t ( l t F x ) = d e t ( l t F X ) v i a Plemelj-Lidskij and we r e c a l l t h a t R = s 21 m and F AR in (*A) y i e l d s , f o r = C = (-m,m) (om) F ( x ) = ( 1 / 2 r ) j m sZlexp Q,

( i k x ) d k = ?21 ( x ) .

Further Tr(prsZ1)

10" $,, ( 2 x ) d x

TrL ( L Q, Fo = Fo; Lf ( x ) = J; ;,,(xty)f(y)dy. f E H2', i n [MClO]). One notes a l s o t h a t i n order t o deal w i t h x v a r i a t i o n one simply replaces sZ1 by ?21 = sZlexp(2ikx) ( o r Q ,

=

rz1= sZ1exp(2%)t o get J o s t s o l u t i o n s f

k

'y

more generally by N

e+

(g = gt is

t h e BA function corresponding t o

c l a s s i c a l r e l a t i o n s based on t h e d e f i n i t i o n of *r e - , w i t h 7 = e-exp(-ikx)

N

--f,

(11.23)

In f a c t

g- i s

. "

t

-

s Z 1 f + = s 11 f -

N-

plr

the BA function f o r

e,

N

t

N

yz1).

r

y

= f+ = e x p f i k x )

Further we have t h e ru via ( a * ) + Zzlet -- s l l

v

zi

N-

Y

sZ1e+ = s l l e - ; f - + s1 2f - =

Zl2.

In c o n s t r u c t i n g the generalized t h e t a functions O(sZ1) = det(1 t prsZ1 res t r i c t e d t o H2') i n [ MC101 and eventually representing t h e BA function a s a quotient of such t h e t a functions one deals w i t h "updated" sZ1 o f t h e form -+ . " %1 = s Z l ( ( w - k ) / ( w + k ) ) where w E upper half plane. Then in f a c t (&A) Ft(w) = )/o(F2,) and sllg-(w) = O(Fz1(wtk)/(w-k))/o(rZl ) . The proof of t h i s

@(Ti1

*;

i n [ M C l O I i s a n a l y t i c a l and q u i t e involved and only p r o p e r t i e s o f deduu h c i b l e from an f* d e f i n i t i o n ( i . e . ; + = Ttexp(-ikx) and $- = f - e x p ( i k x ) ) a r e

used i n t h e proof.

Clearly such t h e t a functions will be t a u functions i n

our determinant constructions ( s e e below) and we will gl've a d i f f e r e n t proof of (U) based o n vertex operator a c t i o n ( t h e proof i s more geometric in a

160

ROBERT CARROLL

sense and quick, modulo a l o n g p r o o f o f t h e WE by d e t e r m i n a n t methods as i n Thus l e t us observe t h a t i f one c o n s t r u c t s t a u f u n c t i o n s by determin-

57).

a n t methods a s i n 57 t h e r e i s a noteworthy r e l a t i o n between v e r t e x o p e r a t o r

on ~ ( x and ) "updating" o f t h e Fourier transform o f the kernel ( c f .

action

Thus f o r KP w i t h F as i n ( 7 . 7 6 ) and T ( x ) ¶ D(x,z) g i v e n by 2 G - ( k ) r ( x ) = T(xl-l/k,x2-1/2k ,...Ii s g i v e n v i a (see t h e p r o o f o f

57 and [ P Z ] ) . (7.78),

Theorem 7.14) G-(k)/xm nk)))ndni

...Lrnd e t ( ( F ( n .J, n k )))ndqi /xm .../," d e t ( ( G - ( k ) F ( q j , =

and f o r F(x,y)

G-(k)exp(S(x,p))

=

= exp(S(x,p))(l-(p/k))

n

F = sZ1 o r more g e n e r a l l y F(x,y)

( 1 / 2 1 ~ ) ~sZlexp(2ikx)exp(ik(y+z))dk z (11 2 4 ) (z'

X F(z,y)

f o r example.

t h i s i n v o l v e s (6.) Now f o r KdV we have

-

where x

zj+1 - Y 2 j t l one g e t s f o r example ( 6 6 ) Fz1(y+z) = =

tZ1 (y+z+Zx)

o r more g e n e r a l l y ( j 2 1 )

= ( 1 / 2 1 ~ ) 1 1sZ1e2 i k x$o(zyk)$o(Y,k)dk = F(zl+x,yl+x,z',y')

j 21); n o t e F(zl+x,yl+x

(zZjtl)for

%

S(y,q))dA

= ($o(x,k),$o(y,k))A

For sZl + sZlexp(2ikx)

f o r j 1. 1.

-

IA exp(S(x,p)

¶ . . . ) i s a d d i t i v e i n t h e f i r s t two

...)).

v a r i a b l e s and we w r i t e t h i s as F(yl+zl+2x,

Thus i n t h e H2+ c o n t e x t

,...) f(yl)dyl

= F x f ( z ) and we r e c a l l t h a t d e t ( l + F x ) =

d e t ( l + F X ) e t c . a s i n d i c a t e d above.

I n any event f o r KdV t h e v e r t e x o p e r a t o r

XF f ( z )

= 1: F(zl+xyyl+x

e q u a t i o n (VOE) above a p p l i e s and d e t ( ( z - ( w ) F ( n j y n k ) ) ) on TrFx = TrFX (F(x,y)

since 2 i

1k2jt1

(y2j+l

Hence %-(w)T(x) =

T-

a s above).

-

But

l/(2j+l)iw2jt1)

= E?(y,k)

+ log((l-k/w)/(~+k/w)).

corresponds a x a c t l y t o a t a u f u n c t i o n c o n s t r u c t e d v i a u+

determinants w i t h updated sZ1 = l/iw,x3+1/3iw3,

as above w i l l be based

yZl(w-k)/(w+k).

S i m i l a r l y C + ( W ) T ( X ) = T(Xlt

...) w i l l correspond t o a t a u f u n c t i o n based on

Thus we have p r o v i d e d a n o t h e r p r o o f o f formulas (6.)

k)/(w-k).

LHE0REl 11.3.

I n t h e c o n t e x t x2j+l

-

y2j+l

Ti1

= ?21(w+

and we s t a t e

f o r j 2 1 t h e formula ( W ) c o r -

responds d i r e c t l y v i a d e t e r m i n a n t c o n s t r u c t i o n s t o v e r t e x o p e r a t o r equations N

$,

U

exp(S)e,

Y -

= eXp(S)G-(W)r/r

and T$-

-

exp(-?)z-

= exp(-?)c+(w)r/-i

which

a r e known t o be v a l i d v i a r e s u l t s i n 17 f o r example.

3EI;IARK 11.4 done.

(HZR0CA FOR KdU ON EHE ClNE).

Thus f o r KdV t h e VOE a r e

L e t us r e c a p i t u l a t e what has been

HIROTA FOR KdV O N THE LINE

1 61

where x2j+l = s2j+1 f o r j 1 ( c f . 5 7 ) . The formula f o r c - ( k ) T / T above can be proved d i r e c t l y from t h e determinant constructions as i n 97 via updating based o n (11.25). The use o f Fx on H2+ a s i n [ M C l O I i s equivalent t o u s i n g Fx i n the dressing context. The kernel f o r dressing i s the basic F(x,n) = ($,(x,k),$o(n,k))A where the flow v a r i a b l e s a r e s,rl (s2j+l = n2j+l f o r j 2 1 ) and when x, appears i n T(XYX3y...) via a lower limit o f i n t e g r a t i o n a s i n (7.42) f o r example t h e o t h e r variables slynl a r e killed i n t h e i n t e g r a t i o n ( n o t the higher o r d e r v a r i a b l e s ) . Then g-(w) is passed under t h e i n t e g r a l sign a s i n t h e formulas preceeding (11.24). Thus we p r e f e r n o t t o t h i n k o f updating v i a exp(2ikx) or exp(2Y) i n [ M C l O ] b u t r a t h e r working w i t h a basic kernel F(x,y) = ( + o ( x y k ) , $ o ( y , k ) ) Ai n t h e dressing context and t h i n k i n g o f updating s21 a s vertex operator action on t h e r e l a t e d t a u functions. In this s p i r i t t h e ( s i n g u l a r ) t h e t a functions CI(;~~) e t c . of [ M C l O ] , defined function: once t h e flow dependence via determinants, a r e our determinant u!t Y i s i n s e r t e d v i a sZlexp(2S). Thus (6+)?21(y+z) = (y+z+Zx, ...) = ( 1 / 2 1 ~ ) Jm ~ ~ ~ e x p ( 2 f ( x , k ) ) e x p ( i k ( y + z ) ) d kcorresponds t o kerF 23( and a t the determinant -+

-m

level this involves d e t ( l + F x ) which i s our tau function. One notes t h a t t h e a d d i t i o n theory o f [ E31 f o r example is equivalent t o the hierarchy ( o r a -+ s u b s t i t u t e f o r t h e hierarchy) and involves t h e updating sZ1 -t sZ1 sZ1 so t h e r e a r e good grounds f o r this approach used i n [MClOI. Our development i n t h i s book i s m r e centered around t a u f u n c t i o n s , VOE's, and determinant methods, e t c . however and hence we have adopted t h e point o f view indicated. -+

Now l e t us combine t h e geometrical c o n s t r u c t i o n s o f Remark 11.1 w i t h the Grassmnnian o f Remark 11.2 above. What we do i s take the proof o f t h e Hiro t a b i l i n e a r i d e n t i t y in Remark 11.1, based o n loop groups over S1, and t r a n s p o r t i t t o t h e goemetry o f t h e Hardy spaces via A n % e n . We then introduce a formal residue c a l c u l a t i o n a t " m " which embodies t h i s geometry, so t h e Hirota formula will have t h e same appearance as before. Our cons t r u c t i o n will give e x a c t l y t h e same motions of W , H , e t c . a s i n a corresponding loop group theory based on S 1 . Thus t h e residue c a l c u l a t i o n i s a r t i f i c i a l b u t t h e geometrical f a c t s expressed through i t a r e genuine. Now l e t = e n t l 3 and vertex operator a c t i o n be expressed f o r KdV via

ROBERT CARROLL

162 Qw = (1

-

(Q E-(w)); t h i s i s the KdV version of t h e q

A/w)(l t A/W)-'

1

5

used

To s e e how this a r i s e s one can represent W 'L H as t h e graph of m -1. H2+ + H2( c f . [ MClO]). Thus consider = ( 1 - p ) ( 1 + r s Z l ) ( 1 + prsel) . f o r KP.

(Ht

Q

where p(1 t r s e l ) = 1 t prsZ1 on H, and (1-p)(l t rsZ1) = (1-p)

He')

--

Then w-w;' = ( l - p ) r s Z 1 ( 1 t prspl)-l i s the map m of [MClOI. Now uprs21 Y dating t o T2, = sZ1exp(2S) involves e + w i t h %; t s21et = sllg- (and JI, + ;

A

= exp(y)zt) while W +. W

where

Gt

= 1 t

prZz1 and

6- =

[l-p)(l

t

rrZl)_(note

s l l is unchanged here). The corresponding map i s m, = ( l - p ) r ? 2 1 ( l t p r s z l ) - 1 A A-1 and this should correspond t o G- = w-w+ of Remark 11.1. Now q i s simply 5 an " a r t i f i c e " to make -cr(q,)= $ w ( v l y c ) i n t h e KP theory ( s e e t SE1 I, Prop. 5.14). Note t h e "contrivance" involving q-' = ( ao bd ) i n (*A): b : z - + ~ 5-k 5 -1 'L l - A / c ) , a - l b : z m k 5-k so a - l b : f ( z ) f ( 5 ) : H- + H., For w - ( l ) 95

(95

-1

-

-+

-+

is b w - ( l ) = f ( 5 ) . I t i s simply a t r a n s f e r device; ;-(I) t h e c r u c i a l o b j e c t (1 +;-(1) E W). Note a l s o i f T ( X ) n, T(exp(1 x i z i ) ) then = f ( z ) one gets a

T(x-l/k,x2-1/2k2,

...)

'L

T(exp(1 z j ( x j - l / j k j ) ) = r ( g - q k ) . T h u s vertex opera-

t o r notation i s just another way of saying T W ( g q or a l t e r n a t i v e l y q i s 5 5 simply another way of expressing t h e vertex operator a c t i o n . For KdV we 1 Q = (1 - A/s)(l t A / < ) - ' Thus we w i l l have t o use ( b ) in the 2 ,context. 5 a b r e c a l l from Remark 11.1 the a c t i o n f o r q , = l - A / < (o and q i l from t h a t we obtain f o r Q = t h e formu1a s ) ¶

(ti)

5

(11.28)

v:

I-,-1 wSmA

S +.

Q

-N

-1 ,m / 5

Q

(i

+.

-1 Thus the 1 t A / < terms cancel and one has v v c a l c u l a t i o n s i n Remark 11.1 a r e a l s o s i m i l a r ) . T h e natural extension now to H2'

a)

A-lA b just a s f o r q

= a

seems to be to replace A : f

-+

5

(other

Af in Remark

11.1 d i r e c t l y by t h e i n f i n i t e s h i f t matrix A = ( ( 6 )) % ((6i,j-l)) with 2lJ $2t A n = ( ( 6 i y j - n ) ) r e l a t i v e t o a basis (say e n ) of L = H @ H2- (see e.g. [ C13;U3]

-

e n is given i n ( 1 1 . 2 2 ) ) .

Then we simply reproduce t h e theory of

Remark 11.1 ( f o r KdV now) and t r y t o f i r s t recover t h e formulas of [ M C l O ]

HIROTA ON THE LINE

163

with some interpertation (subsequently we derive a Hirota formula). Note t h a t the picture looks s l i g h t l y different since 1 $ H2'. b u t 1 i s simply replaced by eo. We observe directly t h a t r - ( w ) ( l + r?21) % 1 + rs21 so for-

*+

mal l y

1 Based on basis vector action i n the S context above we write Qw Then (1 + n/w)-' 'L

z-(w).

=

(1

- n/w)

as i n (u).Now acting in H2+, for G- = (l-p)r?21(l + pryzl)-' one has F* n r l = (1 + pr~21)-1pr?'21 (acting in H - ) . T h u s the BA function g+ o f H = W i n rv [MClO] can be formally expressed as $+'L 1 - w* b u t we only work w i t h t h i s expression i n an operator context. FIow according t o the recipe o f Remark 'L w(1 - ( l / w ) e b W - ~ , ~ W - ~where ) however the dual11.1 one expects $,(?,w) 1 i t y on S played a role (;r = n-' e t c . ) a n d we will a v o i d this recourse t o i 1 : Tjpmemso G:(e-l) = below. Note also ;-(em) = 11, wSmeS implies G:(e ) = 1

-

P

1: G-l

,,,em. Now as mentioned in Remark 11.1 one makes a careful distinction between x as a position variable and 5 or rl as a complex number; t h u s in = (1 +G-)(Ao)IA+g f o r example f a c t Gb,(y.5) i s defined as (1 + ; - ( A o ) ) ] A-+5 with cWl(;,rl) = (1 - E:)(A-l)lA-+rl. Therefore we define now +,n = (l/w) - l ~ ~ -m l , m ~ n To avoid an a r t i f i c i a l unit c i r c l e here in dealing with w we introduce our formal ( a r t i f i c i a l ) geometry a l i t t l e differently a s follows. From (11.22) = -en-l (exercise) so we have immediately ( W ) (11.31)

$ H ( ; , ~ ) = (1

A

-

;:)(e-l)le

en

-

-det(l + p -1 vw-)

-

=

-GH(qYw)

Then for orthogonality expressed v i a an integral over C (an arbitrary c ircle a t " m " now) one can simply write (here y1 i s the same) (+A) JC $bJ(yl.w)$, n A A via ( W ) ( r l y w ) d w = 0. This will be consistent w i t h $ y ( y l y w ) J-$H(;l.w)

ROBERT CARROLL

164 AI

+ ;-)(eO),(l - w*)(e - - 1 ) ) = 0N (note Ej_(eo) E H- and G*(e - - , ) E H t ) . Thus writing jHfor the operator 1 - w_* = 1 - (1 t prFz1 )-'przZl , jH($,wJ o r 5, ((1

A

(v^,w) ar e formed from $H(e -l ) or $,(e-,) by replacing en by wn (with a minus X sign adjustment for ILH(yA'w) explained below). Note here (enye,) = 6mn be( en , em ) % IC en em (w)dw = -IC e n e -in-1 (w)dw = comes in o u r w geometry a t -Ic wn-m-ldw so replace dw by -dw/Zin and take \LT ( C a n a r b i t r a r y c i r c l e a t Hence The w geometry i s thus compatible with ( , ) in H2+ @ H2-. ll-lly

'l-").

-

3, = 1 - GT

-

1 - (1 t pr521)-1pr?21 a n d $,(e-,) -det(l t II-''JW-) for e n w n (y, % exp

One writes

CltE0REUl 11.5.

%

-m-1 ) = W-l,,,w Similarly jW = 1 t 31 + (l-p)rFel (1 + prFzl)-1 and j w ( e 0 ) + (2:)). S -1 at ( Y , , w ) = (1 + X ) ( e 0 ) l e n n n-m-1 = 1 + ws,o w (same y , ) . W i t h geometry dw 'I, (en,em)we have ( l / Z a i ) I j ( $ ,w) "mil determined by -(1/2ai)IC w 83 C H 1 ~ , ~ ( y ~ , w ) =d w0 (note a l/w factor as in (11 .8) or (11 .20) has been removed H, y; W, with @w = $wexp(r(t,w)) by our construction). For diffe re nt y1 et c ., the requirement ~1~ E W a n d $H E H = W'leads t o a formal Hirota formula ( c f . (11.20)) JC ~ w ( y , w ) ~ ~ ( v * ' , w ) e x p ( S ( t - t ' , w ) ) d w = 0 ( y % y1 e t c . ) . -?H(?YW)

-

=

10

OD

-f

?w

I-,

~

A

Q

Q

-

This approach revolves around the underlying vertex operators as fundamenThe contour integral in Theorem 11.5 i s a formal expression i n the w geometry of a genuine perpendicularity W I H. One i s n o t asserting anything a b o u t asymptotic expansions i n a NBH of ( t h i s question, o f asympt o t i c s in Imw > 0, IwI + -, deserves further study however). One notes also t h a t s l l i s fixed a n d determines the "isospectral" manifold in the s p i r i t o f 53,4 while sZ1 (or phase s21) varies. In t h i s s p i r i t o u r formulas o f the form (7.56) or (7.65) giving a ( k ) = l / s l l ( k ) in terms of T provide n o t only a cr i t e r i o n for "spectrality" b u t also a determination of the isospectral manifold in terms of the t a u function.

tal.

OD

-

tH(;,w)

In order t o r e l a t e our functions f o r example with G+(w) we note f i r s t t h at det(1 t prs21) for Q corresponds to det( l+F x) = det( l+F X ). Referring t o Remark 11.1 f o r guidelines with W = (1 + r s )H2+ a n d @ = det(1 + A+ "6 p r s z l ) we have G-(w)(l t r z Z l ) % 1 4- rszl a n d w - G - ( w ) w For '?-(w) % !) 1 ' , Qw w e would expect 1 + p r z i l = a(1 t a-'b(l-p)r?21(1 f prs"21)-')(l + a -.+

h

N

PrTz1) so (11.32)

det(l+pr?21)

= O(zll)

=

detao(?21)det(l+a-1b(l-p

(t

ALGEBRAIC CURVES AND KP

-

165

-

-1 where t h e l a s t d e t e r m i n a n t PI, d e t ( 1 + P vw-) = $H(?yw). 321 i n l o l v e s ‘I -1 ( y ) i n t h e s p i r i t (+A) y; 1-W PI, y;16H2+

*

C o n c e p t x a l l y (11. ( ao db ) ( G w+- ) H2+

Q

(aw+tbw, 2+ ( c f . (11.12) e t c . ) . B u t here Y’; 5-c~)so we a r e t h i n k i n g o f )H dw N rW (yy2) w h i c h i n v o l v e s G+(w) and T + ( x ) . By v i r t u e o f Theorem 11.5 and ReI mark 11.1 t h i s i s a t a u f u n c t i o n a s s o c i a t e d w i t h GI = H and so i t i s n a t u r a l

-

Q

-

f o r $,(?,w)

t o a r i s e i n (11.32).

-det(l-A/w)(l+A/w)-’ diagonal.

%

Thus

EHHEOREl’l 11.6.

We n o t e a l s o t h a t t h e f a c t o r deta = d e t ( a )

1 s i n c e A = ((6iyj-l))

The formula ?,,(?,w)

= O(;i,)/O(?21

(11.32) and s i n c e t h e same 0 q u o t i e n t g i v e s 11.3) we can i d e n t i f y

12. ACGE%BRAZC ~ 7;SH1,2]

i s upper t r i a n g u l a r w i t h 0

g+(w)=

-

) f o l l o w s f o r m a l l y from

c+(w)

( b y [MClOI and Theorem

$H(?,w).

~ AND~KP. €We g$i v e here a s k e t c h o f r e s u l t s o f [MS1,3,5-

on r e l a t i o n s between a l g e b r a i c curves and KP ( c f . a l s o [MS2,4;IM1,

2;AR3,4;Mu2;DU9;N041).

I n p a r t i c u l a r one g e t s an i d e a o f how t h e S c h o t t k y

problem o f c h a r a c t e r i z i n g t h e t o r i c o r r e s p o n d i n g t o Jacobi v a r i e t i e s o f

I w i t h some i n j e c -

Riemann s u r f a c e s i s s o l v e d v i a KP.

We m a i n l y f o l l o w [ M S l

t i o n o f m t e r i a l from [MS5,6;MU21.

One s h o u l d c o n s u l t IMS4-61 f o r connec-

t i o n s o f K r i z e v e r data t o curves and Grassmannians and t h e c l a s s i f i c a t i o n o f commutative a l g e b r a s o f d i f f e r e n t i a l o p e r a t o r s .

We g i v e a few b a s i c d e f i n i -

t i o n s and i d e a s about a l g e b r a i c geometry f o l l o w i n g

HAl;MU2,4,5;GRl

t h e r e i s no space t o g i v e a g r e a t deal o f p e r s p e c t i v e .

I but

Thus t h e t r e a t m e n t

i s o f n e c e s s i t y somewhat s k e t c h y and many d e t a i l s a r e o m i t t e d .

The m a t e r i a l

i s however so i m p o r t a n t t h a t we f e l t such a n incomplete t r e a t m e n t was b e t t e r t h a n none; we have t r i e d t o p r o v i d e enough m a t e r i a l t o g i v e a good idea o f what i s going on and e n t i c e t h e r e a d e r t o c o n s u l t t h e r e f e r e n c e s f o r more (see a l s o Appendix B f o r m a t e r i a l on commutative algebra, sheaves, schemes, etc.).

As f o r a l g e b r a i c geometry t h e r e d o e s n ’ t seem t o be any q u i c k way t o

c o v e r t h e scheme language and pathology.

Thus we o n l y have e x t r a c t e d some

minimal c o l l e c t i o n o f i n f o r m a t i o n from [ HA1 ;MU2,4,5;FUl L e t us make a few remarks here based on [AR4;MS4;SH2].

;MXl

I.

Via K r i z e v e r one

knows t h a t t h e t a f u n c t i o n s f o r Jacobians s a t i s f y t h e KP equations i n H i r o t a b i l i n e a r form and Novikov t h e n c o n j e c t u r e d t h a t Jacobians can be c h a r a c t e r i zed by t h e KP e q u a t i o n s .

Versions o f t h i s were proved by Shiota, A r b a r e l l o -

166

ROBERT CARROLL

-

deConcini, and Mulase ( c f . IAR3,4;MS1-7;SHl,21 Mulase's work involves t h e KP system a s shown i n this s e c t i o n ) . There i s some discussion o f a l l this i n terms o f h i s t o r y , motivation, p r i o r i t i e s , and o t h e r work i n [ AR4;SH2;MS4] ( c f . a l s o [DU9;GU5;NO4;WLl;MU21 f o r o t h e r work). We a r e not concerned w i t h h i s t o r y or p r i o r i t i e s here, o r i n comparing various mathematical refinements. The approach of Mulase has immediate v i s i b l e connection to " s o l i t o n mathemat i c s " and we will t r y t o convey t h e s p i r i t o f this w i t h enough d e t a i l to exh i b i t the e s s e n t i a l f e a t u r e s and make things believable. In a sense t h e main point for us i s t h a t KP has (another) purely a l g e b r a i c content ( i n some way c h a r a c t e r i z i n g Jacobians); we have already seen (mathematical ) o r i g i n s o f KP i n KM algebras and o r b i t s o f t h e vacuum. The "miracle" revolves around around this purely mathematical content and i t s eventual r e l a t i o n s to physical content and o r i g i n s i n water waves, s t r i n g s , 2-D q u a n t u m g r a v i t y , Hamiltonian mechanics, e t c . In an introductory book such a s t h i s we wish t o dwell more on this miracle r a t h e r than deal exhaustively w i t h mathematical refinements. In p a r t i c u l a r we a r e not concerned w i t h reviewing who proved what about t h e Schottky problem or t h e Novikov conjecture b u t will be q u i t e content i n showing why t h e r e should be a connection between a1 gebraic curves and KP equations. We have s e l e c t e d what seems t o be a quick and natural a p proach following [MSl-71. Let us summarize the contents of this s e c t i o n in advance. T h u s l e t K be a n a l g e b r a i c a l l y closed f i e l d ( K C is f i n e for our purposes) and l e t B be a commutative subalgebra of d i f f e r e n t i a l o p e r a t o r s containing a monic Ln o f order n . Then ( t a k i n g e.g. R = KL[xl] and Rc 2, c o n s t a n t s ) one has an i s o morphism B 2 A C Rc((i3-')) where A does not contain negative order elements 1 (powers of i3-l). One w r i t e s A E 4 f o r such A a n d shows t h a t A = S- BS f o r S a gauge operator. Then some r e l a t i o n s t o isospectral deformations L n ( t l , t 2 , ...) o f Ln(0) and Lax operators a r e given ( L ( t ) = L n ( t )l / n - L ( t ) will sat1 i s f y t h e KP equations). The Birkhoff decomposition i s e s t a b l i s h e d and H ( A ) 1 1 is r e l a t e d t o a curve C = Proj(GrA) via H (A) = H (C,OC) (here we spend some 1 time explaining what t h e r e t h i n g s mean). In p a r t i c u l a r dimH ( A ) < i f and 1 hijaj, a = only i f rank(A) = 1 . One has a l i n e a r map f : H ( A ) -+ T ( h i = a / a x , a r e basis elements and t 1 h i j y i , y i a r e coordinates r e l a t i v e t o j 1 h i ) and t h e isospectral deformations L n ( f ( y ) ) can be defined o n H (A). Now Q

DEFORMATIONS

167

g i v e n A one defines XA = {gauge operators S such t h a t SAS-' C D = Rial} and t h e c o l l e c t i o n o f corresponding L = 9 S - l i s denoted by YA; (FA,T)is the KP system defined by A E A . An o r b i t (yA,T)is A maximal i f i t is not "determined" by A' 3 A, A' E A. One s e e s , f o r A E A, t h a t a maximal o r b i t M A 1 % H (A) l o c a l l y and t h e motion i s l i n e a r r e l a t i v e t o t h e l i n e a r s t r u c t u r e of H 1 ( A ) . Moreover for every o r b i t M o f KP t h e r e is an AM E A such t h a t ( X A , T ) contains M a s an A maximal o r b i t . Thus o r b i t s of KP elements A E A . F i n a l l y f i n i t e dimensional o r b i t s A correspond t o open s e t s in t h e genera l i z e d Jacobian Pico(C) ( C g s above) and s i n c e ( s u i t a b l e ) curves give r i s e t o a1 gebras A E A one has a correspondence between Jacobians and KP o r b i t s . In p a r t i c u l a r abelian v a r i e t i e s can be Jacobians i f and only i f they correspond to KP o r b i t s . This can be phrased i n terms of tau functions and theta &

Q

Q

functions and gives a s o l u t i o n to a version of t h e Schottky problem.

W e will t r y to explain a l l t h i s following [MSl-71. One notes t h a t the emphasis i s not on t h e Schottky problem a t a l l ; t h a t a r i s e s as a consequence b u t n o t a t a r g e t . The emphasis i s on commutative algebras o f d i f f e r e n t i a l o p e r a t o r s and a complete r e s o l u t i o n of t h e c l a s s i f i c a t i o n of such algebras i s given in[MS7]. T h u s i n p a r t i c u l a r t h e s p i r i t i s q u i t e d i f f e r e n t from t h a t of [AR3,4;SH1,2]. RmARK 12. I (FORI7AC PIE11DBDZFFERMtZAL QPERAC0W = PSDO, DEFQRFtACZ0ll.5, AND GAUGE OPERACOM)t We begin w i t h [MSl I. Let K be a f i e l d of c h a r a c t e r i s t i c 0 and a l g e b r a i c a l l y closed f o r convenience (one can t h i n k of K = C f o r our purposes). Let KIrx]] = formal power s e r i e s ring, K ( ( x ) ) = quotient f i e l d , R = commutative d i f f e r e n t i a l algebra over K with u n i t 1 such t h a t ( 1 ) f E R implies t h e r e e x i s t s g E R such t h a t ag = f ( 2 ) f E R implies 1 ( l / n ! ) f n E R ( f o r our purposes one can think of R = K [ [ x ] ] w i t h a = a / a x ) . The cons t a n t s i n R a r e denoted by Rc ( a f = 0 ) . Let D = R [ a I = { 1 pn a n , Pn E R , n > 0, f i n i t e sum}, E = R ( ( a - l ) ) = p a n , pn E R, n bounded above}. Note -qP

{I""

t h e Leibnitz rule (*) avf = 1 ; (:)(alf)?-i gives an a s s o c i a t i v e algebra s t r u c t u r e . Let EV = elements i n E of o r d e r i v y E = D B E - l , and Rc((a-')] = constant c o e f f i c i e n t operators i s a maximal commutative K subalgebra of E. Let B c D be a commutative K subalgebra with u n i t ; assume B contains a t l e a s t one monic ( t o p o r d e r c o e f f i c i e n t = 1 - i n general t h i s i s no problem

168

ROBERT CARROLL I n f a c t we can assume (A) B c o n t a i n s an element Ln = an t bn-2

t o achieve). an- 2

+

... + bo.

Now l e t B C D be a comnutative subalgebra w i t h 1 and a monic (A).

Then

such t h a t ( 0 ) A n R [[a-’ 1 I - a - l $1 10) and a K a l g e b r a isomorphism B -+ A. The n o t a t i o n A n ;R a = COI means t h a t A has no n e g a t i v e o r d e r elements and we denote by A t h e s e t o f u n i t a r y

t h e r e e x i s t s a K subalgebra A C Rc((a-’))

K algebras A

+

t umla-’

1

+

K((3-l)) satisfying

C

.... By (7.4)-(7.5). -1 t ... E G = 1 t E

(0).

The p r o o f l o o k s a t L = (Ln)l’n

To see t h i s n o t e f i r s t Ln = Sans-’

Q =

Set S-lPS = (cf. p A

[MS5]).

1;

a

.

I = 0 which i m p l i e s A c Rc((

P E B, [ P y L n ] = 0, so we want t o show [S-lPS,B

am’)).

=

e t c . one knows t h e r e e x i s t s S = 1 Set A = S - l B S and f o r such t h a t L = SaS’ Lemma 7.1,

so [P,L

qmaN-m and w r i t e o u t [Q,anl

n

I

= 0 =

= 0 s [S-’PS,an1

= 0.

1; zy (?iqji)aN+n-m-i -1; (?)qp-i ( q i = q ( i ) ,

This g i v e s a r e c u r s i o n formula n q ’ = -1 P-1 = 0 f o r a l l m . Hence [S PS,a 1 = 0 f o r a l l P E B so

1 ) from which q; C

Rc((a-’)).

Since t h e isomorphism S-lBS does n o t change t h e o r d e r o f op-

e r a t o r s i n E, A has no n e g a t i v e o r d e r elements. T h i s shows t h a t s t u d y i n g commutative K a l g e b r a s (we w i l l a l s o s i m p l y say a l g e b r a f o r K a l g e b r a ) i n 0 can be reduced t o s t u d y i n g subalgebras i n Rc(( i 3 - l ) ) h a v i n g no n e g a t i v e elements.

Now i f A

C

K ( ( a - l ) ) as above one wants

t o go t h e o t h e r way and f i n d S such t h a t SAS-I C D. mally (cf.

T h i s can be done f o r -

[MSl ]) and i n v o l v e s t h e c o n s t r u c t i o n o f an L =

a

t u-la-l

...

t

where f i n i t e l y many n o n l i n e a r ODE h o l d on f i n i t e l y many u - ~ , U - ~ , . . . , U ( f o r some m ) . tion.

-m+l One must however t h e n show t h a t such equations have a s o l u -

T h i s i s perhaps b e s t approached d i f f e r e n t l y ( c f . [ M S l

I). Thus t h e

passage from KP o r b i t s t o a l g e b r a s B and A as above w i t h B = SAS-’ be c a r r i e d o u t below.

To go t h e o t h e r way (A

-+

c u r v e s C = Proj(GrA) as i n d i c a t e d below and t h e n t o a l g e b r a s ;ever

techniques ( c f . [MU21).

C D will

KP) one can go from A t o

B

C

D via K r i -

I n t h i s manner one would a r r i v e a t KP equa-

t i o n s and t h i s i s e s s e n t i a l l y developed elsewhere i n t h e book. L e t us go t o t h e idea o f i s o s p e c t r a l d e f o r m a t i o n o f d i f f e r e n t i a l o p e r a t o r s . An i s o s p e c t r a l d e f o r m a t i o n o f Ln as i n (A) s a t i s f i e s ( 6 )

a L = [Z(y),Ln(y)I y n

f o r some Z E 0.

L e t us use tn as d e f o r m t i o n parameters w i t h R = R“tl,t2,

...11,

E = ~ ( ( a - ’ ) ) , E = D Q E - ~ , and T = ( t l

D

= RVI,

,...I

with a

%

a

Y‘

DEFORMATIONS

169

...

= a + u 3-l + (L a x o p e r a t o r ) s a t i s -1 i s c a l l e d a Lax p a i r i f [ P , L l EE-’. i s a Lax p a i r s i n c e [ L:,L 1 = - [ L r , L l E [E-1 ,€ 11 C E - l g i v e n

A t t h e € l e v e l we work w i t h L = L;In

a L

fying

= [Z,Ll

Y Note ((Ln)+,L)

(y

%

t h e form o f L (Ln = L: if(P,L)

(P,L)

tl).

One shows e a s i l y (by i n d u c t i o n on o r d e r ) t h a t

t Lr).

i s a Lax p a i r t h e n P i s a l i n e a r combination o f Ln w i t h c o e f f i c -

i e n t s i n Rc = i f E

R; a f

=

n anL = [ L + , L ]

(12.1)

01.

The KP h i e r a r c h y i s t h e g e n e r i c system

(n ‘2)

Here one i s i n d e x i n g t h e h i e r a r c h y v a r i a b l e s as tl = x o r y w i t h

2

I n p a r t i c u l a r (+) ( 3 / 4 ) a * ~ - ~= KP equation.

( n o t e L2

%

If a condition L

ak

%

a/atk.

- %a u - ~- 3u-l au-l ) i s t h e c l a s s i c a l D i s imposed one g e t s t h e KdV s i t u a t i o n

(a3u-l

2

E

a 2 + Z U - ~ and t a k e n = 2 i n (12.2) t o g e t KdV).

Similarly L

3

E

D g i v e s t h e Boussinesq e q u a t i o n e t c . To g e t a d e f o r m a t i o n f a m i l y one s o l v e s (12.1) as i n Theorem 7.4 f o r example t o g e t L ( t ) and Ln = Ln.

One t h e n wants t o e l i m i n a t e i r r e l e v a n t d e f o r m a t i o n

2

parameters (e.g. f o r KdV even numbered tn a r e i r r e l e v a n t s i n c e aL / a t z n = 2n 2 [ L + ,L 1 = [LZn.L21 = 0 ) . Now l e t B be t h e s e t o f d i f f e r e n t i a l o p e r a t o r s commuting w i t h L, fore (S E 1

+ E-’

E

D; t h e n A

= S-lBS C K ( ( a ” ) )

s a t i s f i e s LA’n

can be c o n s t r u c t e d a s b e f o r e Note t h a t A has c o n s t a n t co-

= L = SaS-’).

e f f i c i e n t s w i t h no n e g a t i v e o r d e r elements and t h e c o n s t r u c t i o n o f A does not

r e q u i r e B t o be commutative (A = S-lBS,

But A i s commutative, so f o r blyb2

E

etc.),

B w r i t e [bl,b21

[a, ,a2 IS-1 = 0, and t h i s imp1 i e s B i s commutative.

o n l y t h a t [B,Lnl = [SalS-1,Sa2S-1

= 0.

I= S

It t u r n s o u t t h a t t h i s

a l g e b r a A w i l l determine an e f f e c t i v e complete f a m i l y o f i s o s p e c t r a l d e f o r mations o f Ln.

Here i f Ln(ti)

i s a f a m i l y o f i s o s p e c t r a l deformations o f

Ln(0) t h e parameters (ti) a r e c a l l e d e f f e c t i v e i f ( m ) t h e l i n e a r map from t h e t a n g e n t space To(Km) (m <

m

o r m = -) t o E - l d e f i n e d by

( t = 0) E E-l i s i n j e c t i v e ( L = L!,In

as b e f o r e ) .

ai

.+

aL(t)/ati

The i d e a o f i s o s p e c t r a l

d e f o r m a t i o n above v i a KP equations (12.1) i s based on a n a l y t i c ideas o f spectrum e t c . i n 11-7 b u t can i n f a c t be r e l a t e d t o SpecB o r SpecA ( a n a l ge bra ic idea )

ROBERT CARROLL

170

Let us r e c a l l a s u l t s from 57 following [F?S1-71. In p a r t i c u l a r w i t h D a s E-l, and C = 1 + E-l ( t h e t variables a r e present i n D , E , C ) v av: ; v E Z; t h e r e e x i s t s N such t h a t o r d e r ( a v ) > v-N, f o r

REmARK 12.2 (CHE 3IRKH0FF DEC0CtP0$IC10N),

{loa and

5'

few f a c t s and reabove, E = D @ we s e t 2 = A all v>zOl, D =

{I

"X

same order r e s t r i c t i o n on a"}, E = {A E $, A(0) E C, A - l E El, = {A E D^, A(0) = 1 , A-l E $ 1 . Then one has the Birkhoff decomposi-

a';

t i o n expressed via

zx

^Dx

= C.3' w i t h 6 n = 1 . T h u s given u E CHDREI 12,3. a unique S E G and Y E $' such t h a t U = S-lY.

f x there

exists

Proof (following [MS2,31): F i r s t i n a purely formal way, t o f i n d SU = Cx one solves (SU)- = 0 f o r unknown S via

YE

This leads t o a system of equations ( k = - 1 , - 2 , . . . )

- m i Then l e t M = ((lo( i ) a ukti-,,)) = ((Mkm)) ( m , k = - l , - Z , . . . ) and note t h a t because o f the order conditions t h e Mkm determine a n element i n R ( e x e r c i s e ) Since U(0) E 6 t h e ( k , m ) component Mkm(0) i s 1 i f k = m and 0 i f k > m . T h u s M(0) has zeros below t h e diagonal a n d i s i n v e r t i b l e so one can solve (12.4) A a -1 A M? = - U w i t h S = - M U. QED

We r e c a l l a l s o t h e equations (note we use S here f o r P i n 1 7 ) (*) LS = 3, m t = -L_"S, Z = 11 L n d t n y Zc = -1; L_"dt dL = [ Zc,L] = [ Z , L ] , dS = ZcS, dZ = Z A Z, dZC = Zc A Zc, Z = SnS-' + dSS- , R = 1 ; a n d t n , dU = nU where U = 1 S- Y a n d Z = dYY-', and U ( t ) = exp(R)U(O). The notation Zc = Z; i s a l s o used i n comparing notation.

anS

ni

Let us now look a t some algebra followi n g [MS1,3,51 ( t h e o u t l i n e f o r some of this goes back to [MUZ]). The idea i s to p i c k Ln "* L n a s i n Remark 12.1; l e t B C D = t h e commutative subalgebra o f operators P such t h a t [ P , L n l = 0, S E G such t h a t L = 3 S - l , a n d s e t A = REfitARK 12.4

(C0lXWCAC1VE A L G E 3 W ) .

COMMUTATIVE ALGEBRAS S - l B S so A C R c ( ( a - ' ) )

such K a l g e b r a s d e r terms.

w i t h no n e g a t i v e o r d e r terms.

One wants t o check t h a t such A

N; t h e r e e x i s t s P

E

E

basis i n question (exercise

-

NA = r ( N P,Q E A

FA)

.

# K have transcendence degree 1

A has a K l i n e a r b a s i s w i t h i n d i c e s i n NA = { n

A such t h a t o r d ( P ) = n } .

ement Pn o f o r d e r n f o r e v e r y n there exists r

-

€

NA w i t h P = l. Then these Pn formthe 0

c f . [ MS1

I). Next ( c f .

[ MS1,5])

N, r = rank(A), and a f i n i t e subset FA

U

to}.

Now t a k e r

Then NA C r N .

f o r each A E A

N such t h a t (*A)

1 (r = 0

2,

A = K ) so A has a t l e a s t

L e t ordP = m and o r d Q = n w i t h r = GCD(m,n). E

Set

NA f o r e v e r y k >> 0 ( s u f f i c -

Thus p i c k a,b > 0 such t h a t r = am

i e n t l y large).

C

This f o l l o w s b y d e f i n i n g r ( A ) = m i n GCD(ordP,ordq),

m = m ' r and n = n ' r and one wants t o show k r E

To see t h i s p i c k a monic e l -

E

one element o f o r d e r > 0.

rp

L e t A be t h e s e t o f a l l

K ( ( 3 - l ) ) ( t a k e K = Rc = C ) w i t h u n i t and no n e g a t i v e o r -

C

F i r s t any such A

o v e r K. E

A

171

-

bn and one shows t h a t

1. bm'n' ( c f . here a l s o Remark 12.6). To see t h i s w r i t e (**) cm' t s ( 0 5 s i m ' - Euclidean d i v i s i o n a l g o r i t h m ) . Then r p =

NA f o r p

p = bm'n' t

-

r b m ' n ' t rcm' t r s = bm'n t cm t s(am bn) = ( c t as)m t bn(m'-s) = ord ( pc a' S , (m ' s ) b ) E NA. D e f i n e now FA = Cn E N such t h a t r n $5 NA} and i t f o l l o w s t h a t (*A) holds ( n o t e c a r d ( F A ) < b m ' n ' ) . Q

-

Now we can s t i p u l a t e Rc = K = C f o r o u r purposes, and f o r 3

z w i t h a-'

and K ( ( a - ' ) )

with K((z)).

Then A

C

%

a Z we i d e n t i f y

K ( ( z ) ) and q u e s t i o n s o f t r a n -

scendence degree e t c . become r e l a t i v e l y simple. (see e.g. [ FU1;HGl;MQl c f . a l s o Appendix B ) .

Thus l e t V = C ( ( z ) ) and Vn = C [ [ Z ] ] - Z - ~ so Vntl

determines a f i l t r a t i o n (and t o p o l o g y ) i n V .

Set An = A

n Vn ( n o t e Vn

I and 3 Vn 2,

{aan t ... I ) and e v i d e n t l y f o r p 2 bm'n', dimArP/A(P-l)r = 1. But s i n c e we used o n l y P and Q i n t h e argument one has a l s o dimC,P,QIPr/CIP,Q1 ( P - 1 ) r = 1 (C[P,QI"

= C[P,Q]

n Vn and C[P,Q] i s t h e subalgebra o f V generated b y P , Q , l ) .

Hence dimA/C[P,Q] 5#FA < bm'n' so t h e transcendence degree d o f A o v e r C = K i s a t most 2 . Note i f dimA/B = n p i c k u E A/B so t h a t 1, u, u 2 , un

...,

must be l i n e a r l y dependent.

...

t

Hence t h e r e e x i s t s bi

bnun = 0, so u i s a l g e b r a i c o v e r B.

E

B such t h a t bo + b,u

+

Since u i s a r b i t r a r y , A i s a l g e -

b r a i c o v e r B and f o r B = C[P,Q] t h i s means A i s o f t h e same transcendence degree a s B o v e r C, namely d 2 2 .

Next we show t h a t i n f a c t P and Q s a t i s f y

a polynomial r e l a t i o n so d = 1 . To see t h i s assume Q E K [ P l ( i f Q E K [ P l we a r e f i n i s h e d ) . L e t ul, ..., u be a K l i n e a r b a s i s f o r K [ P , Q l r ( b m ' n ' - 1 ) 9

ROBERT CARROLL

172

= P Q ( m ' - s ) b f o r p 2 bm'n' ( c f . (*@)). Then {u l , . . . , u q ) U P Cvpl ( p 2 b m ' n ' ) gives a K l i n e a r basis f o r K [ P , Q l . Since none o f t h e v P can contain only f i n i t e l y many powers of P a r e powers of P and u 1 , . . . , u N q some h i g h power P must be represented by a n o n t r i v i a l l i n e a r combination of b a s i s vectors u 1 , . . . , u q Y a n d V ' S . B u t this i s a nontrivial polynomial r e l a -

and set v

P t i o n between P and Q, a n d K [ P , Q l has transcendence degree 1 over K. quently A also has degree 1 over K.

REflARK 12.5

(ALGE3RAIC CURVE$).

curve now w r i t e ( * 6 ) GrA = 8;

Conse-

In order t o define a n appropriate a l g e b r a i c

(GrA),, An

(GrA), = 8; ( A p / A p - l ) X n - p , where A is a transcendental element of order 1 so a l l elements in ( A p / A p - l ) A n - p have degree n (note (GrAlo = A. = K = C and (GrA), 2 An v i a X = 1 - s p e c i a l i z a t i o n ) . We can t h i n k o f a valuation, namely degree, w i t h deg(A) = 1 , deg This will be P = o r d P f o r P E A of order > 0, and deg(c) = 0 f o r c E C*. useful below. Let us remark here t h a t f o r K = C a l o t of t h e a l g e b r a i c machinery can be simplified b u t schemes and sheaves s h o u l d s t i l l be included and will a l s o give a f l a v o r of t h e general s i t u a t i o n ( c f . [MS1,4,5;MUZI f o r extensions and d e t a i l s ) . This m i g h t a l s o serve a s a window i n t o t h e world of a l g e b r a i c geometry. We will show (following [MS1,51) t h a t C = Proj(GrA) i s a reduced i r r e d u c i b l e complete a l g e b r a i c curve over K = C ( o f genus #FA, not proved). Further t h e r e i s a smooth K r a t i o n a l point ^p on C such t h a t C = SpecA U {;I. Thus C is the 1-point completion of the a f f i n e curve SpecA. ( c f . Appendix B y Remark 12.6, and r e f e r e n c e s ) . Actually we will not prove a l l the d e t a i l s b u t will r e c a l l the appropriate d e f i n i t i o n s and sketch some main ideas of s t r u c t u r e and proof; then h e u r i s t i c a l l y a t l e a s t one should be a b l e t o see w h a t is going on. =

As mentioned before some of t h e background f o r various constructions i n [MS 1 , 4 - 7 1 l i e s i n [MU2,5] and t h e r e i s a nice account of h e u r i s t i c matters i n [MS1,4,5;MU21. There seems t o be no point in t r y i n g t o give an e l a b o r a t e rigorous discussion of C = Proj(GrA), curves, schemes, e t c . following e . g . [ H A 1 1 b u t some of this i s sketched i n Appendix B. Here l e t us sketch t h e Thus ( c f . ( * & ) ) GrA = @ A: and An 'L CP relevant constructions b r i e f l y . E A; o r d P 2 n ) w i t h A' = K = C , a n d An 2 @(AP/AP-') so A n - AR (Ao = K ) . We know A has degree 1 over K ( i . e . is f i n i t e over C[Pl f o r some monic P), so

ALGEBRAIC CURVES

(and as i n d i c a t e d l a t e r C w i l l be 1 dimensional

GrA i s f i n i t e over C [ I , P I

Now f o r Proj(GrA), l e t , i n a more g e n e r i c sense, B be a graded

over K).

r i n g , B = @;

Bc

C

Bd+c,

Bd, where Bd c o n s i s t s o f homogeneous elements o f degree d, B d etc.

L e t B+ =

An i d e a l A

C

B i s c a l l e d homogeneous i f A = @d>O (A

n Bd).

Bd (B+ i s c l e a r l y an i d e a l ) and P r o j ( B ) ={homogeneous prime

i d e a l s P which do n o t c o n t a i n a l l o f B ,, i n B, V(A) = I P

n V(Ai)

173

E

P 2 A}.

Proj(B);

P

0 B+).

For A a homogeneous i d e a l

Since V(A3) = V(A) U V(3) and

V(1

Ai)

=

t h e c l o s e d subsets o f P r o j ( B ) a r e taken t o be s e t s V(A) ( c f . AppenFor P

d i x B).

E

P r o j ( B ) l e t Bp be t h e elements o f degree 0 i n t h e l o c a l i z -

T i s t h e m u l t i p l i c a t i v e system o f a l l homogeneous eleme n t s i n B n o t i n I?. For U C P r o j ( B ) open 0(U) i s t h e s e t o f f u n c t i o n s s : i n g r i n g T-lB where

U

-+

(u means d i s j o i n t u n i o n ) ,

UB,

s(P)

E

Bp,such t h a t s i s l o c a l l y a quo-

t i e n t o f elements o f B (as i n t h e c o n s t r u c t i o n o f SpecA i n Remark B9). (Proj(B),0)

i s a scheme.

One checks t h a t t h e s t a l k 0p ?. Bp.

E B+ i s homogeneous l e t D + ( f )

= tP E Proj(B);

P r o j ( B ) and such open s e t s c o v e r P r o j ( B ) .

4

f

PI.

Then

Further i f f E

Then D + ( f ) i s open i n

For each open s e t ( D + ( f ) , O ] D + ( f ) )

- Spec(B7f)) where Bo( f ) i s t h e s u b r i n g o f elements o f degree 0 i n t h e l o c a l i z e d r i n g Bf. We r e f e r t o Appendix B f o r more o n P r o j ( B ) . 'L

Now ( c f . Appendix B ) C = Proj(GrA) d e f i n e s a complete a l g e b r a i c v a r i e t y , which b e i n g one dimensional o v e r C, i s a curve, and i t w i l l have genus #FA (which we do n o t prove).

L e t us reproduce now an argument from [MS51 t o i d -

e n t i f y C = Proj(GrA) and SpecA U { $ } f o r a c e r t a i n p o i n t

;on

C.

L e t P be

t h e polynomial used b e f o r e i n showing t h e degree o f A o v e r K i s 1. L e t K -1 be t h e f i e l d o f q u o t i e n t s o f A C C ( ( z ) ) = V so K = {a 6; a,6 E A; a C 01 V.

Every element o f K has an o r d e r which i s a m u l t i p l e o f r = ranK A = GCD

t o r d v ; v E A) = mintGCD(ordP,ordQ); e x i s t s monic y y ) ) = K.

E

K-r such t h a t A

C

P,Q

K

C

E

A).

K((y))

Set Kn = K n Vn.

Then t h e r e

K ( ( z ) ) ( K = C),

and A n K((

C

To see t h i s n o t e t h e r e e x i s t s y E K o r o r d e r ( - r ) and such a y

P-aQb w i t h o r d ( P ) = m, o r d ( Q ) = n (P,Q monic) s a t i s f y i n g bn = am

-

bn.

Then y

E

s a t i s f i e s K(y) C K.

-

Q

am = -r o r r

K n V-r C K n K [ [ z ] l and t h e f i e l d K(y) generated by y Next n o t e t h a t y-n i s monic o f o r d e r n r and hence dim

-

= 1. I f v E Knr i s a r b i t r a r y choose c E K such t h a t v co (Knr/K(n-l)r) 0 y-n € )r Continuing, s i n c e y n+l i s monic o f o r d e r ( n - l ) r , t h e r e ex-

-

.

i s t s c1 such t h a t v

-

coy-n

-

cly

-'+'

E

K("-*lr,

etc.

This g i v e s a sequence

174 c

n

ROBERT CARROLL such t h a t v

- 1;

E "p Vm =

cPy-"+'

{Ol.

K C K((y)). Also K ( ( y ) ) Vo = K"yl1 and A C ~n ~ " ~ 1= 1A n ~ ( ( y ) n ) vo = A n vo = K.

C

Now go t o Proj(GrA) = SpecA

U

{;I.

Since (GrA),

1,

GrA[P-l

(1.2.5)

loi t s homogeneous

over GrA w i t h GrA1P-l

2 An we can w r i t e ( c f . Remark 12.4)

= An

{P-kv;

n,

K

c y-n+p so K(y) P C K((y)). Consequently

L e t P be a s above and l e t GrA[P-l I be

t h e graded a l g e b r a generated b y P-' order 0 part.

1;

Therefore v =

k 0; v E A; ord(P- v ) 5 0)

k

( t h i s i s c o r r e c t i n v i e w o f (*4), where o r d e r ( A ) = 1, v i a s p e c i a l i z a t i o n -a b y = P Q has o r d e r -r, y E G r A [ P - l loand thus K [ y ] C G r A [

o f A-l).Since

lo.

P-l

loC

= K[[yl] so K [ y l C GrA[P-' loC 1 T h i s i m p l i e s t h a t t h e ( y ) - a d i c c o m p l e t i o n o f GrA[P- lo= K"y1l

But GrA[P-'

KO C ( K ( ( y ) ) 0 V o )

K"y11. A ( s i n c e K [ y l = K"y11). o f GrA[P-l

a

D e f i n e now

logenerated by

y.

p

= K[[ylly n GrA[P-l

L e t D+(P)

= Spec(GrA[P-'l,)

1,

= maximal i d e a l

so D+(p) i s an op-

en a f f i n e subscheme o f t h e p r o j e c t i v e scheme Proj(GrA) = C. hand A 2 GrA[h-l los p e c i a l i z e d f o r h = 1.

A1

%

[Ac1

A/C so A1 2 A.

h has

Now

lo = {X-pv; p 1. 0; v

E A;

= SpecA i s an open a f f i n e

F i n a l l y one checks t h a t C = SpecA U

Hence SpecA

The c o n s t r u c t i o n i n [ M S l haps more r e v e a l i n g .

U

6 and on

{!I

by n o t i n g t h a t any

SpecA, i s c o n s t a n t ( e x e r c i s e

-

{$I has no m i s s i n g p o i n t s i n t h e complete c u r v e C.

1, based on CMU21, i s somewhat l e s s formal and per-

One s i m p l y covers C d i r e c t l y b y 2 a f f i n e open s e t s

SpecA and D+(P) = Spec(GrA[P-l 1), -a b Then from t h e above c o n s t r u c t i o n s y = P Q E GrA[P-' loserves

(schemes) D + ( X ) (as above).

This simply involves l o o k i n g

Hence D + ( A )

r a t i o n a l f u n c t i o n on C, r e g u l a r a t c f , [MS51).

= K = C and

o r d e r 1 ( o r degree 1 ) so b y d e f i n i t i o n s G r A ord(A-Pv) 5 0).

a t ( a l l ) elements o f A when A = 1. subscheme o f C.

On t h e o t h e r

To see t h i s n o t e A.

= Spec(GrA[X-l loIx=l)

-

as a l o c a l c o o r d i n a t e f o r D+(P) and C i s a 1 - p o i n t c o m p l e t i o n o f SpecA by a smooth K r a t i o n a l p o i n t o f C (; = K where K(x) =

O,/mxy

t i o n vanishing a t

y = 0; a r a t i o n a l p o i n t x s a t i s f i e s K ( x )

mx = max i d e a l i n

t; which

defines

P).

ax,

and here y i s a r a t i o n a l func-

We r e f e r t o Appendix

B

for further

m a t e r i a l on curves ( c f . a l s o [ HAl;MU51).

REFlARK 12.6 (THE ROLE OF C0H0n0L0CU)- Now f o l l o w i n g [ MS1 ,5 I one d e f i n e s H 1 ( A ) (A n C[[a'' I 1 -2-l = COlposited above) as I-cohomology d e r i v e d from

COHOMOLOG Y

We t a k e h e r e r = 1 f o r s i m p l i c i t y .

Qb = a-'

... and

+

C[[yll = C[[z]l;

o n l y i f r = 1 and [ M S l

175

Note i f one assumes r = 1 t h e n y = P-a 1 i t t u r n s o u t t h a t dimH (A) < i f and

I, which we f o l l o w now, b a s i c a l l y deals w i t h t h i s s i -

1 t u a t i o n ( f o r more general s i t u a t i o n s see IMS51). Now one shows t h a t H (A) 1 1 % H (C,Oc) where C = Proj(GrA) as above. To see t h i s compute H (C,0,) via t h e a f f i n e c o v e r i n g C = D+(X) (12.7)

(r

%

2 r(D+(P) -

H1 ( C , O c )

sections

-

0

H

U

-

DP) t o

o b t a i n ( w i t h some abuse o f n o t a t i o n )

F.OC)/(r(D+(h)yOC) + r(D+(P),OC))

t h i s formula w i l l be discussed below).

compute t h e r i g h t s i d e i n K ( ( 2 - l ) ) % C((z)) s i n c e from K[y] -1 A K [ [ y l l one o b t a i n s K"y11 = G r A I P lo( 2 K"z11 f o r r = 1 ) .

(**I

r(Dt(P)

r(Dt(A),Oc)

-

c

;,Oc)

K((y)) = K ( ( z ) ) ,

= A C K((z)).

Now one can C

Thus f o r r = 1

r(D+(P),Oc) C K"y1l

Hence from (12.7)

G r A 1 P - l lo C

= K"z11,

and

( c f . (12.9) and remarks below

f o r proof)

Thus H ' ( A )

Note here t h a t (12.7)

i s a cohomology group o f a c u r v e C.

M a y e r - V i e t o r i s t y p e r e s u l t ( c f . [ 6x1 ;IN1 1 and Appendix B ) f o r D+(P) D+(P) n D + ( ~ ) N D+(P) H0(D+(P),0,) 8 H0(D+(X),Oc)

= C,

-

U

is a

D+(h )

A

p, and a cohomology sequence 0 HO(C,OC) Ho(D+(P) *p,Oc) H1 (C,Oc) -+ 0. The passage

-

+

-f

-f

-f

t o (12.8) f o l l o w s from [MS51 and we o n l y s k e t c h t h e i n g r e d i e n t s below.

Thus

1

= H (D+(P),Oc) = 0 and i t i s known e.g. t h a t f i r s t one wants H1(D,(h),BC) k H (M,F) = 0 f o r k : 1 whenever M i s an a f f i n e scheme and F i s q u a s i c o h e r e n t

I). T h i s i s discussed w i t h some d e t a i l s i n Appendix B and a t some l e n g t h i n [HA1 1. Next f o r U C C a n a f f i n e open s e t (U * D+ ( c f . [HAl;DQl;SERl;SLl

( P ) h e r e ) one d e f i n e s U as t h e c o m p l e t i o n a l o n g p ( c f . I H A 1 7 ) which i n t h e P Note t h a t one d e f i n e s Ho(U-^p,LU) p r e s e n t s i t u a t i o n means U = SpecK"y11. P lim o lim o = 3 H ( U y L u I3 0,(n)) and Ho(U -$,L ) = -+ H (U ,i; B 0 ( n ) ) where e.g. P UP P UP UP Ou(n) i s d e f i n e d v i a s e c t i o n s fi = (t!/t!)f. o v e r Ui fl U.C U where t h e ti J 1 . l J a r e c o o r d i n a t e f u n c t i o n s i n P' Cw'- [O) ( c f . [ SERl I). I n t h e p r e s e n t s i Q

t u a t i o n e.g.

Ho(U-p,CU)

o r d e r poles a t

p*.

%

r e g u l a r s e c t i o n s o f i;,, d e f i n e d on

Then v i a lMS51one has Ho(U-$,Ou)/Ho(U,OU)!Y

U-6

with f i n i t e

H0(Up-3,0

UP

)/

176

ROBERT CARROLL

0 ). Now one has r(D+(h),OC) ‘L A and ( c f . LMS.51) HO(Up,Oup) 1 K [ [ y l l P’ UPo w i t h H (U -p,O ) K((y)). The passage from (12.7) t o (12.8) i s t h e n P UP d e d u c i b l e ( c f . a l s o Remarks B9-11). Ho(U

I1

R m R K 12.7

(KP B R K I W ) ,

Now go back t o (**) w i t h A = S - l B S C R c ( ( a - ’ ) )

L e t L n ( y ) be a f a m i l y o f i s o s p e c t r a l deformations o f Ln(0) (y = (y, One w r i t e s from (12.8)

and L = L!,ln.

etc.

,. . . ) )

( r e f i n e d i n a more o r l e s s obvious

manner) (12.9)

I1

H1(A) = K ( ( 3 - l ) ) / ( A + K“a-’

3-l)

1 K[a I - a / ( A / K [ [ a - l 11)

l7i

(where K [ a l a b terms i n K P I w i t h no c o n s t a n t t e r m ) . L e t ( * m ) hi = h.. 1.l 1 a J E K [ a l . a , i = 1,2 be a b a s i s o f H ( A ) and y = (yl ) be t h e c o o r 1 d i n a t e system o f H ( A ) r e l a t i v e t o t h i s b a s i s . Consider t h e ( c a n o n i c a l ) 1 f(y) = t with t = hijyl. L e t B be l i n e a r map (A*) f: H (A) + T: y j t h e commutative a l g e b r a o f d i f f e r e n t i a l o p e r a t o r s commuting w i t h Ln E D and A = S - l B S C K((a-’)) where S E 1 + E - l i s determined v i a L = LA’n = 9 s - 1

,...

,....

1;

-f

.

L e t L ( t ) be t h e s o l u t i o n o f anL = a L / a t n = [L:,Ll

w i t h L n ( t ) = L(t)l/n

E

s t a r t i n g a t L ( 0 ) = LA/n 1 Then t h e f a m i l y L n ( f ( y ) ) d e f i n e d on H (A) i s a n

D.

e f f e c t i v e complete f a m i l y o f i s o s p e c t r a l deformations o f Ln ( f : y

t as above).

+

f(y) =

Here e f f e c t i v e i s d e f i n e d i n ( m ) and t h e r e m a i n i n g f a c t s a r e

proved l a t e r i n Remark 12.14, a f t e r some f u r t h e r ideas a r e developed. For now we make a few o b s e r v a t i o n s l e a d i n g t o Lemma 12.8 below.

Thus g i v e n

A

= ZS;

S-’(O)BS(O)

i n g a t S(0) w i t h S

€

G e t c . ( t v a r i a b l e s have been i n t r o d u c e d i n G ) .

one can see t h a t S ( t ) A A ( t ) - l c a l l U ( t ) = exp(tla

GX

D, we t h i n k o f S determined v i a dS

as above, B C

+ t2a2 +

C 9 ( c f . here

...) S ( O ) - 1

i n Theorem 12.3 ( w i t h exp(1;

(1 2.10)

S ( t )AS( t)-’ =

C

‘L

P).

Then

To see t h i s r e -

( c f . Remark 7.3) and s i n c e Y ( t ) E

commuting w i t h A ) one has

Y ( t)U( t)-’AU( t )Y (t)-’ = Y ( t ) S ( O)AS( O ) - l Y ( t ) - ’

w/ i c h belongs t o Y ( t ) D Y ( t ) - ’ o r d e r so S ( t ) A S ( t ) - l

tiai)

97 where S

start-

C

5.

But elements i n S ( t ) A S ( t ) - ’

have f i n i t e

D, a s s t a t e d .

XA = IS E G;SAS-l c D] i s “ t i m e ” i n v a r i a n t under f l o w s Hence t h e s e t (u) 1 S ( t ) and one d e f i n e s (A@) XA = CL = SaS- ; S E XA) c G/Gc (here Gc ‘L { S E G; N

KP ORBITS

[ S , a l = 0 ).

177

Recall from (7.5) t h a t ( w i t h o u t o t h e r s t i p u l a t i o n s ) S (% P)

w i t h L = 3 S - l i s o n l y determined up t o C such t h a t [ C , a l e n t i f i e s L w i t h t h e e q u i v a l e n c e c l a s s o f such subdynamical system o f KP d e f i n e d by A

s.

= 0.

Thus one i d -

(r,A ,T)

Now one c a l l s

the

A and an o r b i t ( L ( t ) ) o f (XA,T)

E

is

c a l l e d A-maximal i f i t i s n o t c o n t a i n e d i n any s m a l l e r system ( j r A ” T ) ( A ’ 3

A

-

U

N

n o t e A C A ’ i m p l i e s XA 3 XAl ) .

I

F i r s t f o l l o w i n g [MSl

Given L ( t ) c o r r e s p o n d i n g t o an A maximal o r b i t i n (G/Gc,T)

trmmA 12.8.

S ( t ) s a t i s f y i n g dS = ZS;

-

( c f . Lemma 7 . 1

21;

Zt =

and

L i d t n h e r e ) t h e n A can

be recovered v i a (12.11)

A = { s - ~ ( o )a(L(O));S(O);

a(a)

E

Take an a r b i t r a r y a ( a ) E A C K ( ( a - l ) ) .

Proof:

( 0 ) ) = s(o)a(a)s(o)-’

[ ~ ( L ( o ) ) + , L ( o ) I = 01

K((a-1));

Since S(O)AS(O)-’

= a(L(O))+ ( n o t e a ( L ( 0 ) )

2,

a(a)

E

= 0.

K ( ( 3 - l ) ) w i t h [a(L(O))+,L(O)I

S(O),al = 0 and hence y

= 0.

Conversely l e t

It f o l l o w s t h a t [S(O)-’a(L(O))+

= S(O)-’a(L(O))+S(O)

s y m n e t r i c a l g e b r a generated by y o v e r A.

c D; A n K I 1 a - l

s(a(a))s-’

1 I - a - l = 101). Hence [ a ( L ( O ) ) + , L ( O ) I = [ a ( L ( O ) ) , L ( O ) I

C D, a ( L

E

K((a-l)).

L e t A ’ = A [ y ] be t h e

Then A ’ E A w i t h S ( O ) A ’ S ( O ) - ’

C D

N

( e x e r c i s e ) so L ( t ) i s c o n t a i n e d i n (XAlyT) f o r A ’ 2 A.

A maximal by assumption we must have A ’ = A, so y

E

Since t h e o r b i t i s

QED

A.

N

Now l e t MA be a n A maximal o r b i t o f (XAyT) d e f i n e d by A

+

t i o n o f dL = [ZL,Ll

s t a r t i n g a t L(0)

has e v o l u t i o n s (A&) anL = [ (SanS-l)+,Ll

E

MA.

E

For any b a s i s

where dS = ZS;

A with L ( t ) a solu-

an o f K[a

1-3 one

and f o r any a

E

A one

One shows t h e n has a s t a t i o n a r y e v o l u t i o n [ (SaS-l)+,L] = 0 b y Lemma 12.9. 1 t h a t H ( A ) r e p r e s e n t s t h e e f f e c t i v e e v o l u t i o n as d e f i n e d i n ( m ) , namely (A+)

Y c T, 0

E

jective.

Y,

i s L e f f e c t i v e i f t h e map To(Y) 3 (ai)+(aiL)

E

E-l i s i n -

This i s c o n t a i n e d i n

1 1 The image Y = f ( H ( A ) ) C T o f f: H ( A )

LElXiA 12.9.

-f

T d e f i n e d by (A*) i s

a n L e f f e c t i v e parameter space o f T o f maximal dimension. Proof: ayi =

1‘;

L e t hi =

lyi

hija/at.

t h e b a s i s hi). (12.12)

J

E

h. .a’ E K [ a l be a b a s i s o f H1(A) ( c f . (12.9)) w i t h a / ’J 1 To(Y) (yi a r e t h e c o o r d i n a t e s o f H ( A ) w i t h r e s p e c t t o

The KP h i e r a r c h y i n terms o f yi i s g i v e n by

aL/ayi

= [

(shis-’)+,~]

178

ROBERT CARROLL

( s i m p l y use (Ad)). Suppose t h e r e e x i s t s a K l i n e a r r e l a t i o n ( A m ) 0 = ciaL/ayilt=O Then v i a KP one has (**I 0 = [ (SclN cihi S- 1 ) + y L l l t = O .

.

C1N BY

1

L e m 12.9 l c i h i + s u i t a b l e non p o s i t i v e o r d e r terms E A which i m p l i e s cihi = 0 as an element i n H1 (A) ( c f . (12.9) an element i n A i s 0 i n H1 ( A ) ) .

-

1

T h i s means ci = 0 and hence Y = f ( H ( A ) ) i s L e f f e c t i v e (spy -+ aL/aylt=O i s injective). E To(T)

-

To(Y).

F i n a l l y t o show maximal dimension t a k e any a/ay = By d e f i n i t i o n ( i . e .

and hence i t belongs t o A / K [ [ a - l (12.13)

aL/aylt,o

by construction)

1

k . a j = 0 i n H (A) J Then by Lemma 12.9

I 1 n A.

= 0

= [ ( S c k j a J- S- 1 )+3Lllt,o

1 so Y = f ( H ( A ) ) has maximal dimension r e l a t i v e t o L. Now f o r MA

1 k J1 .a/at j

QED

a n A maximal o r b i t l e t aL/atn t=O be a n element o f TL(o)(MA). . I

(MA) d e f i n e d by

.

(*A)

a/ay

E

To

Hence one can conclude t h a t MA 1 (coordinates o f H (A))

i s l o c a l l y isomorphic t o H ' ( A ) and we'can t a k e yl,... as l o c a l c o o r d i n a t e s o f MA n e a r L ( 0 ) . d e s c r i b e d v i a yi

i s linear i n t

E

Hence t h e t i m e e v o l u t i o n o f L ( t ) E MA

T.

This shows rv

Every A maximal o r b i t MA i n (XA,T) o f t h e KP system d e f i n e d 1 by A E A i s l o c a l l y isomorphic t o H ( A ) and t h e dynamical system o n MA c o r 1 responds t o l i n e a r motions r e l a t i v e t o H ( A ) c o o r d i n a t e s .

CHEBR€R 12.10.

We emphasize here a g a i n t h a t we have been d e a l i n g w i t h A known w i t h SAS-'

C D.

Thus s t a r t i n g w i t h L o r Ln

2,

E

A f o r which S i s

Ln t h e KP machinery p r o -

v i d e s e v e r y t h i n g and C = Proj(GrA) w i l l e v e n t u a l l y produce a Jacobi v a r i e t y . The d e t a i l e d s t e p s going t h e o t h e r way a r e n o t a l l p u t t o g e t h e r here. t h e S c h o t t k y problem one would a l s o want t o go from a c u r v e -,

A

-,

point

KP o r b i t .

-, w i t h z

A n K[[z]]

-,

For

Jacobi v a r i e t y

The i d e a here i s t o t a k e a c u r v e C w i t h a smooth K r a t i o n a l = 0

= r(C,Oc)

'L a,

and c o n s i d e r T ( C - m , b )

= K and ( w i t h z

s t i l l need S however such t h a t SAS-'

+

8-l)

C

D.

-+

K ( ( z ) ) w i t h image A. Then 1 ) ) belongs t o A . We

A c K((a-

The correspondence i n d i c a t e d i n

[MU2 I between KricYever data and commutative s u b r i n g s o f K [ [ t l I [ d / d t I w i l l p r o v i d e t h i s connection.

We o m i t t h i s f o r now.

Now l e t L s a t i s f y dL = [ Z;,Ll

correspond t o a n o r b i t M o f KP and l e t S be a

KP ORBITS

179

Then L defines an onto l i n e a r map h : Define ( 0 6 ) BM = a n = a / a t n a L / a t n It,O. E Kerhl C D. Then we have

gauge o p e r a t o r s a t i s f y i n g dS = ZiS. To(M) TL(o)(M) by (‘0) cnLt; n c E K;lrinitecnan +

n CEiIEtA 12.11. Proof:

+

BM is a commutative subalgebra of D and AM = S-lBMS E A .

Let a/ayi = l r i n i t e c i j a / a t . be a basis o f Kerh, and a s s o c i a t e t o a / J

a y . an element Zi = 1 c . . L j E D. Since aL/ayi Itz0 = 0 we have aLn//ayi Itz0 1 1J tn = 0 f o r a l l n , and hence aL+/ayi Itz0 = 0. This implies aZi/ayj Itx0 = 0 for a l l i , j , and hence via (7.3) o r ( 7 . 8 )

z ~ , z ~ I =I o~ == az./ayi ~J

(12.14)

-

ltZ0

a z i / a y j ltz0

Since BM = K I Z 1 , Z 2 y . . . ] we know i t gives a commutative subalgebra BMItz0 D a t t = 0. Now consider AM = S-’BMS = K[S-1ZlS,S-1Z2S.... I. we o b t a i n

Since a S / a t j

= -LJS

(12.15)

-

S-’ZiS = S-’1 c 1J . .(Lj

s-l 1 c . .as/at 1J

j

=

Lj)S

=

1 c 1J . .aj

1 c 1. J.S-’LjS t

t

s-’as/ayj

We want t o show S-lZiS pendent o f t . (1 2.16)

E K((a-l)). F i r s t one checks t h a t S-laS/ayi i s indeTo s e e this one computes ( a n = a / a t n )

a n ( s - l a s / a y i ) = -s-lanss-las/ayi

+ s- 1a / a y i ( a n s )

=

s-lLnas/ayi

- s-la/ayi(L:s)

+ s- 1a n ( a s / a y i ) =

s-1 L nas/ayi -

Similarly a l l higher d e r i v a t i v e s i n t vanish a t t = 0. [s-’as/ayi,a 1 = (12.17)

o

at t =

o

[s-’as/ayiYa I =

=

n s-1 L-as/ayi s-lL:as/ayi

=

o

Next one sees t h a t

since

s- 1[ a s / a y i s -l ,sas-’

= s-l (aL/ayi

IS = s - l a / a y i

(sas-’ 1s

1s

and aL/ayi Itz0 = 0. B u t s i n c e S-’as/ayi is independent of t we can conclude 1 t h a t [s-’as/ayi,a] 5 0. Consequently ( a + ) s-lzis = 1 c i j a j t S- as/ayi i s an element of K ( ( a - l ) ) f o r each i . Therefore BH i s a commutative algebra -1 i n D and AM = S BMS E A . QED

180

ROBERT CARROLL

Hence e v e r y o r b i t M o f KP determines an a l g e b r a A, L ( O ) E M ) and M i s c l e a r l y a n A,, m a x i m 1 o r b i t o f can be s t a t e d as

E

A ( n o t depending o n

(rA

T) ( e x e r c i s e ) .

Every KP o r b i t M corresponds t o a unique A

CHEP)REm 12.12.

This

MY

E

A such t h a t

(rA,

T) c o n t a i n s M a s an A maximal o r b i t .

RRllARK 12-13 (U0SPECCRAC DEF0RmAEZ0IU). These r e s u l t s can be used t o emb e l l i s h some e a r l i e r d i s c u s s i o n i n Remark 12.7. Thus g i v e n Ln E D, w i t h s, B, and A as i n Remark 12.8, we want t o show L n ( f ( y ) ) i s a n e f f e c t i v e com1 p l e t e f a m i l y o f i s o s p e c t r a l d e f o r m a t i o n s o f Ln where f: y + f ( y ) = t: H ( A ) +

T.

T h i s can be e s t a b l i s h e d by showing t h a t t h e o r b i t M c o r r e s p o n d i n g t o

L ( t ) i s A maximal. s a t i s f y dS = ZS; C

I t i s t h u s s u f f i c i e n t t o show t h a t A = AM.

and s e t B ( t ) = S ( t ) A S ( t ) - l

Let S ( t )

so B ( t ) = B a t t = 0.

Then BM

For t h e converse t a k e Q ( t =)

B ( t ) s i n c e elements o f BM commute w i t h L ( t ) .

s ( t ) a ( a ) s ( t ) - ' E B ( t ) (a(a) E A C K ( ( a - l ) ) ) . Since B ( t ) C D one has Q ( t ) = a ( L ( t ) ) = a ( L ( t ) ) + ( c f . Lemna 12.9) and we can w r i t e (om) a ( L ( t ) ) + = I 1f i n i t e an(L(t)')+

(an c o n s t a n t ) .

But from t h e KP equations

( a ( L ) = a ( L ) + i m p l i e s [ Q , L ] = 0). means Q

E

But by d e f i n i t i o n o f BF1 i n ( 0 4 ) t h i s

BM and hence B ( t ) = BM.

RETtARK 12-14

(REfllARW ON JAC0BZ OARZECZS).

about Jacobians. R = K [ [ x ] ] (K = C).

One s h o u l d say something now

F i r s t l e t us s k e t c h t h e " d e s i d e r a t a " f o l l o w i n g [ M S l

1. Take

L e t M be a f i n i t e dimensional o r b i t and l e t L = L ( t ) be t

a s o l u t i o n o f dL = [ ZL,L] w i t h S ( t ) a gauge o p e r a t o r s a t i s f y i n g dS = ZLS. We g e t t h e n an a l g e b r a A E A as i n Theorem 12.13 say and A i s o f rank 1 s i n c e M i s f i n i t e dimensional.

T h i s g i v e s B = SAS-'

E 1);

D has a ( n a t u r a l )

l e f t R module and ( v i a B C D o r A ) a r i g h t module s t r u c t u r e ( i . e . E

D for

^D

E

D). The rank o f D o v e r R QK A i s 1

6

= rankA ( e x e r c i s e ) .

+

6SAS-

1

There

i s t h e n a corresponding r a n k one s h e a f L ( t ) ( o r l i n e bundle) o v e r SpecR XK C determined by L o r e q u i v a l e n t l y by A ( C ( t ) i s determined by t h e module s t r u c ture o f follows.

D induced by R BK A ) .

Here ( c f . [HA1

Think o f N = Gr(D) w i t h r =

I) t h e sheaf i s determined as

R IK A module s t r u c t u r e determined a s

n.

above ( s u i t a b l y graded). The sheaf N o v e r P r o j ( r ) can

be d e f i n e d a s f o l l o w s .

JACOBIANS

181

For prime p E r l e t N 0 degree elements in No S'lN, S % homogeneous P (P1 elements f 4 p. For U C Proj (r) open define the group i ( U ) = Is: U +UNp (p E U ) such t ha t s ( p ) E N and s i s locally of the form n/f, n E N, f E r, P n,f homogeneous o f the same degree) ( c f . Appendix B ) . Then make N i n t o a sheaf v i a the obvious maps (c f. HA1 I). In particular for any homogeneous w r d y N over Dt(f) 2 (Nyf))"via Dt(f) 2 Spec(r7f)). Finally ref E rt = call C = Proj(GrA) so w i t h grading via A one can t h i n k of SpecR XK CcProj ( r ) ( c f . [HAl;MU51). This i s somewhat cavalier b u t here we will n o t t r y t o spell o u t the d e t a i l s now ( c f . [MSl 1 and Appendix B). Now l e t m be the maximal ideal of R = K"x11 generated by x a n d l e t L o ( t ) be the r e s t r i c t ion of L ( t ) t o (m X T ) X C 2 T X C . This i s a deformation famne bundle on C with parameters in T a n d one has a formal map (&*) 1 1 1 T H ( C y O E ) . Let dimKH ( A ) = dimKH ( C , O c ) = g with y1 ..YY 9 H 1 ( A ) as before and f: y + t the canonical map defined by t = -+

Y .

j

Composing the maps f a n d (&*) we have a local isomorphism (U) Hence L ( t ) E M + t o ( t )E H'(C,0;) injectively a n d M H1(C,O;). I 1 1 + H (C,0;) as an open s e t . This says t h a t M C Pico(C) = H (C,0;) = H C,0,)/ 1 H ( C , Z ) as an open s e t . Since Pico(C) has a natural (induced) linear structure the flows on M defined by the KP system a r e s t i l l linear r elative t o the l i n e ar structure of Pico(C). Finally one identifies Pico(C) with a generalized Jacobian variety. We have discussed PicS, Pic's, e t c . in the cont e x t of Riemann surfaces i n §4,5 and in Appendix B; thus in terms of divisors, 1 ine bundles, e t c . the background i s present a n d for generalizations a few remarks a re given in Appendix B ( c f . however [ HA1 ] for more on t h i s ) . In particular we recall from Remark B4 t h a t Pic's 2 JacS 2 DivoS/ { ( f ) } . n n- 2 t t a n , L = P 1 l n , and RrmARK 12-15. Following [MS41 for P = a t a 2 a AL = {Q E D; [ Q,L I = 01 one thinks of a relation SpecAL Spectrum L (definition of the analytic spectrum - c f . [MS4 I for philosphy). Then one shows SpecAL i s a curve. Let us also remark that [SEl 1 i s a remarkable source of ideas for studying the connections of curves and Kricever data t o Grassmannians a n d concrete soliton problems. Some indication of t h i s i s given in 119 b u t the whole paper [ SE1 I should really be absorbed ( c f . also [ EH1 I ) . Relations between a1 gebraic geometry a n d PDE abound today.

...

182

ROBERT CARROLL

13, lNtR0DUCC10N t 0 SAC0 tHE0R1J. W e will follow here a t f i r s t I OH1 1 ( c f . a1 so [ D1; DF1- 3; HI1 - 3; FF1,2; NMl - 3;MYl; PR1 ,2; SE1 ;S J1; S2,4; TA1,3; TE1; U2 ,3 I ) . Some notation will be changed here in order t o conform t o [ O H 1 1 b u t t h i s e will construct the universal Grassmann manishould cause no problems. W fold (UGM) of Sat0 and develop the language of Maya and Young diagrams, P l k k e r coordinates, t a u functions a n d Hirota equations. The D module point of view i s also sketched.

REElARK 13,l ( F l I l t E Dll!iENSl0NAt t0N3CNlCC10M), W e consider PSDO or micro(called gauge operators differential operators W = 1 + wla-l + w2a-' + before and denoted by P or S ) where a . ~ a / a xand one observes (Leibnitz)

...

(13.1)

anf(x) =

r n-r ( n ( n - 1 ) ...(n - r + l ) / r ! ) a r f / a x a

1"0

...

This defines a n also for n < 0. Thus e.g. a-lf = fa-' - f'a-' + f " a q 3 ( c f . also e a r l i e r comnents in 17). An inverse for W e x i s t s in the form W-l 2 = 1 + v1 2-l + v2a-' + where (*) v1 = - w l , v 2 = -w2 + w1 , v3 = -w3 + 2wl - w13 , .. I n [OH11 for convenience one r e s t r i c t s attention t o w* W, = 1 , w n ( x ) d n (wo = 1 ) and we follow t h a t here. The machinery i s essen-

..

'aWi

.. .

-

t i a l l y the same for m = and t h i s will be discussed l a t e r extensively. Consider therefore the ODE (A) Wmamf= (3" + y a m - ' + + wm)f = 0 which has Assume the f J are analytic so m l i n e a r l y independent solutions f1 ,.. . ,fm. f j ( x ) = 5,j + S1x j + @,;x2 + .... Then the rank of the X m matrix

...

-

(13.2)

5 =

"0

1 El

'0

*"

c12

...

'7

i s m and ( 0 ) Wmam(l,x,x2 / 2 ! , ...) E = 0. For any m X m invertible R, :R also s a t i s f i e s ( a ) so :i s only unique u p t o a factor R a n d one writes z E G M ( m , m ) = {- X m matrices of rank m}/GL(m,C) which i s a kind of Grassmann manifold (more on general Grassmannians l a t e r ) . Let now A be the s h i f t operat o r represented by an i n f i n i t e matrix w i t h 1 on the superdiagonal ( i . e . A i Y i + , = 1 ) a n d zerox elsewhere. Then (broken notation) (13.3)

eXp(XA) =

1

x /2.

...

FIN I TE DIMENSIONAL PICTURE

183

( p u t the r i g h t side under the l e f t s i d e ) a n d H(x) = exp(xA)E i s an m X m k j 1 matrix with columns ( a f ) ( j = l,...,m; k 0 ) . Given f ,...,$" now one can also determine by writing ( A ) in the form ( 0 ) wl(am-lf) + t wmf

W i

...

- ( a m f ) f o r f = f' ,...,fm. This i s a system of l i n e a r equations with unknowns wk and can be solved by Cramer's rule f o r W k ' Thus ( c f . also (13.27)) =

(13.4)

W,

=

l 1 f amfl p

... p

... a

/Am; A

l p ,-1 f

m

=

... am-1 f l

1

. .. am-'P

(where a - j i s always t o be p u t in the rightmost position when expanding the numerator i n (13.4) - c f . [OH1 I ) Now assume the w . a r e also functions of ( t l y2 ty . . . ) so f J = f J ( x y t ) (note in J the following x a n d tl will play similar roles in certain respects and event u a l l y the distinction will be dropped). Let H ( x , t ) evolve now via (13.5)

H(x,t)

=

e

en(tyA)E; q ( t , A ) =

One expands exp(xA)exp(n) =

1"0 pnAn

1;

tnAn

now via Shur polynomials now t o get

...

where v + v1 t 2v2 + = n (note we use pn here instead of Sn as in §8 since Sn i s used below f o r the symmetric group). I t i s e a s i l y checked (exe r c i s e ) t h a t a k p n = pn-k ( p = 0 for n < 0; a k = a / a t k ) and in particular This allows one t o write H ( x , t ) as an m X m matrix with enaxPn = Pn-1. t r i e s h i ( x . t ) ( j = l,...,tn; n L O ) , having the properties h i ( x , t ) = a n h i ( x , n t ) = a:hi(x,t). Thus h i s a t i s f i e s ( a n - a ) h = 0, n = 1 , Z ,..., with i n i t i a l value h ( x , O ) = f j . Further (+) w m ( x , t ) a m h i = 0 ( j = l,...,m) a n d repeating the analysis above one obtains a formula (13.4) for Wm(x,t) with now the f j The notation a n h i = a n h0j = h i i s useful here, a n d we replaced by h:(x,t). ( j = l,..., will see below t h a t the determinant A: ( o r Am) with e n t r i e s m; n = O , . . . , m - 1 ) plays the role of a t a u function ( c f . (13.111, (13.27)).

hi

Note from h i = a n h i = a n h i and ( 6 ) one has (anwmam + Wmaman)hi = 0 which i s an ODE of order n+m having in particular m l i n e a r l y independent solutions h:. Hence there must be a factorization possible of the form anWmam +

184

ROBERT CARROLL

+ Wmaman = BnWmam where Bn i s a d i f f e r e n t i a l operator o f order n ( e x e r c i s e such f a c t o r i z a t i o n s will be a l s o considered l a t e r ) . Apply a -m Wm-1 from t h e

-

r i g h t now t o o b t a i n (13.7)

B~ =

anwmw,l

+ wma n wm-1

n -1 Since a W W-l contains only a-k, k > 0, we must have Bn = (WmanWm )+. T h u s n m m t h e “time” evolution of Wm is determined by the Sat0 equation (m can be remo v ed here )

anw

(13.8)

= B,W

- wan;

B~ =

(wa n w -1 )+

Note t h a t this is the form taken by the dressing P = e r a t e s on functions o f x = x1 alone w i t h W We can drop t h e d i s t i n c t i o n between x and tl now > 2 . Now define L = waw” = a + u2a -1 + and

i-’expressions

...

i n t h i s we obtain

u 2 = - a w l ; u3

(13.9)

equation (7.74) when i t op1 + K depending o n ( x , t ) . b u t t h i n k of (13.8) f o r n p u t t i n g the e a r l i e r W and

=

- a w 2 + w l a w l ; u4

=

- a w 3+w 1 aw 2+W2 a1w - w12 aw1 - ( a w l )

2

D i f f e r e n t i a t i n g L by tn and using (13.8) one finds now a L = anW W-l + Waa, n w-l = (B,W - wan)aw-l - waw- 1 (B,W wan)w-l = BnWaW-l WaW-lBn. Hence

-

-

anL = [ B n , L ] which is our standard Lax equation. Also evidently L n = WanW-’ so Bn = L+n a s usual. As before i n 97 we a l s o o b t a i n the ZS equation anBm amBn

= [

Bn,Bml ( c f . [ OH1 I )

Next one considers t h e l i n e a r system w i t h wave functions $ s a t i s f y i n g ( m ) L$ = A$, an$ = Bn$, a n A = 0. S e t t i n g q0 = W -1 $ one has a$, = A$o so $o = g(i,A)exp(Ax) where ^t ’~r ( t 2 , t3 , . . . ) now b u t t ( x , t 2 , t 3,...). Hence $ = (1 + 1 wiA-i)gexp(Ax) and we may a s well suppose g i s a n a l y t i c i n i t s a r g u ments. One writes L~ = B~ - B: (B; so (**I an+ = ( L +~ B;)$. Since L - j = ,-j ju2a-j-‘ + ... from (13.1) we see t h a t a - j = 1 :J Jn ( t ) L - n a n d consequently (*) becomes an$ = ( L n + v L- 1 + vA2 L-‘ + ...)$, f o r s u i t a b l e nl v ( t ) . Now L j $ = A’$ so t h e r e r e s u l t s an$ = ( A + vnl/A + ... )$ or anlog$ nj ; A J t .J + to + 1; vj(t)A-’ ( t o = h + v n l / x + .... I t follows t h a t log$ = 1 = c o n s t a n t ) a s a Laurent expansion a t A = and consequently = exp(1; v j Aj)exp(to + 1 ; A j t j ) (tl x ) . Expanding t h e exp(1 v J. A - j ) terms and Q

Q

-

-L!)

r.

+

Q ,

MAYA AND YOUNG DIAGRAMS requiring agreement with

$ =

(1

t

185

wih-i)gexp(hx) a t

t^

= 0 we get

( t l 'L x ) . This gives a derivation o r m t i v a t i o n for the rule used before in e.g. 57 in a n a d hoc manner.

)I

= Pexp(E)

RF311ARK 13.2 (nA1JA ANI) g0UN6 DIAGRAns). Now f o r the tau function which will be Am in (13.4) f o r the W, s i t u a t i o n , b u t with e n t r i e s h i ( j = l,...,m = column indices; n = O,...,m-l = row indices), where a n h i = a n h i = h.! This T can be written as (*A) .(t) = det(Zo exp(n(t,A))E) where 2; i s an m X mat r i x with 1 in the ( i , i ) position for 1 5 i 5 m and 0 elsewhere. Again tl will play the role of x. The pn then a r e given by (13.6) with x and vo a b sent and we expand .r(t) in (*A) as a sum of products ( c f . [OH1 I)

(1 3.11 ) p1

0 1

....

=I

pkl

...

pkrn

'kl-1

..-

pkm-l

...

.-. 1

Pkl'm

.

. ..

.

E1krn

Pkm-mtl

" -

Em km

.*

...

< km runs over a l l possible combinations of m nonwhere 0 5 kl < k 2 < negative numbers ( i n f i n i t e in number). For each s e t of numbers k = ( k l , k 2 , _ _ aiagram. une _.Ir / A \ - 7 7 ..., Km \ one aeTines a- Laivaya puzs teririi p a n i c l e s i n ~ n di i i cell s numbered 5 -m and ( B ) each cell numbered kl -mtl , k2-mtl ,.. ,k,-mtl Thus given e.g. (2,3,5,7) we can draw I.

A--

J:

_1_,?1..__

1

__._12-7--

2 -

.

(13.12)

X

X

-5

-4

-3

-2

x

x

-1

0

X

1

.

X

2

3

4

5

The vacuum s t a t e corresponds t o X in a l l c e l l s numbered 5 0. Then there i s a connection between Maya diagrams a n d Young diagrams. If a cell i s occu. * .. .plea assign T ana i t empty assign + ; tne aiagram surrounaea OY sucn lines i s the Young diagram. Thus for (13.12) one has .

I

I

.

I

I .

I

I

I

- L

- 2

186

ROBERT CARROLL 1

(13.13)

2

3

0'

-1 -2

"

-3

I'

4

-

- '

5

-

v

"

-4"-3-

-2-

Thus t h e vacuum s t a t e corresponds t o

r.

One r e c a l l s here t h a t Young d i a -

grams a r e used t o c l a s s i f y t h e i r r e d u c i b l e r e p r e s e n t a t i o n s o f t h e symmetric group ( c f . [ W l ; W Y l

I)

ponds t o (8.26).

Thus f o r t h e m a t r i x o f p

i n (13.11) c o r r e s and t h e d e t e r m i n a n t composed o f p j i n t h e r i g h t s i d e o f (13.11) we j w r i t e S y ( t ) where Y r e f e r s t o t h e c o r r e s p o n d i n g Young diagram ( i . e . i n d e x Then

t h e p d e t e r m i n a n t as Sy and i n d e x t h e 5 determinant as S Y i n (13.11)). (13.11) becomes i n an obvious n o t a t i o n (13.14)

1

Sy(t)Sy $5Y9 w i t h summation o v e r t h e i n f i n i t e c o l l e c t i o n o f Young diagrams w i t h l e s s t h a n =

T(t)

m t l rows. One can do much more w i t h Young diagrams and some f a c t s a r e summarized i n t h e appendix t o [OH11 ( c f . [ LW1;WYll f o r more d e t a i l s ) .

Thus l e t Sm be t h e

p e r m u t a t i o n ( o r symmetric) group on m numbers; one can c l a s s i f y a E Sm v i a c l a s s e s (la' ,Za2

,...,ma"')

example (11 ,Z1 )

%

3) w h i l e (13)

%

w i t h a1 t 2a2

(1,2,3)

(1,2,3)

-t -f

(1,3,2),

(1,2,3),

+

...

(1,2,3) 1 and (3 )

+

Thus f o r S3 f o r

(3,2,1 ), and (1,2,3)

-t

%

= m.

ma,

(1,2,3)

+

(2,3,1)

-f

(2,1,

and (1,2,3)

The p a t t e r n should be c l e a r . I r r e d u c i b l e r e p r e s e n t a t i o n s o f Sm (3,1,2). Xm 2 0 s a t i s f y i n g a r e c h a r a c t e r i z e d by p a r t i t i o n s [XI w i t h X1 L X 2

-f

,..

1 hi

= m ( c f . [ LW1;WYl

f o r S3, 2+1+0

%

[2,1,0]

%m)

I). 'L

One a s s o c i a t e s t o each [2,1 ]

%

p;1+1+1

...,

%

[ X I a Young diagram (e.g.

3 [l]

%

; 3+0+0

%

[ 3,0,0]

,...

and one maps [ X I cf (klyk2-1, k m - m + l ) t t (k, ,km) ( i . e . 3 k.-mtl so e.g. [2,1 I t , (0,1,2) e (0,2,4); [ l1 cf (l,l,l) ++ (1,2,3); j J I f a E p = (laf ,2a5,...,mam) then tt (0,0,3) e+ (0,1,5)). %

[3]

A

-f

where

xY(a)

[31

i s c a l l e d a c h a r a c t e r (depending o n l y o n t h e c l a s s p t o which a

PLUCKER R ELAT 10NS

187

belongs) and I Y I i s t h e number of c e l l s i n t h e Young diagram. orthogonality r e l a t i o n (Peter-Weyl theory)

There i s a n

which can a l s o be w r i t t e n as ( * a ) 6 y y l = ( l / m ! ) l h x y x Y ’ w i t h h = m!/la P P P P 2” .mamal !. . a m ! . Now an a r b i t r a r y a n a l y t i c function f ( t ) can be w r i t t e n

..

(13.17)

.

f(t) =

1; 1

cm ( a l y . . . , a m ) t y l (2t2)aa...(mtm)ad

+ 2a2 + ... + m m = m. B u t from (13.15)-(13.16) one (l/m!)l h X Y t a l (2t2)aa...(mtm)a‘so t h e r e i s a 1-1

w i t h inner sum over a1

knows ( * I ) ) S y ( t ) = P P P l l i n e a r transformation between Sy f o r I Y I = m and monomials t ~ 1 ( 2 t 2 ) a a . . . ( m tm)am w i t h a1 +.?a2+. . +mcc, = m. This implies t h a t (*+) f ( t ) = 1 c S ( t ) . ’Y a , Further, putting t h e formula above f o r h i n t o (*&) gives S y ( t ) = 1 x p t l ... P a ! so t h a t ( r e c a l l yt = (alyL2aZya3/3 ,... ) m

.

tp/al!... (13.18)

sY(Tt) =

1P xPY ( a y ’ ...aamm)/lal...maln,l!...am!

and S y ( z t ) S y l ( t ) = 6 y y l w i t h c y = S y ( ? t ) f ( t ) l t = o i n

(*+I.

REmARK 13.3 (PLUCKER REtAtI0W AND C N l FrtnrCCZ0W). Going back now to (13. 1 4 ) one derives next t h e P l k k e r r e l a t i o n s f o r t h e cy. Generally one embeds Grassmannians i n p r o j e c t i v e space a n d t h e image is described via such r e l a t i o n s ( c f . [ GR1 ] - we will say more about this l a t e r ) . In the present s i t u a t i o n take an example m = 2; pick k w i t h kl < k 2 < k3, and notice t h a t (1 3.1 9 )

Expanding one has

+

188

ROBERT CARROLL

which g i v e s c o n s t r a i n t s among t h e S y . 1,2,3)

I f e.g.

one takes (k,klyk2,k3)

Other examples a r e g i v e n i n 1 OH1 ] f o r (0,1,2,4),

... <

and nl < n2 <

where 6 = m + i - j , ni+l,...,nm+l) (13.22)

(0,l ,3,4),

(0,2,3,4),

t h e eventual r e s u l t i s , f o r each c h o i c e o f kl < k 2 <

(1,2,3,4);

Y1

... <

and km-l

nm+l,

(kl

%

,... ,kj,niYkj+ ,,... k),

and Y2

(nl

%

,...,n i - 1 '

w i t h k . < ni < kj+l. For a l l combinations o f ( k v ) and (n,) J g i v e s an i n f i n i t e number o f c o n s t r a i n t s on t h e Ey, c a l l e d t h e P l k -

ker relations.

On t h e o t h e r hand i f f ( t ) =

1 Sycy

P l k k e r r e l a t i o n s then f i s a t a u f u n c t i o n ( i . e .

11) =

= (0,

t h e n (13.20) says

cy

and t h e

s a t i s f y the

f w i l l have t h e form (13.

(*A).

F i n a l l y one shows t h a t t h e t a u f u n c t i o n s a t i s f i e s PDE h a v i n g t h e same form T as t h e P l i c k e r r e l a t i o n s . Thus w r i t e T(t+S) = det(soexp(n(t,A))exp(n(s,A)) Then expand (*.) T ( t + s ) = 1 S y ( t ) c y ( s ) E) and s e t E(s) = eXp(n(S,A))E. where t h e s y ( s ) have t h e same form as i n (13.14) (or (13.11)) and s a t i s f y t h e P l k k e r r e l a t i o n s w i t h parameter s .

Apply

Sy(yt)

t o (*.)

t o get

Put t h i s i n (13.22) t o o b t a i n t h e n ( c f . [OH1 I )

These a r e t h e b i l i n e a r r e l a t i o n s f o r t h e t a u f u n c t i o n , one f o r e v e r y c h o i c e o f k,

< k2 <

... <

km+l

and nl

< n2 <

.. . <

nm+l

as b e f o r e .

Thus t h e y c o r -

respond t o t h e i n d i v i d u a l H i r o t a equations, n o t t h e formula (8.36) which conSome o t h e r c a l c u l a t i o n s i n [ OH1

t a i n s a l l such equations.

1 (which we o m i t )

g i v e a d i r e c t d e r i v a t i o n o f e q u a t i o n s l i k e (13.24) i n terms o f t h e H i r o t a b i l i n e a r operation. (13.25)

Thus f o r each ( k ) , ( n ) a s above

I.m+l (-1) i- 1Sy

N

I

(%~t)Syn(-%at)~-~= 0

THE INFINITE GRASSMANNIAN

189

Some f u r t h e r c a l c u l a t i o n s based o n [ FF1 ,21 ( c f . a l s o [ NM1,21) a l l o w one t o w r i t e from (13.4)

(A*)

w

=

j

( ~ / T ) S ~ ( - TSO~ ) tTh a t ,

u s i n g (13.10),

p r o o f o f t h e v e r t e x o p e r a t o r e q u a t i o n (VOE) i s o b t a i n e d i n t h J, = e ‘0

(13.26)

+

E Xntn t(tl-1/k,t2-1/2i2,.

..

f o r t h e d e t e r m i n a n t Am i n (13.4) w i t h e n t r i e s hJk t h e n .r(x,t) t h e form ( x b e i n g r e i n s t a t e d ; t = (tl,t2,

1

w .(x,t) J

(al ,...,a

Let

m

U s i n g (13.16) -1 EVx) = 1 + = S.(t)),

J

= ( - 1 ) j 10,l

(x,t) =

IO,l,...,

y...ym-j-lym-j+ly...ym~/

) = n ( a .-a .) be t h e Vandermonde d e t e r m i n a n t and one checks

(or

J

(*A)),

1

p l u s c a l c u l a t i o n s based on t h e r e l a t i o n s nY(1+ E nm ) (so $ j

$.(E)xJ’ = 1 ; s j ( t ) x j f o r tn = ( l / n ) ( E y + 1 J one a r r i v e s a t a formula

1’

(1/TI s

gi (yt

(13.29)

w j = (-1

where

r e f e r s t o (1,1,...,1,0,...,0),

to

(”)

k,,kz,....kml i n (13.11) has

and

(13.27)

(E)

...,t,))

form

)/T(t)

We do n o t g i v e a l l t h e d e t a i l s b u t remark t h a t i f one w r i t e s

m-1

another

(*A)

...

)T

1 a p p e a r i n g j times.

This leads

(see [ OH1 I f o r d e t a i l s ) .

REIURK 13.4

(CHE ZWXNlCE GRA$$mANNZAN).

We f o l l o w now [ T A 3 ] and i n d i c a t e

f i r s t some f i n i t e dimensional c o n s t r u c t i o n s .

C o n f i r m a t i o n o f some o f t h e

c a l c u l a t i o n s c o u l d be a n o n t r i v i a l e x e r c i s e so one must r e g a r d t h e present a t i o n as p a r t l y h e u r i s t i c . (m,n)

= Fr(m+n,m)/GL(m)

m ) ; rankg = m l . (m).

The f i n i t e dimensional Grassmann m a n i f o l d GM

where Fr(m+n,m) = ( 5

‘L

The q u o t i e n t corresponds t o 5

We assume complex e n t r i e s .

(m+n) X m m a t r i c e s = Mat(m+n, -t

C.h,

5

E

Fr(m+n,m),

subspaces o f Cm+n = V and Fr(m+n,m) ‘L m-frames 5 = (So,...,Sm-l) p+n (gj = m l :-g’i .ei connects 5 w i t h ( ( 5 . . ) ) E Mat(m+n,m)). 0 1-1 ) 1J frame g = ( 6 ,..., g one can w r i t e (13.30)

5’ A

... A C m - l

=

e

h

E

GL

c o n s i s t s o f m dimensional

Thus GM(m,n)

A

... A e

Um-i

in V = Given an m-

ROBERT CARROLL

190

where 0 5

a.

<

... <

u

m- 1 < m+n.

The coefficients

(*a)

5 Uo...

Um-1

=

det

0 5 i , j < m-1, a re called Plicker coordinates (see [GRl 1 f o r a dis-

cussion). They s a t i s f y the PlG'cker relations (13.31) below a n d any element of AmV whose coefficients s a t i s f y such relations i s factorizable l i k e '5 A ... A ( i . e . corresponds t o a point of G M ( m , n ) ) . Note the P l k k e r coordinates a r e t o t a l l y antisymmetric a n d the Plicker relations are

f o r each choice of ( a ) (0.') as above ( c f . ( 1 3 . 2 2 ) ) . This definer a n embedd i n g of G M ( m , n ) into P(AmV) = (AmV - {Ol)/GL(l). Note the coordinates ref er t o the ambient space in which GM i s embedded. One should check t h a t the P1u"cker coordinates a r e unchanged under a map 5-,5h ( h unimodular) so t h a t they a r e r e al ly well defined functions on G M ( m , n ) = Fr(m+n)/SL(m) rather t h a n on GM. In fa c t & GM i s a principal G L ( 1 ) bundle (determinant bund l e ) . Plicker coordinates represent sections of the dual bundle a n d give an embedding of GM into AmV - I01 (also called PlGcker embedding). Thus Iy

-+

Now GM can be covered by a ffine coordinate patches GM(m,n)u = 15, # 0) where ...,u m-1 ) with uo < ... < um-1 (we will write u t ) . On GM,one has a f f i n e coordinates Wu = (5u(j-+i) / 5 (I), i E uc, j E u , where i E uc means i # m-i 6 u k for any k and j E u means j = u k for some k; here Sa(j-+i) means 0 ohj Thus in 5u(j-+i)y replace j in u by i; i f j u or if i '(. .ukLIiuk+, 1. The map 5 -+ Wu gives a 1-1 correspondence GM a r i s e s twice 5 = 0. $ , , (j+E 1 t! ( u = complement u i n ~0,1,...,m+n-1l, row index from a ). Mat(uc,u) 2 C If e.g. u = 4 = {O,l, ...,m-ll any 5 E Fr(m+n,m) representing GM+ has an inv e r t i b l e upper square ( ( 5 . .)), 0 5 i , j < m. Multiplying by the inverse from 1J the right one can normalize 5 t o 6 i j = s i j y 0 5 i , j < m (we show details l a t e r for t h i s ) . The remaining e ntrie s a r e then identified w i t h the af fine coordinates W so 5 = ( 1 ) modulo G L ( m ) . Over each such coordinate patch 4 41 one t r i v i a l i z e s GM via 1~ GM r G M u X GL(1) 2Mat(uc,u) X GL(1): cniod SL(m) u = (uOy

.

-+

-

+

( m o d GL(m),S,)

-+

(WU,Eu).

PLUCKER COORDINATES

191

The a c t i o n o f GL(m+n) and g1(m+n7) o n t h e P l i c k e r c o o r d i n a t e s i s s p e l l e d o u t as follows.

Work w i t h t h e l o c a l t r i v i a l i z a t i o n j u s t i n d i c a t e d and w r i t e W

= ((wij)), 0 5 i < n, 0 5 j < m, wij = tu(j+i)/S4. gS, = (gt), = guulE,l (uy s t r i c t l y i n c r e a s i n g

1

(g,.u.l L

J

)Iy0 5

i , j < m.

For A

E

For g 5

E

@

GL(m+n) w r i t e

u t t ) where guuI = d e t

gl(m+n) w r i t e ~ ( A ) s , = DE(exp(EA)Su)

1 aij5a(i+j)y

0 5 i,j < m+n. One checks t h a t 6 ( A ) preserves P l k k e r r e l a cv Now w r i t e W = ((w. .)), t i o n s and hence defines a v e c t o r f i e l d on GM(m,n). Q IJ One computes (exercise) 0 5 i < n, 0 5 j < m, where wij = E , ( ~ + ~ ) / c ~ . =

(13.33)

S(A)wij

1 wikakpwpj

=

6(A)tg = and wij j

<

( 0 5 k,p < mtn);

1 a J..w. .c 1 1J 4

( 0 5 i < m+n, 0 5 j < m )

i s extended o u t s i d e o f t h e o r i g i n a l i n d e x range as wij

m ) and wij

(m 5 i , j < m+n-1).

= -6ij

"horizontal", i.e.

so one w r i t e s

c o n t a i n s o n l y wijy

..

= tjij

Thus t h e & ( A ) a c t i o n on wij

...,

(A&)

8,6(A)

( 0 5 i,

stays

= 1 wikakp ...,

w .a/aw. I f we w r i t e + f o r (m,m+l, mtn-1) and - f o r (0,1, m-1) t h e n pJ A -- = ( ( a i j ) ) , 0 I t f o l l o w s t h a t 6 ( A ) a c t i o n has < i,j < m. A = (A+,A++ t h e form ( e x e r c i s e ) 6(A)W+ = A-,

(13.34)

w(A,W

0

+ A+,W4

) = T r ( A -- + A

This means f o r m a l l y ( c f . (13.35)

-

& ( A ) = s,6(A)

WQA--+

- W Q A -+ W 4 ,

*

6 ( A ) F 4 = w(A,w4)Sg;

W )

Q

(A&))

+ w(A,Wg)EQa/aEQ

For t h e i n f i n i t e Grassmann m a n i f o l d

or

UGM ( u n i v e r s a l Grassmann m a n i f o l d )

N

one w r i t e s UGM = Fr(Z,NC)/SL(NC) and UGM = Fr(Z,NC)/GL(NC) where Z = i n t e C

cv

w i l l be c o n s t r u c t e d as a d e t e r m i n a n t

gers, N = O,l,...,

N

bundle o v e r UGM).

Here Fr(Z,NC) = ( 5 =

= -1,-2,....(UGM

((C..)), i E Z, j E NC, and t h e r e 1J e x i s t s m > 0 such t h a t cij = tiiij f o r i 5 j and i < -m w h i l e t h e r e e x i s t s n w i t h r a n k ((s..))= m when -m 5 i < n and -m 5.j < 0); SL(NC) = { ( ( h . . ) ) = 1J 1J h, i,j E NC, and t h e r e e x i s t s m > 0 such t h a t hij = 6ij f o r i 5 j and i < -m w h i l e ( h . . ) ) E SL(m) when -m 5 i , j < 03, and GL(NC) i s l i k e SL(NC) except 1J t h a t ( ( h . . ) ) E GL(m) f o r -m 5 i,j < 0. I t i s i n s t r u c t i v e t o draw p i c t u r e s 'J

ROBERT CARROLL

192 of these a s in [ S4;TA31. (13.36)

Thus e.g. f o r F r ( Z , N C )

* I

0

* * 1

*

-m-1

* *

L

element of Fr(mtn,m)

I

*

*

*

I

n-1

* -m

-1

For P l k k e r coordinates take a semiinfinite sequence of integers u = ( u i ) , 1i m i E NC, ui = i f o r almost a l l i , a n d define 5, = d e t ( ( E u i j ) ) i , j E N c = mw d e t ( ( ~ b))-m
i 0 ( - l ) i t;( . . . u - 2 a t ) E(...uf-l

Is;+,

...

,C ;i

= o

Affine coordinates are defined in the same way as before via patches UGM,= 2 M a t ( u c , u ) with gmodGL(NC) -+ Wu = ( S u ( j + i ) 1 5 ) where now 15, # 0; u For 9 = ( . . ,-3,-2,-1 ) the NC X NC ‘u(j+i) = lk<06uMj‘( u,+,iuk+ )* part of 6 i s invertible and we normalize t o S i j = t i i j y i , j E NC (see below 1 for remarks o n such normalization). Then as before 5 = ( ) mod G L ( N C ) on w+ UGM+

...

,...

.

13.5 (GR0W ACCZ0W AND C M FWCCZ0W). Now f o r group action define G L f ( Z ) = {g = ( ( g i j ) ) such t h a t g i j = 6 i j unless i , j E (-m,-rntl, ..., n - 7 ) for some m,n with ( ( g . . ) ) E G L ( m + n ) for -m 5 i , j < n l . For Lie algebra action

REmARI(

1J

take g l f ( Z ) = {A = ( ( a i j ) ) such t h a t for some m , n a i j = 0 unless i , j E (-my -m+l,...,n-l)I. Then ( c f . [TA3])

TAU FUNCTIONS (13.38)

193

0 A =

GL(m+n)

,o

n- 1

g l (m+n 1

n-1

cy

Then GLf can a c t on UGM and UGM v i a a c t i o n o n P l k k e r c o o r d i n a t e s a s (A+) 95,

=

1 guu15,1

where gUu1 = det((g,.,!)),

(0' f )

C aijEu(i+j)

DE(eXP(EA)cuIE=O

as before.

i,j

L

NC, and 6(A)c,

E

=

J

However one wants a l a r g e r group

( c f . 58) and i n p a r t i c u l a r i t i s necessary t o i n c l u d e m a t r i x r e p r e s e n t a t i o n s

( ( i s i , j + l ) ) and a % A = ((6iyj-l)) i?I here one i s t h i n k i n g o f a b a s i s ei = ( a x ) so ei = ei-l

o f operators l i k e x = (i+l)a-i-2

K =

%

(we w i l l d i s c u s s t h i s p o i n t o f view l a t e r ) .

= (i+l)ei+,

X

one c o n s i d e r s g l ( Z ) = I A = ( ( a , . ) ) ; 1J

ment must be made i n d e f i n i n g &(A) n .

(13.39)

6(A) =

f o r i , j E Z; -1-i and Kei = xax

lz aij(6(Eij)

-

= 0 f o r j-i >

aij

(as i n 18 f o r

-

m).

Thus

Then an a d j u s t -

see e.g.

(8.7)).

Set

o ( i < 0)6..M) 1J

) c , = Sa(i+j)y O ( i < 0) = 1 f o r i < 0 and 0 otherwise, and ME, = 1J 5, (M 'L S,a/ag, i s t h o u g h t o f a s t a n g e n t t o t h e f i b e r s o f UGM - n o t e M(s,/s E , , ) = 0. a s a s t i p u l a t i o n ) . The M t e r m i s o f t e n r e f e r r e d t o as a quantum

where 6 ( E .

h,

correction.

Now proceed as b e f o r e i n (13.35) %

w..)),

= s , $ ( ~ +/~5 ,$)

i

1J

E

Z, j E N';

wij

4-

.

Mat(N,NC) X GL(1) we g e t c o o r d i n a t e s (W form M = 5

,$

a/ac

4-

a,G(A)W

t o decompose ;(A).

Consider

Mat(N,NC) c a r r y i n g a f f i n e c o o r d i n a t e s W4, = ( (

t h e c o o r d i n a t e p a t c h UGM

From t h e t r i v i a l i z a t i o n I T - ~ U G M2

5 ) f o r which M i n (13.36) has t h e

,$, ,$

has t h e same form as i n (13.34) and s,s(A)

=

b u t t h e v e r t i c a l component changes so t h a t

a,Y(A)

ly

6 ( A ) = IT$(A)

(13.40)

+ G(A,W,$)M; G(A,W,$)

has removed t h e "divergence p a r t " TrA

Ttius

= Tr(A- W ) + $

--

from w(A,W

4

) i n (13.34).

Now c o n s i d e r f o r m a l l y a group a c t i o n b y g(E) = e x p ( 1 E ~ EA e x~p ( g l ( Z ) ) o v e r 1 Thus 5 E UGMbhas t h e form 5 = ( ) as b e f o r e and use k s i g n s f o r 4 UGM

0'

zx z

w+

m a t r i c e s so g(E) = ( g - - g-+) w i t h gs = ( g - g-+'+). Use 9+- g + + (g-- + g-+W,$)-' t o p u t t h i s i n n o r m a l i z e d form g.5 = ( where gW = (g+gw ,$ g++W,$)(g-g-+ W ,$ )-'. T h i s i s f o r m a l l y meaningful (c%. [ TA1,21); e.g.

blocks o f

+

(g--

+

gf- + g++w*

+

+ g-+W,$)-l does e x i s t i n t h e sense o f formal power s e r i e s , e t c .

I f one

ROBERT CARROLL

194

looks a t g ( c ) = exp(EA) i n this context and expands i n

E,

t h e c o e f f i c i e n t of

+

is 6 ( A ) W from (13.34) a s desired. Next w r i t e g l ( Z ) = n- t h + n t where nt = {A = ( ( a . . ) ) ; a 1J i j = 0 f o r t ( i - j )2 01 and h = A = ( ( a1. J. ) ) ; a i j = 0 f o r i i j . These a r e Lie subalgebras of g l ( 2 ) and the i n f i n i t e s i m a l a c t i o n via 6 can be exponentiated. For exp(n-) (n- % lower t r i a n g u l a r ) t h e r e is no problem, any g ( e ) = exp(1 €,A,) can a c t o n Fr(Z,NC) as i n (A+). Note by t r i a n g u l a r i l y g ( E ) u o l = o unless u ~ u ( i' . e . o. > a! f o r a l l i E N'); thus the 1 1 sum over u ' has only a f i n i t e number of terms. For n t n, upper t r i a n g u l a r , E

-

i n f a c t (A+) has t h e same form and one of 6 ( 1 E,A,), b u t g(E)S must be taken the E, (note g ( E ) u o l = 0 unless u 5 (I' t i o n gives diagonal matrix a c t i o n g . s u n N , which is f i n i t e . Now consider

+

w i t h t h e exponent 4 i n t h e ring o f formal power s e r i e s i n f o r n + ) . For exp(h) d i r e c t computa-

can i d e n t i f y

+=

m

(13.41)

g,(x)

= el1

(n

g(E)s

where

gii)s,

iE

-n

n

'nA

n runs over

E

expfn,);

g - ( y ) = el; ynA

E

exp(n-)

where An = ( ( ~ 5 ~ i , ~j E~ Z ~(thus ~ )A )= (~ ( ~ 5 ~ %~ a x~) . ~ Then ~ ) )gt(x)u,I =

x u l / u ( x ) and g - ( y ) o u l

nomials

= x u l / u ( y ) where

= det((pu,-u

(Am)

xu,/,

represents r e l a t i v e S h u r poly-

( x ) ) ) , i,j E NC.

One knows ( c f . [MA1 I ) t h a t

{xo(x) = X , / ~ ( X ) ; a l l u l form a basis i n t h e r i n g of formal power s e r i e s i n x so t h a t (.*I f ( x ) = 1 f,x,(x) (o t ) w i t h f o = ~ ~rr( a ~ ) f ( (xm a x) =~ (~a = ~ XI ' Jia ,...)). Further x u l / u ( x ) = x u ( ? x ) x o l ( x ) and hence ( u t ) xz

1 x,(x)x,(Y)

(13.42)

= exp(C

nxnYn)

(note t h e c o e f f i c i e n t s on the r i g h t in an expansion

1 anxo(x) are

a n =@ !)

('8;)exp(1 nxnyn)Ix=O= x,(y)). One can then compute t h e a c t i o n g + ( x ) g (y)s 0 and g ( y ) g t ( x ) s u ( c f . [ TA31 f o r formulas). In p a r t i c u l a r

g,(x)C+

(13.43)

=

1

x,(x)C+

o t

( d l r e c t l y from (A+) w i t h ( g t ) + u l = x o l / + = x u l ) . This is i n f a c t a tau funct i o n and gives the o r b i t s on UGM ( w i t h coordinates (wij.r)) of dynamical flows generated by 6 ( A n ) , n = 1 , 2 , to R = GL;1 Note

r

i n 18.

g,(x)t

9

...; thus

T

= exp(C7 t n A n ) ) S

0

i s analogous

is a generating function f o r the PlGcker coordinates

D MODULES

195

Y

Also a function f a s in (**) i s a t a u since from (a*) 5, = x , ( a x ) T l x = O . function i f a n d only i f i t s coefficients f, s a t i s f y the P l k k e r relations. k Further from T ( x + x ' , S 4 ) = T ( x ' , ~ ( x ) ) = 1 xU(x')5,(x) with 5 ( x ) % x,(ax,) 4 4 T(X+X'y$), one sees t h a t the P l k k e r relations (13.37) for S,(x) have the form of quadratic differential equations (or equivalently Hirota equations). Note also formally t h a t T represents a section of the dual determinant b u n rv dle associated t o the determinant bundle UGM +. UGM. From the formulas for infinitesimal action of ?(An) one can read off the evolution equations determining the flows. Thus one obtains (exercise)

The w i j equations describe dynamical flows on UGM a l o n g the vector f i e l d s n These then determine T from the f i r s t equations and 6 ( A ), n = 1 , 2 ,

-

....

t h i s amounts t o a l i f t i n g of flows o n UGM t o UGM. I t i s the vanishing of an anomaly term c(Am ,A n ) = 0 (m,n = 1 , 2 , . . . ) t h a t makes t h i s possible ( a n d hence makes i t possible t o define the t a u function). Here c(A,B) i s the - A-+B+-)) involved i n central exKac-Peterson cocycle (c(A,B) = Tr(B-,A,N tension gl(Z) of g l ( Z ) a n d will be discussed l a t e r . We only remark here t h a t (with M a s in (13.39) e t c . ) with [ 6 ( A ) , M l = 0.

REmARK 13.6

(0.1

[ 6(A),6(B)1 =

6([B,Al) + c(B,A)M

A Sketch of the D module point of view i s appropr i a t e here ( c f . [ TA1,3] and see also [ S4;U21). We will mainly follow [ TA1,

IT,

31.

(x)

(D n0DWEsd).

=

T h u s consider PSDO P = p n ( x ) a n a n d use the Leibnitz rule ( 0 0 ) anf 1: ( L ) f ( k ) ( x ) a n - k . The collection of such P i s called E and has a

{I-,

s p l i t t i n g E = D fIi E-l ( c f . § 7 , 1 2 ) where D = p n a n l and E-' Q, -1 pna n l . Let ( )+ o r ( )- denote projections onto 1) o r E-l. Then define the KP hierarchy as usual via W ( t ) = 1 + 1 ; w n ( t . x ) a i n a n d Bn = (Wa"W-')+ ( c f . (13.8)). This gives r i s e t o an i n f i n i t e system of evolution equations for the w,,. To connect this t o UGM consider Wi(t) = ( a L w ( t ) - ' ) + w ( t ) ( i = 0,l ,...Iand look k a t D = D, and € = E0 as algebras with coefficients from 0 = C [ a wn, n 1, k 2 0 1 generated over C by the coefficients of W . Then W i ( t ) E €0 a n d forms a n 0 generator of the l e f t D0 module D O W ( t ) = li,oOWi(t) C EO. The theory of the KP hierarchy can be reformulated as deformations of such D modules

196

ROBERT CARROLL

and the affine coordinates w i j a r i s e a s the coefficients of the W i above v i a (13.45)

wi(t)

=

a Xi - I -1-, wij(t,x)a J

We discussed this also, indirectly, in 112 b u t l e t us give a few more det a i l s here following [TAl 1. Some related algebra connected t o holonomic quantum fields appears in 514. Thus take R t o be a simple differential ring, i.e. a ring w i t h derivation a : R + R satisfying a ( f g ) = ( a f ) g + f ( a g ) and l e t C = constants (ac = 0 for c E C ) . Think of R a s a ring o f functions over C. Then a differential operator w i t h coefficients i n R i s a linear combination P = 1 a v a V ( f i n i t e sum) where v = (vo, v a n d av = (aolVo ( a l l V t ( as-l )vs-' Set IvI = vo + .+ 8-' = and v+k = (vo+ko,. . . , ~ ~ - ~ + ).k ~ The - ~ set D = D vs-l c-l of a l l such differential operators forms a noncommutative ring with 1 ava B 1 b a v = 1 c pv where cv = 1 (m)a 8 kbv+k-m. In case of confusion one writes V V a v . f = operator product E D and a ( f ) = operation o f a v on f E R . A l e f t module i s a n additive group M equipped w i t h a n action of D from the l e f t . The l e f t R submodules ( 0 6 ) Di = R + R a + ... + R a i ( i 0 ) give D a natural f i l t r a t i o n and one concentrates o n l e f t D submodules S o f D satisfying the s p l i t t i n g condition ( 0 6 ) D = S @ Dm-l where m > 0. Thus D = Ei + Dm-' a n d S n Dm-' = 0. Then under the s p l i t t i n g condition ( 0 6 ) there i s a unique R generator system IWi,i ) m o f 9, ti = li,mRWiy o f the form Wi = a i -

...

...,

(I)(k;)...(is-')y

.

..

k w

-

(om) wi+l aWi - Wi,m-lWm = 0 The existence of such a n R generator system in fact characterizes l e f t D submodules o f D with the s p l i t t i n g property. To see t h i s note t h a t one can i n deed obtain such generators by decomposing the monomial a i i n t o the sum o f a n element of S and a n element of Dm-' according t o the s p l i t t i n g D = S B Dm- 1 , the f i r s t component then giving Wi as required. This also shows the uniqueness of such a generator system. Further the l e f t side o f (om) i s chosen t o l i e i n the intersection S n Dm-' a n d hence vanishes. Reversing the argument one obtains the characterization statement. Further (om) recursively determines Wm+l , Wm+*, ... in the form o f a differential operator .W,. Hence S i s generated over D by a single element as S = DWm and t h i s characterizes l e f t D submodules o f D, the generator Wm being an arbitrary

w.1J.a' satisfying the "structure equation?'

197

D MODULES

monic operator of order m. Having obtained a family of l e f t D modules of D m e considers time evolutions S ( 0 ) = S S ( t ) a s deformations. A simple such evolution would i n volve (&*) S ( t ) = Sexp(-tF) for F € C[ a 1 = constant coefficient differential operator. B u t exp(-tF) is o f i n f i n i t e order so one looks a t $ = D [ [ t l l = 11; t n A n , A n € D a n d thinks of time evolutions (&A) % ( t )= $Sexp(-tF). More precisely formulate everything within the framework o f formal power series i n t with R [ [ t ] l as basic ring ( a i s extended via a ( t ) = 0 ) . Then instead In t h i s setting consider of DR = R[al one uses DR = R"tll[al (R = R"t11). DR submodules S ( t ) o f DR t h a t s a t i s f y (04) with D replaced by DR. S k h a DR submodule S ( t ) has a unique system o f R generators W i ( t ) = a L 'ij . As a DR module s ( t ) i s gen( t ) a J with the coefficients lying i n R = R"tl1. erated by a single element W,(t). -+

Let us derive an infinitesimal version of (&A). While $ i s made u p of differential operators with t dependence, S i t s e l f i s independent of t and a / a t induces a C linear map: 39 5s. Twisted by exp(tF) i t gives r i s e t o a C linear map: % ( t ) $$(t)sending P E & ( t ) a(Pexp(tF)/at exp(-tF) = a t P If P i s of f i n i t e order ( i . e . a member o f S ( t ) ) so i s the image of t PF. t h i s C linear map. Thus one obtains the following infinitesimal version o f the time evolution law (6.) IatP + PF; P E S ( t ) l C S ( t ) . Applying (6.1 t o the generators Wi ( t ) we obtain the evolution equations -+

-+

-+

where b . . ( t )€ R (the right side being actually a f i n i t e sum). Another 1J equivalent expression o f these equations i s due t o the DR generator Wm(t)o f s(t) which yields time evolution governed by a single equation (13.47)

atWm(t) + W m ( t ) F = B ( t ) W , ( t )

where B ( t ) E DR. The b . . ( t ) a n d B ( t ) a r e uniquely determined by the equa1J tions themselves. For example comparing the a j terms in (13.46) one finds f an, f n E C . An explib . , ( t ) = ln>Owi,j-nfn where f n is given via F = 4 0 n 1J c i t formula-for B ( t ) will be given l a t e r in a more general context.

198

ROBERT CARROLL

The e v o l u t i o n e q u a t i o n s above can be w r i t t e n i n a more compact m a t r i x form.

To t h i s end s e t

which enables one t o s o l v e them i n c l o s e d form. (13.48) (rl

5 = S(S) = ((w..)), 1J

O
rl

m5i<-,O<j<m -

= ((-w..)), 1J

= rl(S)) where i and j denote t h e i n d i c e s o f rows and columns r e s p e c t i v e l y

and t h e wij

o u t s i d e t h e o r i g i n a l range m 5 i <

my

0 5 j < m a r e supplemented

= 6 (0 5 i , j < m-1); w . . = -6ij (m 5 i,j < ?J Note t h e f o l l o w i n g r e l a t i a n (&+) n(aJ)ogsm = (W)im
a s ( 4 4 ) wij

ij

e v o l u t i o n equations a r e s a t i s f i e d b y 5 = ( ( t ) and n-=

m)

( c f . Remark 13.2).

We show now what

n(t).

The f i r s t obser-

v a t i o n , which i s an immediate consequence o f t h e c o n s t r u c t i o n o f q. says t h a t t h e e v o l u t i o n equations (13.46) a r e e q u i v a l e n t t o t h e m a t r i x system (6m)

atq

= B rl F

-

rlF(A) where BF and F(h) denote t h e m t r i c e s

i n t h e l a s t m a t r i x ) . Now i n o r d e r t o t r a n s f e r t h i s m a t r i x sys( 0 5 i,j < tem f o r 17 i n t o one f o r 5, we n o t e t h a t 5 and rl a r e connected v i a Q E = 0. B u t i f a m a t r i x 5' = ( ( 5 1 . ) ) o f t h e same s i z e a s 5 s a t i s f i e s t h e r e l a t i o n 1J 06' = 0, t h e n t h e r e i s a unique m a t r i x A = ( ( a . . ) I , 0 2 i,j < m y w i t h which 1J 5' can be w r i t t e n a s 5' = EA. To see t h i s d i v i d e t h e m a t r i c e s E , n , s ' i n t o 1 where W = ( ( w . . ) ) , m 5 i < my b l o c k s as 5 = ( w ) y rl = (-W,l), 6' = ( ' I - - ) , El+1J O(j<m, 5' +-(--)- - ( ( 5 i j ) ) , m 5 i i m ( 0 5 i < m), 0 5 j < m and 1 means u n i t m a t r i c e s o f a p p r o p r i a t e s i z e s . The r e l a t i o n Q E ' = 0 becomes WE;-

5:-

= 0.

Therefore 5' = EB w i t h B = 5 1 - which i s e v i d e n t l y unique.

Note t h a t we use ++,

--,+-, -+

as usual f o r v a r i o u s blocks; t h u s e.g.

A_-

0 5 i,j < m y e t c . I t f o l l o w s t h a t (0.) w i l l be e q u i v a l e n t t o t h e m a t r i x system (+*) at5 = F ( A ) S cAF; AF = (1 f w < i , j < m. &,, n i+n, j), O Indeed suppose ( W ) i s s a t i s f i e d . D i f f e r e n t i a t i n g 115 = 0 w i t h r e s p e c t t o t = ((aij)),

-

and u s i n g ( 6 . ) one has n(atE

-

F ( A ) S ) = 0.

t h a t (+*) i s s a t i s f i e d f o r some AF.

The remarks above t h e n ensure

To see t h a t AF has t h e form i n d i c a t e d

one d i v i d e s each m a t r i x i n (+*) i n t o b l o c k s A++, upper h a l f p a r t .

Then 0 = ( F ( A ) S )

One can t h u s d e r i v e (**)

-- -

A -- , e t c . and compares t h e

AF which p r o v i d e s t h e d e s i r e d form.

from ( 6 m ) and p a r a l l e l reasoning g i v e s t h e converse.

Now g i v e n Cauchy data E(t=O) one can s o l v e t h e e v o l u t i o n equations as

KP AND DEFORMATION THEORY (13.50)

199

S ( t ) = etF(”)E(t=O)h(t)-’; h ( t ) = etF(”S(t=0)

where exp(tF(A)) =

1:

tnF(A)’/n!.

--

To see this note t h a t t h e matrix S ( t ) =

e x p ( t F ( A ) ) S ( t = O ) s a t i s f i e s atS = F ( A ) S , S(t=O) = S(t=O). From this one can r e a d i l y derive (+*I for S ( t ) = S ( t ) h ( t ) - l w i t h A F unidentified; however AF will turn o u t automatically to have the form indicated ( e x e r c i s e ) .

REmARK 13-7 (KP AND DEF0RmACl0N UtE0Rg).

Now go to the KP hierarchy in t h e spSrit of deformation theory of D modules. Let ( R , a ) be a s above with N = IO.1, . . . I and NC = I-1,-2 ,... 1 a s i n Remark 13.2. Let E denote PSDO o r micn r o d i f f e r e n t i a l o p e r a t o r s o f t h e form E = ER = {Iz ana ; al E R , a n = 0 f o r n > some m (which may vary)}. As before one w r i t e s 1 ana -1 b n a n = cnan , i k = 0 ‘n = 1 ( k l a i a bn+k-f and note t h a t the sum i n cn i s f i n i t e ( r e c a l l f o r i < 0 ) . E is then a ring with D a subring and one w r i t e s f o r formal adj o i n t (1 a n a n ) * = 1 (-a)’aa, g i v i n g r i s e t o an anti-automorphism of E . The l e f t R submodules Ei = a n a n E E; a n = 0 f o r n > i } give a f i l t r a t i o n o f E. I t i s easy t o s e e t h a t a PSDO is i n v e r t i b l e ( i . e . has an inverse i n E ) i f a n d only i f t h e leading c o e f f i c i e n t i s i n v e r t i b l e i n R . We w r i t e again E = D @ E-’ w i t h ( )& denoting projection onto t h e f i r s t (+) and second ( - ) components. T h i n k now o f the KP hierarchy a s made u p o f a s e t of evolution equations describing an i n f i n i t e number of simultaneous time evolutions o f -2 + . . One has 3 equivalent representaa monic PSDO W = 1 + wla-’ + w2a t i o n s f o r t h e KP hierarchy, Lax, Zakharov-Shabat, and a t h i r d which i s c a l l e d the W representation i n [ TA1 l. Thus (Lax) a n L = [ Bn,L l, n = l , 2 , . ., Bn = (Ln)+, L = WaW-’, and (2-S) anBm - amBn + [ B m y B n I = 0, while f o r t h e W representation one considers evolution equations f o r W (+@) anW = BnW - Wan ( i . e . the Sat0 equation (13.8) - c f . a l s o §7,12). One notes here t h a t ( 0 . ) i t s e l f , under the requirement t h a t Bn i s to be a d i f f e r e n t i a l operator, uniquely determines the r e l a t i o n of Bn t o W; indeed from ( 0 . ) t h e Bn a r e writt e n as Bn = WanW-’ t anWW-l and t h e ( )+ p a r t of both s i d e s gives Bn = (Wan W-l)+ ( w h i c h simply i s a restatement o f the d e f i n i t i o n o f Bn above).

(i)

. .

.

As D modules now one considers l e f t D submodules 51 of property ( 0 4 ) E = .$ @ E-l ( d i r e c t sum) a n d proceed a s ( 0 6 ) t h e r e i s a unique R generator system IWiy i 2 01 a i - -1 ’J.aJ’ w i t h the s t r u c t u r e equations Wi+l - aW.i

1-,w.

E w i t h the s p l i t t i n g

before. First under of .$ of the form Wi = - w i,-1 Wo = 0. The

200

ROBERT CARROLL

e x i s t e n c e o f such an R g e n e r a t o r system, conversely, c h a r a c t e r i z e s l e f t D

E

modules o f

Secondly as a c o r o l l a r y S t u r n s

w i t h the s p l i t t i n g property.

D by a s i n g l e element, i . e . S = DW, which a l s o c h a r a c t e r i z e s l e f t D submodules o f E, t h e generator Wo b e i n g an a r b i t r a r y monic element o f E o f o r d e r zero (cf.§7,12). o u t t o be generated o v e r

Now as b e f o r e i n t r o d u c e an i n f i n i t e s e t o f t i m e e v o l u t i o n s v i a

an w i t h t i m e

v a r i a b l e s tn. L e t S ( t ) be t h e r e s u l t o f these t i m e e v o l u t i o n s ; k ( t ) i s now

DR submodule o f ER t h a t s a t i s f i e s t h e s p l i t t i n g c o n d i t i o n (*d) w i t h E = ER r e p l a c e d b y ER (R = R [It)]). According t o ( 6 * ) i n f o r m a l l y k ( t ) = S e x p ( - t 3 1 .) and a more r i g o r o u s f o r m u l a t i o n comes from (6.) as ( * * ) { aP + n Pa ; P E k ( t ) ) C k ( t ) , n 3 1. W i t h t h e R generators W i ( t ) o f k ( t ) one can

a

-tt2 -..

w r i t e ( W ) i n t h e form o f e v o l u t i o n equations ( d e t e r m i n i n g bijk) (13.51)

+ Wi(t)an

anWi(t)

=

1

anWo(t) + W o ( t ) a n = Bn(t)Wo(t)

bijnWj(t);

.i
W = Wo.

E

R

( n o t e however Bn = (Wa'W-')+

so t h e c o e f f i c i e n t s come

T h i s i s n o t h i n g b u t t h e W r e p r e s e n t a t i o n o f t h e KP h i e r a r c h y w i t h One can thus r e p r e s e n t t h e KP h i e r a r c h y as d e f o r m a t i o n s o f D moi -1 Note here (13.45) and a s s o c i a t e d remarks; e.g. Wi % ( a x W )+W.

dules.

From t h e c o e f f i c i e n t s w . . ( t ) o f t h e 1J

R generators we c o n s t r u c t now t h e f o l -

l o w i n g m a t r i c e s ( c f . (13.48) (13.52)

S ( t ) = ( ( w . . ( t ) ) ) (i€Z,j€Nc);

and supplement t h e w C

N ); w . . ( t ) 1J

n ( t ) = ((-wij(t)))

1J

= -fiij

(i€N,j€Z)

o u t s i d e o f t h e o r i g i n a l range a s w. . ( t ) = 6ij

(llj E N).

Again w r i t e A++,

f o r i,j E NC, e t c . and s e t An = ((6i,j-n)). one can r e w r i t e (13.51)

(i,j E

1J

Am-, e t c . w i t h A --

=

((aij))

F o l l o w i n g t h e p r e v i o u s argument

i n terms o f S ( t ) and q ( t ) y i e l d i n g t h e two equiva-

l e n t m a t r i x systems (13.53)

anS = A n S ( t )

-

S(t)An(t);

A n ( t ) = ((AnC(t)))--;

aq ,

= Bn(t)q(t)

-

n(t)An;

Bn(t) = ((n(tbn))++; n 2 1

The m a t r i c e s E ( t ) and n ( t ) may be t h o u g h t o f a s "frame m a t r i c e s " r e p r e s e n t i n g a p o i n t (moving w i t h t ) o f UGM.

I n t SE1

I

t h e same f a c t i s approached

KP HIERARCHY

201

from a somewhat d i f f e r e n t p o i n t o f view (see 511).

The approaches a r e dual

t o one another; i n [ S21 t h e 5 r e p r e s e n t a t i o n i s t h e fundamental p i c t u r e whereas i n [ SE1 1 t h e argument based o n BA f u n c t i o n s may be t h o u g h t o f a s prov i d i n g a r e a l i z a t i o n o f t h e II r e p r e s e n t a t i o n .

I n t h e case o f R = C [ [ x l l and

a = a / a x f o r example a complete system o f BA f u n c t i o n s i s g i v e n v i a formal Laurent s e r i e s (13.54)

i n a new formal v a r i a b l e X ( s p e c i a l parameter)

qi(x,t,h)

=

lz wijXjexp(xh

+

1"1 t n X n ) ,

i> 0

T h e i r l i n e a r combinations w i t h c o e f f i c i e n t s i n R form a t o 9.

D module isomorphic

I n terms o f m a t r i c e s t h e r e l a t i o n t o S t a k e s t h e compact form

which shows how t o connect t h e s e t t i n g o f [ SE1 1 w i t h t h e p r e s e n t s i t u a t i o n . ( c f . a l s o §11,18) From [ S4 I we i n d i c a t e another way o f l o o k i n g a t t h e correspondence between D modules and v e c t o r spaces which should c l a r i f y t h e 5,n,wij ( c f . (13.481, Consider t h e v e c t o r space €/Ex = V (as a l e f t E m d u l e w i t h (13.52), e t c . ) . <
EC n E-i-la

i E Z, where Ei was d e f i n e d e a r l i e r .

Think o f ei as a column

v e c t o r w i t h 1 i n t h e ith p o s i t i o n and 0 elsewhere so t h a t Xei = (i+l)ei+l and aei = ei-l b e f o r e (13.39),

(which i s what we have been d o i n g a l l a l o n g (13.49),

etc.).

l e c t i o n o f V o f t h e same s i z e as U4 = De-l r e f e r r e d t o as t h e o r i g i n i n UGM.

c f . K and

A

Now t h i n k o f UGM a s { v e c t o r subspaces U

V such t h a t dim(U n V o ) = dim(V/(U+Vo)) <

or 5

-

-1.

C

Thus r o u g h l y UGM i s t h e c o l -

= ~ & < O c v e v ac

E C),

which i s

A g e n e r i c s i t u a t i o n i n v o l v e s V = U 8 Vo

# 0 i n a p r e v i o u s terminology.

The s e t o f such g e n e r i c p o i n t s forms 4 an ( a f f i n e ) open dense subset UGM*of UGM. Then ( W = Wo n, li b e i n g g i v e n a s above) l e f t D submodules S

E a r e i n 1-1 correspondence w i t h p o i n t s UEUGM' U4> (S + U ) and 5 = { A E E; AU C U > ( U +. 9 ) . 4 4 An advantage o f w r i t i n g e v o l u t i o n e q u a t i o n s i n m a t r i x forms (e.g. (13.53)) v i a U = W-lU

= I v E V;

Sv

C

C

i s t h a t one can s o l v e them i n a c l o s e d form ( c f . (13.50)). t h e f o l l o w i n g s o l u t i o n formulas

Thus one o b t a i n s

ROBERT CARROLL

202 ( ( t ) = el;

(13.56)

h ( t ) = (el:

n ( t ) = k(t)-'

tnAnc(t=O)h(t)-';

n tnA S(t=O))-- j

k ( t ) = (n(t=O)e

-1:

n(t=O)e-17 'nAn; tnAn

++

t l a ...) =

To make sense o u t o f t h i s one w r i t e s weight(t:'

I1 nv and n m

intro-

duces a f i l t r a t i o n i n t o R [ [ t l l v i a (+=) R [ [ t l l v = { l i n e a r combinations o f monomials i n t w i t h w e i g h t 2 v l .

T h i s f i l t r a t i o n p l a y s t h e r o l e o f a norm

measuring t h e convergence o f i n f i n i t e s e r i e s i n R [ [ t l ] ( c f . a l s o §12). basic property i s : as n

-f

Given a sequence a n E R [ [ t l l v

t h e i n f i n i t e sum

a,

1:

,n

= 1,2

,..., w i t h

A vn

-+

-

an converges and d e f i n e s an element o f

To see t h i s c o n s i d e r t h e c o e f f i c i e n t o f a monomial i n t o). R[[t]lmin(v,,n I t has t h e form o f a n i n f i n i t e sum o f arising i n the i n f i n i t e series an.

1;

elements i n R.

Under t h e assumption however o n l y a f i n i t e number o f terms

can s u r v i v e i n t h a t sum which has t h e r e f o r e a meaning w i t h i n R.

The l a s t

statement i s e v i d e n t because t h e c o e f f i c i e n t s o f monomials l e s s t h a t min(vn, n 2 1 ) a l l vanish.. 1 Now go t o h ( t ) and s p l i t t;(t=O) i n t o two p i e c e s as c ( t = O ) = ( o ) (S(t=O))+-,

and a p p l y e x p ( 1 t,,An)

0

+ (w), W

=

t o each t e r m t o g e t m

(13.57)

h ( t ) = ho(t)+hl

( t ) ; ho = (e

tn")

-- ; hl

= (el,

t An 0 n )(w))--

The f i r s t t e r m h o ( t ) has a unique i n v e r s e s i n c e i t i s upper t r i a n g u l a r whose diagonal p a r t i s t h e u n i t m a t r i x so one t r i e s t o c o n s t r u c t h ( t ) - ' as (m*) h(t)-l =

1;

(-l)'h0(t)-'(hl

(t)ho(t)-l)n

and ask whether t h e m u l t i p l i c a t i o n

o f m a t r i c e s i n each term on t h e r i g h t makes sense, and whether t h e s e r i e s converges.

To t h i s end one notes t h a t ( 1 ) hoij(t)

. . ( t )= 01 J

(2) h hlij(t) ho

+

E

0 f o r i > j ( 3 ) hoij(t)

R[[tll-i

f o r a l l i,j.

E R[[tllj-i

are invertible i n R [ [ t ] ]

for a l l i,j f o r a l l t (4)

Hence t h e s e t o f a l l NC X NC m a t r i c e s h =

hl w i t h f a c t o r s s a t i s f y i n g (1)

-

( 4 ) forms a group under m a t r i x m u l t i -

p l i c a t i o n and t h e i n v e r s e i s g i v e n by (=*).

To see t h i s focus

t i b i l i t y o f each h ( t ) , t h e o t h e r p a r t b e i n g e a s i e r t o check.

on the inverFrom ( 1 ) - ( 3 )

h ( t ) i s i n v e r t i b l e w i t h an i n v e r s e s a t i s f y i n g t h e same p r o p e r t i e s . 0

Then

one can show t h e convergence o f t h e i n f i n i t e s e r i e s t h a t occur i n t h e evaluat i o n o f each e n t r y o f t h e nth power o f t h e m a t r i x hl(t)hO(t)-'. product o f t h i s estimate the ( i , j )

As a by-

e n t r y o f t h e nth power t u r n s o u t t o l i e

KP HIERARCHY

203

,...

on the r i g h t s i d e is i n R [ [ t l l - i + n . Hence the infinite sum over n = 1,Z entrywise convergent and f u l f i l l s condition ( 4 ) . One can thus make sense of h ( t ) - l w i t h the estimate ( ( h ( t ) - l ) l i j E R [ [ t l l - i - l f o r a l l i , j . From t h i s one deduces t h a t t h e matrix product on t h e right i n (13.56) makes sense a s a matrix w i t h e n t r i e s l y i n g i n R [ [ t l l . rl

Similar reasoning can be used f o r

i n (13.561.

For extensions of the D module point of view t o BA modules of g-dimensional p r i n c i p a l l y polarized abel ian v a r i e t i e s see [ NW1-3 1. One d e a l s w i t h commut a t i v e r i n g s o f p a r t i a l d i f f e r e n t i a l operators w i t h matrix c o e f f i c i e n t s and higher dimensional analogues of KP. There a r e many developments here involving a l g e b r a i c geometry, D modules, PDE, e t c . ( c f . a l s o [ MS4-6;S2,4;TAly3,41)

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205

CHAPTER 3

PHYSICS 14. H@)I;ON0WC QWIClIII FZEtDS. In a modern s p i r i t one goes back here t o [ S1, 31 and t h e r e a r e e x c e l l e n t review a r t i c l e s i n [Dl:J5,7,8;S5] ( c f . a l s o [ 54, 6;MW1,3;MBl;KADl;NYl;PM1-3;TY1,2]). The s u b j e c t involves a mixture o f f i e l d theory, monodromy, Painlev6 equations, s t a t i s t i c a l mechanics,. . , which somehow adheres and provides a nice introduction t o various mathematical techniques useful i n conformal f i e l d theory (CFT), quantum inverse s c a t t e r i n g (QIS), and o t h e r a r e a s ( c f . here a l s o §15,16,18 f o r r e l a t e d m a t e r i a l ) . Thus we r e f e r t o §l5,16,18 for motivation and simply present here ( w i t h some a t tempt a t coherence) a c o l l e c t i o n of techniques and ideas. One should remark however t h a t a f t e r some exposure t o what is going on t h e milange o f material seems quite n a t u r a l . Indeed t h e tau function T = ( $ ( a l ,L1 ). . . $ ( a n . L n ) ) i s a f a m i l i a r o b j e c t i n quantum f i e l d theory (QFT) a s a r e t h e ideas of normal ordering, Wick theorems, e t c . ( c f . here [BMl;CWl;GGl;LDl;RRl I ) . In f a c t the " c l a s s i c a l " use o f such vacuum expectation values in dealing w i t h Green's functions and Feynman propagators reveals already c e r t a i n underlying comb i n a t o r i a l m a t e r i a l . Here we have t h e special f e a t u r e of f i e l d operators lying i n t h e C l i f f o r d group which encodes t h e combinatorial information of i n t e r e s t i n this s e c t i o n ( o r a t l e a s t much o f i t ) . We will say more about this l a t e r . The connection t o monodrorny and Riemann-Hilbert (RH) problems i s f a s c i n a t i n g a s is t h e connection of monodromy to Cauchy-Riemann and Dirac operators ( t o be sketched b r i e f l y i n 422). We will see i n 120 how t h e i n t r o duction of Koba-Nielsen v a r i a b l e s o f strings connects hierarchy v a r i a b l e s and momentum v a r i a b l e s , s t r i n g amplitudes to tau functions, e t c . ( c f . a l s o 515,16 for o t h e r connections o f hierarchy v a r i a b l e s t o physics). The vertex operators of physics correspond to t h e vertex operators of 58 f o r example and one can c a r r y a l l of these correspondences t o Riemann surfaces (RS) where where e.g. Fay's t r i s e c a n t formula (generalized) f o r t h e t a functions corresponds t o an equivalent expression f o r tau functions i n a general boson-

.

ROBERT CARROLL

206

fermion correspondence (see §20,21 ) . A d e r i v a t i o n of t h e Hirota b i l i n e a r i d e n t i t y i n terms o f f r e e ferrnion operators i s given a t t h e end of this sect i o n . Hirota b i l i n e a r d i f f e r e n c e equations a r i s e in t h e string context a n d will be mentioned i n §20,21. (lI0NODRNt!! AND CHHE RH PR03tEIJ). We will e x t r a c t here from [ 55; S5] to get some basic material displayed and r e f e r to [Dl;J5-8;Sl-3,5] for f u r t h e r d e t a i l s . The amount o f d e t a i l i s i n f a c t enormous ( c f . [ 531) and one does not expect t o prove everything i n a sketch o f this s o r t . However [ J5;S51 a r e w r i t t e n very c l e a r l y , so some understanding should be conveyed via our “condensation”. Let us r e c a l l f i r s t t h a t two closed (continuous) paths y i : I X , y i ( 0 ) = a , y i (1 1 = b a r e homotopic (y, 2 y 2 ) i f t h e r e e x i s t s a continuous map F: I X I X such t h a t F(x,O) = y l ( x ) and F ( x , l ) = y 2 ( x ) . When a = b we speak o f closed paths. The c o l l e c t i o n o f equivalence c l a s s e s [ y ] o f closed paths a t a i s denoted by n,(X,a). One has a group s t r u c t u r e [ y ] ” a][ 8 1 defined via y ( x ) = a ( 2 x ) ( 0 5 x 5 L,) and y ( x ) = ~ ( 2 x - 1) (L, 5 x < 1 ) w i t h a - l ( x ) = a ( 1 - x ) and a ( x ) = constant i s t h e i d e n t i t y . One sees t h a t n,(X,a) = nl(X,b) and this group is c a l l e d n,(X) (fundamental group). Now we consider a f i r s t order system o f d i f f e r e n t i a l equations R€l!MRK 14.1

-+

-+

(14.1)

dY/dx =

1,” ( A v / ( x -

av))Y; A V

=

constant m X m matrix

having x = a l y . . . , a n ,m a s i t s regular s i n g u l a r i t i e s (note t h a t x % s p e c t r a l v a r i a b l e e v e n t u a l l y ) . Let Y = Y(x) denote the fundamental matrix s o l u t i o n of (14.1) normalized a s Y(xo) = I where xo E C - (a l , . . . , a n 1 . For any closed path y i n C - {al ,...,a n ) w i t h i t s endpoint xo t h e a n a l y t i c continuat i o n yY(x) of Y(x) induces a l i n e a r transformation (*) Y(x) yY(x) = Y(x) Here M = yY(xo) is a constant matrix depending on y only t h r o u g h i t s MY Y homotopy c l a s s [ y ] . I t i s c l e a r t h a t M = M M Thus t o each system Y, YY. Y, Y r (14.1), and i t s s o l u t i o n matrix, t h e r e i s a representation of the fundament a l group, t h e monodromy representation -+

.

.

(14.2)

nl(C

Since n1 ( C xO

Yk

-

-

P

{al y . . . y a n l ; x o )

.

-+

GL(m,C); Iyl -+My

,..

{al ,. . , a n l ; x o ) is a f r e e group w i t h n generators y1 .,y n’ Q a k , t h e monodromy group p(n,) is generated by the n

MONODROMY

207

matrices Mv M (v = l,...,n). One knows t h a t the Mv are e n t i r e functions yv o f the Ak. The Riemann problem refers t o the inversion o f these e n t i r e functions, namely ( A ) Given a l ,.. . , a n and M1 ,. . , M n E G L ( m , C ) find a system o f differential equations (14.1 ) whose monodromy representation p i s the preNote here t h a t t h i s problem (21s t scribed one : p ( [ y v 1 ) = Mv ( v = l , . . . , n ) . Hilbert problem) amounts t o showing t h a t a l l linear representations o f n l ( C - { a l , . .,a n I ) are monodromy representations arising from a differential equation. The word holonomy i n the t i t l e of 114 a r i s e s from considering representations o f IIl. On a RS one can relate t h i s t o the existence o f CR (or Dirac) operators and work in the moduli space ( c f . §17,22). For other points o v view see [ KX1;MJl I. One can phrase matters in terms o f connections, sheaves, Pfaffian systems, e t c . and we will refer t o some o f t h i s l a t e r . In order t o specify the solution uniquely i t i s necessary to refine the d a t a {a l , . . . y a n ,M 1 Mn) s l i g h t l y with an additional condition ( 0 ) The eigenAm = Av do n o t d i f f e r by integers. Then the norvalues o f A1,...,An, malized solution Y(x) can be shown t o have the following properties ( & ) ( 1 ) Y(xo) = I ( 2 ) Y(x) i s (multivalued) analytic and invertible for x 9 sly..., a n ,am = a. ( 3 ) a t x = a V Y has the form Y(x) = ? ( x ) ( x - a v ) L v where ? ( x ) i s holomorphic and invertible a t x = a V (for x = m replace x-a V by l / x ) . Here the exponent matrices Lv a r e related t o the monodromy t h r o u g h M v = exp(2ni L v ) so t h a t they constitute a refined n o t i o n o f the M v . I t i s also admissable f o r xo t o be one o f the branch points a v . In t h a t case condition ( 1 ) s h o u l d be replaced by ( 1 ' ) ^v"(xo) = 1 (1 % I ) .

.

.

,...,

-I?

Conversely,as noted by Riemann,such a matrix Y(x) ( i f i t e x i s t s ) should necessarily be a solution o f a differential equation o f the form (14.1). InI t i s single valued (the monodromy cancels), hodeed consider (dY/dx)Y-'. lomorphic except for x = a, ,.. .,an,= by (21, and has a local expression AY V ( x ) ( L v / ( x - a v ) ) ~ v ( x ) - l + holomorphic function a t x = a v (resp. holomorphic a t x = a ) ; hence i t must be a rational function In ( A / ( x - a v ) ) w i t h Av = Res (Y'Y-') a t x = a v which equals ~ v ( a v ) L v ~ V ( al1v.) - I t i s also unique since the ratio Y1 ( x ) Y 2 ( x ) - l of two such matrices Y1 , Y 2 i s necessarily constant Y1 (xo)Y2(xo)-' = 1 . Thus (+) differential equations Y ' = 1 ; (Av/(x-av))Y with singularity data a k y A k correspond t o multivalued matrices Y(x) with monodromy d a t a a k y L k (1 5 k 5 n ) . Then one can pose the Riemann problem

=

208

ROBERT CARROLL

i n t h e s t r i c t sense: Given a,,

...,,a,

and L,,

...,Ln

fdnd Y(x) s a t i s f y i n g (6)

L e t t h e unique s o l u t i o n be denoted by Y(x) = Y(xo,x)

(l’), (21, ( 3 ) .

= Y(xo,

Note t h a t a d i f f e r e n t c h o i c e o f n o r m a l i z a t i o n p o i n t xo g i v e s r i s e

x;akyLk). t o a s i m i l a r i t y t r a n s f o r m a t i o n f o r Av = Av(xo), namely = Y (xo ,x; ) - l A V (xo ) Y (xo ,xC,

xd I-’Y (xo ,XI ; Av(xd

Riemann suggested s t u d y i n g Y(xo,x;ak¶Lk) x,al,.

..,an

(m)

Y(xg,x)

= Y(xo,

1.

as a f u n c t i o n o f n+2 v a r i a b l e s xo,

..,an

and a s k i n g how Y depends on x0,al,.

when t h e monodromy i s

An answer by S c h l e s i n g e r was t h a t i n terms o f Y(x) it i s n e -

kept invariant.

c e s s a r y and s u f f i c i e n t (assuming

( 0 ) )

f o r t h e monodromy t o be preserved

t h a t Y(x) s h o u l d s a t i s f y a system o f l i n e a r PDE (14.3)

aY/axo =

-1;

(AV/(xo-aV))Y;

aY/aav = ( - ( A v / ( x - a v ) ) + AV/(xo-aV)Y

I n terms o f t h e c o e f f i c i e n t s Av = Av(xo;ak) t h e c o n d i t i o n becomes (14.4)

aAv/axo =

+ l/(xo-a,,,))

1 [ Am,Avl(-l/(xo-am)); (m # v ) ,

(where m 4 v i n t h e sums).

=

aAv/aam = [Am,AVl(

1 [Am,AV1

(l/(av-am)

wv

i n p a r t i c u l a r t h e c h o i c e xo =

(14.5)

aY/aav = -(Av/(x-av)Y; aAv/aam =

l/(xo-am))

(m = v )

The S c h l e s i n g e r equations (14.4) a r e t h e i n t e -

-

g r a b i l i t y c o n d i t i o n f o r (14.3) and (14.1). (14.4);

-

-l/(av-am)

I f xo = a v one m o d i f i e s (14.3)-

s i m p l i f i e s them t o be

- 1 [Am,Av]/(am-av)

REmAW 24.2 (f0RiXlI;# V I A Q U ” f 2

(m 4 v )

aAv/aam = [A,,Avl/(am-av)

FZELDS),

(m = v, m iv i n t h e sum)

The Riemann problem can be p u t

i n t o v a r i o u s f o r m u l a t i o n s among which i s t h e H i 1 b e r t approach v i a boundary v a l u e problems (RH problem).

Choose any s i m p l e c o n t o u r

t h e branch p o i n t s aly...,an,-;

M(F)

= M1

L e t Y+(x),

...Mn

al<=J

r

-M(f)

passing through = Mn

-a

an-2

n-1

D

M(5) = Mn-lMn Y-(x) be t h e branches o f Y(x) i n (D+) o r i n ( D - ) which a r e a n a l y -

t i c c o n t i n u a t i o n s o f each o t h e r through a p o r t i o n o f

r,

say between an and

-.

QUANTUM FIELDS

209

Then the monodromy property of Y(x) i s equivalent t o the following relation between the boundary values of Y+(x) (*) Y,(E+) = Y-(C-)M(E), 5 E r - {al, a n , ~ l where , Y+(E') = l i m Y+?x) ( x E D+) and M ( 5 ) i s a step function X-tE whose values are indicated above. Thus the Riemann problem amounts t o finding two matrices Y+(x) holomorphic and invertible in D*, growing a t most polynomially a t x = a l , an,-, and related on r t h r o u g h (*). This R H problem admits a description in the language of QFT. To see how t h i s works assume for convenience t h a t a l < < a a l l l i e on the real l i n e . n

...,

....

...

Let $*(')(x), $ ( j ) ( x ) ( j = 1 , . ..,m; x E R ) denote free fermion f i e l d operat o r s in one dimension, satisfying the following anticommutation relations ( [ a , b ] + = a b + ba) (*A) [ $ * ( " ( x ) , Q ( " ) ( x ' ) ] + = 6j j ' 6 ( x - x u ) ; [ $ * ( j ) ( x ) , $ * ( j ' ) (x')]+ = 0 = [$(j)(x),$(")(x')It ( c f . g8 a n d see here also Remark 14.3 a n d Remark 20.1). I t i s enough for the moment t o say t h a t these a r e operators parametrized by x E R in a sui ta ble sense, acting on some Hilbert space F. The l a t t e r space (resp. F*) has a distinguished vector Ivac) (resp. ( v a c l ) . For an operator 0, the inner product of 0lvac) with (vacl i s called the vacuum expectation value of 0, and i s denoted by ( 0 ) = ( vacl0( vac) . We will assume t h a t

The main point i s the + i O prescription in ( 1 4 . 6 ) which guarantees t h a t , for any 0, the expectation values ( 0 q ~ ( ~ ) ( x )(01j*(~ ), )(x)) (resp. ( $ ( i ) ( x ) O ) , ( q ~ * ( ~ ) ( x ) 0a)s) functions of x are analytically prolongable to the lower o r upper complex half planes ( I m x < 0 or Imx > 0 ) (see below for more on t h i s ) . Now l e t v be a f i e l d operator satisfying the rules

where (**) M(x) exp(2siLv) ( x < (14.8)

M1(x) ...Mn(x); Mv(x) = exp(2siLve(-x+av)) = 1 ( x > a,,), = a,,). Let Y ( x ) be m X m matrices whose ( j , k ) elements are =

Y + ( x ) ~ ~= 2 ~ i ( x - x o ) ~ $ * ~ ~ ~ ( ~ o ) $rP);~ k Y-jk ~ ( ~ =) P% ~'P$'k)(x) / ~

)/( Ip)

21 0

ROBERT CARROLL

where

'L

= 2 r i ( x - x o ~ J I * ( j ) ( x o ) and xo

i n f a c t these f u n c t i o n s ,Y = 1.

E

R i s a n o r m a l i z a t i o n parameter.

Then

a r e s o l u t i o n s t o t h e RH problem s a t i s f y i n g Y,(xo)

Indeed as mentioned above each element o f Y-(x) can be prolonged ana-

l y t i c a l l y t o Imx < 0 and Y+(x) has an a n a l y t i c c o n t i n u a t i o n t o Imx > 0. Note can be r e w r i t t e n a s Y+(x) = -~IT~(X-X~)(I)(~)(X) "9. J.) t(hxaot ) Y~, ) /i(n~ ((14.8) jk ) t h e remainder term i s p r o p o r t i o n a l t o 6(x-xo) by and

$*

(*A)

i s c a n c e l l e d by t h e f a c t o r 2 r i ( x - x o ) i n f r o n t ) ; t h e c o n t i n u a t i o n t o Imx < 0 v i a t h e x - x ' + i O t e r m was i n d i c a t e d above. s u r e t h a t Y,(x)

F i n a l l y t h e r e l a t i o n s (14.7) en-

The n o r m a l i z a t i o n c o n d i t i o n i s seen from

= Y+(x)M(x).

($*(~)(X~)$(~)(X)IP = )6 j k ( i / 2 ~ ) ( l / ( x o - x + i O ) ) ( ~ )

x

. 0

Now one wants t o f i n d

+ h o l o m r p h i c terms a t x

=

IP. Note a l s o t h a t one must check t h a t t h e m a t r i x

Y+(x) c o n s t r u c t e d v i a (14.8) s a t i s f i e s ( 6 ) and f o r t h i s one needs a c l o s e r

-

l o o k a t t h e s t r u c t u r e o f IP. One sees t h a t i f

v,v'

satisfy relations o f the

form (14.7) c o r r e s p o n d i n g t o M(x) and M ' ( x ) r e s p e c t i v e l y t h e n t h e p r o d u c t

vv'

has t h e same p r o p e r t y w i t h t h e m a t r i x M(x)M'(x).

I n f a c t the collection

o f such 9 form a m u l t i p l i c a t i v e group G ( t h e C l i f f o r d group) and t h e c o r r e s pondence

v

-+

M(x) g i v e s a r e p r e s e n t a t i o n o f G.

In particular it i s suffi-

c i e n t t o c o n s t r u c t 'p f o r n = 1 s i n c e t h e general case o f ( * a ) i s o b t a i n e d as But t h e Riemann problem w i t h two branch p o i n t s x = a, a p r o d u c t IP~...IP,,. a d m i t s an elementary s o l u t i o n (x-a)

v

p l i c i t l y the operator

L . T h i s w i l l enable one t o c o n s t r u c t ex-

f o r t h e Riemann problem.

REmARK 14.3 (CHE CLIFFORD GROW, FREE FERIIIIOW, AND NORVIAL BRDERINC). o p e r a t o r c p w i l l have t h e f o l l o w i n g p r o p e r t y . T($(j)(x))ip; T($*(j)(x))

The

If we w r i t e (*6) v ~ ) ( j ) ( x ) =

I P $ * ( ~ ) ( x )= T ( $ * ( j ) ( x ) ) ~ we see from (14.7) t h a t T ( ~ ) ( j ) ( x ) l , a r e a g a i n l i n e a r combinations o f f r e e f i e l d s .

To examine t h e

a l g e b r a i c s t r u c t u r e one l o o k s a t t h e s i t u a t i o n i n t h e case o f a f i n i t e numThus l e t W be a complex v e c t o r space o f even

b e r o f degrees o f freedom. dimension N, and l e t

(

,

)

be a nondegenerate symmetric b i l i n e a r form o n W.

L e t A(W) be t h e C l i f f o r d a l g e b r a determined by W w i t h t h e d e f i n i n g r e l a t i o n (*+) ww' + w ' w = ( w , w ' )

E C

f o r w,w' E W ( t h e r e a r e more d e t a i l s below).

Thus elements o f A(W) have t h e form a = < i < N).

- k -

elements g

lo

i,

...

. ..

N c ...ikwi, wi, ( 1 5 il 5 The C l i f f o r d group G(W) i s by d e f i n i t i o n t h e s e t o f i n v e r t i b l e

E

A(W) such t h a t

the l i n e a r transformation T 9

T ( w ) = gwg-' E W f o r any w E W. C l e a r l y 9 belongs t o t h e o r t h o g o n a l group O(W) = I T E

(*a)

CLIFFORD GROUP

21 1

= ( w , w ' ) for w,w' E Wl. Note adX w = [ X , w l E W for X = w E W; also exp(X)wexp(-X) = exp(adX)w E W. Some other basic facts about GL(W) a r e (1) An operator g belongs to G = GL(W) i f a n d only i f g = wl...wk for wI E W such that ( w . , w . ) t 0 (j = I , . . . , k ) ( 2 ) For J J g E G one defines a complex number n r ( g ) E C by n r ( g ) = gg* where g* i s de; 1 ci, il,wi ... w . for a as above. Then one has n r ( g l fined via a* = 1 I 2 g 2 ) = n r ( g l ) n r ( g 2 ) and nr(cg) = c n r ( g ) f o r c E C a n d g k E G ( 3 ) G i s a disj o i n t union o f G, = G n A, and G = G n A- (A, even partitions in a = 1 1 c ( ~ ) w... ~ w. , above; A- 2, odd partitions) ( 4 ) For g E G one can specify T lk 9 E O(W) via T ( w ) =&gwg-' (refining ( * m ) ) where f corresponds t o g E G,. I n 9 any event one has the exact sequence o f group homomorphisms (A*) 1 + G L ( 1 ) .+ G(W) + O(W) 1 ( g + Tg: G ( W ) .+ O ( W ) ) . Thus g i s uniquely determined from T u p t o a multiplicative constant. To make the correspondence e x p l i c i t we 9 need the idea o f normal ordering.

GL(W); ( T ( w ) , T ( w ' ) )

1 ajkwjwk

E A and

..,

-

-+

Thus l e t W = V* 8 V be a decomposition o f W into a direct sum o f ordered p a i r s via ( V * , V ) such that (.A) Cvf',v;> = 0 for vf', E V* a n d ( v 1 , v 2 ) = 0 for any v l , v2 E V . Thus V* a n d V generate in A(W) the Grassman algebras (exterior algebras) A ( V * ) and A ( V ) respectively. There exists a unique isomorphism of l e f t A ( V * ) and r i g h t A ( V ) modules called the normal ordering (A*) A ( W ) = A ( V * ) A A ( V ) .+ A ( W ) = A ( V * ) A ( V ) ( A : A : ) such t h a t :1: = 1

vi

-+

(we will emphasize below t h a t A ( W ) a n d A(W) a r e n o t isomorphic as algebras and note t h a t we are n o t yet using the decomposition V * 8 V here). In fact there will be considerable discussion o f A ( W ) , G ( W ) , normal ordering, e t c . i n 514,15,17,18,20,21 a n d we will often repeat definitions or constructions for convenience in reading. There i s also a certain confusion introduced by the use of different notations in the l i t e r a t u r e . We note that the Clifford group i s very important i n building t a u functions so no apology for excess i s needed. The notational confusion i n the l i t e r a t u r e a r i s e s from t o o many * symbols. T h u s with W = V* t l V , V* 'L creation operators, and V 'L annihilat i o n operators; one places v* t o the l e f t o f v i n normal ordering. However i n many o f the free fermion notations there will be JI: and Q n b o t h serving a s say annihilation operators for different indices n so s p l i t t i n g s W = Wcr 8 W will be more appropriate. I n the end i t doesn't matter o f course an since using commutation or anticommutation rules a n d Wick's theorem an

21 2

ROBERT CARROLL

o r d e r i n g w i t h c r e a t i o n o p e r a t o r s on t h e l e f t can always be achieved.

I n any

A(W) t h e t e r m o f degree 0 o f t h e c o r r e s p o n d i n g element i n A(W) 2 w i t h r e s p e c t t o t h e g r a d i n g A ( W ) = C 8 W @ A W I3 i s denoted by ( a ) E C event f o r a

E

...

and i s c a l l e d t h e vacuum e x p e c t a t i o n v a l u e o f a. f o r : : i s seen v i a Wick's theorem.

w1w2 = ( w w

1 2

) -t

wl...wk

(14.9)

...,kl

({l,

:w1w2:,(w1w2w~ =

Cm,

For wlY w2,..

-

= (w1w2)w3

The general p r e s c r i p t i o n E

( w w )w + ( w 2 w 3 ) w 1 1 3 2

1 sgn

,...,m r l

W one w r i t e s w1 = :wl:,

m m

U

...,n s l ,

{nly

ml

<

...

,...,

... < mry

n

n, c

+ :w w w

1 2 3:'

... wn .. . y

... <

n

S'

r =

..

< ik; Iil j,, .,i k, j,> = (1,. ..,2k}). The (il< j, ik < j k ; i1 < l a t t e r formula t e l l s us how t o compute ( a ) when ( w w ' ) f o r w y w ' E W i s known.

Note t h a t i f a

E

A- t h e n ( a ) = 0.

F u r t h e r comments a b o u t : : and normal o r -

d e r i n g w i l l be g i v e n i n Remark 20.4 ( c f . a l s o [FK2,51). t h a t a v a l u e ( a ) can a l s o be d e f i n e d as f o l l o w s .

We mention here

One d e f i n e s Trace on A(W)

= A as a C valued f u n c t i o n c h a r a c t e r i z e d by ( 1 ) T r ( a b ) = T r ( b a ) ( 2 ) T r ( a ) =

0 if a

E

A- ( 3 ) T r ( 1 ) = 2L2N ( N i s even).

Then f o r g E G+,

(Tr(g))'

= nr(g)

z+

+

T ) and g i v e n go E w i t h T r ( g o ) = 0 one can d e f i n e ( a ) O = Tr(ago)/ g T r ( g o ) which g i v e s an e x p e c t a t i o n v a l u e r e l a t i v e t o go. Given a b a s i s w 1' wN o f W and a b i l i n e a r form ( , ) one can d e f i n e N X N m a t r i c e s J and K

det(1

...,

= ( w w ) and K = ( w . w ). f o r go E G+. L e t To 2, T be t h e m a t r i x jk j' k jk ~ k o Then one can show t h a t K = 591 + To)-' and representation r e l a t i v e t o w j' ) such t h a t Conversely g i v e n ( one n o t e s t h a t ( W W ' ) ~ and ( W ' W ) ~ = ( W , W ' ) .

via J

( w w ' ) and ( w ' w ) = ( w , w ' )

t h e r e e x i s t s go

I n o r d e r t o compute : : (A

:A:

-+

E

G+ such t h a t ( w w ' ) =

(WW')~.

i n (A*)) i n t h e p r e s e n t s i t u a t i o n (where

t h e decomposition W = V* @ V i s n o t y e t used t o d e l i n e a t e c r e a t i o n and ann i h i l a t i o n o p e r a t o r s ) one can d e f i n e l e f t and r i g h t d e r i v a t i v e s i n A ( W ) v i a n j-1 Rw(w wl) = (-1) wn...(w.w)...wl and Lw (w1 wn) = n (-1) j - 1 W1". J

,...

c1

w1 ( wJ w . ) n. '. .where ~

...

,...,wn

E

-

W C A(W).

l1

Then we d e f i n e t h e normal p r o d u c t

-

Especially r e c u r s i v e l y by :wA: = w:A: :Lw(A): o r :Aw: = :A:w :Rw(A):. 2 one has f o r ? € A W % 41 R . w.w as i n (14.12) below w : e x p ( r ) : = : ( w + L w ( 8 ) Jk J k Now l e t g be an element o f G, exp(5): and :exp(?):w = :(w+Rw(a))exp(5):.

NORMAL ORDERING

21 3

such t h a t ( g ) # 0.

L e t T be t h e m a t r i x r e p r e s e n t a t i o n o f T w i t h r e s p e c t t o 9 so gw = 1 w gT Assume g i s o f t h e form g = ( g ) : e x p j Jk' (p"): , = La1 R . J, .J, , where R = R. Then from t h e formulas above 1 + R K = Jk J k T - RtKT o r R = (T-1 )(tKT + K 1 - l . Thus one can compute t h e normal p r o d u c t t h e b a s i s wl,...,wN

-

f o r m o f g i f one can i n v e r t tKT + K.

For c a l c u l a t i o n s based on t h i s see

(14.1 2 ) below. Now each decomposition W + V* @ V a l l o w s us t o r e a l i z e A(W) as a n " o p e r a t o r a l g e b r a " o f f r e e fermions.

For t h i s one i n t r o d u c e s two v e c t o r spaces (Fock

spaces) on which A(W) a c t s from t h e l e f t o r f r o m t h e r i g h t (4) F = A(W) mod A(W)V and F* = A(W) mod V*A(W). sidue class o f 1 (14.10)

€

F = A(W)lvac) = A(V*)lvac);

A(W) 2 EndC(F) = endC(F*).

and

v i a F* X F

-+

C: ((vacIal,a21vac))

number o p e r a t o r N (v*

E

V*),

I f we denote by I v a c ) (resp. ( v a c l ) t h e r e -

A(W) i n F (resp. F*) t h e n

E

( N ) = 0.

F* = (vaclA(W) = (VaClA(V)

These v e c t o r spaces a r e dual t o each o t h e r -+

(ala2)

F u r t h e r t h e r e e x i s t s a unique

I N,v*l

A(W) w i t h t h e p r o p e r t i e s [ N,vl = - v ( v E V ) ,

= v*

F ( r e s p . F*) can be decomposed i n t o a d i r e c t sum o f

eigenspaces Fk ( r e s p . F i ) c o r r e s p o n d i n g t o t h e eigenvalues k = 0,

...,+N o f

By choosing a b a s i s v'! E V* ( r e s p . v . E V ) t h e k - p a r t i c l e s e c t o r s Fk J J (resp. F i ) a r e spanned by t h e " s t a t e v e c t o r s " o f t h e form c V* j, j, J I " ' v? (vac) (resp. < v a c ( l c ' v. v ) where j, < < jk, and V*Fk = Jk jl J~ j, Fk+l w i t h VFk = Fk-l (F-l - 0 % FLaN+l = 0). I n t h i s sense elements o f V*

N.

..&

...

...

1

...

(resp. V ) a r e c a l l e d t h e c r e a t i o n (resp. a n n i h i l a t i o n ) o p e r a t o r s o f f r e e fermions.

One g e t s t h e r e f o r e a m i n i a t u r e v e r s i o n o f QFT.

Note ( c f . [ML2]) t h a t one f r e q u e n t l y d e f i n e s A(H) = C(H) = C l i f f o r d a l g e b r a o f H,

H a r e a l v e c t o r space o f dimension n w i t h i n n e r p r o d u c t ( , ), a s t h e

a l g e b r a generated b y 1 and x

E

H w i t h [ x , y l + = 2(x,y)

( i . e . C ( H ) = T(H)/J

where T(H) i s t h e t e n s o r a l g e b r a and J i s t h e two s i d e d i d e a l generated by elements x B y y 5 x

-

-

(x,y);

a l t e r n a t i v e l y J i s generated by elements x B y +

Given a n orthonormal b a s i s e

Z(x,y)). by 1 and t h e elements ei

. ..ei

1'

...,en

o f H, C(H) i s spanned

... < i < n ) ; n o t e e: = 1 and PnIn (n) = 2 . A ( r e d u c i b l e ) r e p r e O P

(1 < il <

0.e = - i . e . f o r i = j. Thus dimC(H) = 1 j J 1 s e n t a t i o n o f C(H) o n A ( H ) can be c o n s t r u c t e d as f o l l o w s .

R e c a l l t h e con-

21 4

ROBERT CARROLL

...

t r a c t i o n operator i x is defined by ( c f . Appendix A ) i ( e . /1 e. ) = x 11 1P A A A e A ... A e (2 means e is d e l e t e d ) . The rejk iP presentation of c H ) is obtained via x -+ y ( x ) = x A t i x where x A y = -y A x , i i = - i i and (x,y) = x A i t i x A . For n even, n = 2m, one deX Y YX’ Y Y f i n e s C(H)C via a k = % ( e k + i e k m ) , a i = %(ek - iek,,,,), so t h a t [ a * , a . l = 1 J t 6 i j (and o t h e r commutators a r e 0 ) . Then t h e fermionic Fock space F is C(H)C modulo t h e l e f t ideal generated by t h e a n n i h i l a t i o n operators a k ( F has a b a s i s a* a? and 1 = vac i n F ) . Obviously this corresponds t o t h e s i t u a il ’P t i o n above w i t h W = V * B V e t c . b u t we emphasize t h a t V* here N c r e a t i o n ope r a t o r s ( c f . Remark 20.1 and s e e a l s o [ W41 f o r a discussion of f r e e fermions and a1 gebraic curves).

...

-

...

(t0WCRUCCl0N 0F 0PERAMRd). F i r s t one goes t o a formula which recovers g E G(W) from T E O ( W ) . Choose any b a s i s w1 ,.. . ,wN of W and s e t 9 (note t h e d i s t i n c t i o n between ( w j , w k ) and ( w w ) )

REJWRK 14.4

j k

(14.11)

J = (((w.,w ))) = K t

J k

t K; K

( ( ( w w ) ) ) ; E,

=

j k

=

J-’K;

E- = J -1 t K

Some of t h e s e constructions have a l r e a d y been indicated i n Re( j , k 5 N). mark 14.3 b u t the r e p e t i t i o n will be i n s t r u c t i v e . I f E, + E-T i s i n v e r t i b l e 9 2 2 Rjkwjwk E A (W); ( g ) = nr(g)det(E, i.E-Tg); (14.12) g = ( g):eb:; p = R = ( ( R j k ) ) = (Tg ) ( E t

+

E-TS)

-1 J -1

2 one writes w.w i n A (W) when no confusion can a r i s e ) . Note J k here t h e f a c t o r 4 i n i s placed i n the exponent b. We have s e t n r ( g ) = gg* = g*g E C f o r g E G ( W ) w i t h * t h e unique antiautomorphism of A ( W ) such t h a t w* = -w (w E W; see remarks a f t e r ( 1 4 . 9 ) ) . Note t h a t exp(+p) = 1 + $0 2 t (1/2!)($) t i s always a f i n i t e s e r i e s in A ( W ) . More generally the c l o s u r e o f G ( W ) i n A(W) i s characterized a s ( j , k = l,...,N;

...

(14.13)

-

G(W)

{c:w ’...wmeb:;

c

E

C; w

1 ,...,w m

E

W;

p

2

E A (W)

( s e e [ S3,51 f o r proofs here - one i s applying Wick’s theorem t o the Clifford g r o u p ) . A few d e t a i l s from S51 will be useful l a t e r and i n §15 ( c f . [ SO1 I ) N Let w = 1 C . W . be an a r b i t r a r y element o f W. S e t t i n g = : e x p ( b ) : we 1 J J

CONSTRUCTION OF OPERATORS

21 5

c a l c u l a t e t h e normal p r o d u c t r e p r e s e n t a t i o n o f w y a n d <w by u s i n g t h e rule = :ww l . . . wm: + 1 : (-l)k-l<ww k > :w l ... A ...wm :. The result i s

w:w l . . . wm:

wG=

:w'eL,:; w' =

I1

jy

5''

= t(Cir...ych),

a r e determined To a t TI. This 9 g t amounts t o setting "c' = T ?', i.e. 1+RK = ( 1 - R K)T Solving f o r R gives 9 g' (14.12). Products o f elements o f t h e form g i n (14.12) a r e c a l c u l a t e d as t-. follows ($ = (w ,,..., w,)). Let g v = < g v > : e x p ( + p v ) : , p v = W R ~w ( v = I , ..., n), and s e t

t

=

t(cT,...yc;)

J

(14.15)

R = diag((Rk)); A =

-- .. . - .

-tK

0

-tK

:--- t K

K

l

K

t Assuming t h a t 1 - R A = (1-AR) is i n v e r t i b l e we h a v e (A+) - , . . . g n = ( g , . . . g n ) :exp($(GR1 . . .n % ) : ; ( gl . . . g n ) = ( g l ) ...( g n ) d e t ( l - RA)'; R12,.,, = (v,v = l , . . . , n ) where ( )vv d e n o t e s t h e ( p , v ) block a c c o r d i n g ((l-RA)-'R) vv to t h e p a r t i t i o n i n (14.15).

Go now t o t h e c o n s t r u c t i o n o f a n o p e r a t o r 9 . One p r o c e e d s n a i v e l y i n g o i n g from t h e d i s c r e t e t o t h e c o n t i n u o u s ; r e p l a c e sums b y i n t e g r a l s a n d a p p l y t h e f i n i t e t h e o r y a b o v e . As a v e c t o r s p a c e W o f free f e r m i o n s we c h o o s e t h e , t s p a c e o f 2m t u p l e s o f f u n c t i o n s w = ( w f ( x ) , . . . , w ~ ( x ) , w l ( x ) , . . . , w m ( x ) ) w i t h inner product (14.16)

(w,w') =

dx(wt(x)w!(x) 1: im " J J

By t h e i d e n t i f i c a t i o n w =

t

w.(x)w*'(x)) J

I_Idx(w?(x)$*(j)(x)

J Set here $*(j)(p)

J

t

wj(x)$(j)(x))

(14.16) is

e q u i v a l e n t t o (*A). = dxexp(ixp)$*(j)(x); $(j)(p) = d x e x p ( - i x p ) $ ( j ) ( x ) w h e r e ? * ( j ) ( - p ) , j ( j ) ( p ) a r e c r e a t i o n or a n n i h i l a t i o n operators according a s p < 0 o r p > 0 respectively, s a t i s f y i n g

,z

(14.17) [ $A*

J

[ ~*("(p),~*(")(p')lt

= 0 =

[;(j)(p),$ h ( j I )( p ' ) I t ; S j j , 6 ( p - p ' )

( p ) , $ ( j ' ) ( p l ) ~ + ; j * ( j ) ( - p ) l v a c ) = 0, $ ( j ) ( p ) l v a c ) = 0,

for p >

o

=

21 6

ROBERT CARROLL A

and ( v a c l $ * ( j ) ( - p )

= 0, ( v a c / J , ( j ) ( p ) = 0 f o r p < 0.

I n t h e basis J,*(j)(x),

$ ( j ) ( x ) t h e m a t r i c e s (14.11) a r e now i n t e g r a l o p e r a t o r s w i t h m a t r i x k e r n e l s (14.18)

E+(x,x')

= ( ? i / 2 a ) ( l / x - x 1 k i 0 ) 12,,

Thus one has chosen a decomposition W

V* @ V such t h a t

s i s t s o f 2m t u p l e s , whose components w;(x),

wi(x)

V) con-

V* (resp.

a r e boundary values o f

holomorphic f u n c t i o n s i n t h e upper (resp. l o w e r ) h a l f p l a n e Imx > 0 (resp. One l o o k s f o r a n element 9 o f t h e C l i f f o r d group G(W) correspond-

Imx < 0).

i n g t o t h e orthogonal transformation (14.19)

T9: ($*(x),?(x))

+

($*(x),?(x))

rM;)-' 1 M;x)

1 and T ( x ) = ( J , ( x ) ,..., q m ( x ) ) . Now i f 11 1 t h e r e e x i s t i n v e r t i b l e m a t r i c e s ,Y s a t i s f y i n g (*.) E-Y E = 0, E+f E- = 0, +1 Y_ = TY, t h e n R i n (14.12) i s g i v e n by (a*) R = (Y;' - Y: )(E,Y-+E Y,)J-'. t- -1 Here one chooses Y+- t o be m u l t i p l i c a t i o n by m a t r i x f u n c t i o n s d i a g ( Y+- ( x ) , -1 t h e n say t h a t Y+(x), Y, - ( x ) s h o u l d be bounY+(x)) and t h e c o n d i t i o n s -

where + ( x )

= t$*'(x)

,...,J,*m(x))

d a r y values o f holomorphic f u n c t i o n s o n Imx

k

0 and t h a t Y-(x) = Y,(x)M(x)

Thus once t h e RH problem i s s o l v e d an o p e r a t o r

f o r x E R.

y

satisfying

(14.7) can be found e x p l i c i t l y as (14.20) (Y;'(x)

-

9 = (9 )

e x p ( . f I l I dxdx'$(x)R(x,x')t$*(x')):;

Y:'(x))

1/2s) (i Y_ ( X )/ I (x-x'+iO)

-

R(x,x')

=

iY+(x' )/ ( x - x l - i o ) )

As remarked e a r l i e r t h e case o f o n l y two branch p o i n t s a,m a d m i t s o f an e l L Correspondingly one has t h e f o l l o w i n g

ementary s o l u t i o n Y+(x) = (x-a+iO) formula f o r

9

.

= 9 ( a r L ) ( w i t h normalization ( 4 ) = 1)

lI dxdx'$(x)R(x-a,x'-a,L)t$*(x')):; -L L -iaL iaL /(x-x'+io) - i e /(x-xl-io)); - 2 i ~ i n ( n ~-) xx -' ( 1 / 2 n ) ( i e (14.21)

9 (a,L)

= :exp(lz

R(x,x',L) x\

o

=

(x > 0)

L L and x- = 1x1 f o r x < 0.

By a p p l y i n g t h e p r o d u c t formula (A+) we may d e r i v e 1 an i n f i n i t e s e r i e s e x p r e s s i o n f o r Y i n g e n e r a l . Using (14.14) ( w :w'exp (%p): ) =

1 ( w w ' ) and t h e Neumann s e r i e s expansion (1

-

R A ) - l R = R + RAR +

...

CONSTRUCTION OF OPERATORS

Here t h e i n n e r sum i s o v e r vly...,v (1-6

mV

)(i/x+am-x'-aV-iE

mv 0) w i t h

E,,"

from 1 t o v and Amv(x,x') k-1 = sgn(m-v).

Now one can j u s t i f y t h e formal procedure. be shown t h a t (14.22)

21 7

= (l/ZV)

Making use o f ( 0 6 ) below i t can

i s convergent and a n a l y t i c f o r complex ( x o y x ) E (C-r,,,)

X ( C - r V ) p r o v i d e d t h a t t h e Ln a r e c l o s e t o 0 ( t h i n k here o f

- A

L

rm

am

)

(0.)

m '

...

and t o be p r e c i s e one argues f i r s t f o r t h e case h a l > > han; the orig2 i n a l c a s e is a t t a i n e d as a l i m i t i n L convergence). As an e x e r c i s e we n o t e t h a t i f a l l t h e eigenvalues o f L l i e i n t h e s t r i p ]Rex\ < gral operator (06) f ( x ) 2 bounded i n L (--,O;dx)m.

-+

ic

Local b e h a v i o r o f Y a t x = a, (14.23)

where

11

Y(xo,x)

= (x-a,)

+ then the

(dx'/2V)lxl L( i / x - x ' t i E ) l x ' l - L f ( x ' )

(E

inte0) i s

of t h e d e s i r e d t y p e i s v e r i f i e d w i t h (*+) as

- L m AY ( x .x)(x-an)LV; mv o

;m,(xo,x)

= 6,,

-

2ai(x-xo)

i n v o l v e s k # m and p i v and xo (resp. x ) a r e supposed t o l i e i n -

s i d e t h e c o n t o u r s C, and C, r e s p e c t i v e l y i n (*.).

Local behavior a t x =

m

can be checked through an argument u s i n g t h e e s t i m a t e 12 ( x , x ) l = 0 ( 1 / mv o I n t h i s manner f o r small L, 41 XI ) (1x1 -+ - ) which f o l l o w s from (14.22). QFT p r o v i d e s a method o f s o l v i n g t h e Riemann problem.

(Gill FWCCIQ)N$), One t u r n s now t o t h e vacuum e x p e c t a t i o n v a l u e 9(al,L,) . . . y ( anyLn)) i t s e l f ; t h i s can be i n t e r p e r t e d as a k i n d o f t a u

REmARK 14.5 T

=(

21 8

ROBERT CARROLL

f u n c t i o n and we d i s c u s s i t s r o l e and p r o p e r t i e s l a t e r ( c f . a l s o Remark 7.7). Note t h a t i n t h e absence o f d e f o r m a t i o n parameters t = ( t l y t 2 y . . . ) and xo r e p r e s e n t t h e d e f o r m a t i o n v a r i a b l e s . derivative o f

T

t h e av

One shows t h a t t h e l o g a r i t h m i c

i s e x p r e s s i b l e i n terms o f a s o l u t i o n t o t h e S c h l e s i n g e r ’ s

equations (14.5) v i a (14.24)

d 1og.r =

4 1 Tr

-

AmAv (dam

-

da,)/(a,

a“)

m#v The r i g h t hand s i d e r e p r e s e n t s a c l o s e d 1 - f o r m f o r any s o l u t i o n o f (14.5). This f a c t can be g e n e r a l i z e d t o t h e case a d m i t t i n g i r r e g u l a r s i n g u l a r i t i e s o f a r b i t r a r y rank ( t h e r e a r e some remarks o n t h i s below). t h e r e a r e s e v e r a l approaches.

F i r s t t h e formula (.+)

l o g r a s a Neumann s e r i e s & l o g det(1-RA) = n e l (RA)’(x,x’) less.

To d e r i v e (14.24)

enables one t o express

-41; Tr(RA)’/n.

However t h e k e r -

i s s i n g u l a r on t h e diagonal x = x ’ and i t s t r a c e i s meaning-

Nevertheless i t s d e r i v a t i v e does make sense as a convergent s e r i e s

On t h e o t h e r hand (14.28)

Y,(x)

=

1i m (xo-av) L v Y(xo,x) xo+av

A

= Y

vv (av ,x)(x-a,,)

L

V

i s t h e s o l u t i o n o f Riemann’s problem w i t h t h e n o r m a l i z a t i o n (6) ( 1 ’ ) . we s e t (‘*)&YJ;l (14.29) ) : A

=

1;

= Lv; Y,(’)

Akv)/(x-am)

t [ Y1(v),Lvl

then =

1

m# v

Am(v ) /(av-am);

If

TAU FUNCTIONS

21 9

Since A, w i t h d i f f e r e n t normalizations a r e s i m i l a r t o each o t h e r ( s e e Tr(Ap)AAV1)= Tr(AvAm) is independent of xo a n d we have (14.24).

(m))

Secondly one can c a r r y o u t the above procedure a t t h e level of f i e l d operat o r s . D i f f e r e n t i a t i o n o f (14.21) y i e l d s

(1 4.30) d a v ( a ,L) = :aa+p (a,L)e% (a ' L ) : Here ?,(a)

a

= 2 n i :$-L

(a )Lt?*L (a):* 1

= ( $ - L ( a ) , . . . , $ -mL ( a ) ) ,

:exp$p(a,L): a n d t h e row vectors $-,(a) = ($*L(a),...,$*;(a)) a r e given by

Ip(a,L)

(14.31)

( a ' L):

=

$-L(a) = - ( l / n ) L t dxd(x)SinrLlx-al-L-l; $*,(a) = ( l / n ) L t dxj;*(x)Sinn t L-lx-al tL-l

a

Moreover one has t h e formal o p e r a t o r expansion a

-L

A

(14.32)

$ ( x ) d a , L ) = ( ~ - ~ ( a , L+ l v - L - l ( a , L ) ( x - a )

...) ( x - a I L ;

t

which imp1 i e s (14.33)

t v v ( a y , x ) = 1 + Yl(")(x-a,)

t

...; (Y1( L O ) j k

-

Note a l s o t h a t Y, i n (14. = 2 n i ( x - a , k g ( a l ,L1 1.. .@('I(% ,Ly). .. IP(an,

Hence (14.27) follows from (14.30) and (14.33). 28) can be expressed via Y,(x)

~ , ) $ ~ ( )/( x ) v(al ,L,

1..

jk .v(an,Ln) ) .

Thirdly i t is a l s o possible t o s t a r t from t h e formula (14.12) b u t we omit this ( s e e [ S3,51 f o r d e t a i l s ) .

($COKE'$ I!IUCCIPLZERd, DEFORmACZON P A M E C E R Z , ECC.). We will not cover much more here o f the penentrating work on monodromy preserving transformations, deformation equations, s i n g u l a r i t y s t r u c t u r e s , Stoke's m u l t i p l i e r s , Painleve/equations, e t c . i n [ D2-4;FL5;J1,4-6,9,10;ITl;MWl ;OT1 ;S3, 51. Some material i n these d i r e c t i o n s based on [MBl;PMl ,2,4,5;TY1,2] will be looked a t in §22; the Ising model e t c . is b r i e f l y discussed in515,

REmARK 14.6

ROBERT CARROLL

220

a p p l i c a t i o n s t o 2-0 quantum g r a v i t y occur i n 116 (with monodromy and S t o k e ' s m u l t i p l i e r s displayed), and RH problems a r i s e a l s o i n §9. For o t h e r work on t h e s e and r e l a t e d t o p i c s s e e [ AB6-8;AD4;CJl;F013;FL3,8;El ;MBl;ITZ;NEI1-3; RM1 ;SK1 ;STE1 ;VM1;WIly21. For d i r e c t connections of monodromy t o CFT s e e Let us r e c a p i t u l a t e . One defines a Riemann problem of finding d i f [ BJ1 I. f e r e n t i a l equations f o r Y having a prescribed monodromy ( a n a n d M n o r L n given). T h i s problem is shown t o be equivalent t o a R H problem which can be expressed and solved i n terms o f f i e l d operators IP having commutation prope r t i e s based on t h e Mn or Ln (and a n ) . Such operatorscpbelong to the C l i f ford group w h i c h is examined b r i e f l y . Then t h e appropriate cp a r e constructed i n (14.20) via s o l u t i o n s Y, - of the R H problem o r a l t e r n a t i v e l y one can use (14.21) t o generate a s o l u t i o n of t h e R H problem via f i e l d operator cons i d e r a t i o n s ( c f . a l s o (14.8). Going t h e o t h e r way one uses monodromy and R H ideas t o solve physics problems i n §15,16. I t i s a l s o pointed o u t t h a t a tau function can be defined as a vacuum expectation and i s represented i n (14.24) (we will c l a r i f y this l a t e r ) . If one looks a t t h e references i n d i cated many o t h e r i n t e r e s t i n g and i n t r i c a t e r e s u l t s and ideas about monodromy e t c . have been s t u d i e d . For example one can look a t d i f f e r e n t i a l equations (14.3) w i t h mre complicated pole s t r u c t u r e ( e . g . A ( A ) = 1; - a v )k +1 t IYmAa,-kAk-l - use here A f o r x ) and monodromy data c o n s i s t s ( c f . [ S5;J5] f o r d e t a i l s ) of Stoke's m u l t i p l i e r s Sv connection matrices C v , and

12Av,-k/(A-

jy

V

(t-jcr 6 a BI y 1 ( a , ~ (m. VJ v v Then i n order t o have constant monodromy data ( S C , T o ) t h e Av,-k must sat-

exponents of monodromy (-asymptotic

behavior) TV.

'L

jy

i s f y c e r t a i n deformation equations extending t h e Schlesinger equations (14. 4 ) and i n v o l v i n g a v and t_" (1 5 p ( rv, v = 1 , . . . y n , ) a s independent dePya a r e constant and t h e o t h e r t -V will eventually formation variables ( t h e ja correspond t o hierarchy v a r i a b l e s - c f . here §15y16). Various techniques a r e used, including a tau function analogous t o (14.24). Questions which a r i s e involve e.g. ( 1 ) i n t e g r a t i o n of t h e deformation equations ( 2 ) studying t h a t s o l u t i o n s o f the deformation equations should the Painlev; property have a t most poles a s i d e from c e r t a i n c r i t i c a l v a r i e t i e s ( 3 ) showing t h e

ticr

-

function i s a n a l y t i c on t h e universal cover of C N - c r i t i c a l v a r i e t i e s ( N i s the number of deformation v a r i a b l e s a i y t V ). Note t h e s i t u a t i o n w i t h t k -P& V 'L t-ja i n some order corresponds t o looking a t a system w i t h A - A ( A , t ) and

STOKE'S MULTIPLIERS

221

requiring constant monodromy; the deformation v a r i a b l e s t k a r e however a l l p u t in t h e exponents o f nonodromy T_" The corresponding typical isospecj' t r a l type equations will be of the standard hierarchy form akY = BkY and combined w i t h a,Y = AY one has the isomonodromy equations ( a c t u a l l y i t i s t h e various compatabil i t y equations which should be c a l l e d the i s o s p e c t r a l o r isomonodromy equations - see [ IT1 1 f o r a good discussion of a l l t h i s ) . Questions such a s ( 1 ) - ( 3 ) a r e a t l e a s t p a r t i a l l y solved and we do n o t t r y t o come u p t o d a t e on these matters ( c f . [ D2-4;FL5;ITl;J1,4-6,9,10;S3,5;MBl I). Note t h a t i f the hierarchy v a r i a b l e s t i a r e absent the deformation equations involve a, and xo a s i n (14.4) and a r e r e l a t e d t o Y s a t i s f y i n g (14.3). W i t h t h e t k present via T I a s above t h e t k automatically e n t e r the tau function j and one has vertex operator equations for t h e ( J o s t ) matrices Y. One a l s o produces Hirota formulas e t c . For special s i t u a t i o n s ( w i t h t r i v i a l monodromy) one can d i r e c t l y r e l a t e t h e tau functions t o theta functions 'a l a KriEever

...,

( c f . 15) and Y corresponds t o t h e BA (matrix) function. Note however a l , 1 an,- E P - I b l , ,bN1 where the bi a r e branch points defining the appropr i a t e RS ( c f . S4,5). T h i s i s discussed i n [J61 f o r example ( c f . a l s o [IT1 I ) . 1 The points n - l ( a v ) (IT: RS P ) correspond t o p o i n t s a t m and t h e BA funct i o n s a r e constructed accordingly. The determine growth around bv here and the hierarchy v a r i a b l e s a r i s e r e l a t i v e t o p o i n t s a v .

...

-f

Ti

I t will be i n s t r u c t i v e here t o give an example of a l l this f o r NLS ( c f . [ I l l ; J 6 ] ) . We r e f e r to §9,10 f o r d e t a i l s on NLS and use here a d i f f e r e n t notation following CJ61. Thus t h e NCS equation can be w r i t t e n EXAIZfE 14.7,

(14.34)

aq/at2

=

2 2 -+(a q / a t l

t

2 2 2 2 2q r ) ; ar/at2 = +(a q / a t l + 2qr )

I f q and r s a t i s f y (14.34) t h e following equations a r e compatible (14.35)

8,log.r 2 = qr;a1210gr 2 = L , ( q r l - r q l ) ; a 221 0 g r =ki(qr11-2q1r1+q,1r+2q 2 r 2 )

Hence we can introduce a new dependent v a r i a b l e

T.

To define t h i s consider

222

ROBERT CARROLL A

A

We d i v i d e Y(x) i n t o two p a r t s F ( x ) and D(x) (em) Y(x) = F(x)D(x);

+ F2x-' +

Fix-'

..., D j

..., F j

1 ) = 0 on diagonal ; D(x) = 1

(j

( j 2 1 ) = diagonal m a t r i x .

F(x) = 1 t

+ D1 x - l + D2x"

+

Denote by d t h e e x t e r i o r d i f f e r e n t i a t i o n

w i t h r e s p e c t t o tl and t2. Suppose t h a t Y(x) s a t i s f i e s (&*) dY(x) = n ( x ) 1 Y(x) w i t h Q ( x ) r a t i o n a l i n x on P w i t h i t s o n l y p o l e a t x = m . Then n ( x ) i s u n i q u e l y determined by t h e formula Q ( x ) = F ( x ) d T ( x ) F ( x ) - l Q(x) i s determined b y T-2, O(x)F(x)

-

T-l,

We r e w r i t e (a*) as ( W ) dF(x) =

F1 and F2.

F ( x ) A ( x ) ; A(x) = d T ( x ) + d l o g D ( x ) and s e t F1 = (:

t h e c o e f f i c i e n t s o f dtl

F2, F3,...

m o d ( l / x ) . Thus

in t h e o f f diagonal p a r t o f

i n terms o f tl d e r i v a t i v e s o f q and r.

:).

Equating

( W ) one can s o l v e

For example n

The diagonal p a r t o f ( W ) determines A f X ) and t h e r e s t o f t o t h e NLS e q u a t i o n s .

(&A)

i s equivalent

Next, u s i n g (14.34) one can show t h a t ( & a ) dA(x) = 0

and h ( x ) g i v e s us an i n f i n i t e number o f l o c a l c o n s e r v a t i o n laws. and ( 6 0 ) t h e d e r i v a t i v e s o f DlY 02,

e t c . (aIl/at2

-

D;.

... a r e

determined.

From ( W )

For example

and a 1 2 / a t 2 a r e d i s p l a y e d i n [ 561) where Il =

D1 and I2 = 202

Now one extends t h e d e f i n i t i o n (14.24) i n d e f i n i n g a form w =

m = l,...,ny

1 wmy

) with

wm = -x=a

(14.39)

for

$"( x ) )

Tr(^vm( x)-'aX?d m

where l o c a l l y Ym(x) log(x-a,) dTV

-

n,

or for v =

dT:log(x-aV)

F(x)exp(?(x))

a,

Tm(x)

or d'T

= dT

1-; -

(?(x)

n,

T:.xj/(-j)! J dTolog(l/x)

:1

Tv.(x-av)-j/(-j)!

+

T'iog(l/x)) 0

t TO"

and d ' T V =

( d denotes d i f f e r e n t i a t i o n

r e l a t i v e to t variables). I n t h e p r e s e n t s i t u a t i o n ( 6 6 ) w = Tr(YldT-l) + 2 I-,Yl)dT-2). Then one can show t h a t w i s c l o s e d (dw = 0 ) so we can

Tr((Y2

-

i n t r o d u c e a new dependent v a r i a b l e 35).

T

v i a dl0g.r = w o r e q u i v a l e n t l y v i a (14.

We mention here a l s o [ IT1;561 f o r more examples.

RECMK 14.8

(0PERACO)R CHE0Rg f O R KP),

T h i s area began i n t h e c o n t e x t o f

papers on t r a n s f o r m a t i o n groups f o r s o l i t o n h i e r a r c h i e s ( c f . [ D1-4;51,4;MWl;

OPERATOR THEORY

223

One uses t h e f r e e f e r m i o n o p e r a t o r s here i n v a r i o u s ways ( c f .

S3;U1,4-6]).

a l s o §8,17) and we w i l l i n d i c a t e some o f t h e a l g e b r a f o l l o w i n g [ D1;SOl;SATl3;TA3;GB1 ;TK51 which 1 eads one t o t a u f u n c t i o n s , v e r t e x o p e r a t o r s , H i r o t a There w i l l be some d e l i b e r a t e r e -

formulas, e t c . and e v e n t u a l l y t o s t r i n g s .

p e t i t i o n o f d e f i n i t i o n s and n o t a t i o n as s p e c i f i e d e a r l i e r . t h e o p e r a t o r a l g e b r a A generated by $n and $;

[$m,$nl+ = [$is$;]+=

(14.40)

0;

(n

Z) w i t h

= AmYn

Set f o r n E Z,V = $ C$,

( n o t e a l s o 1 E A).

E

Thus c o n s i d e r

V* = @ C $ i ,

W = V @ V*,

and use

t o p r o v i d e a d u a l i t y between V and V* w i t h dual t h e p a i r i n g ( $ m $*) = 6 ' n m, n basis and I ) : Now . c o n s i d e r q u a d r a t i c o p e r a t o r s $,$: s a t i s f y i n g ((I+)

-

The $ $* w i t h 1 span an i n f i n i t e = 6 ' 3 $*, 6mnl~ml$,*. nrn m n m n dimensional L i e a l g e b r a $(V,V*) w i t h c o r r e s p o n d i n g group G(V,V*) =Cg E A;

[ J ,m$*,q n

m n

there e x i s t s g-l; cf.

(*.)).

gVg-l = V; gV*g-'

An o p e r a t o r g

E

G(V,V*)

= V*l

(subgroup o f t h e C l i f f o r d group

-

induces l i n e a r t r a n s f o r m a t i o n s on GL(V]

and GL(V*) v i a (14.41)

Wn

1 $mgamn;

=

$F;g =

As an example one c o n s i d e r s

((I.)

-

1 g$,*;lanm g = 1

-

(fim + Gin)(sjm + $n)($: + $;I, aij -- 'iij v a c ) i s c o n s t r u c t e d as usual, where $,lvac)

+

(J,;

-1 $i)($, + J,) g = -1

+ ($,,

The Fock space F = A1

+ 6jn).

= 0 ( n < 0 ) and $;lvac)

= 0 (n

T h i s i s a l e f t A module and t h e r e p r e s e n t a t i o n o f A i s c a l l e d t h e

> 0).

S i m i l a r l y one d e f i n e s a r i g h t A module v i a ( v a c I A =

Fock r e p r e s e n t a t i o n .

= 0 (n

F* where (vacIJ,,

This g i v e s a b i l i n e a r

< 0).

2 0 ) and (vacj$,* = 0 ( n

pa ir ing (14.41') where

(

( v a c l a l 5 a21vac) +(vaclala21vac)

v a c l l lvac

gree 0 ( c f .

)

(A*),

= (ala2)

= 1 ( r e c a l l ( a ) = vacuum e x p e c t a t i o n v a l u e (A&),

etc.).

Note however i n (Aa),

Q

t e r m o f de-

e t c . V* - c r e a -

(A&),

t i o n o p e r a t o r s e t c . and cannot be i d e n t i f i e d w i t h V* here ( c f . Remark 20.4 f o r more e x p l i c a t i o n ) . (14.42)

(vaclgw?

Again as an example f o r wi

...w*wn n "

.wlIvac)/(vaclglvac)

E

V, w;

= det((w

E

jk

V*,

))

g€G(V,V*)

ROBERT CARROLL

224

jk Jk

= (vaclgw*.w

J k

Ivac)/( v a c l g l v a c ) .

One w i l l show t h a t t a u f u n c t i o n s

( n < 0 ) - n o t e we want ( c f . 513 113 - we w i l l n o t

98 C$, CJI, UGM = G(V.V*)lvac)/GL(l) G(V,V*)lvac)/GL(l)

I n parV C V), and one can w r i t e 9 b e l a b o r t h e m a t t e r o f ideneverything i s equivalent).

t i f i c a t i o n here b u t a l i t t l e t h o u g h t shows t h a t $;)($, As a n o t h e r example l e t m < 0 5 n, v = (1 - ($; (JI; + $:)($, $;I($, As + $,))lvac) ($J; qn))lvac) = C$, ) tB C$, , ( t h e f i r s t sum f o r k < 0, k 9 m ) . -$*$ I v a c ) . Then V ( v ) = ( @ 8 B mn More g e n e r a l l y iif f (14.41) holds t h e n V i s r e p r e s e n t e d by t h e m a t r i x A = 9 E Z, n < 0 ) modulo A Q AP, P E E GL(Van), where Van = BC$, @C$, (n(0). ((amn)) (m,n E % AP, $C$, Q

For t h e t aa uu f u n c t i o n we i n t r o d u c e h i e r a r c h y v a r i a b l e s x1 ,x2,. H(x) =

1;: & 1'

x~$,$;+~ X~JI,$;+~

=

+

1;

xpAP xpAp ( c f . below).

.. and

s e t (+*)

Then e v i d e n t l y H ( x ) l v a c ) = 0

- -$:l$-k-l -$!l$-k-l b u t ( v a c l H ( x ) 9 0 ( n o t e $-k-l$!l = etc.). The x e v o l u t i o n o f an o p e r a t o r a i s d e f i n e d a s a ( x ) = exp(H(x))aexp(-H(x)) and H(x) E i(V,V*) w i t h A n A

exp(H(x)) E E G(V,V*) (" 2, formal c o m p l e t i o n as i n §12,13 and H a c t i o n here will d H ii nn og(V,V*)). lV.V*\\. NNote n t ~tt h aatt fnr [A \ \ I( csiui npnerrrdliia g nn f o r AA = [((6m+l,n)) o nnaallI) w i l l be vv ii aa aadH ; x AP t h e a c t i o n o f adH(x) on V i s r e p r e s e n t e d as 1 A p as i n d i c a t e d i n ( W ) . 7;; S-(x)kn. I tt Now r e c a l l t h e Shur polynomials d e f i n e d v i a e xxpp( f( 7k xx-kP) kp) = 1 Sn(x)kn. P f o l l o w s t h a t exp(H(x)) a c t i o n on V i s r e p r e s e n t e d by L.L.

(14.43)

el

xphP

2,

... 1

--.

S,(x) 11

s2(x)

*.-

The a c t i o n on V* i s c o n t r a g r a d i e n t . exp(H(x) )$,,exp(-H(x) (14.44)

1

etc

(14.45)

T(X,V)

'; 1;

E

$G(x) =

1 $;+,Sp(x)

G(V,V*))vac)

= ( v a c l g ( x ) l v a c ) = ( vacleH(X)glvac)

i s ddetermined b y vv (so T((XX, V, )V ) is e t e r m i n e d by

xpAp

%

x1 x2 00 x1 x, 0 x1

.... -

This y i e l d s i n p a r t i c u l a r f o r $,(x)

.

$,(XI = 1 $n-pSp(x);

D e f i n e now f o r v = g l v a c )

s1( x )

... ...

....-

= gg l v a c ) ) ..

As example A s aan n example

\

...

...

, =

TAU FUNCTIONS

225

f o r m,n 0 and correspondingly = 6mn f o r m,n < 0 ) . (thus ($*$n m > = In order to evaluate (14.45) one can r e f e r t o general r e s u l t s i n [ S3] which here reduce t o t h e following. Recall (14.41) and s e t ( ( d m n ) ) = ( ( a m n ) ) - ' . Let * be the antiautomorphism o f A such t h a t w*= w f o r w E W and then n r ( g ) by c2 = nr(g)det((dmn))m,n,o/det((amn)) 9 9 (m,n < 0 i n ( ( a m n ) ) ) . Then T ( X , V ) = c det((1, Sm-n(x)amn,)) ( n , n ' < 0 ) . We 9 will discuss the ( ) construction f o r t h e q n , $ i l a t e r i n more d e t a i l ; one should look a t constructions based on W = Wcr 8 Wan ( c f . Remark 20.1 and remarks a f t e r ( 1 4 . 8 ) ) . = gg* = g*g

Define c

i s a constant.

One now develops e s s e n t i a l l y t h e same material as i n 58 f o r o r b i t s of the vacuum and t h e boson fermion correspondence b u t i n a d i f f e r e n t way based on t h e f r e e fermion f i e l d operators ( c f . [ D1 I ) . T h i s approach is very productive when c a r r i e d i n t o CFT and s t r i n g theory a s i n SOl;SATl-41 (as well a s f o r g i v i n g perspective and i n s i g h t i n t o t h e whole complex o f ideas surrounding tau functions, t h e KP hierarchy, Grassmannians, Hirota equations, e t c . c f . a l s o 513). Thus r e f e r r i n g t o 113 consider Young diagrams a s i n (13.13) and w r i t e this a s Y = ( f l , . . . , f s ) where each row i is o f length f i . Note here i n 513 f o r an i r r e d u c i b l e representation of Sm characterized by [ X I = A > 0; 1xi = m}, one s e t s f l = x l , . , f m = A m' Characx2 2 {xi;+ m-

..

...

t e r s can be defined a s before and Y = ( m )

%

xy

=

Sm(x) w i t h Y

(l,...,l)

= Q ~ ( X ) = (-l)"'s,,,(-x). Set a l s o (+A) xmn(x) = (-1)"'C s - (-X)S~-~(X)= P#* P m ( - X ) S ~ - ~ ((Xc f). (14.46) and note Sk = 0 f o r k < 0 ) . For a (-l)mt S

xy

'1p s - l

Young diagram

P-m

n1 +I>

(14.47)

U

one has ( m =

(exercise

-

inl,

...,mk;

n

=

n1

Y . .

c f . (14.461, (14.42),

., n k )

(+A))

so t h e c h a r a c t e r polynomial x Y ( x ) %

226

ROBERT CARROLL

T(X,V)

f o r s u i t a b l e v.

Take now deg$,

= 1, deg$;

space o f charge

5 nk

< mk < 0

<

= -1

m operators.

... <

n

(degree

-

charge) and l e t A(m) be t h e sub-

...$GRqnk..

Now v e c t o r s ,$;

.qn, Ivac),

ml

<

. ..

form a b a s i s o f A(O)/vac) w h i l e t h e c h a r a c t e r p o l y -

,... I

B = C[x 1 ,x 2 B v i a (+*) a l v a c )

nomials c o n s t i t u t e a b a s i s o f an isomorphism A ( 0 ) l v a c

-+

-+

( c f . a l s o 58).

Thus one has

(vaclexp(H(x))alvac).

isomorphism can be extended t o a l l Alvac) as f o l l o w s .

This

Define a s t a t e o f

(+&I (nl = 0 ) . I n t r o d u c e a parameter z as i n § 8 t o map i s o m o r p h i c a l l y C[xl,x 2,...;z,z -1 1 v i a ( + 4 ) i:a l v a c ) (mlexp(H(x))alvac)zm.

charge n v i a

...q~;-~ Alvac)

-+

-+

lz

This can a l s o be expressed n i c e l y i n terms o f Young diagrams ( c f . [ D1 1).

-1

Next one expresses t h e a c t i o n o f A on C[x ,x o p e r a t o r s ( c f . 58).

Set

$(k) =

1 via differential

cz $nk1 n2'.; $ * (.k') = cz $ i k - n . *

z y z

The x-evolu-

t i o n i s d i a g o n a l i z e d by t h i s i n t h e form e H ( x ) $ ( k ) e - H ( x ) = eE('Yk)$(k);

(14.49) (e(x,k)

%a

=

1,*

n xnk ).

e H ( x ) $ * ( k ) e - H ( x ) = e-c(xYk)$*(k)

-

As i n 58 ( w i t h some abuse o f n o t a t i o n ) t a k e ? = (al, -1 ) ) w i t h X*(k) = expf-E(x,k))exp

...) and X(k) = exp(c(x,k))exp(-c(a,k

2: (c(2,k-l)). (14.50)

Then

( m l e H ( X ) $ ( k ) = km-lX(k)(m-l I e H ( x ) ; ( m l e H ( X ) $ * ( k ) = k-mX*(k)(m+l IeH(') I n words: The i n s e r t i o n o f f e r m i o n o p e r a t o r s w i t h i n b r a c k e t s

(exercise).

d i f f e r e n t i a l operators acting outside. (S(k)f)(x,z)

= zf(x,kz)

and ( S * ( k ) f ) ( x , z )

D e f i n e o p e r a t o r s ( c f . (8.21 ) ) = kz-'f(x,k-'z).

-1

o f $ ( k ) and $ * ( k ) on A l v a c ) = C[x1,x2,...;z,z $ ( k ) = X(k)S(k) and $*(k) = X*(k)S*(k).

T h i s can be used t o c a l c u l a t e N

s o i i t o n s o l u t i o n s t o KP f o r example ( c f . [ D1 t i o n s w,w* =

T(x,v),

as usual (w = XT/T; v = glvac)).

w*

=

Then t h e a c t i o n

1 can be i n t e r p e r t e d v i a (=*)

I).

Now d e f i n e t h e wave func-

X*T/T) f o r some g i v e n

T

f u n c t i o n (say

T

The H i r o t a b i l i n e a r i d e n t i t y now t a k e s t h e form

( n o t e from (14.50) w(x,k) exp ( H ( x ' f I$* ( k g 1 va c )/ k-r ( x

=

( 1 lexp(H(x))S.(k)glvac)/T(x)

and w*(x',k)

= (-1

1

HIROTA BILINEAR IDENTITY (14.51)

227

(1/2+i)j'c w(x,k)w*(x',k)dk =

= ( v a c l e H ( X ) g l v a c so ) X(k)r = X ( k ) ( OleH(X)g/O)= ( 1 leH(')$(k) Now use (14.41) (contragradient of g10) e t c . ) . + gJrng-' and ~r; -+ gJrig-') t o rewrite the sum i n (14.51) a s

(recall

(14.52 )

T(X,V)

cz

(1

IeH(x)g$mI vac X -1 I e H ( ' )g$; I vac )/ ~

( )Tx( x I )

JIi

(note $ng = 1 gqmamn a n d carrying the amn over t o we have 1 amn$;g = g$;) B u t e i t h e r JrmIvac) = 0 or $;lvac) = 0, giving the Hirota bilinear identity

I ww*dk = 0. This i s probably the neatest derivation of t h i s important formula available. One can now produce Hirota equations e t c . a s in §7,8 b u t we omit detail s . 15. IklNG lil0DEI; AND %0$EGAS, We develop here a few ideas based on the Ising model and inpenetrable Bose gas. In particular the monodromy picture

emerges along with t a u functions a n d t h i s provides some good examples related t o 5 1 4 . We refer especially t o [ BAXl ;BOGZ;BULl;GMl ;IT5-7;KF1-4;KAPly2; KHAl ;KH4; IG1; KAOl ;J5,9,11 ;MW3;MZ1; NR1; PM2-6; S5; TY1-3;WVl 1. Related work on Q I S appears in 123. RrmARK 15.1 ( S I N G B0DEI; - %MZC IDEAS AND l?I0N0DR0n!J)+ We begin with some comments o n the Ising mdel following [ J5,11;MW3;S51. In particular i t would be h a r d to improve on [ S 5 ] for a short survey so we follow t h i s ; note however the free fermion picture i s developed more in [ J4;MW3] ( i n terms of p . , q . below). The 2 dimensional Ising model w i t h nearest neighbor interacJ J tion i s an exactly solvable model in s t a t i s t i c a l mechanics. Consider a rec-

tangular l a t t i c e of s i z e N X N . To each s i t e ( j , k ) there i s attached a r a n dom variable u (spin) taking the values kl. The probability t h a t a specijk

f i c configuration u = {ajk], (1 5 j 5 M , 1 5 k 5 N, takes place i s given by Ziiexp(-BE(o)) (8-l = kT, k = Boltzman constant, T = temperature) where

(cyclic boundary conditions) with Ei > 0.

The normalization constant i s (*)

228

ROBERT CARROLL

- B E ( u ) (sum over a l l u = 51). ?4N = Iu jk point spin correlation functions

Here we a r e concerned w i t h the n

The Ising model admits a variety of approaches. One of the standard methods is t o introduce the transfer matrix V and the spin operator s = V k s.V- k M jk J both acting on a linear space of dimension 2 and t o rewrite (15.2) as (15.3)

(u

j, k l

...uj n kn)

= Tr(sj,

,...sjnkn VN)/Tr(V N )

(kl 5

... 5 k n )

(cf. [BAXl;Jll I ) . Denoting by 'vac) the unique normalized eigenvector o f V corresponding t o i t s largest eigenvalue one has for large M,N (15.4)

( u j , k,

...ujnkn)

-

(vacls

j,

4

... sj, kn (vac)

Onsager (cf. [ON1 I ) observed in effect t h a t i t i s possible t o associate free fermion operators p q . in such a way t h a t s j k y V belong t o the Clifford j' J group (cf. 814). This i s the key t o the solvability o f the problem. We do n o t give explicit formulas for p q . here; for expressions in terms o f spin j' J operators see [ J11;MW3] ( c f . also [ IZ1 I). In the limit of a n i n f i n i t e l a t t i c e M,N -+ m y there appears a c r i t i c a l ternperature Tc which sharply separates the ordered ( T < Tc) a n d disordered ( T > Tc) phases. The behavior o f (*) and (15.2) in the c r i t i c a l region T 2, T, i s o f particular interest. We will concentrate upon the scaling l i m i t (A) 2 + k v2 -+ m; constant IT-T I . ( j , k ) = ( a i , a t ) fixed ( v = l , . . . , n ) . T -+ Tc; j v c v v I t is possible t o take t h i s limit a t the level o f operators. Apart from a scaling factor proportional t o IT-Tc11/8 the spin operator s tends t o the jk following two dimensional fields with r e l a t i v i s t i c covariance (here one makes a "Wick rotation" and considers the region where x = ( x0 ,x 1 ) = cons t a n t IT-Tcl(-ij,k) i s real; : : denotes normal order a s usual). Thus for (15.5)

T I Tc, I#I,(x) = :ebF('):;

(15.6)

pF(x) =

fm m

T,

Tc, 4 F ( x ) = :$o(x)ebF(X):

lm( d u / 2 s l u l ) ( d u ' / 2 s l u ' I m

)(-i(u-u')/(u+u'-i0))

ISING MODEL

229

( x 0 +x1 )/2). In t h e above $ ( u ) s i g n i f i e s t h e c r e a t i o n ( u < 0 ) or a n n i h i l a t i o n ( u > 0 ) operator w i t h energy momentum ( p o , p l ) = ( & ( u t (J, =

+

J , ( u ) ; x- =

u -1 ),+jn(u-u -1 1).

I t obeys t h e canonical anticommutation r e l a t i o n s

(e) [

J,(u),

$ ( u ' ) I t = 2slu16(u+u'). Accordingly t h e n point c o r r e l a t i o n functions (15. 4 ) (with a s c a l e f a c t o r removed) tend r e s p e c t i v e l y t o ( $ F ( a l ) . . . $ F ( a n ) ) o r F F ( 4 ( a , ) ...$ ( a n ) ) a s T + Tc 7 0. The f r e e fermions p . i q a r e scaled t o J j give t h e massive f r e e r e l a t i v i s t i c Majorana f i e l d

From t h e e x p l i c i t formula (15.5) i f follows t h a t $,(x)

the C l i f f o r d group. lations

and $ F ( x ) b e l o n g t o

In f a c t they s a t i s f y , w i t h $ -+ ( x ) , the commutation re-

In w h a t follows we will show how the scaled n point F functions ( $ ( a l ) ( a n ) ) ( 4 = $F o r ) a r e c a l c u l a t e d via monodromy preserving deformation theory (see e.g. [ S1,3;PM2-6;TY1-3] f o r more d e t a i l ) .

f o r x-a space-like.

...+

+

The monodromy problem comes i n when one considers instead o f

(

$(al

1..

t h e following wave functions which c a r r y enough information t o give (15.9)

wFv(x) =

t

.+(an)) T~

etc.

( w ~ v + ( x ) Y w F v - ( x )wFvk(x) ); =

Hereafter we allow t h e variables x,a, ... t o be complex, in p a r t i c u l a r t o run 2 in the Euclidean space {xo ( = - i x ) E i R , x1 E R } . The notation z = -x-, z* t t = x , a v = -a;, a: = a i s used. Note t h a t due t o the square root i n (15.10) V

ROBERT CARROLL

2 30

t h e Euclidean continuation o f wF(x) i s a double valued function changing i t s s i g n when prolonged around the branch p o i n t x = a i n t h e Euclidean region. This was to be expected from the commutation r e l a t i o n s (15.8) ( c f . Remark 14.2). Since ( v a c ( $-+ ( x ) ( r e s p . $-+ ( x ) l v a c ) ) contains only the positive ( r e s p . negative) frequency p a r t t h e expectation value ( $ +- ( x ) $ F ( a ) ) ( r e s p . ( $ F ( a ) $5 ( x ) ) ) is a n a l y t i c a l l y extendable t o t h e upper (resp. lower) half Euclidean plane Im(xF-a') < 0 ( r e s p . Im(x*-aF) > 0 ) . For xo = a', x1 > a' they a r e a n a l y t i c continuations o f each o t h e r while f o r x1 < a' they d i f f e r by sign. This shows t h a t w F ( x ) has a monodromy -1 around x = a . For general n (15.9) is expressible a s an i n f i n i t e s e r i e s ( c f . 514) + + -1 (15.11) (+n~)-%~,,*(x) = I" ( d u / 2 a u ) ( O + i u ) -+%e -Im((x'-a;)u+(x - a v ) u )

+

0

e ( uo )e ( -EVo v,ul 1. . .e ( - E vk e

-Im((x--a-

- )uo+(avo-a

"I

VO

u (o+iuo

1'9i( uo+ul )/ (uo -ul +i0 )

( i ( u k+u )/ ( u k - u + i 0 )

+ + -1 + + -1 ) ~ ~ + . . . + ( a-a;)u + ( x -a ) u +...+(a - a , , ) ~ ) ) VK

where cmn = -1 (m < n), = 0 (m = n ) , ( u < 0 ) . Similarly t h e tau function

vo 0

= T~

Vk

1 (m > n) and e ( u ) = 1 ( u > 0 ) , = 0 = ( $ F ( a l ) . . . $ F ( a n ) ) can be w r i t t e n

The s e r i e s (15.11)-(15.12) a r e convergent f o r Im(-xF+a;) > 0 ( m = l , . . . , n ) and s u f f i c i e n t l y l a r g e Im(-a?a;), . . , I m ( - a ~ - l + a ~>) 0. ( a l s o via covariance under a: -+ exp(+ie)a' one can cover t h e region where a, i s real 1. The above V argument on t h e double valuedness o f w F v ( x ) a p p l i e s to the general case a s well.

.

Consider now w ( x ) = w r e s t r i c t e d t o t h e Euclidean region, where w s a t i s Fv f i e s t h e Euclidean Dirac equation m

w

=

0 outside o f al

,...,a n

ISING MODEL

231

+

This is a d i r e c t consequence o f the Dirac equation (a/ax )$+(x)

=

m$-(x);

(a/ax-)$-(x) = -m$+(x) f o r t h e Majorana f i e l d s $ -+ ( x ) . Secondly w has t h e monodromy property ( 6 ) w changes s i g n when prolonged around a l Y . . . , a and n f i n a l l y we have the growth c o n d i t i o n s (+) !w = O(l/(lz-avl’) a s z -+ a”, v =

-

l , . . . , n , and ( w l = O(exp(-Zmlzl)) a s I z I m. P r o p e r t i e s (15.13) (+) f o l low e i t h e r from (15.11) o r d i r e c t l y from (15.9). To s e e (+) a t the o p e r a t o r l e v e l we need the following s h o r t d i s t a n c e o p e r a t o r expansion obtained by applying (1 4.1 4 ) -+

(15.14)

F

(+m)-’$[x)$,(a)

= %$

(a).wo + (-l/m)(a/aa-)+

%+F(a)*wg + (l/m)(a/aa + ) +F (a)*wT + (-i/m)(a/aa-)+,(a)-w,

+

... +

+ +

Here wk = wk(-x-+a-,x -a ),

wi

... ;

... +

(+zm)-%(x)$F(a) = ( i / ~ ) + ~ ( a ) . +w ~ t

k

+

...

+-a + )

a r e s o l u t i o n s o f the Dirac modified Bessel f u n c t i o n ) + (m 2 zz*/(k+%)!) + )

= w*(-x-+a-,x

(mz)k-’((l/(k-%)!) =

(a).w, t

(-i/2)$F(a).wt + (-i/m)(a/aa )$,(a)-w?

equation w i t h t h e l o c a l behavior ( I v (1 5.1 5 ) wk ( z , z*)

F

[ (mz)k+f((l/(k+$)!)

...

+

(m 2 zz*/(k+3/2)!)

+ ...)

I=

where z = rexp(%ie), z* = rexp(-%ie). I t can be shown t h a t any l o c a l s o l u t i o n w o f (15.13) having the monodromy -1 and growing a t most a s O ( l / l z -N c*w*(z-a,z*k k has a unique expansion a s w = l~Nckwk(z-a,z*-a*) + 1” One notes a l s o (u ) azwk = mwk-l , azw; = m ~ ; , ~ , az*wk = mwktl , a z* w*k = mwc-l, MFwk = kwky MFw; = - kwc where M F denotes the i n f i n i t e s i m a l g e n e r a t o r w i t h s p i n L2 o f the Euclidean r o t a t i o n around the o r i g i n (**) M F = zaz - z*

a*).

az*

+ %(: -:).

RERARK 15.2

Thus i n p a r t i c u l a r (15.13)

-

(+) hold.

(DZFFERENCZAL EQllACZ0W FOR DAVE FU#CCZ0W).

-

One can show t h a t

the p r o p e r t i e s (15.13) (+) a r e s u f f i c i e n t t o c h a r a c t e r i z e the wave funct i o n s i n terms o f a system o f l i n e a r PDE. F i r s t one notes t h a t the v e c t o r o f functions s a t i s f y i n g (15.13) - (+) i s f i n i t e dimensionspace Wa ,,..-,an a1 and i t s d e s c r i p t i o n l e a d s t o e.g. T ~ . 2 (15.16) IF(wywI) = +m J f idz A dz*(wtwl* + w -w’*) = IF(wI,w)*

232

ROBERT CARROLL

(w,w' E wa ). P r o p e r t y ( 6 ) guarantees t h a t t h e i n t e g r a n d i s s i n g l e , ,a7 valued w h i l e (+) i n s u r e s t h e convergence. Hence (15.16) d e f i n e s a p o s i t i v e

,.. .

d e f i n i t e H e r m i t i a n i n n e r p r o d u c t on Wa I

,---,an

.

Using t h e Euclidean D i r a c

+ w w'*)

e q u a t i o n (15.13) one has imdz A dz* (wtw;*

= id(w+w'*dz)

so t h a t

o n l y t h e boundary t e r m c o n t r i b u t e s t o (15.16) and (15.17) (15.18) for w

E Wa I .,an T h i s i m p l i e s dimWa

,..

s i d e s 0 ) i n Wa

.

I n p a r t i c u l a r i f cAvl(w)

,...,an

,.. .,an

.

= 0 for v = 1

,.. .,n

t h e n w = 0.

< n and t h a t t h e r e a r e no bounded f u n c t i o n s (be-

-

Actually W(a)

i s e x a c t l y n dimensional, and making

use o f some f u n c t i o n a l a n a l y s i s one proves t h e e x i s t e n c e o f n elements

...,Cn (15.19) ,*-a:)

of W

(a)

w i t h normalized l o c a l behavior

wCm = 6 mvw o (z-av,z*-a;)

+ amvw 1 (z-a,,z*-ac)

+

z * - a(Z~+) a, (1 ~ ~ ~ a ~ ~ ) w ~ ( z - a ~+ , ...

m,v

n

t

Wcl

... + ~w,(:

z-a,,

= T,...,n)

s =1

In terms o f t h e c o e f f i c i e n t s Bmv t h e p o s i t i v e d e f i n i t e n e s s o f (15.16) i s e q u i v a l e n t t o (**)p=((Bmv)) i s n e g a t i v e d e f i n i t e and h e r m i t i a n , so B = -exp(EH) f o r a unique H = tH*. Moreover by c a l c u l a t i n g - / / i d ( w c m + (awcm+/ v we see t h a t ( * a ) a = ( ( a m v ) ) i s az)dz) = f f id(wCm-(awCm-/az)dz*) f o r m

*

symmetric. S i m i l a r l y (*i) -1 f id(wCmtwCvtdz) t (*+I B B = I, i . e . H = - t ~ .

= f f i d ( wCm-'Cv-

dz*) g i v e s

n o t e t h a t t h e Euclidean cv D i r a c o p e r a t o r (16.13) commutes w i t h MF i n (**). Hence MFwCv s a t i s f i e s (15. Nowto d e r i v e t h e d i f f e r e n t i a l equations f o r w

1 3 ) as w e l l as (6) - (+) e x c e p t f o r t h e growth c o n d i t i o n s a t On t h e o t h e r hand

azWCv

and az*wcv

z

= al,.

..,an.

share t h e same p r o p e r t i e s i n c l u d i n g t h e

with F Cv l i n e a r combinations o f azwCm, az*wCm and wcm and a p p l y i n g t h e f i n i t e dimens i o n a l i t y argument above one o b t a i n s a system o f l i n e a r d i f f e r e n t i a l equagrowth c o n J i t i o n O ( l / l z - a v l 312 ) .

t i o n s o f the form

By matching s i n g u l a r i t i e s o f M w

MODEL

ISING

233

Using t h e r e c u r s i o n formula ( m ) t h e c o e f f i c i e n t m a t r i c e s A,B,F

A = diag((a.)),

(15.21)

J

B = -G- 1A*G;

F = (a,mA);

are

G = -13 -1 = e-2H

The c o e f f i c i e n t s o f w g and w1 i n t h e l o c a l e x p a n s i o n o f ( l 5 . 2 0 ) g i v e respect i v e l y (*m)

GFG-’

= -(a*,mA*)

and

(A*)

(C2,mA)

w i t h C2 = ((cbv)(wFm))) (m,v = l,...,n).

+

a

-

Fa

-

mA* + G- 1mA*G = 0

I n p a r t i c u l a r t h e diagonal o f (A*)

gives (15.22)

a

ss

1

(fsvfvS/m(as-av))

siv

Since t h e Euclidean wave f u n c t i o n s w

Fv

+ ma; -

(G’lmA*G)SS

belong t o W

... ,an

a,

(+m)-hFS

combinations o f t h e b a s i s w

they are l i n e a r

(h) = c w The s h o r t d i s sv C V ’ cv tance expansion (15.19) enables us t o c a l c u l a t e t h e l o c a l b e h a v i o r o f wFs

and one o b t a i n s ( c f . (15.18))

F F t h e Euclidean c o n t i n u a t i o n o f ( ~ $ ~ ( a ~ ) . . .( @ a s ) ...I$( a v ) v I$F(an))/(~F(al)...~F(an)). S e t t i n g T = ( ( i e s v $S C = ( ( c s v ) ) and comp a r i n g t h e l o c a l b e h a v i o r o f (u) we have 1 - T = C, -1 T = C Since 1

...

tSv denotes F

where

)Iy

-

= 1 + G - l i s i n v e r t i b l e one o b t a i n s (A@)

2G(l+G)-l ‘Fv

,

,...,an So f a r t h e p o s i t i o n o f branch p o i n t s al,aT

,...,an,”;

as z

-+

a

V‘

-

= Tanh(H); C =

and t h e w

V

were r e -

Next l e t us c o n s i d e r t h e i r dependence on as,

It i s easy t o see t h a t t h e i r d e r i v a t i v e s awCv/aas,

a.; p r o p e r t i e s (15.13)

-

I n p a r t i c u l a r C i s i n v e r t i b l e and hence t h e

= exp(-H)(Cosh(H))-l.

a l s o span t h e v e c t o r space Wa

garded as f u n c t i o n s o f z,z*.

T = (l-G)(l+G)-’

.

awCv/aa:

share t h e

( + ) w i t h t h e growth c o n d i t i o n a t most O ( l / } ~ - a ~ 1 ~ / ~ )

Thus t h e same argument as above (matching o f t h e s i n g u l a r i t i e s )

y i e l d s t h e f o l l o w i n g system o f t o t a l d i f f e r e n t i a l e q u a t i o n s f o r wcv = (-dA (a/az)

-

G-’dA*

G(a/az*)

+

0)

2 34

ROBERT CARROLL

Here d denotes t h e e x t e r i o r d i f f e r e n t i a t i o n w i t h r e s p e c t t o av,a;, dw =

1;

((aw/aav)dav + (aw/aa;)da;)

o=

and

namely

( ( o s v ) ) i s a m a t r i x o f 1-forms

r e l a t e d t o F = ( ( f s v ) )i n (15.21) v i a (A&) B s v = -fs,((das-dav)/(as-av))

s # v and 0 f o r s = v.

The Euclidean D i r a c e q u a t i o n ( 1 5 . 1 3 ) t e q u a t i o n s

for (15.

20) and (15.24) c h a r a c t e r i z e ( t o w i t h i n a f i n i t e number o f i n t e g r a t i o n cons t a n t s ) t h e wave f u n c t i o n s w ~ , ~ . . . , w ~ as ~ functions o f the t o t a l s e t o f v a r i a b l e s z,z*,ak,a;

(1 5 k 5 n ) .

As t h e i n t e g r a b i l i t y c o n d i t i o n f o r them ( i . e .

c r o s s d i f f e r e n t i a t i o n terms) we o b t a i n a system o f n o n l i n e a r t o t a l d i f f e r e n t i a l equations, t h e d e f o r m a t i o n equations f o r t h e c o e f f i c i e n t s F,G, (15.25) Here

*

o*

dF = [ e , F l = ((o;,,))

+ m2 ([dA,G-lA*GI

here d i f f e r s i n s i g n f r o m

tF = -F;

tF* = GFG-’

I

( = -F*);

(A&)

(note t h e d e f i n i t i o n o f

S31) and F,G a r e s u b j e c t t o t h e symmetry (A+) t G = G - l = G*. I n d e r i v i n g (15.25) we have

I f Pv(az,az*)

used t h e f o l l o w i n g f a c t .

dG = -Go + o*G

+ [A,G-’dA*Gl);

i s t h e complex c o n j u g a t e o f

as

i s a d i f f e r e n t i a l o p e r a t o r w i t h con-

1;

P v ( a z y a z ~ ) w c v = 0 then Pv(az,az*) E 0 modulo s t a n t c o e f f i c i e n t s and i f 2 2 (a /azaz* m ) . T h i s i s proved by examining t h e s i n g u l a r i t i e s . As a n ex-

-

ample w r i t e (15.25) i n t h e s i m p l e s t n o n t r i v i a l case n = 2. I n view o f t h e

-:)

Euclidean c o v a r i a n c e we may s e t mA =

= - f ( d o / o ) ( P -:);

Then (15.25) reduces t o

(Am)

+* =

f * = f;

f = %o(d+/do);

o r e q u i v a l e n t l y t o a Painlev;

= mA* (0 = mla,-a21

and

J,

2 2 d $/do + (d+/do)/o = 2Sinh(2+)

equation e f the t h i r d k i n d f o r q = exp(-$)

namely (**) d2q/do2 = ( l / q ) ( d n / d o ) 2

-

( l / o ) d n / d o + n3

-l/q.

It i s i n s t r u c t i v e t o n o t e t h a t t h e d e f o r m a t i o n t h e o r y developed here i s i n -

c l u d e d i n t h e general scheme f o r ODE ( c f . 914).

Let

be a formal Laplace t r a n s f o r m a t i o n t o s o l v e t h e Euclidean D i r a c e q u a t i o n (15.13).

Then t h e systems (15.20) and (15.24) a r e r e w r i t t e n i n terms o f a n

n component

COI

umn v e c t o r

Gc

=

t

A

(wcl,.

..)S,

as

I S I N G MODEL

1

a u*w c = (G- mA*G/u2

(15.28)

-

F/u

235

n

-

mA)wC; dGC = (-G-

1

n

mdA*G/u t O-umdA)wC

I n o t h e r words, we a r e d e a l i n g w i t h t h e d e f o r m a t i o n t h e o r y o f l i n e a r o r d i n a r y d i f f e r e n t i a l e q u a t i o n s (15.28) a t u = 0 and u =

having i r r e g u l a r s i n g u l a r i t i e s o f rank 1

The system (15.25) i s n o t h i n g b u t t h e d e f o r m a t i o n equa-

m.

t i o n s f o r (15.28).

REmARK 15.3

(C0RREUWZ0N F1INCEZ0W).

closed expression f o r dlogrF.

Now i t i s r e l a t i v e l y easy t o o b t a i n a

The key i s t h e s h o r t d i s t a n c e expansion (15.

1 4 ) and from (15.9) and (15.14) i t f o l l o w s t h a t t h e second c o e f f i c i e n t s o f t h e l o c a l expansions o f wFv+(x) a r e g i v e n by ( c f . (15.18)) (15.29)

= (i/m)(aaV)CF/TF;

(M)-'c!v)(w~v)

1:

T h i s shows t h a t d l o g r F =

(a), (A*). (15.30)

(%)-'E/')(~F,,)

= -(i/m)aa*TF/TF Y

( ~ ) - ' i ( - i m c , ( v ) ( w F v ) d a v t imc,(v)(wFv)da:)

o r by

etc. dlogTF =

+ n . cvs(asvmdav Il -

(Ba*)svmda;)

= $Tr(l-T)amdA

t conjugate.

t t Using (15.22) and n o t i n g t h a t Tr(TamdA) = Tr(mdA a T) = -$Tr(T6) we have 2 (*A) d l o g r F = $Tr(To O*T) t & where ( 0 0 ) w = -+Tr(FO O*GFG-') t m Tr

-

(d(AA*)

-

G-'A*GdA

-

-

GAG-'dA*).

F i n a l l y t h e f i r s t term o f (*A) i s i n t e g r a -

t e d as 'idlog d e t CoshH by u s i n g (15.25).

The l - f o r m

(0.)

i s shown t o be

c l o s e d f o r any s o l u t i o n of t h e deformation e q u a t i o n s (15.25). o b t a i n t h e formula ( 0 6 ) T~ = c o n s t a n t ( d e t CoshH)' notes a p r i m i t i v e of

(0.).

':"

(15.31)

($F(al

exp(t,/ w ) where

(0.)

de-

except f o r t h e e x a c t

F ...@ (av)...@F(an))

(A*)

Iw

One remarks t h a t t h e c l o s e d 1 - f o r m a s s o c i a t e d

w i t h (15.28) i n t h e sense o f 114 c o i n c i d e s w i t h 2 d i f f e r e n t i a l t e r m m dTr(AA*). F o f ( E u c l i d e a n ) ( 9 (a )...@ (as) The r a t i o s o b t a i n e d d i r e c t l y from

Summing up we

as

to

(L+)'GF= i [ ( i T a n h ( H ) ) ) s v ( s + v ) and

F F F I...@ fa, I...@ (av2) ...9 (a, I

~ ~ - P f a f f i a n( i ( ( T a n h ( H ) ) ) v v , )

5

)...@F(an))

T~

are

=

( v , v ' = v1,...,vS) F

I n p a r t i c u l a r t h e (Eucl i d e a n ) c o r r e l a t i o n f u n c t i o n

T~

has a n e x p r e s s i o n (*=) -rF = c o n s t a n t - ( d e t i S i n h ( H ) )

4 exp(4fw).

=

(

@ (al

1..

F

.$ (a,))

For n = 2

236

ROBERT CARROLL

( 0 6 ) and

reduce t o

(Om)

e($k e dt(t,t)(Q2 - S i n h 2 $ ( t ) ) ) ' ~ = j c o n s t . Cosh(U)j

(15.32)

S i n h ($$ )

'I

The continuum model ~ ' ( x ) i s o f equal i n t e r e s t from t h e f i e l d t h e o r e t i c a l p o i n t o f view.

I t i s a n o n t r i v i a l massive model o f r e l a t i v i s t i c f i e l d theo-

r y whose n p o i n t f u n c t i o n s a r e known e x a c t l y i n a c l o s e d form.

RmARK 15.4 DR0lXg).

(B03E

BM

-

BWZC

IDEM, DE&ERmZNANC.S,

We g i v e now ( f o l l o w i n g [ J 9 , l l

t e d t o a n i m p e n e t r a b l e Bose gas.

W.l FUNC&IOW, AND m0N0

-

1) a b r i e f s k e t c h o f some ideas r e l a -

Monodromy comes up and some o t h e r connec-

t i o n s w i t h i n t e g r a l equations which w i l l be discussed l a t e r .

The c e n t r a l

p o i n t i s t h e computation o f t h e N p o i n t c o r r e l a t i o n f u n c t i o n u s i n g some t y p e o f n o n l i n e a r PDE (which i n c l u d e s t h e Painlev;

equation of the k i n d ) . We 2 s t a r t w i t h t h e NLS i n 1-D (6*) i$t= -$$, (6*) a s a c l a s s i c a l t c$*$ e q u a t i o n i s a v e r y t y p i c a l example o f a s o l i t o n e q u a t i o n (discussed i n 59)

.

b u t we c o n s i d e r i t now as a quantum e q u a t i o n .

C o n s i d e r + a s a quantum f i e l d

o p e r a t o r s a t i s f y i n g t h e equal t i m e commutation r e l a t i o n

((A)

t ) ] = 6(x-XI).

Consider a f i n i t e

T h i s corresponds t o an N body problem.

[ $(x,t),$*(x',

box 0 2 x 5 L ( n o n r e l a t i v i s t i c problem) and c o n s i d e r (4*) i n a second quant i z a t i o n d e s c r i p t i o n , corresponding t o a f i r s t q u a n t i z a t i o n problem w i t h N 2 H a m i l t o n i a n ( 6 0 ) HN = a /axi 2 t C I ~ < ~ ~ ( X ~( N- Xbody . ) problem Hamil-

-$Il

J

tonian with 6 function potentials).

The e x a c t meaning o f t h i s H a m i l t o n i a n i s t h a t one has an e i g e n v a l u e problem (66) a 2 2 = EqN w i t h boundary

conditions ( l 5. 33)

J"l

-IN 1

(ai = a/axi)

x.=x .+o=$~k.=x.-O; (ai-a j xi=x 1

J

1

J

.+O -(ai-a j )$N I xi=x .-o = 2c$N J J

+O a t t h e end o f (15.33) (assume here $N(xl, ji s symmetric and c f . [F4,5;THlJ f o r other discussion).

where $N i s e v a l u a t e d a t xi = x

. ..,xN)

The e q u a t i o n (6*) i s a l m o s t a f r e e e q u a t i o n and one can e x p e c t t h a t i n s i d e t h e box t h e s o l u t i o n has t h e e x p o n e n t i a l form (15.34)

QN(x;k;c)

= Z i l p Ap(k,c)eikP(l)x,t

f o r a l l xi such t h a t 0 5 x1 5

... 5 xN 5 L

...

t

( x = (xl

P(N)xNIP

E

,. ...xN;

k = kl....,kN).

SN

BOSE GAS

237

I t was shown i n ILF1 1 t h a t (15.34)

gives a s o l u t i o n f o r a r b i t r a r y N (note

ZN

%

i s a n o r m a l i z a t i o n constant, ki

c o n s t a n t momenta, and t h e sum i s o v e r

a l l permutations P w i t h c o e f f i c i e n t s Ap g i v e n by ( W ) Ap(k,c)

-

(l/ic)(kp(j)

=

-

II

(1 j<j I n o r d e r t h a t t h e boundary c o n d i t i o n s be s a t i s f i e d

kp(jl))).

i t i s necessary and s u f f i c i e n t t h a t one has c o e f f i c i e n t s o f t h e form ( W ) up

t o some p r o p o r t i o n a l i t y c o n s t a n t w i t h t h e ki values l i m i t e d by t h e c o n d i t i o n (6m) (-1 )N-lexp(ik.L)

J

boundary c o n d i t i o n s ) .

= njZj(l-(l/ic)(Kj-kjI

))/(l+(l/ic)(k

j

-k

) ) (periodic

.in

I t i s a l s o known t h a t f o r r e p u l s i v e i n t e r a c t i o n , i . e .

c > 0, t h e p o s s i b l e c h o i c e s o f k g i v e t h e complete s e t o f e i g e n v e c t o r s f o r t h i s problem.

The above e i g e n f u n c t i o n s were c o n s t r u c t e d i n

o f t h e Bethe ansatz, which i s discussed l a t e r . case h e r e o f c =

-.

LF1 I by means

We t r e a t o n l y t h e s p e c i a l

Then t h e boundary c o n d i t i o n s reduce t o o n l y one equa-

t i o n (+*) $N = 0 a t xi = x c r o s s each o t h e r ( i . e .

We have t h e n a system o f bosons which cannot j' i m p e n e t r a b l e ) . I n t h i s case t h e wave f u n c t i o n r e -

duces t o t h e s i m p l e form (15.35) Thus

$N =

I$N,FFI;

$N,FF

= det ((e'

j x j ) ) / ( N/LN)'

i s j u s t an a b s o l u t e v a l u e o f a d e t e r m i n a n t and t h e c o n d i t i o n (+*) i s

o b v i o u s l y s a t i s f i e d by t h e d e t e r m i n a n t a l f o r m which i s o f course antisymme-

t r i c (i.e.

a f r e e f e r m i o n wave f u n c t i o n ) b u t w i l l be symmetric upon t a k i n g

the absolute value.

Thus i n a sense t h e problem reduces t o a f r e e f e r m i o n Recall

problem and t h i s i s more apparent i n t h e language o f QIS (see 123).

t h a t t h e c l a s s i c a l NLS i s r e l a t e d t o a s c a t t e r i n g problem o n t h e l i n e v i a i n v e r s e s c a t t e r i n g w i t h r e f l e c t i o n c o e f f i c i e n t R = R(k,$,$*), as a f u n c t i o n o f t h e f i e l d s .

t h o u g h t o f now

U s i n g GLM techniques one r e c o v e r s

the fields The R

$ v i a R and i n t h e quantum s i t u a t i o n t h e r e a r e s i m i l a r procedures.

a r e now o p e r a t o r s which can be f o r m a l l y w r i t t e n i n terms o f s e r i e s i n v o l v i n g t h e f i e l d s $,$*

and one can show t h a t R s a t i s f i e s a s i m p l e commutation r e l a -

t i o n which i n t h e p r e s e n t case (any c ) i s (+A) R ( k ) R ( k ' ) = - ( ( l - ( l / i c ) ( k - k ' ) ) /

(1 i ( 1/ i c ) (k- ' ) ) R ( k ' ) R ( k ) . When c = m one o b t a i n s t h e n (+*) [ R ( k ) , R ( k ' ) l + = R(k) i s a f r e e fermion and i n f a c t t h e formula t o r e c o v e r $ and $*

0 (i.e.

from R shows t h a t $ ( z ) belongs t o t h e C l i f f o r d group (see below).

Hence one

m i g h t expect t h a t i n t h i s example t h e monodromy t e c h n i q u e can a l s o be used. The problem

s thus t o compute t h e n p a r t i c l e reduced d e n s i t y m a t r i x

2 38

ROBERT CARROLL

(1 5.36) Thus one has a 2n p o i n t c o r r e l a t i o n f u n c t i o n where

(N/L = po f i x e d ) .

$ONL

r e p r e s e n t s t h e ground s t a t e wave f u n c t i o n , po t u r n s o u t t o be t h e d e n s i t y o f p a r t i c l e s , and $NL(x;c)

i s t h e n o r m a l i z e d ground s t a t e N p a r t i c l e wave func-

I f p0 = 0 t h e problem i s t r i v i a l s i n c e

tion.

obtains 6 functions.

$ONL

i s t h e n t h e vacuum and one

The non t r i v i a l case i s t h a t o f f i x e d f i n i t e p o (which

i n t h e r e l a t i v i s t i c QFT corresponds t o t r e a t i n g t h e f i l l e d D i r a c sea). L e t

us n o t e t h a t (15.36) can be w r i t t e n as (1 5 - 3 6

pn =

YN;C)$NL(X; I n what f o l l o w s c = exp((Zni/L)(j

-

. . .&L

N! / (N-n) !

1

m

Y

* * *

Yx,!,YYn+l

dyntl

.. .dyN$*( x1 ,. . .,xn, yntl ,. ..,

.YYN;c)

9 . .

i s assumed and i n t h a t case $NL = I(l/(N!LN)’)det

4(N+l))xk)j,k =

I

ly...,N

and (15.40) below w i l l r e s u l t .

We w i l l i n d i c a t e t h e Fredholm i n t e g r a l e q u a t i o n t e c h n i q u e here ( f o l l o w i n g [ Jll

u(x)

I).

F i r s t suppose we have a Fredholm i n t e g r a l e q u a t i o n o f t h e form ( + 6 )

- XI

K(x,x‘)u(x’)dx’

= f ( x ) (1-D case).

The F d e t e r m i n a n t i s d e f i n e d

a s usual v i a ( c f . 17)

We a l s o w r i t e t h e rth m i n o r as ( c f . 57)

Now l e t (++) k ( x , x ’ )

-

= Sin(x-x’)/(x-x’).

density matrix with coupling c =

Results o f [ L G l ] s t a t e t h a t the

can be w r i t t e n as a Fredholm m i n o r d e t e r -

m i n a n t o f t h e form ( w i t h k e r n e l (++)) (15.39)

pn(X,X’,m)

where al

<

... <

= k(-z)-nA

...,

( x : y * * ” x ? ;2/a); I x,, X”

aZn i s a r e o r d e r i n g o f xl,

...,xn,Xi,...,xA

and I i s t h e

239

BOSE GAS

..

.

union of closed nonoverlapping i n t e r v a l s I = [ al ,a,] U . U [ a?,,-, 'a2,,] ( f o r l a c k of space we omit any d e r i v a t i o n of t h i s ) . Note t h a t t h e f r e e f e r mion d e n s i t y matrix a r i s e s as indicated above, namely

Equation (15.40) is only t h e f i r s t term of t h e Fredholm minor ( w i t h X = 0 ) . The case h = 2 / r corresponds to impenetrable bosons. We will t r e a t now t h e case o f general X and will derive the nonlinear PDE s a t i s f i e d by the Fredholm minors and determinants. To obtain d i f f e r e n t i a l equations we consider f i r s t t h e resolvant kernel. For t h e special case of f ( x ) = K(x,x") i n (+6) t h e resolvent u ( x ) = RI(x,x1') is obtained via (15.41)

(xo

=

R I ( ~ . ~ " y X= ) 1 ;

x, x

P +1

=

k(x0,xl)

...k ( x P' x P+l )dx,...d x P

x") and the Fredholm minor i s r e l a t e d t o this by

A I ( xx;'...'x; 'y.''yxn

(15.42)

XPiI ...II

;A) = (-1 )'A1(A)det( ( R I ( x j y x i ; X ) ) )

Now we modify this d e f i n i t i o n s l i g h t l y in order t o g e t some a n a l y t i c funct i o n s . In (15.41) x and x" a r e a r b i t r a r y complex numbers a n d we change (15. 41) by i n s e r t i n g (x-a,). . (x-a2,,)) contour over a and one e x t r a c t s

.

t h e functions (++) h I ( x ) = (1/2Ti)10g((x-al) . . . ( x - a 2 n - l ) / between t h e kernels in (1 5.41 ) and r u n n i n g t h e i n t e g r a l s CI containing I w i t h x and x" o u t s i d e C I . Then l e t x" m -f

a function of x, namely

Consider a1 so

2i (xo-x1 ) ) h I (xl (Si n ( x l -x2 )/ ( x1 -x2 1). ( H = h ( x )exp(tix,)). I P

..h I (xp-l

( S i n(xp-l - x p ) / (xp-l -xp) )H

Summing (15.44) o n t h e upper indices one o b t a i n s

240 (+m)

ROBERT CARROLL R+yI(xyA) -

= R

+ f, I

(x,A)

I n a l l t h e above formulas t h e p o i n t

+ R;yI(x,A).

-

x l i e s o u t s i d e t h e c o n t o u r C r y t h e f u n c t i o n hI(x) c l o s e t o aiy

and t h e f u n c t i o n S i n ( x - x ' ) / ( x - x l )

has s i n g u l a r i t i e s f o r x

i s regular.

I n o r d e r t o see

t h e b e h a v i o r near t h e ai one i n t r o d u c e s y e t a n o t h e r f u n c t i o n +I

well.

(x,A) which fY1 i n c l u d i n g x as

N

i s t h e same as R~yI(x.A)

b u t w i t h an i n t e g r a t i o n c o n t o u r CI

Then

.-+I

R-f,I (x,A) and R+yI(xyA)

a r e holomorphic a t x = ai and t h e s i n g u l a r i t y s t r u c -

t u r e o f R ? I T i s - e v i d e n t from (15.45).

One t h e n d e f i n e s

For x c l o s e t o ai t h e f u n c t i o n Y has t h e l o c a l decomposition Y(x) = AS(x)((x-a,)

(15.47)

...(x-a2n-l)/(x-a2)

...(x-a,,))

L ; L = ( 0o k i l l

where $ ( x ) i s holomorphic and i n v e r t i b l e ( d e t S = 1 ) and t h e power must be understood as

(*+I

aL = exp(L1oga) = ( o

liiApga). Thus t h e s i n g u l a r i t y

s t r u c t u r e o f Y i s c o m p l e t e l y determined by (15.47) and we can a l s o compute t h e monodromy around t h e p o i n t s ai which i s g i v e n by t h e m a t r i x e q u a t i o n (mA)

Y(x)

-+

Y(x)(b )';'

(+

' I ,

i odd;

-

' I ,

When z

i even).

-,Y,

has a v e r y

s i m p l e s t r u c t u r e (ma) ym(x) = ( I +

exp(-ix) ). The two ma1 1 t r i c e s Y and Ym a r e connected by t h e r e l a t i o n ( ~ 6 Y(x) ) = Ym(x)(, ) . By

an argument s i m i l a r t o t h a t o f Riemann i n t h e standard d i f f e r e n t i a l equat i o n s c o n t e x t these p r o p e r t es l e a d t o f i r s t o r d e r ODE w i t h r e s p e c t t o x and

PDE w i t h r e s p e c t t o ai (15.48)

dY/dx =

( c f . 114).

(Ifn (Aj/ x - a . ) ) J

+ A,)Y;

aY/aa

j

(-Aj/(x-a.))Y J

T h i s l o o k s l i k e t h e Riemann S c h l e s i n g e r equations except t h e r e i s an a d d i t i o n a l t e r m Am which accounts f o r t h e e x p o n e n t i a l b e h a v i o r a t

m.

Changing t h e v a r i a b l e a t x = m v i a x = l / t t h e f i r s t e q u a t i o n (15.48) b e 2 comes -dY/dt = ( A m / t + ) Y w i t h a double p o l e a t t = 0 ( i r r e g u l a r singu-

...

BOSE GAS l a r i t y o f rank 1).

-

241

I n t h e general t h e o r y o f i r r e g u l a r s i n g u l a r i t i e s one con-

siders singularities a t x =

-

f o r an e q u a t i o n o f t h e form dY/dx = (Arxr-’

t

xr-’ t ...)Y where t h e s e r i e s ( ) i s convergent a t ( c f . 114). When Ar-l r < 0 one has a r e g u l a r p o i n t . For r = 0 i t i s a r e g u l a r s i n g u l a r i t y and for r

> 0 one

has an i r r e g u l a r s i n g u l a r i t y o f r a n k k.

I n t h e present s i t u a -

t i o n r = 1 and we suppose t h a t t h e eigenvalues o f t h e l e a d i n g m a t r i x A,

are

distinct.

Then one can f i n d t h e formal s e r i e s s o l u t i o n i n t h e form ( W )

N

+ Ylx-’

Y(x) = G ( l

t

...)x Dexp(Tlx

gonal m a t r i c e s and G, Yi

-

+

... +

are matrices.

Trxr) where D, T1,.

The s e r i e s i n

(m+)

..) Tr

are dia-

i s n o t always

convergent i f r” z1 b u t i t does r e p r e s e n t t h e s o l u t i o n i n t h e f o l l o w i n g At x =

sense.

i f there i s a s i n g u l a r i t y o f rank k then there e x i s t 2 r

s e c t o r s such t h a t i n each s e c t o r S . t h e r e i s one and o n l y one s o l u t i o n t o J t h e d i f f e r e n t i a l equations w i t h s e r i e s ( m + ) a s an a s y m p t o t i c expansion. We have 2 r s o l u t i o n s Y.(x) and t h e y must be r e l a t e d v i a Yjtl(x) = Y.(x)S. where J J J t h e S . a r e c o n s t a n t m a t r i c e s . For r a n k r one has Y2r+l = Y1S2rS2r-1...S1 J so t h e m u l t i p l i e r m a t r i c e s S c o n s t i t u t e a n o t i o n o f monodromy f o r r > 0. j

G e n e r a l l y ( c f . 114) w i t h p o i n t s av e t c . one w i l l have a c o l l e c t i o n o f such c o n n e c t i o n m a t r i c e s S! and t h e a v w i t h T: J m a t i o n parameters ( c f . a l s o 116).

(1

v 5 n,

a)

w i l l be t h e d e f o r -

Now r e t u r n i n g t o (75.48) t h e r e s u l t i n g d e f o r m a t i o n e q u a t i o n s w i l l have a j = 1,. ..,n) and i n Consider 2n t a u f u n c t i o n s ( T Tj’ j y t r o d u c e t h e Poisson b r a c k e t s ( m D ) { ‘ t . , T 1 = { T - j , T - k } 0, { T j y T - k } = 6 j k . J k Then we d e f i n e t h e Hamil t o n i a n 1 - f o r m

H a m i l t o n i a n form.

The d e f o r m a t i o n e q u a t i o n i s t h e n g i v e n b y (**) e q u a t i o n (**) (15.50) U(X)

drij

= {T?~,W).

When n

=

2

has t h e s i m p l e form

2 2 ( x d o/dx ) = - 4 ( x d ~ / d x - 1 = x ( d / d x ) l o g p x);

pl(x-x’,-)

-

o)(xdo/dx

t

(do/dx)

2

-

u);

= p(x-x’)

where t h e d e n s i t y m a t r i x i s now t r a n s l a t i o n i n v a r i a n t and depends on one parameter.

Equation (15 50) i s e q u i v a l e n t t o t h e Painleve‘ e q u a t i o n o f f i f t h

24 2

ROBERT CARROLL

k i n d . F i n a l l y w i s r e l a t e d to the Fredholm determinant ( c f . [ J 9 , l l

I)

(15.51 )

( t h e l a t t e r expression f o r j = k and t h e former f o r j # k ) . T h u s t h e Fredholm determinant is by d e f i n i t i o n a t a u function f o r t h e corresponding monodromy problem and i t is a l s o t h e expectation value o f C l i f f o r d operators. These a r e t h e general f e a t u r e s o f t h e deformation problem. In t h e most general case t h e tau function can be shown t o be t h e Fredholm determinant o f an i n t e g r a l equation which a r i s e s i n solving t h e Riemann problem. Further the Clifford operator point of view can be generalized t o the most general c a s e e r e c a l l here o f an a r b i t r a r y number of i r r e g u l a r s i n g u l a r i t i e s ( c f . 914). W t h a t i n t h e s o l i t o n p i c t u r e more general elements of t h e Clifford group come into play.

For the Ising model only special elements of the Clifford g r o u p

(spin operators o r t r a n s f e r matrices) occur; f o r s o l i t o n s , and generally f o r evolution of vacuum expectations, one deals w i t h vertex o p e r a t o r s , hierarchy v a r i a b l e s , and vacuum orbits (e.g. (Olexp(H(x))g]O)% T ( x ) ) .

REmARK 15.5

(CBRRELACZBN FllNCClBW, Gtl3 E9MClBN3, CAll FUNCEZBW).

We f o l -

low here some work i n [ IT61 r e l a t e d t o the Bose gas of Remark 15.3 ( c f . a l s o

H = if (az$+az$ + c$+$+$qJ - hqJ+$)dz be t h e Hamiltonian f o r a 1-D n o n r e l a t i v i s t i c Bose gas w i t h C $ ( z ) , $ + ( y ) l = 6 ( z y ) and h a chemical p o t e n t i a l . Only the case o f impenetrable bosons is considered w i t h c = m. Then ( c f . [ LFl ; L G l ; Y l I ) a t zero temperature t h e thermal equilibrium s t a t e is the ground s t a t e o f t h e Hamiltonian, representing a Fermi zone. All t h e s t a t e s o f p a r t i c l e s w i t h momenta k , -q 5 k 5 q a r e f i l led where q = h4 is the Fermi momentum. A t temperature T > 0 the thermal equilibrium d i s t r i b u t i o n o f p a r t i c l e s i s given by t h e Fermi weight (**@) w ( k , h , T ) = (1 t exp(c(k)/T))-' where E ( k ) = k2 - h i s a p a r t i c l e energy. The gas d e n s i t y D is ( * 6 ) D = (1/21~)jI w(k,f,T)dk and t h e k i n e t i c energy den2 s i t y E i s given a s (**+) E = ( 1 / 2 1 ~ ) L zk w ( k , h , T ) d k . The chemical potential h determines t h e gas density a s a monotone increasing function of h . A t T = 0, D = 0 a t h = 0 and 0 < D < m a s 0 < h < m . A t T > 0, D = 0 a t h = -mand [ IT5,7;KF1-41 and 523).

Let

(*A)

GLM EQUATIONS 0 < D <

m

as

m

< h <

w.

At T

>

243

0 t h e mean v a l u e o f an o p e r a t o r

6

i s de-

f i n e d as (15.52)

(

8)T = Tr(6exp(-H/T))/Tr(exp(-H/T))

I n p a r t i c u l a r t h e two p o i n t f i e l d c o r r e l a t o r (**.) ( $ + ( z ) $ ( - z ) ) ~ (where d i s t a n c e z i s r e a l ) i s a r e a l valued f u n c t i o n o f

t o c o n s i d e r t h i s f o r z > 0 which i s assumed now. f o r c o r r e l a t o r s o f d e n s i t i e s (*A*)

Hence i t i s s u f f i c i e n t

121.

The g e n e r a t i n g f u n c t i o n a l

e x p ( a Q ( z ) ) )T w i l l a l s o be considered.

(

Here Q(z) i s t h e number o p e r a t o r on a n i n t e r v a l o f l e n g t h z (*u) Q(z) = $+(y)$(y)dy and a i s a f r e e parameter. One o b t a i n s then t h e 2 - p o i n t den= L2azaa( 2 2 exp(aQ(z)) s i t y c o r r e l a t o r as (*A*) ($+(z)$(z)$+(O)$(O)),

)Tla=O.

The v a l u e o f t h e g e n e r a t i n g f u n c t i o n a l a t a =

( X ) ) ) ~ I ~ =has - ~ also

P(z,t,h)

= (exp(aQ

a p h y s i c a l meaning, g i v i n g a p r o b a b i l i t y t h a t t h e r e a r e

no p a r t i c l e s o f t h e gas on t h e i n t e r v a l [ 0,z librium.

(*A&)

-m

I

i n t h e s t a t e o f thermal e q u i -

P i s c a l l e d t h e emptiness f o r m a t i o n p r o b a b i l i t y .

t h e c o r r e l a t o r s (+*.),

(*A*),

and

(*A&)

One shows t h a t

a r e generated by t h e same l i n e a r

i n t e g r a l o p e r a t o r and some c o m p l e t e l y i n t e g r a b l e PDE f o r t h e s e c o r r e l a t o r s a r e g i v e n below. One w r i t e s now x =

24.

Then (**.)

and t = h/T.

becomes

(**.f

($‘(Z)$(-Z))~

= T’g(x,t)

and (*A*) o r (*A&) a t x,t

f i x e d do n o t depend on temperature ex-

plicitly.

A l l o f t h e c o r r e l a t o r s mentioned can be expressed i n terms o f t h e

l i n e a r i n t e g r a l operator K defined v i a

(*A+)

K(X,u) = (o(h))S(Sinx(h-u)/h-u))(o(u))L2 a s (*AM) @ ( A ) = w(A,t,l)

= (1

+

Kf(A) =

/I K(A,u)f(u)du,

where

and t h e Fermi w e i g h t @ ( A ) i s g i v e n

2

exp(X - t ) ) - ’ . For d i s c u s s i o n and d e r i v a t i o n

o f e q u a t i o n s such as those below we r e f e r t o Remark 15.2 and [ I T 7 ; L f l ;LG1;

It

J5,9;KF1-4;THl;Yll.

is convenient t o d e s c r i b e p r o p e r t i e s o f t h e o p e r a t o r

K i n terms o f f u n c t i o n s f+(A,x,t,y) o f t e n b e i n g suppressed ( t h u s f +-( A )

and f-(A,x,t,y),

t h e dependence on x,t,y

f+(h,x,t,y)). -

These f u n c t i o n s a r e de-

%

f i n e d as s o l u t i o n s o f t h e l i n e a r i n t e g r a l e q u a t i o n s (15.53)

f +-( A )

-

Y~I K(A,v)f+(u)du -

= e+(A) -

Here Y i s a r e a l parameter and t h e f u n c t i o n s e+(A) a r e (*.*) exp(+ihx).

The 2 x 2 m a t r i x o f p o t e n t i a l s ( ( B

pm

(x,t,y)))

e -+ ( i ) =

( p = +,-;

(o(x))’

m = +,-)

244

ROBERT CARROLL

will be important where

I t is a real symnetrical matrix w i t h 2 independent matrix elements B++, B+where B+:

= B,,

Now (**=)'can (1 5.55)

= B--

and BT-

=

B+-

=

B-+-

be represented a s g(x,t)

=

L4B++(Xlty~)A(~yty~)~y=2/n

where A is a Fredholm determinant A(x,t,y) = det(1 - yK). We show below t h a t (15.55) can be obtained from the well known Lenard formula (15.57) below by means o f t h e Fourier transform. On t h e o t h e r hand t h e expression f o r (*A*) is ( c f . [ KF2,3] - we do not give the d e r i v a t i o n here) (15.56) where the c r i t i c a l value y = 1/71 (a = -a) corresponds t o t h e emptiness f o r mation p r o b a b i l i t y (*A&) P ( z , h , T ) = A(%x,t,y)ly=l/n. Thus the c o r r e l a t o r s a r e represented i n terms o f t h e operator K. Let us now d e r i v e t h e represent a t i o n (15.55). A r e s u l t of [ LG1 1 is (1 5.57)

g(x,t)

=

h(x,-x)det(l

-

Y ~ T ly=2/* )

which we will assume. Here O T i s a l i n e a r i n t e g r a l o p e r a t o r w i t h a d i f f e r ence kernel a c t i n g on ( - x , x ) , (OT$)(S) = /_X,oT(S-n)$(n)dn.where OT(s-n) i s m exp(i(5a Fourier transform o f the Fermi weight (*A=) ( i . e . BT(S-n) = $im q)A)e(A)dA); p(x,-x) is the special value o f the resolvent p ( s , n ) (**A) p ( S , U ) - Y ~ ~ ~ ~ T ( S - S ' ) ~ ( =S OT(S-n). ',~)~S' Now f o r t h e equivalence of (15.57) and (15.55) consider t h e i n t e g r a l $ ( S ) exp(-iAs) yJ-' 0 (S-S')$(S')dC' = @ ( S ) . Using Fourier transforms $(C) = (~(x))T%f ( X ) d A and one has t h e equation f o r f(A)

_/I

(15.58)

f(X)

- yl:

K(A,u)f(u)dv

= F(A);

F ( A ) = (1/2n4'(A))fmm eiXs@(S)dg

where K ( A , v ) here i s exactly t h e kernel o f (*A*). u n i t a r y ) t h a t t h e Fredholm determinants o f K a n d

OT

I t follows then (F i s u a r e t h e same, det(1-yK)

GLM EQUATIONS = det(1

- y oT )

p(x,-x)

=

=

245

A(x,t,r) and making a Fourier transform i n

(*.A) one gets which completes the proof of (15.55) ( g i v e n ( 1 5 . 5 7 ) ) .

(2y)-’Bt,

We sketch a few o t h e r results now from [ IT61.

Recall t h a t c o r r e l a t o r s (**.)

(*A*), (*A&) a r e expressed i n terms o f Fredholm determinants A(x,t,y) and t h e function B,+(x,t,y) a t special values of t h e parameter y . One shows 2 more i n [ IT61, namely a t any Y(*.*) B,+2 = -axlog(A) and d i f f e r e n t i a l equa2 t i o n s f o r B,, and A a r e obtained. The PDE i n x , t f o r B,, i s (*.&) atB++ = 1 + %ax((axatB,,)/B,+). I n i t i a l conditions a r e given by asymptotics a t x + 0 and t fixed, namely B,+(x,t,y) = y d ( t ) + ( y d ( t ) ) 2 x + O(x 2 ); d ( t ) = lI ~ ( h ) 1i m dX, and by t h e requirement t h a t t+-m B+,(x,t,y) = 0. B++ decreases a s t -m and can be expanded i n t o a Taylor s e r i e s i n e x p ( t ) as t -m and x is fixed. The equation (*a&) can be obtained from t h e Sine-Gordon equation by means o f deformations proposed in [ BUR1 I. The Fredholm determinant A and hence t h e c o r r e l a t o r s (*A*), (*A&) a l s o s a t i s f y PDE valid f o r any y . Indeed p u t t i n g u ( x , t , y ) = logA(X,t,y) one has (*.+I ( a t a2,I2 = -4(a:o)(zxa a u + x ( a a u)* - 23,~) where the i n i t i a l conditions a r e u = - y d ( t ) x y 2 d i ( Xt ) x 2 / 2 x2 1i m + O(x ) and t+-m o ( x , t , y ) = 0. As t + -- ( w i t h x f i x e d ) u i s a decreasing function of t and can be expanded i n t o a Taylor s e r i e s i n e x p ( t ) . The d i f f e r e n t i a l equations f o r B,, and u a l s o allow one t o obtain asymptotics of c o r r e l a t o r s i n various regions of t h e parameters b u t we will n o t dwell o n asymptotics here ( c f . [ IT61 f o r d e t a i l s ) . Finally f o r T = 0 t h e i n t e g r a l t o p e r a t o r K i s reduced t o the i n t e g r a l KO a c t i n g on (-q,q) ( q = h 2 ) w i t h the kernel Ko(A,p) = (A-p)-lSinx(h-u). The operator O T becomes t h e operator Oo a c t i n g on ( - x , x ) w i t h kernel o0(E-q) = ( E - q ) - ’ S i n q ( E - q ) . The Fredholm determinant A a t T = 0 depends only on a product 7 of x and tS, A ( x , t , y ) ( T = O L = a O ( T y y ) ; = xt’ = z h 2 and one g e t s f o r r o ( T ) = r(alogAO(T,y)/ar) ( 8 a / a T ) ( m i ) 2 = -4(Tud - u o ) ( 4 r u d + (u~)’ - 4 0 0 ) , reproducing a result of(J91. -+

-+

-

To derive (*a&) and (**+) above one begins w i t h t h e i n t e g r a l equations i n d i cated above f o r f+(X), namely (15.59) Here e ,-( h ) (15.60)

-(h) f+

-

Y~I

K(A,p)f+ -(u)dp =

e -+ ( h )

= (O(A))’exp(+ixx) and K ( h , p ) can be represented a s

K(X,IJ)

= (Zi(A-~))-’(e+(X)e-(p)-e-(h)e+(u));

K(A,u) = K ( u , X )

ROBERT CARROLL

246

I t s h o u l d be n o t e d t h a t t h e r e s o l v e n t k e r n e l R(A,p)

(15.61) (i.e.

R(A,p)

(1

- yK)R

-

y / z K(A,A')R(A',p)dA'

d e f i n e d as

= K(A,u)

= K) may be r e p r e s e n t e d i n a form s i m i l a r t o (15.60),

To prove t h i s one m u l t i p l i e s b o t h s i d e s o f (15.61)

by (A-p), w r i t e s t h e f a c -

t o r ( A - u ) under t h e i n t e g r a l i n t h e second t e r m as A-u = ( A - A ' ) and t h e n uses (15.59)

(as w e l l as equations c o n j u g a t e t o t h e s e ) .

t o t h e d e r i v a t i o n o f (*a&) and (*a+).

t (Al-p),

Now t u r n

The main idea i s t o c o n s i d e r (15.59)

a s a GLM t y p e e q u a t i o n f o r some i n t e g r a b l e system, t h e f u n c t i o n @(A) the role o f a reflection coefficient. i s f i x e d as (*A*) @ ( A ) = (1

-

namely

playing

Though @ ( A ) f o r i m p e n e t r a b l e bosons

+ exp(A 2 - t ) ) - ' a l l t h e c o n s i d e r a t i o n s w h i c h f o l -

l o w a r e v a l i d a l s o i f 0 i s an a r b i t r a r y f u n c t i o n o f t h e d i f f e r e n c e A * - t c r e a s i n g as A 2 - t

+

(i.e.

de-

s a t i s f y i n g (*am) (2xat + a A ) @ ( A ) = 0,01,2-~,

=

0). The dependence o f s o l u t i o n s o f (*a&) and (*a*) on an a r b i t r a r y o(A) ent e r s i n t h e i n i t i a l values c o n t a i n i n g t h e f u n c t i o n d ( t ) = l O(A)dA. One sees e a s i l y t h a t f o r a g i v e n d ( t ) t h e f u n c t i o n @ o f (*am) i s d e f i n e d uniquel y and v i c e versa.

One notes a l s o t h a t t h e idea o f o b t a i n i n g d i f f e r e n t i a l

e q u a t i o n s f o r s o l u t i o n s o f i n t e g r a l e q u a t i o n s was developed i n v a r i o u s ways i n [ KRE1;SNl

I.

L e t us d e r i v e now (*a&) f o r B++(x,t,y).

The f o l l o w i n g r e l a t i o n s can be ob-

t a i n e d from (15.59) (15.63)

axF(A) = ( t h o 3 + Q)F(A);

(2Aat + a A ) F ( A ) = ( i x o 3

-

iatV)F(A);

(x,t,y) ( p = +,-; m = +,-) were i n t r o d u c e d e a r l i e r and i t i s Pm t o be s t r e s s e d t h a t t h e y do n o t depend o n t h e s p e c t r a l parameter A. To Potentials B

prove t h e f i r s t e q u a t i o n i n (15.63) one f i r s t d i f f e r e n t i a t e s (15.59) to get

in x

GLM EQUATIONS

24 7

Then t h e o p e r a t o r (1- K1-l s h o u l d be applied t o both sides. Noting t h a t t h e kernel o f axK is proportional t o t h e sum o f two 1 - D projectors, 2axK(A,p) = e+(A)e-(p) + e-(A)e+(u), one e a s i l y c a l c u l a t e s t h e a c t i o n of (1- K)-’ on t h e To c a l c u l a t e t h e a c t i o n on t h e r i g h t s i d e one uses the r e l a t i o n

l e f t side.

(1- K ) ( l + R ) = 1 and (15.62) f o r t h e resolvent kernel. The proof of t h e second equation i n (15.63) is based e s s e n t i a l l y on properties (*a=) of @ ( A ) . I t i s more cumbersome and is r e f e r r e d t o Remark 15.4 below. The r e l a t i o n s (15.63) a r e v a l i d f o r any A and can be regarded a s a zero curvature r e p r e s e n t a t i o n f o r some nonlinear evolution equations f o r t h e potent i a l s B. The compatibility condition (*&*I [ a x - iXu3 - Q,2ha t + a, - i x u 3 + i a t V l = 0 l e a d s t o the following system o f equations f o r B++, (15.65)

ataxB+-

=

2

a t ( B + + ) ; atB+,

B,-

= x + %(axatB++)/Bt,

Using the f a c t t h a t B -+ 0 a s t -m one can transform t h e f i r s t e q u a t i o n 2v i n t o (*&A) axB+- = B++ ( t h i s can a l s o be derived d i r e c t l y from (15.54) de-f

f i n i n g t h e B u s i n g only t h e f i r s t equation i n (15.63) f o r a x F ) .

Eliminating

f o r B++ a s desired. To see now B+- from (15.65) - (*&A) one a r r i v e s a t (*@&I s e e how (*a+) f o r u = log(A) is derived l e t us c a l c u l a t e t h e d i f f e r e n t i a l do = (axu)dx

(15.66)

+ (ato)dt. aXo = - B + ;

The p a r t i a l d e r i v a t i v e s turn o u t to be ata = -xatB+-

+ $((atB+-)

2 - %(atB++)‘

To o b t a i n t h e x d e r i v a t i v e is simple s i n c e aXo = axlog det(1-yK) = - y T r ( ( l YK)-’a,K) which by means o f t h e formula f o r a x K above i s reduced t o (15.66). The equation f o r ato i s referred to Remark 15 4. To o b t a i n (***I i t i s now s u f f i c i e n t t o s u b s t i t u t e B,- = - a x u and B++ = (-a;o)’ i n t o (15:66). One notes a l s o t h a t t h e e q u a l i t y of mixed d e r i v a t ves a a u = ataxo plus (*&A) x t leads t o (*a&) f o r B++.

RMtARK 15.6.

We give here t h e d e r i v a t i o n of

he second equation in (15.63)

and t h e equation f o r a t u i n (15.66) following [ IT61.

Consider t h e i n t e g r a l

equations (15.59) and d i f f e r e n t i a t e e x p l i c i t l y w i t h respect t o t t o get

24 8

ROBERT CARROLL

Taking i n t o account (15.61)-(15.62) (15.68)

-

2Aa t f +-( A )

(a

f- (A f+(u )

one sees t h a t from (15.67) f o l l o w s

= 2A(ato(h)/20(A))f+(h) -

(U 1/20 (11

If+ - (11 )dV

+

+ ( 2 y / i ) f I (f+(A)f-(u)

2 ~ l IR (A ,U )IJ (3

(P ) / 0 (V

-

If+ - ( 11

D i f f e r e n t i a t i n g (15.67) i n h and i n t e g r a t i n g by p a r t s one has a l s o f +-( A ) = (aA0(A)/20(h))f+(h).ixf+(A) -

(15.69)

+ Adding (15.68)-(15.69) (15.70) (15.71

Upm(x,t,u)

f+

and r e c a l l i n g t h a t (.?hat + a,)@ = + i x f +- ( h )

(2Aat + a,)f,(A)

I

ZY~I R (A ,u 1( au@(u 1/20 (u 1

=

-

iyf+(A)U-

,- + +

( u )du = 0 one g e t s iyf-(A)U+

,+

lz fp(u)fm(u)(atO(ll)/O(ll))du

= ati_z e ( u ) f m ( u ) d u which a l l o w s one Pm P The d e r i t o r e w r i t e (15.70) i n t h e form o f t h e second e q u a t i o n i n (15.63).

Using (15.67) a g a i n one proves t h a t U

v a t i o n o f t h e r e l a t i o n (15.66) f o r ato begins w i t h ato = a t l o g d e t 1-yK) -Tr((1-yK)-’atK) (15.72)

a p

which by means o f (15.67) and (15.62) can be w r i t en as

= -ylf R(X,X)(atO(A)/O(A))dA

( f + have argument A).

= L a i y l I (ato/O)(f-ahf+-f+axf_)dx

I n view o f (15.69) f o r aAf,

and (15.71) one o b t a i n s

t h e second e q u a t i o n i n (15.66). We conclude w i t h a few remarks based on c o n c l u s i o n s o f [IT61.

Thus t h e i m -

p e n e t r a b l e Bose gas i s t h e s i m p l e s t n o n t r i v i a l quantum i n t e g r a b l e system which can be o b t a i n e d by q u a n t i z a t i o n o f a c l a s s i c a l n o n l i n e a r system, namely t h e NLS e q u a t i o n .

It i s remarkable t h a t a f t e r q u a n t i z a t i o n t h e c o r -

r e l a t i o n f u n c t i o n s a r e d e s c r i b e d by a new c l a s s i c a l i n t e g r a b l e system.

The

e s s e n t i a l p o i n t i s t h e p o s s i b i l i t y o f r e p r e s e n t i n g c o r r e l a t o r s i n terms o f t h e l i n e a r i n t e g r a l o p e r a t o r , t h e Fredholm d e t e r m i n a n t p l a y i n g t h e r o l e o f t a u f u n c t i o n o f t h e new i n t e g r a b l e system.

As r e p r e s e n t a t i o n s o f t h e c o r -

r e l a t o r s o f t h i s k i n d a r e known a l s o f o r o t h e r quantum models ( c f . [ KF2,31) one hopes t h a t t h e scheme d e s c r i b e d above i s g e n e r a l .

It should be n o t e d

2-D QUANTUM GRAVITY

249

t h a t the c l a s s i c a l i n t e g r a b l e system obtained gives t h e complete descript i o n of c o r r e l a t o r s and we r e f e r t o [IT61 f o r d e t a i l s about asymptotics e t c . 16. SmE RESRARKZ( ON 2-D Q1IANCUiY CMUZ.eY AND KM. There has been a l o t of a c t i v i t y i n recent months o n 2-D quantum g r a v i t y and t h e KdV hierarchy ( c f . [ BAN1 ;BREl ;AG7; DJ1-3; DK1; DG3-5; FG4,5; KAZl ;MX1 ;MF5,6; KR17; PV1-3; SCHl ;V4 ;WT4 I). One can e x t r a c t some o f the mathematical f l a v o r and some substance from e.g. [ DG3;FG3,4;MF5,61 w i t h o u t requiring g r e a t physical i n s i g h t or background. One sees many o f t h e f a m i l i a r faces a r i s i n g again: tau functions, monodromy, BA functions, c o r r e l a t i o n functions, f r e e fermions, strings, Riemann surfaces, Kac-Moody algebras, e t c . We r e f e r here to §21,22 f o r background on strings and some r e l a t e d formulas, b u t we f e l t i t was b e t t e r t o i n s e r t this s e c t i o n here i n view of t h e connections t o monodromy. REWRK 16.1 (EIACRIX f10DEtS7 PARCICI0N FllNCCZ0NS7 C0RRELACZ0N FMCC10W). We follow rllainly [ MF5,6 I unless otherwise s t a t e d and concentrate on the rel e v a n t mathemtical s t r u c t u r e , using physical terminology when i t seems helpful i n motivating o r understanding the mathematics. One deals here w i t h t h e l a r g e N limit o f matrix models involving p a r t i t i o n functions

(1 6.1 )

ZN

%

1 ll d h 1. 8 2

e- N C V ( A i )

where V ( A ) is a polynomial a n d A i s a Vandermonde determinant (*) A ( A ) = The mathematics (and even t h e physics) here is r e l a t e d t o t h a t n(hi hj). discussed i n §14,15 b u t we will not t r y t o discuss g r a v i t y (see the r e f e r ences listed above). The physical ideas involving gravity, strings, CFT, e t c . i n t h e background here a r e q u i t e i n t r i c a t e and we a r e not competant t o discuss them i n t e l l i g e n t l y . Some introduction t o CFT and strings is however given i n 117-22. In any event f o r h i = ai/N ( i = 1 , 2 say) one writes ~ ( 1 , M ) f o r the p r o b a b i l i t y t h a t no eigenvalue f a l l s i n the range I = [ h 1 , X 2 1 . Then one knows from [ J9] t h a t (A) .r(al-a2) = .r([al/N,a2/N1,N) i s t h e t a u function f o r t h e isomonodromy problem associated to t h e Painlev6 5 equation (P5). The a n a l y s i s o f (16.1) can be c a r r i e d o u t via orthogonal functions ( 0 ) $,(A) = Pn(X)exp(-%NV(X))where t h e Pn a r e orthogonal polynomials f o r 2 wI Xi . i t h Hermite funcdhexp(-NV(X)). As a n example here consider V ( X ) = % tions

-

2

250

ROBERT CARROLL

(16.2)

$,(A)

= (N/2a)L(N/n!)4(An

...)e- NAL/4

t

t

1:

One can use second q u a n t i z e d wave f u n c t i o n s ( 6 ) $ ( A ) = $,$)an and $(A) = $,(Alan t. w i t h { $ t ( A ) , $ ( A ' ) l = 6 b - A ' ) where t h e ground s t a t e i s t h e Fer-

1;

m i sea w i t h t h e f i r s t N l e v e l s f i l l e d . and n

Q,

I n t h e example problem f o r l a r g e N

O(N)

(16.3)

qZn(y/N)

'L

(-1 ) ' ( N / h

2 N) %C o s ( y ( ( 4 n t l )/2N)');

(y/N) = (-1 ) n ( N / 2 ~2 N ) %S i n ( y ( (4n+3)/2N)') @2n+l ( t h e dominant c o n t r i b u t i o n t o t h e c o r r e l a t i o n f u n c t i o n s e t c . comes from x = n, N

Q,

1).

(16.4)

One has ( i n general ) a D a r b o u x - C h r i s t o f f e l formula ) ~ , ( ~ 2 N) ) 1 = (rntl

(

t )2($K-l

( ~ ) 1+ N ( A ~ ) - $ N +( ~A ~ ) J I ~) )( /A( ~A ~ - A ~ )

L L where f o r even V ( A ) (+I A $ ~ ~ (1A = (rZntl ) %2n+l + (rZn) %2n-l and (1) L t = (r2nt2) 2$2ntl + (rZntl ) F, ( c f . [ DJ3;GKl ;BAN1 ;BRE21 and see [ C24 1 f o r

Darboux-Christoffel ideas). (16.5)

(

t

NI? (y)$(y')l N)

-f

S e t t i n g j ( y , N ) = (l/N')$(y/N)

one o b t a i n s

(l/~)Sin(y-y')/(y-y')

One checks t h i s e x p l i c i t l y i n t h e example (16.3) ( c f . [MF6]). (m)

a,(p)

= N'(-l)

n t N (a2,, 4

-

i$2ntl)

A

where an = an+N and p = (n-ZN)/ZN.

The sum o v e r n

Q,

jm dp

-1

Now d e f i n e

+

and a 2 ( p ) = N'(-1)ntN($2n

iszntl)

and assuming t h e

main c o n t r i b u t i o n comes from t h e NBH o f t h e Fermi l e v e l one extends t h i s t o dP. Thus p h y s i c a l arguments l e a d t o t h e a s s e r t i o n t h a t l a r g e N 1 i m i t g i v e n by

i:

(16.6)

j ( y ) = eiYII

al(p)eiyPdp

t e-jYI:

a2(p)eqiYPdp

= eiYq

has a good

+ e-iv$2

(qi = I J ~ ( . v ) ) so t h a t t h e Fermi sea becomes t h e ground s t a t e d e f i n e d by (**) ai(p)IO)

t

= 0 f o r p > 0 and ai(-p)IO)

f i n d s t h e n ( c f . [ I Z 1 1)

= 0 f o r p > 0.

Referring to

(A)

one

STRING EQUATIONS AND KdV where normal o r d e r p u t s

I = [a,,a,]

j

A t o the r i g h t o f $

t

.

251

As N

-+ m

with X

i

= ai/N

and

one o b t a i n s

z(1) = ( O l : e x p ( - y l I

(16.8)

$'$):lo)

where y = 1 and t h e normal o r d e r e d exponent i s e v a l u a t e d by expanding and p o i n t s p l i t t i n g a l l i n t e g r a l s ( t o e l i m i n a t e c o n t r i b u t i o n s from 6 f u n c t i o n s i n the 2 point correlation functions

-

For Y = 2 t h i s e x p r e s s i o n

c f . 917).

i s a c o r r e l a t i o n f u n c t i o n i n t h e t h e o r y o f t h e 1-D Bose gas ( c f . 515 and [ IT5-7; J9;KFl , 2 ] ) .

(kCRLNG EQIlACZ0M, RS0LVANC3, AND K a V ) .

REIIIARK 16.2

DJl-3;DG3;FG4,5;MF5,61)

p h y s i c s arguments ( c f .

Now w i t h o u t g i v i n g any

we make a few p h y s i c s comF i r s t f o l l o w i n g [ FG41 one can

ments and t h e n w r i t e down v a r i o u s e q u a t i o n s .

quantum g r a v i t y can be f o r m u l a t e d as a double s c a l i n g l i m i t o f

say t h a t 2-0

a m a t r i x model which l e a d s t o n o n p e r t u r b a t i v e " s t r i n g " equations ( n o n l i n e a r d i f f e r e n t i a l equations i n t h e s p e c i f i c h e a t u ( x ) , where x i s t h e s c a l i n g variable

-

o r renormal i z e d cosmological c o n s t a n t ) .

a general massive model i s ( c f . a l s o tGK2;NBl (16.9)

X =

1;

The s t r i n g e q u a t i o n f o r

I)

(k+$)tkRk(U(X,tj))

where t h e tk a r e c e r t a i n "sources" and t h e Rk a r e t h e G e l f a n d - D i k i i d i f f e r e n t i a l polynomials (16.10) ( I

%

ax).

R

0

=

4; R1

=

- 4 ~ ; R2 = ( 1 / 1 6 ) ( 3 ~2

-

u");

...

One r e c a l l s here t h e n o t a t i o n o f t h e formal c a l c u l u s o f v a r i a -

1;

t i o n s from [ 621. Thus (*A) a x = uPtla/auP (up % u(')), a X up = uptl, 6/ k k k 6u = ( - 1 ) ( a x ) 6/6u ( r e c a l l from 16, D 1 F(u+E@)dx = I (6F/6u)@dx e t c . ) . Now t h e Rm a r i s e from R(x,G) = Rm(u)/cm" % ( x l ( - a 2 t u t r ) - l I x ) where

1;

1;

i s t h e r e s t r i c t i o n o f t h e r e s o l v e n t k e r n e l R(x,y,c) t o t h e diagonal 2 s a t i s f i e s (-ax + u + r)R = 0, R(x,y,c) = R(y,x,~), and Rx x = y (R(x,y,c) R(x,c)

has a jump d i s c o n t i n u i t y o f 1 a t x = y v a r i o u s equations; f l o w is now ( a k

%

i n p a r t i c u l a r (*.) a/atk)

-

c f . 13).

(6/6u)Rmt,

The f u n c t i o n s R = -(mt$)R,.

The !th

satisfy KdV

ROBERT CARROLL

252

aku = axRk+l(u)

(16.11)

and f o r k = 2 (t2 = 32/9, t

P

= 0 for p #

2, i n ( 1 6 . 9 ) ) w i t h x

= u2

-

u" one

has t h e Painleve' 1 ( P l ) equation. The p a r t i t i o n function Z ( x , t l , ...) ( x = t o ) will turn o u t be be z2 ( x , t l , t 2B...) where 'I i s a tau function f o r the KdV hierarchy ( t h i s will be a special uniquely determined tau function f o r 2 which - u = 2a 10gT s a t i s f i e s (16.9)).

Another paint o f view developed i n [DJ1-3;MF5,6] argues t h a t g i v e n L = Dq u

q-2

Dq-'

... + uo a s

t

+

t h e continuum l i m i t o f a m u l t i p l i c a t i o n operator A :

f hf, t h e contimuum l i m i t o f a h s h o u l d be of t h e form p = L y / q . Then t h e d i f f e r e n t i a l equation (*&) P , L l = 1 should define nonperturbative 2-D q u a n t u m g r a v i t y coupled t o t h e ( p , q ) minimal model o f CFT; s i m i l a r l y the equa-f

t i o n s f o r massive models coupled to 2 dimensions should be o f t h e form (*+) [ P , L ] = 1 where P = 1 t L f j q , t h e t representing the masses in t h e theory. P P In order t o w r i t e t h i s i n matrix form (following [ DRl;IM1,2] and 59,lO) one defines f i r s t ( i n a typical s i t u a t i o n o f t h e (2m-1,2) model equations)

L = - a x + (10 h + u 1; pm = (Am

(16.12) t A"'R~;

=

where P-l

+c;;

B,,

(A+u)c,

-

A;;

P =

+ A m-1 R~ + 0 x-l(j+L,)tjRj -a,-+~(j++)tjPj-lt(o 0

cm =

R~ + AR,-~

+

...

= 0, the R . a r e Gelfand-Dikii p o t e n t i a l s and J

(16.13)

[P,,LI

(% = 1 h e r e ) . [

'"1;

Cm- A,

2am + P,,LI

=

(o0

-kRA+I)

The mth KdV flow i s then t h e compatibility condition (*.) = 0 and the orthonormal wave functions $ s a t i s f y

A

P$(A,x,t.) = 0; L$ = 0; ( 2 3 . t P.); J J J

(16.14)

Compatibility o f t h e l a s t two equations gives

= 0 (*m)

and c o m p a t i b i l i t y o f t h e

f i r s t two equations gives t h e ( c o l l e c t i o n o f ) massive (2m-1,2) s t r i n g equat i o n s . Note f o r example f o r P, = - a A + Pm + ( 0o X + ZR,,, ) the c o m p a t i b i l i t y [ P,,LI

-4x.

t o t h e massless (2m-1,2) s t r i n g equation Rmtl = KdV and (16.9) as follows. ( k + + ) t a R from (16.9) and hence, d i f f e r e n t i a t i n g in tmand k x k

= 0 i s equivalent

In p a r t i c u l a r one checks t h e consistency o f

First 1

=

1;

MO NODROM Y u s i n g (16.11)

(A*)

0 = (m+%)axR,

+

1;

253

(kt%)tkamaxRk= (mt+)axRm t

1 (k+t,)

tkamak,l LJ = (m+t,)axRm+ 1 ; (kt%)tkak,l axRm+l = (m+%)axRm + 1 ( k + t , ) t k5: k a x Rm+l (note amak,] = ak-l a m and we have s e t here sk % ak,l i n the spirit ck = 1 ; ~ : ~ + ' ) ( 6 / 6 u l ) w i t h a k u = sk+lu so 5 k = R; - axRm = a k m l u via (16.11). B u t (A*) says (u) a x ( 6 / 6 u ) = 1 (k+t,)tkck which is equivalent t o 1 = 1 (j+%)

t j a x R j = 1 ( j + t , ) t j a j - l u = 1 (jt4)t.E.u = (6/6u)u. We do not t r e a t the f o r J J ma1 c a l c u l u s of v a r i a t i o n s i n g r e a t d e t a i l here s i n c e one expects i t t o be covered f u l l y i n t h e forthcoming book t DK4 1. We consider now monodromy ideas a s i n 514. One focuses on t h e A equation P ~ I= 0 i n (16.14) and f o r s i m p l i c i t y we begin w i t h

REmARlC 16-3 (i!tOW0DR0l3!J).

(A@) a,$ = A ( A ) + . Near regular s i n g u l a r points w i t h simple poles of A one looks f o r formal s o l u t i o n s $ ( A ) = j(A)exp(Mlog(A-a)) where j i s a formal

power s e r i e s i n (A-a), whereas near i r r e g u l a r s i n g u l a r points where A ( A ) % A-r/(A-a)rtl + = An(A-a)"', w i t h A-r diagonal i z a b l e , one has formal solutions

...

i s invertible. One t h i n k s o f a NBH o f Qk = { d k < arg(A-a) < e,) such t h a t i n each region t h e r e i s a unique true s o l u t i o n t o (A@) asymptotic t o (16.15). On Qktl n Qk one has $,,+I = QkSk f o r Stoke's matrices Sk as before. This i s discussed i n [MF5,61 i n some d e t a i l and we only r e f e r here t o very special s i t u a t i o n s ( c f . a l s o §14,15 and references t h e r e some material is d e l i b e r a t e l y repeated). For example the A equation PJI = 0 i n (16.14) r e q u i r e s f i r s t a transformation i n o r d e r t o get an i n v e r t i b l e matrix i n a leading position so w r i t e f o r A = 5 2

where T,L a r e diagonalizable and a divided i n t o s e c t o r i a l regions

;0

-

and t h e leading s i n g u l a r i t y i n t h e r e s u l t i n g W equation now m u l t i p l i e s u3 = ( 0 -1O ) .

There r e s u l t s

254 where

+

ROBERT CARROLL

H,'

u t A.

(= x,t

T h i s i s proved u s i n g t h e r e s o l v e n t R o f - a 2

0 1 = 4u and u1 = (1 o).

For f i x e d S t o k e ' s data (= monodromy) one now deforms t h e m o d u l i

l,...)

and expressions ax$$-'

am$$-'

and

can t h e n be e v a l u a t e d v i a

a s y m p t o t i c s ( n o t e d$$-l i s a m a t r i x o f holomorphic 1-forms whose o n l y , r a tional, singularity i s a t

It t u r n s o u t t h a t such isomonodromy c a l c u l a -

m).

t i o n s t h e n y i e l d d i r e c t l y t h e r e m a i n i n g equations i n (16.14) so t h a t u s a t i s f i e s t h e KdV f l o w equations and t h e s t r i n g equation.

REEIARK 16.4

(CAU FWCCZ0W).

The t a u f u n c t i o n a r i s e s as expected from 514.

Thus t h e r e w i l l be a c l o s e d 1 - f o r m

o n t h e space o f d e f o r m a t i o n parameters and

i s d e f i n e d v i a (A&) w = d1og.r.

'I

- a x2 1og.r a s d e s i r e d and

One o b t a i n s immediately from (16.17) t h a t u =

correspond t o a p a r t i t i o n f u n c t i o n f o r t h e m a t r i x model.

T

will

Various p r o p e r t i e s

o f t h e S t o k e ' s m a t r i c e s a r e i n v e s t i g a t e d i n [MF5,61 a l o n g w i t h BA framings, quantum Riemann surfaces, e t c . b u t we o m i t t h i s . view i s discussed a l s o , a l o n g t h e l i n e s o f 914. l a r p o i n t s t h e s o l u t i o n t o an m X m equation i z e d by i t s monodromy.

morphic and i n v e r t i b l e f o r X (A-av)) near av where

Roughly f o r r e g u l a r singu= A(,)$

can be c h a r a c t e r -

L e t av be t h e s i n g u l a r p o i n t s w i t h d i a g o n a l i z a b l e

Then $ i s c h a r a c t e r i z e d by

r e s i d u e s Lv.

a,$

The f r e e f e r m i o n p o i n t o f

3'

E

P'

-

(A+)

,. ..,an)

{al

(1) @(Ao)

(3) $(A)

= 1 (2) $(A)

i s holo-

= SV(X)exp(LVlog

i s holomorphic and i n v e r t i b l e i n a NBH o f a.,

Con-

v e r s e l y any such J, s a t i s f y i n g (&+) determines a r a t i o n a l m a t r i x A = $,$-' w i t h a t most s i m p l e poles.

Hence one has

a l o c a l problem o f d e t e r m i n i n g

f u n c t i o n s h a v i n g t h e c o r r e c t monodromy and i t s s o l u t i o n v i a c o r r e l a t i o n f u n c t i o n s o f s u i t a b l e o p e r a t o r f i e l d s $i(av)

based on f r e e fermions i s discussed

i n [MF5,6] ( c f . a l s o 514).

If t h e av a r e moved, p r e s e r v i n g t h e monodromy, one f i n d s as i n 514 (16.19)

a,

0

@ =

-1

(Av/(ho-av))$;

a

a,

$ = (-(Av/(h-av))

t (Av/(ho-av)))p

For a n a p p r o p r i a t e c h o i c e o f Av t h e c o m p a t i b i l i t y equations f o r (16.19) t h e S c h l e s i n g e r equations as i n 114. g i v e n v i a ( c f . 514)

The corresponding t a u f u n c t i o n i s

give

CONFORMAL FIELD THEORY (16.20)

.

-1 XRes =a

dlOgT(al ,.. ,a ) =

( d i n a v ) and i n f a c t

n

(Am)

255

T

=

(

$1 (al ). ..$,,(an)) ( t h e s i ( a i ) a r e r e l a t e d to

those in 114 a n d we do not give f u r t h e r d e t a i l s h e r e ) . vation of

(A=)

There i s a l s o a d e r i -

via CFT d i r e c t l y ( c f . [ KZ3;MF5,61).

For i r r e g u l a r s i n g u l a r points [MF5,6] follows [MWlI ( c f . 114). Thus one works i n s e c t o r i a l domains i n each of which t h e r e i s a fundamental s o l u t i o n o f a$, = A$ w i t h asymptotics a s i n (16.15). The a n a l y t i c continuation o f say I+ ( i n Q l ) will have asymptotic expansion (**) J , ~% (Ip,,$'(A-a)')(Aa ) Lexp(T)(S1. ..Sk-l)-l i n Q k and $ can be characterized by :A+) again w i t h ( 3 ) replaced by t h e requirement t h a t expansions (**) hold.

Again s o l u t i o n s

v i a c o r r e l a t i o n functions a r e constructed i n [ MF5,61 along with t a u functions b u t we omit d e t a i l s . 17, C0NTQIRElI: FZELD CHE0Rg (CFC). T h i s is a theory s t i l l i n development (along w i t h strings, quantum g r a v i t y , e t c . ) and we make no attempt t o be up t o d a t e . There were t a l k s a t t h e ICM meetings i n Kyoto, August 1990, rel a t e d t o this s u b j e c t , by e.g. B. Feigin, A. Schwartz, G . Segal, A. Tsuchiya, One can c i t e here e . g .

which go f a r beyond anything we discuss i n 117,18. [ FHl;MF7;V31

b u t one must wait f o r o t h e r papers t o appear.

Two dimensional

CFT i n t e r a c t s w i t h string theory (e.g. c l a s s i c a l s o l u t i o n s of str in g theory a r e conformally i n v a r i a n t 2-0 f i e l d t h e o r i e s ) and some o f t h e mathematics i s i n t i m a t e l y connected to t h e "sol i t o n mathematics" which has been developed already i n t h i s book; tau functions, Grassmannians, vertex o p e r a t o r s , t h e t a functions, e t c . a l l will appear.

We will e x t r a c t here from [ GS1 ] i n an a t -

tempt t o give some background material f o r ( c l a s s i c a l ) CFT ( c f . a l s o [BAN"; LS1;PKl;PVl

I).

We do n o t dwell much on physics o r philosophy and r e f e r t o

BFl;BAN2;BH1-5;GSl;LSl;MF7;PKl;PVl;V3] f o r general discussion. Thus some statements o r formulas which can be motivated via physics arguments a r e sim-

[

ply accepted a s d e f i n i t i o n s i n developing t h e mthematical constructions. A more axiomatic mathematical treatment, following [ KM1 ] i s given i n 518. We only develop a l i t t l e " c l a s s i c a l " material before going on t o Riemann surfaces a n d eventually 118.

-

Mainly we want t o d i s p l a y ( i n a more o r less l o -

gical c o n t e x t ) formulas such a s (17.6), (17.10)

Ward i d e n t i t i e s , (17.11 ),

256

ROBERT CARROLL

(17.13) Virasoro algebra, (17.141, and to say a few words about highest weight s t a t e s , conformal blocks, Verma modules, etc. 17.1 (BACKGRBUND mACERZAt). We extract here from [ GS1;LSl;PKll and refer t o these references for further discussion. Consider d dimensions with f l a t metric g = nuv of signature ( p , q ) and d S 2 = g d x' d xV . A change 'V of coordinates x x ' yields g g b V ( x ' ) = (axa/ax")(ax / a x ' v ) g a , ( x ) . The PV conformal group (by definition) involves g'PV ( x ' ) = n ( x ) g 'V ( x ) so the angle v . w / ( v 2w 2 ) 4 i s preserved (v.w = g LJVvpwv) and Poincar6 or Lorentz transformax' + t i o n s a r e conformal since g' = g . A t the infinitesimal level, for x' E' one o b t a i n s ds2 * ds2 + ( a cV + avE')dxpdxV so conformality requires ( * ) LJ a E" + a v E U = ( 2 / d ) ( a - ~ ) g(the ~ ~ constant i s obtained via a trace calculav tion u s i n g gwy). T h i s implies n ( x ) = 1 + ( 2 / d ) ( a . E ) a n d (gPVD+( d - 2 ) a l l a V ) ( a e E ) = 0. We will concentrate mainly on d = 2 w i t h g LJV = 6 'V 1i n which case 2 t h i s equation becomes the Cauchy Riemann ( C R ) equations (A) a l E = a2c , One writes then E ( Z ) = s1 + ie2 and : ( I ) = E~ - i E 2 where z, a l c 2 =,-a2€ z = x f ix Then 2-d conformal transformations coincide with analytic maps z -+ f ( z ) , 2 f ( 2 ) w i t h i n f i n i t e dimensional local algebra ( c f . below). 2 Here n = / a f / a z l and ds2 = d z d r R dzdz. Consider z z + ~ ~ ( w2i t h) E~ = -n+l ); evidently t h i s gives r i s e t o infinitesimal -zn" (and En = -2 generan +1a, and P n = -z -n+l a-.Z One regards now z a n d Z a s independent tors P n = -z variables; they are not conjugates unless specified. Then clearly [ P m y a n ! = (m-n)Lm+,,, [PmyPnl=(m-n)im+ny and [ a i ,P n n I = 0. We emphasize that z and z are independent coordinates and note t h a t P n i s part o f a local conformal algebra ( n o t global). In 2 dimensions the global conformal group i s the group G = SL(2,C)/Z2 2 SO(3,l ) of invertible conformal transformations o n P 1 = Riemann sphere and t h i s is generated by globally defined infinitesimal generators P -1' P 0 ' P 1 ,?-l,Foytl. The elements o f G can be represented by Mljbius transformations z + (az+b)/(cz+d), f + (Sf+i)/(ET+a) ( c f . also comments aft e r (17.15)) REIIIARI(

v

-+

-+

-+

-

'i

.

-+

-+

-+

-

- -

-

Now go back t o d dimensions and write lax'/axl = l/(detg;")' = n'3id. A QFT (quantum f i e l d theory) w i t h conformal invariance should have fields A i ( x ) with a subset $i of "quasiprimary" fields satisfying ( A i 'L dim$i) ( 0 ) O j ( x ) -+ lax'/ax( A j /d $ .J( X I ) for transformations x x ' via G = O(p+l,q+l) = con-+

CONFORMAL FIELDS

257

formal group. The theory should be covariant under such transformations in the sense t h a t c o r r e l a t i o n functions s a t i s f y ( A R - A k )

There should a l s o be a vacuum 10) invariant under G . These requirements i m pose strong r e s t r i c t i o n s on 2 and 3 p o i n t c o r r e l a t i o n functions and some a r gument i n [GSlI shows t h a t i n p a r t i c u l a r (17.2)

(

$,(xl ) 0 2 ( x 2 ) ) =

(A

[ 9 2 0/ 4 2

1

= A

2

= A

(A1 # A 2 1

where r12 = Ix1-x21. There is however more l a t i t u d e in dealing w i t h N point c o r r e l a t i o n functions f o r N 4. For 2 dimensions now ( r e c a l l z and 2 a r e independent and one only i d e n t i f i e s ? and z*- * i s used for conjugate sometimes-when so s t a t e d ) we t h i n k of ds 2 2 = dzd? .+ f f ds and generalize this transformation law to ( 6 ) O(z,z) + h-hZZ ( f Z ) (fZ) $ ( f ( z ) , 3 a ) where h,E a r e real valued (6 # h* unless s t a t e d ) ; h -Ti The r u l e ( 6 ) defines a primary f i e l d t h i s says $(z,Z)dz dz is invariant of weight ( h , i ) . A primary f i e l d will be automatically quasiprimary ( i .e. s a t i s f i e s (*)) and o t h e r f i e l d s a r e c a l l e d secondary (they may o r may not be quasiprimary). The 2 point function G 2 ( z i , z i ) = ( $ l ( z , y Z , ) $ 2 ( z 2 , z 2 ) ) will 2h -25 turn out now t o have the form c12/z12 z1 ( z 1 2 = 1z1-z21 ) which f o r bosonic f i e l d s w i t h s p i n s = h-h (1 7.2 1).

=

0 and A =

h+t

takes the form c12/(21212 '(cf.

Consider now a1 and uo a s space and time coordinates i n Euclidean space so s,: = oo ? ia' a r e the l i g h t cone coordinates (ao ? i n Minkowski metric). Left and r i g h t moving massless f i e l d s i n Minkowski space % Euclidean f i e l d s which a r e holomrphic o r antiholomorphic respectively. To el iminate i n f r a red divergences one compactifies the space coordinate = crl + 2-n) so one 1 0 has a cylinder i n ( a , u ) coordinates. Now consider 5 + z = exp(5) = exp 1 (ao + i a ) (cylinder plane) so t h a t past and f u t u r e (ao % %) go to 0 and m respectively i n z . Equal time surfaces uo = constant a r e c i r c l e s of cons t a n t radius and time reversal a' + -ao % z + l / z * (we will use * f o r conjugation usually since z' can have another meaning). Since d i a l a t i o n z -+

2

(2

-f

ROBERT CARROLL

258

e“z time t r a n s l a t i o n uo uo t c1 t h e d i a l a t i o n generator i n z can be regarded a s t h e Hamiltonian. The Hilbert space i s b u i l t up on constant radius surfaces and this procedure i s known a s r a d i a l quantization. A d t l dirnensional theory (xo time) w i t h an exact symmetry has an associated conserved c u r r e n t j’ such t h a t auj’ = 0. The conserved charge Q = I d d x j o ( x ) generQ

-+

Q

a t e s , via 6 € A = E [ Q , A I , t h e i n f i n i t e s i m a l symmetry v a r i a t i o n i n any f i e l d A . The s t r e s s energy tensor TVu is generally symmetric and divergence f r e e and i n conformal t h e o r i e s a l s o t r a c e l e s s ( s i n c e one r e q u i r e s conservation 0 = a - j = T’ f o r t h e d i a l a t i o n c u r r e n t j = Tpvxvy corresponding t o xu + x’ t u u Ax’). Generally local coordinate transformations a r e generated by charges constructed from THv. C l a s s i c a l l y TaB represents the response of a system t o changes i n the metric. For example ( c f . [ LS1;PVl I and 520) g i v e n a string w i t h world s h e e t W and metric h ( 0 , ~ )on W t h e Polyakov a c t i o n i s S = -L,T Jd20h4 h a B a a x v a g x Y ~ u v= -L,T I Y d oh’ haBrag ( h = - d e t ( h ), T tenP a8 s i o n , nuv % d i a g o n a l ( - l , l , 1 ) % f l a t background ( b u t this could be generMd = d-dimensional Minkowski space). Then Tag is dea l i z e d 1 x’(u,T): fined via T = - ( /Th’)6sp/6haB = L,aaxpaBxu - Lh - hY6a x’a x and t h e ac“8 &gB Y 6lJ t i o n m i n i m i z i n g c r t e r i o n i s Ta8 = 0 ( p l u s a a ( h 2 h a8x’) = 0, which corresponds to a minimal area equation). For i n f i n i t e s i m a l conformal transformai s t r a c e l e s s we have t i o n s j = T ~ , ~where E c v~ s a t i s f i e s (*); s i n c e T ’V automatically a - j = ’TL (a.E) = 0. In 2-D ( z = x t i y ) w i t h g z z = g2z = 0 and g zz- = g-zz = % one has ( e x e r c i s e ) TZz = %(Too - 2iTlo - T l l ) , TzT = %(Too + 2iTl0 - T11 ), and Tir = Tzz = % ( T0% t T11 ) = %T’u = 0. T h i s leads via ga’aaTuv = 0 t o (*) T(z) = TzZ(z) and T ( T ) = Trz(Z) a s t h e only nonvanThen T and 7 generate local conformal transformai s h i n g components of T “B’ t i o n s i n z,? and i n r a d i a l coordinates I j o ( x ) d x % I j r ( e ) d e . Thus one considers conserved charges o f the form ( m ) Q = ( 1 / 2 s i ) 6 ( d z T ( z ) E ( z ) + d z T ( T ) E ( f ) ) ( t a k e 6 d z and $ d ? i n the same d i r e c t i o n U ) . ‘1,

...,

-+

’

REmARK 17.2 (RADZAC ORDER, C0RRECACIOW FUNCCI0W, UARD Z D E N C Z C I S ) . Now the v a r i a t i o n of a f i e l d cp(w,G) i s given by an equal time commutator w i t h the charge Q of

(m)

via say

CORRELATION FUNCTIONS

259

(we accept t h i s without discussion a s a d e f i n i t i o n ) . One r e c a l l s here t h a t the “Greens” function ( A1 (xl ,tl ). .An(xn, t ) ) in Minkowski space has a Eun clidean form via A ( x , t ) .+ exp(HT))A(x,O)exp(-Hr)) ( t = i r ) a n d convergence of ( ) i s assured only for r j z T ~ + ~ Thus . one must expect t o do some radial ordering in (17.3) in order t o have the integrals make sense. Thus one writes

.

(17.4)

R(A(Z)B(W))

=

[

A(z)B(w)

(121 >

Iwl)

B(w)A(z) (121 < I w l ) sign i s used with fermion operators). Then define I [dxB,AIET % ( i , - 62)dzR(B(z)A(w))s$c dzR(B(z)A(w)) where C i s a small c i r c l e Iz-w] = E (draw a picture 1Q u , d ) . Thus t o evaluate (17.3) I lzI>lwI - r I ~ I < I %~ +Ic (with radial order imposed) a n d (17.3) becomes ( r a d i a l l y ordered products can be considered as say meFomorphic)

(a

-

+-2e/:&

where the l a s t l i n e i s n o t a r e s u l t o f calculation b u t simply the required r e s u l t for f ( z ) z + E ( Z ) with ( 6 ) . This leads one t o deduce ( h e u r i s t i c a l l y ) t h a t ( t o get the correct answer in (17.5))

,.,

(17.6)

R(T(z)$(w,i))

=

(h/(z-w)

2

)$(w,$) + (l/(Z-W))aw$(WyiT)+

R(?(?)$(w,G)) = ( h ( ? - i ) 2 ) $ ( w y i i )

+ ( l / ( Z - ~ ) ) a m $ ( w y i+4 )

...;

...

One now drops R in (17.6) a n d considers (always) the operator product expansion a s shorthand f o r radially ordered product. I t i s instructive here t o read [GSl 1 for folklore a n d physics. In 2-D CFT one can always take a n orthogonal basis $i of f i e l d operators with normalized 2 p o i n t functions ( 1 7 . 2 ) of the form (17.7)

( 4 i ( z , Z ) $ j ( w , Z ) ) = (&../(z-w)2 hJ)(l/(Z-G)2’j) 1J

One should also mention the idea of Ward i d e n t i t i e s which a r e general ident i t i e s s a t i s f i e d by correlation functions which r e f l e c t symmetries in a

260

ROBERT CARROLL

t h e o r y ( c f . IBERl;WD1-51).

G e n e r a l l y as i n (17.1) one expects

under M8bius t r a n s f o r m a t i o n s z

-+

w,

z

-+

w.

Now w i t h f i e l d o p e r a t o r s a t t h e

p o i n t s wi c o n s i d e r

( $ ( ~ Z / ~ ~ ~ ) E ( Z ) T ( Zl,~l)...$n(~n,Wn)) )$~(W

(17.9)

($k

Q

$k(wk,zk))

=

1;

< + ( w l,iil)...

where 6E$(w,G) = % ( d z / 2 ~ i ) ~ ( z ) T ( z ) $ ( w , W ) = (E(w)a

t

hae(w))

$(w,G) ( c f . (17.5) - r a d i a l o r d e r i n g i s assumed). Here t h e o r i g i n a l c o n t o u r i s t o surround t h e w . and t h e i n t e g r a l breaks up i n t o a sum o f i n t e g r a l s J around c i r c l e s e n c l o s i n g each w T h i s l e a d s t o t h e u n i n t e g r a t e d formula j' ( c f . (17.6))

(conformal Ward i d e n t i t y

-

holomorphic p a r t ) .

R e c a l l here T i s t h o u g h t o f

i n t h e c o n t e x t o f general l o c a l conformal t r a n s f o r m a t i o n s .

Then (17.10)

says t h a t t h e c o r r e l a t i o n f u n c t i o n s a r e meromorphic f u n c t i o n s o f z w i t h s i n g u l a r i t i e s a t the operator p o i i t i o n s w

REmARK 17.3 (IIESCENVENC FZELDS,

m\

j'

CENb02, 1sZRAs0RP) AfXE3RA).

Now f i e l d s

a r e grouped i n t o f a m i l i e s {$,,I c o n t a i n i n g a s i n g l e p r i m a r y f i e l d $n and an i n f i n i t e number o f descendent f i e l d s (e.g.

a$,).

These correspond t o i r r e -

d u c i b l e r e p r e s e n t a t i o n s o f t h e conformal group and t h e p r i m a r y f i e l d The descendent f i e l d s may n o t obey ( & ) o r (17.6).

e s t weight.

?r

high-

For example

one can show t h a t (17.101) ( c f . [ GS1

T(z)T(w)

I).

%

$ c / ( z - w ) ~ t (2/(z-w)*)T(w)

t

(l/(z-w))aT(w)

The t e r m c i s c a l l e d t h e c e n t r a l charge and i t s v a l u e depends

VIRASORO ALGEBRA t h e p a r t i c u l a r t h e o r y b e i n g considered. and c =

261

S i m i l a r arguments a p p l y t o T(S)‘i(w)

= 0 can be achieved i n h e t e r o t i c s t r i n g t h e o r y f o r example by cou-

p l i n g t h e system t o 2-D quantum g r a v i t y v i a a d d i t i o n o f ghosts ( c f . [ GS11 ). The (holomorphic) f r e e boson s t r e s s energy tenson ( w i t h a c t i o n S = ( 1 / 2 1 ~ )

J axaz, _x

= $(x(z) t

Z(’i))

%

l e f t and r i g h t movers

pagators ( x ( z ) x ( w ) ) = -log(z-w), (17.10~) ~ ( w ) = -h:ax(v)ax(w):

(X(Z)X(G))

%

xL(z)

= -log(:-W))

+ x R ( ? ) and

pro-

i s defined v i a

1i m 2 = -t2z+w ( a x ( z ) a x ( w ) + l / ( z - w )

where x ( z ) i s n o t a conformal f i e l d b u t a x ( z ) can be i n t e r p e r t e d as a (1,O) conformal f i e l d .

T(z)ax(w) = -$:ax(z)ax(z):ax(w) = -$ 2 2 = (ax(w) + (z-w)a x ( w ) / ( z - w ) + . % ax(w)/(z-

Here ( f r o m (17.10”)

ax(z)( a x ( z ) a x ( w ) ) ( 2 ) t

...

..

...

a2x(w)/(z-w) + and [ f ( d z / Z n i ) T ( z ) ~ ( z ) , a x ( w ) ] = 4 ( d z / Z a i ) s ( z ) 2 2 2 + a x(w)/(z-w) t ) =aE(w)ax(w) + E ( w ) X ~ ( W ) ( c f . (17.5) (ax(w)/(z-w)

-w)‘

t

...

with h = 1). For f r e e fermions w i t h $ ( z ) ( r e s p . ?(?)) corresponding t o l e f t (resp. r i g h t ) ( $ ( z ) and b(?) % l e f t = oxax + o a = and r i g h t m o v i n g c h i r a l i t y ) . The 2-D D i r a c o p e r a t o r i s Y Y 0 a, ia,) = (-o a 0 1 0 -i a o) ( o x = (l o), cry = (i o ) ) . One normalizes so (a,+ i a y o Then (*A) T ( z ) = % t h a t (**) $(z)$(w) = - l / ( z - w ) and $ ( r ) $ ( i )= - l / ( ? - G ) . c h i r a l i t y one has an a c t i o n S = ( 1 / 8 n ) l ($%JJ + $a$)

-

: $ ( z ) a $ ( z ) : and ?(?) = $:$(?);J(?): s a t i s f i e s (17.9) w i t h c = Z = $ and a r e primary f i e l d s o f weight ( L , , O ) ,

(O,+).

$,T

One makes now a L a u r e n t e x t e n -

s i o n o f T i n t h e form

-

( t h e o p e r a t o r s Ln,Ln a r e c a l l e d modes) and f o r m a l l y ( * a ) Ln = ( l / Z a i ) $ T(z)dz w i t h

Zn

= ( 1 / 2 a i ) 9 zn+’f(T)dT.

proceed as i n (17.5)

Now t o e v a l u a t e [ 6 dz, 9 dw] we

( f i x w f i r s t ) t o o b t a i n e.g.

zn+’

ROBERT CARROLL

262

3 n+l

Note t h e f i r s t residue involves (1/3!)aZ z

z=w

=

( 1 / 6 ) ( n + l ) n ( n - l )wn-'.

I n t e g r a t ng t h e l a s t term by p a r t s and combin n g gives ( e x e r c i s e ) (1 7.1 3 )

One a l s o checks t h a t [ Ln,Tml = 0 and we have thereby constructed t h e Vira= 0 t h i s is the c l a s s i c a l VIR and i n general soro algebra (VIR). For c = 1 i t is a central extension o f t h e P n algebra of vector f i e l d s on S .

+

Now on t h e real s u r f a c e z = z* an a d j o i n t is defined via A(z,T) = A(l/T, l / ~ ) ( l / , ? ) ~ ~ ( l / z In ) ~and ~ . o u t s t a t e s a r e defined via ( * b ) [ A i n ) = lim A ( z , i ) l O ) (2,: 0 ) . For ( A o u t l one reparametrizes, z = l/w, and s e t s (**) -+

(olr(w,w).

L.

1 im A(w,i) = A ( l / w , l / w ' ) ( - w - 2 ) h ( - ~ ) - 2 ~ ( c f . ( 6 ) ) w i t h ( Aoutl = w,'wto There i s then a natural a d j o i n t r e l a t i o n ( A o u t I = I A i n ) + and one often 1i m 2h 1i m w r i t e s ( A o u t ] = z,T* (OIA(z,Y) which means of course z,zm ( O I A ( z , z ) z TZh. I t turns o u t t h a t in CFT one can a s s o c i a t e a unique f i e l d w i t h each s t a t e , s i n c e the number of f i e l d s and s t a t e s w i t h any fixed conformal weight t mt2 i s f i n i t e dimensional ( c f . [ G S I I ) . For T t h e r e l a t i o n s T ( z ) 1 Lm/? and T ( l / Z ) ( l / Z 4 ) = 1 Lm/?42-m-2 y i e l d (*m) = L-,. Generally one requires

Li

-m-2

10) t o be regular a t z = 0 so only terms with m 5 -2 a r e T(z)lO) = 1 Lmz t allowed. T h i s implies (A*) L,IO) = 0 ( m 2 - 1 ) and ( u s i n g (*.)-(*+)) ( OIL, f = 0 (m 1. - 1 ) ; from L, = L-,, we have a l s o Li10) = 0 ( m 5 1 ) and ( O I L , = 0 (m

Similar r e s u l t s hold f o r the

< 1).

In.

Consider now t h e s t a t e I h ) = $(O)lO) generated by a holomorphic f i e l d 4 of weight h. From (17.6) and (17.11) one has (17.14)

[

so [L,,+(O)I Lolh) = h l h ) by a primary h -+ h . Such l h ) (ni > 0)

-

(hlL

= 0

M hl

Ln,$(w) 1 =

#

(dz/Zri)z"'T(z)$(w)

=

h ( n t 1 )wn$(w)+wn++'a$(w)

0 f o r n > 0. Hence ( r e c a l l Lolo) = 0 ) I h ) s a t i s f i e s ("1 a n d L n l h ) = 0 ( n > 0 ) . More generally an i n s t a t e Ih,F)created f i e l d $ ( z , Z ) o f weight ( h , i ) will a l s o s a t i s f y (a) with L s t a t e s a r e c a l l e d highest weight s t a t e s and s t a t e s L-n ...L -nk a r e descendent s t a t e s . The o u t s t a t e s ( h l evidently s A t i s f y a n d 0 = ( h l L n ( n < 0 ) . S t a t e s ( h l L ...L n R ( n i > 0 ) a r e c a l l e d =

-+c,

n,

VIRASORO ALGEBRA descendents of t h e o u t s t a t e < h l .

263

An easy c a l c u l a t i o n using this informa-

t i o n w i t h (17.13) y i e l d s (17.15) < h l L f n L - n l h ) = ( h l [ Ln,L-n]Ih) = 2n( h l L o l h ) + (c/12)(n3-n)( h l h ) = ( 2 n h + (c/12)(n3-n))( h l h ) For n l a r g e this implies c > 0, and f o r n = 1 , h 0 i s required s i n c e t h e l e f t s i d e is p o s i t i v e (except f o r ghosts which will be only b r i e f l y i n t r o duced l a t e r ) . For c = 0 one can show t h a t t h e Virasoro algebra has only t r i v i a l u n i t a r y representations ( c f . [GSl I ) . Note t h a t a f i e l d 0 w i t h conformal weight (h,O) is purely holomorphic s i n c e from (17.14) adapted t o T ( z ) one g e t s [ L - l y $ ] = a$ b u t arguing a s i n (17.15) one f i n d s IIL 1$10)11 = 0 so a$ = 0 ( n o t e L - 110) = 0 from (A*) e t c . ) . Now examine I h ) = $(O)lO) i n terms of modes by w r i t i n g f o r $ a holomorphic primary f i e l d o f weight ( h , O ) (Aa) -n-h w i t h 9, = ( 1 / 2 a i ) $ z h + n - l $ ( z ) d z . Regularity of $ ( z ) = CnEZ-h@nz 10) a t z = 0 requires t h a t $,lo) = 0 f o r n 2 -h+l a n d I h ) i s created by $ - h ( I h ) = +-hlO)). Similarly (Ol$,, = 0 f o r n 5 h-1. Note f o r h < 0 t h e r e a r e which a n n i h i l a t e n e i t h e r 10) nor ( 0 1 . These will not correspond t o modes u n i t a r y s i t u a t i o n s since h < 0 b u t involve ghosts (e.g. t h e c-ghost has h -1 and t h e zero modes c - ~ ,co, c1 do not a n n i h i l a t e t h e vacuum). Now (17.16) ( 1 / 2 a i ) (p w

[Ln,$,,] h+mtn-1

=

(l/Zvi)

b!

dww

h +m- 1

=

(h(n+l)wn$(w) + wnt

( h ( n + l )-(h+m+n))$(w) = (n(h-1)

-

m)+,+,,,

Hence [ L o ,$m ] = -m+ m which i s c o n s i s t e n t w i t h L0 ( h ) = Lo$-, 0) = h l h ) . We = (n+h)L-nlh). see a l s o from (17.13) t h a t ( A 0 L o L - n l h ) = (nL-, + L-.,Lo)lh Note here again from (17.11) t h a t the Ln a c t as generators o f a l l possible conformal transformations and comparing (17.15) w i t h (17.5) and ( 6 ) we s e e t h a t Ln Q E ( Z ) = zn+'. In p a r t i c u l a r Lo, L e l , L1 generate i n f i n i t e s i m a l transformations 6z = a + BZ + y z 2 and generate SL(2,R); adding To ,T1, T-, we get SL(2,C) ( c f . here remarks before (17.1)). Such transformations can 1 be represented by Mblbius transformations on P a s before. REmARK 17.4 (StAtZSCZQ, C0RRECACZ0N fllNetCZ0W, AND C0W0wIIAI; Bt0CKk)g r o u p f i e l d s $,, i n t o families {$,,I a s indicated above ( c o n s i s t i n g of

Now

ROBERT CARROLL

264

descendent f i e l d s ) ; acting on the vacuum these descendent f i e l d s create descendent s t a t e s . One will see next t h a t the Ward i d e n t i t i e s give different i a l equations determining the correlation functions of descendent f i e l d s in terms of primaries. The conformal families ('L irreducible representations of VIR) organization allows one t o develop the theory via Greens functions A o f the primary f i e l d s . Let us write now ( L - n $ y n > 0, denotes descendent fields) (17.17

= i0$/(Z-W)

z-w) + Here $ i s primary and L n l h ) = 0 for n > 0 i s (17.18)

?-.,$(w,K)

= (1/2ni

2

t

...

implicit.

T(z)$(w,~)dz/(z-w)n--'

Then

= $-n

a n d these a r e called Virasoro descendents ($-n has weight ( h t n , ; ) ) .

The

calculation (?-2*l)(w) = (1/2n ) 6 T(z).l dz/(z-w) = T(w) shows t h a t T ( w ) i s always a level 2 descendent of the identity operator 1 . One orders these n

f i e l d s by conformal weight so (A+) level 0 'L h 'L 9; level 1 'L htl 'L L,l$; n 42 6 "3 level 2 'L h+2 'L L-2$ a n d L - l $ ; level 3 'L h+3 'L 3 + y L,1L-2$, L l $ y ..., level N 'L h t N 'L P ( N ) f i e l d s where P ( N ) i s the number of partitions of N into positive integer parts. Translated into s t a t e language we have what i s called a Verma module, i . e . a representation of VIR determined by a highest

7- ...?- k!,+j

weight s t a t e , e.g., 'L h consistingoffields jy k, [ K1,2] for the algebraic point of view). Note here t h a t (17.19)

l/n;

(l-qn) =

?-'$

),

etc. (cf.

N lm P(N)q 0

(P(0) = 1 ) and = a$ so a$ E {$I, along with a l l other derivatives. Take now the Ward i d e n t i t i e s (17.10), l e t z -+ wn, expand in powers of z-w,,, a n d use (17.17) t o obtain

CONFORMAL BLOCKS

265

Generally, say f o r orthogonal primary f i e l d s a s i n ( 1 7 . 7 ) , t h e r e will be an operator product expansion (summation over p and I k k l )

...-

Ckrkl -= AL m k , . . . L A- k , L -“i L-- $ There a r e r e l a t i o n s C Ii jkp i ) -- C i j p I kL P‘ where t h e C i j p a r e t h e operator product c o e f f i c i e n t s f o r priBij Tij m r y 4 (theA c o e f f i c i e n t s depend on ( h i y i i , c y E ) and can be determined by P conformal invariance). I t follows ( c f . [GSl I ) t h a t complete information t o specify a 2-D CFT i s provided by t h e conformal weightshi,ii o f t h e Virasoro highest weight s t a t e s and t h e C between t h e primary f i e l d s t h a t c r e a t e ijp them. To determine t h e C i j k and h one needs dynamic information and various symmetries can a l s o be exploited. For example d i f f e r e n t ways o f c a l c u l a t i n g say 4 point c o r r e l a t i o n functions must be equal ; i n t h i s approach one lumps together c o n t r i b u t i o n s belonging t o t h e conformal f a m i l i e s {+ 3 a s : F ( P I X ) P The F!m(plx) a r e known as conformal blocks and serve t o d e t e r F!m(plx). 1.J 1.J mine any c o r r e l a t i o n function. A

where

$

REmARK 17.5 (CE0lltECRZt QllANCZZACZ0N AND CHHE 30REC WEZL CHE0REEI). In 818 we will give a semiaxiomatic treatment of CFT following [ KM1 I b u t t h a t i s n o t t h e l a s t word; t h e r e a r e many ways o f looking a t t h e matter and we want t o mention here a few ideas from [AS1-3;GW1;PL1;RI1;RP1;STO1-3]. The point i s only t o i n d i c a t e a few formulas and r e s u l t s t o help smooth t h e passage from §17 t o 118 (and l a t e r t o §20,21). There i s a l s o a connection to t h e r o l e of conformal blocks. Thus following CV33 one r e f e r s t o t h e idea o f geometric quantization which c l a s s i c a l l y provides a quantum system corresponding t o a c l a s s i c a l phase space r w i t h a symplectic s t r u c t u r e w ( c f . [ C1;PLl;WOl I ) . Here one goes t h e o t h e r way; t h e quantum mechanical Hilbert space H i s known (= space o f conformal blocks - see below) and one wants t o f i n d t h e under-

266

ROBERT CARROLL

l y i n g c l a s s i c a l phase space (r,w). Thus suppose r i s a G h l e r manifold with i d h l e r form w of t h e form ( A m ) w = w . . ( z , z ) d z A dZJ with w i j ia.7.K. 1J 1 J There i s some gauge freedom K + K t a ( z ) t Z ( 4 ) of course. Define Poisson ij ij Quantibrackets via (a*) I f , g l = w ( a i f a . g - a i g a . f ) where w wjk = A i k . J J zation means we replace E , 1 by commutator brackets a n d represent t h e commutator algebra via operators i n a Hilbert space. Hence consider covariant d e r i v a t i v e s (*A) v i = a i t a i K and 0 = a w i t h curvature (0.) [ v i , T . ] = j jJ - i w i j and [ v i , v . ] = [Oi,F.] = 0. Then (0,V) determine a connection on a J J holomorphic l i n e bundle w i t h f i r s t Chern c l a s s w. The Hilbert s t a t e s a r e defined (via geometric quantization) t o be s e c t i o n s o f L a n n i h i l a t e d by ha1 f o f the d e r i v a t i v e s . For example one chooses H as the space o f holomorphic sections H = = 01. Then t h e wave functions a r e l o c a l l y holomorphic functions $ ( z ) on r , which transform as $ + e x p ( a ( z ) ) $ . The inner product requires V* = v so

-

-

w i t h measure determined by w.

There is a general recipe i n geometric quantization based o n t h e f a c t t h a t a l l unitary representations o f a compact group can be obtained by q u a n t i z a t i o n o f i t s coadjoint o r b i t s . To see how this works, l e t U be a unitary representation of G on a f i n i t e dimensional vector space H (**) g: I $ ) + U(g)

I $ ) ( I $ ) E H ) . One wants to represent t h e I $ ) as wave functions f o r a q u a n t u m mechanical problem. Thus choose a Cartan subgroup T C G and l e t I h ) H be a highest weight s t a t e , a n n i h i l a t e d by p o s i t i v e r o o t s , a n d s a t i s f y i n g ( 0 6 ) U ( h ) l A ) = exp(ihe)lA) ( h = exp(ieH) E T and A(H) % A h e r e ) . One cons t r u c t s a complete basis o f H by a c t i o n on 11) w i t h negative roots a n d t h e so c a l l e d coherent s t a t e s U ( g ) l A ) form an ( o v e r ) complete b a s i s o f H ( c f . [ PL1;KLAll - for r e l a t i o n s t o tau [AS1-3;811,21), The wave function % E

)1

H is

(a*) $ ( g ) =

$(g-’g’)

a n d via ( 0 6 )

Hence $ a section function. Q

Then G a c t s on $ ( g ’ ) via (am) ( U ( g ) $ ) ( g ’ ) = one sees t h a t $ ( g h ) = e x p ( - i A e ) $ ( g ) ( s e e below). of a l i n e bundle over G/T and will lead t o a wave

(AIUt(g)l$). $

To s e e t h i s one notes i n general ( c f . [ C42;Hfl 1) t h a t , i f V ( G )

= G

XH V

BOREL WEIL THEOREM

267

(vector bundle over G / H ) w i t h points ( g , v ) - H = { ( g h , h - l v ) l , f o r V a vector space and H C G a closed Lie subgroup, then s e c t i o n s o f V ( G ) correspond t o functions $: G -+ V s a t i s f y i n g $(gh) = h-’$(g). The correspondence is expressed by (1*) v ( g H ) = ( g , $ ( g ) ) - H . Further the representation ( p ( g ) $ ) ( g o ) = $(g-’g0) corresponds t o ( F ( g ) F ) ( x ) = g.;(g-lx) ( x = a ( g o ) ) ; note g - ( g ’ , v ) . H = ( g g ’ , v ) - H . We observe here t h a t f o r $ ( g ) = ( h l U t ( g ) l $ ) = ( U ( g ) l A ) * , I $ ) ) we have $(gh) % t A I U t ( g h ) l $ ) w i t h U ( g h ) l X ) = U(g)U(h)[X)= Ulg)exp(iAe)(x) so U(gh)lA)* ‘L exp(-iAe)( A I Ut( g ) and $ ( g h ) = exp(-ihe)$(g). We r e c a l l a l s o ( c f . + [PRlI) t h a t G / T > GC/Bf where B is t h e Borel subgroup generated by T and the p o s i t i v e r o o t s ; f u r t h e r A : T +. S1 extends uniquely t o a holomorphic homomorphism A : B+ + C* and one will think now o f V % C and Gc XB+ C a s t h e homogeneous l i n e bundle o f i n t e r e s t . Actually one considers now (4.) yhol ( 9 ) = $ ( g ) / A ( g ) over r % GC/Bf 2 G / T (coadjoint o r b i t space) where h ( g ) i s u the wave function (A) (highest weight s t a t e ) . Then $hol (gb) = F h o f g ) a s desired ( s i n c e A(gb) and $ ( g b ) will b o t h have t h e same m u l t i p l i e r A(b) E C*). This complex o f ideas i s known a s t h e Borel Weil theorem ( c f . [ PR1 I ) . T h u s s t a r t i n g w i t h coherent s t a t e s U ( g ) l X ) i n H one a s s o c i a t e s wave funct i o n s T h o l ( g ) on a phase space r ( i . e . one c r e a t e s r ) . For the symplectic form w one represents tangent vectors t o r by elements E E ?= Lie algebra o f G and uses t h e Kostant-Kirillov form ( 6 0 ) w g ( e 1 , ~ 2 )= A g ( [ ~ l , ~ 2 1 ) where A s = g - l h g ( n o t e h E (Kt)* c *; and A g % Ad*(g)A - c f . Appendix A and [ PL1; ML21 f o r various points of view). h u s a representation space H l e a d s to (r,w). This i s useful i n dealing w t h conformal blocks e t c . a s in 1 V31, where 2-D quantum g r a v i t y a r i s e s a s t h e s c a l a r product on the conformal 61 ocks. Q ,

AND CAAU FAUNCEl0)Nd. Naturally mathematicians w a n t t o c r e a t e b e a u t i f u l , a l l encompassing, a b s t r a c t t h e o r i e s b u t this i s not always poss i b l e or a p p r o p r i a t e . Thus t h e s p i r i t i n this book has been to d i s p l a y various points of view, i n t e r a c t i o n s between a r e a s i n physics and mathematics, e t c . I t i s compelling however to give a sketch o f a t l e a s t one semi-axioma t i c approach t o C F T on Riemann surfaces following [ KM1 1. This presentation uses a l o t o f the material we have already emphasized in o t h e r s e c t i o n s and we will repeat some d e f i n i t i o n s and r e s u l t s . In p a r t i c u l a r t h e tau function 18. mORE 01 W k

268

ROBERT CARROLL

emerges i n a s i g n i f i c a n t way.

Thus some f a m i l i a r i t y w i t h §8,12,13,17,19-21

w i l l make t h e development seem q u i t e n a t u r a l .

T h i s s e c t i o n w i l l a l s o serve

t o b r i n g t o g e t h e r i n a u n i f i e d way many ideas discussed i n a more fragmented manner elsewhere i n t h e t e x t .

KM1 I i s e x t e n s i v e , d e t a i l -

The development i n

ed, and r i g o r o u s and we w i l l o n l y s k e t c h m a t t e r s (a number o f t h i n g s a r e o f course proved elsewhere i n t h e t e x t ) .

We r e f e r a l s o t o [AG1-6;BFl;EGlY2;

BABl ;DE2; FE1 ;FU1; GAD2; FW1 ,2 ;IH1 ;IV1 ;KI1; KC1 ;KZ1 ;LS1 ;MD1 ;MUK1 ;NK1; SE2; S12; SW1-5;TS1-3;VIl

;WT11 f o r r e l a t e d work.

Thus t h e e n t i r e development i n 118

i s based on [ KM1 1 and we do n o t g i v e f u r t h e r s p e c i f i c r e f e r e n c e .

There w i l l

be much r e c a p i t u l a t i o n o f d e f i n i t i o n s and r e s u l t s i n d i c a t e d o r proved a t o t h e r p l a c e s i n t h e t e x t b u t t h i s s h o u l d be i n s t r u c t i v e .

(Urn, MA IN AND Y0UNP; DIAGRAIW, EEC.),

REmARK 18.1

The u n i v e r s a l Grassmann

m a n i f o l d (UGM) o f Sat0 i s discussed i n §13 and we r e c a l l a s needed v a r i o u s i d e a s ( c f . a l s o §8,12,14).

There i s a l o t o f a l g e b r a i c s t r u c t u r e here and

we w i l l t r y t o d i s p l a y o n l y t h e e s s e n t i a l .

One f e a t u r e i s t o r e f o r m u l a t e

t h e p r e s e n t a t i o n o f [ D l 1 i n t h e s p i r i t o f CFT;

H under B: F

+

t h e image o f UGM i n

again

H can be c h a r a c t e r i z e d by a c o n j u g a t e p a i r o f wave f u n c t i o n s

and t h e H i r o t a equations f o r t a u f u n c t i o n s

(F,H, e t c . t o be d e f i n e d ) .

l e t V be a v e c t o r space o v e r C w i t h a f i l t r a t i o n ... Z ) such t h a t Us"V = V, n F m V = {Ol, dimCFmV/Fm+'V

E

3

FmV

3

Fm+'V 3

Thus

...

(m

= 1, and t h e t o p o l o g y

determined by IS"V1 as a system o f NBH o f 0 i s complete.

4

P i c k Zh = Z +

=

In++] as an i n d e x s e t ( n E Z ) and choose e' E F'-V ' - F'% so t h a t any v E V has an expansion v = v e' ( - - < < j l < w , l ~ E Zh) and FmV = { v = lm
1

'

?

A canonical example here i s V = c(d<)',

= n and em+'

The f i l t r a t i o n comes from t h e " v a l u a t i o n " w(sn(ds)')

series. = sm(dc)'

(dz)'

= C ( ( r ; ) ) = f i e l d o f formal L a u r e n t

i s a b a s i s ( c f . Remark 5.3 f o r (dc)').

t h e n w(zm(d.z)')

I f z c = 1 and V =

C((2-l))

= -m-1.

= W ( C - ~ -(dc)') '

One takes now V * = HomC(V,C) as t h e t o p o l o g i c a l dual w i t h f i l t r a t i o n FmV ker(V*

-+

(FmV)*) and w r i t e s FmV* = F-,V*

= ( c l v ) = ^v(v) i s used.

so

6'+v'o

(v^'

E

= 6;.

C).

^em+'

E S"V*

-

A dual b a s i s $' FmtlV*

One w r i t e s a l s o e

and any

'

= e-'

and

w i t h FmV = F-mV. E

PV* has

t h e form

so (e^'le')

=

The symbol (?,v)

i s t h e n determined v i a (2'le")

e^ ' = ^e"

*

= = 6:

1 G

=

m w 1$' and (e'lew)

MAYA DIAGRAMS

269

We r e f e r now t o 113 f o r Young and Maya diaqrams and say here t h a t a subset M

C

Il.r

Zh i s a Maya diaqram i f b o t h M n Zh,O

E

Zh, p > 01 and MC = complement M ) .

and Mc n Zh,O

n ZhC0) = charge o r E u l e r c h a r a c t e r i s t i c o f M. charge p and < pl

Zh,

-f

a r e f i n i t e (Zh,o

-

Set a l s o K(M) = #(M n Zh,O) Let

m

mP

=

#(Mc

= Maya diagrams o f

= U fl For M E fl t h e r e i s an i n c r e a s i n g f u n c t i o n p : 111 E P’ P Zh such t h a t M {...,p(p-3/2), v(p-+)l. Note ~ ( v =) v f o r v

< e m ( t h e n o t a t i o n and c o n s t r u c t i o n s a r e c l e a r e r i n §13 b u t we keep here t h e

notation o f [KMl;KIl;NKlI

-

{u(v)

f o r easier referencing

-

c f . a l s o 58).

The s e t

v > 0, v E Zhl i s f i n i t e and determines a Young diagram Y(M)

513) w i t h degree d(M) =

1 ’(v) -

u = # boxes.

One w r i t e s

(cf.

Q (resp. yd) f o r

t h e s e t o f Young diagrams ( o f degree d ) and notes t h a t #gd = p ( d ) = number o f p a r t i t i o n s o f d. C

For f i x e d (charge) p,UGMP = s e t o f c l o s e d subspaces U

V such t h a t k e r n e l and cokernel o f t h e n a t u r a l map f: U

-

i t e dimension and p = dim k e r ( f ) diagrams

-

UGM = UUGMP and U i s s e m i - i n f i n i t e ) .

U t h e n dim FmU/?”U

5 1 and M(U)

V/FoV have f i n -

-+

dim c o k e r ( f ) ( c f . 513 f o r d e t a i l s and Note t h a t i f FmU = FmV n = 11 i s a Maya

= {m++ E Zh; dim FmU/Fmt’U

diagram o f charge p i f and o n l y i f U E UGMP.

One can p u t a “good“ s t r u c t u r e

o f i n f i n i t e dimensional complex m a n i f o l d o n UGMP f o r example (which we o m i t s i n c e f o r o u r purposes i t i s n o t r e a l l y needed i n showing how t h e a l g e b r a

is

f i t s together). The (holomorphic) t a n g e n t space t o UGM a t U v i a TUUGM 2 Hom(U,V/U). A frame 5 E U (U E UGMp) i s a b a s i s o f U such t h a t t h e r e e x i s t s uo E Zh w i t h c’ E F”-’U +L r 11 i Po and 5’ F e’ modulo F’ ‘U; one w r i t e s 5 = e”€,; ( v The s e t o f frames i n UGMP i s denoted by FUGMP.

sp-’1

RrmARK 18.2

E

then given

5 = I . . . , € , ~ - 3 / ,2 Fps’U,

..

Now f o r each Maya diagram M = { . , f o r e’(p-’) A e p(p-3/2) The Fock

(CHE PLUCKER EiIBEDDtNG),

p(p-3/2),u(p-t,)}

w r i t e eM o r

space (charge p ) i s F = ll

P

IM)

M

KmP

for a l l

Zh, r E Zh ).
Ce

....

and

F = @Fp. D e f i n e charge and mass op-

e r a t o r s by J o ( a ) = pa ( a E f ) and M(a) = da i f a = exp(M) w i t h d(M) = d. P Then [ JOYMI= 0 and s e t t i n g F ( d ) = { a E F; J o ( a ) = pa; M(a) = d a l one has ’ M F = 8,lTd F p ( d j ( n o t e F p ( d ) = 8 Ce w i t h x(M) = p. d(M) = d and dim F ( d ) =

-

45,

M i s c a l l e d energy and on F (d),Lo P n D e f i n e a dual Fock space v i a = A ep(p-3/6) A p(d)

Lo =

C$.

t

cM N ...

One w r i t e s here (cM,e ) = ( M ( N ) = 6M.

= Identity.(L,p2

2y(p-+) Then a E

!d ) ) . A

= (MI with F =

F

can be w r i t t e n

270 a =

ROBERT CARROLL

1 ( g M I a ) e M = 1 aMeM and

- - -ep-3/2

one s e t s I p ) = ep-'

... w i t h

A

A e p-3/2

(pi =

A ep-l;.

f

L e t GL (Zh) denote i n v e r t i b l e l i n e a r t r a n s f o r m a t i o n s g: g(e')

=

e'g:

of V

such t h a t g t = 6; except f o r a f i n i t e # o f v , v .

L e t P- % l o w e r t r i a n g u l a r i n f i n i t e m a t r i c e s w i t h 1 on t h e diagonal and GLf(Zh) be generated b y GL f (Zh) A e v(p-3/2) A -f ge V ( P - $ ) and P-. Then GLf a c t s on f v i a G(g): M M N A geU(p-3/2) I\ I n t h i s d i r e c t i o n one w r i t e s G(e ) = GNe ( N E m ) M M A Q A w i t h GN = (eANIG(e ) ) w e l l d e f i n e d . Also t ( g ) on f i s determined v i a ( G ( a ) l

...

1

....

G(a')) = ( a l a ' ) . Next, f o r 5 =

{...,C p-3/2),5p-4}

E FUGMP one has

A 5 = tp-+A 5p-3/2

-

A

... E

FX = f {Ol g i v i n g a (holomorphic) map FUGMP -+ f L e t GL(Zh 1 be semiP P P' I. i n f i n i t e m a t r i c e s A = ((a:)), r,s E Zh o f t h e form A = ( *' ) ) , where w
6'

cv

a l s o SL(Zh

) = { A E GL(Zh ); detA = d e t z = 11. s r

Then s e t UGMP = FUGMp/SL

(Zh ) (where (5A)r = 5 a s ) and n o t e t h a t (gc)A = g(5A) f o r g E GLf(Zh) and

c o m p a t i b l e w i t h GLf(Zh); (note

here P ( f

%

Fi/C* i s t h e p r o j e c t i v i z a t i o n o f F

P UGMP i s a p r i n c i p l e C* b u n d l e ) .

CMP -t

The map P: UGMP -+ P(F,)

tJ

c a l l ed t h e Plucker embedding ( b a s i c a l l y f o r 5 = S ( U ) does n o t depend on 5).

E

FUGMP, 125 i n P(Fp)

I t i s i n j e c t i v e and dPU: TUUGMP

i n j e c t i v e ; t h e image o f P i n P(F,)

P

is

-+

Tp(,,)P(F

P

) i s also

i s a c l o s e d m a n i f o l d d e f i n e d by t h e P l k -

k e r r e l a t i o n s ( c f . here §8,13 and [ S Z I f o r d e t a i l s ) .

RRllARK 18.3 C((c)) =

(RZEIXANN SURFACE$, m0DUCZ SPACE, KRZCEVER DACA).

n {In >>--an< ; an

E

C l ; 8 = C"c11

en} (m i s t h e unique maximal i d e a l i n 0 ) .

=

11; anen}; and m I n [ KM1;KIl;NKl

A

W r i t e now K = = c0 =

an

] a g r e a t deal o f

a l g e b r a i c machinery i s m e t i c u l o u s l y i n d i c a t e d , sometimes w i t h o u t p r o o f s b u t a t l e a s t w i t h r e f e r e n c e t o p r o o f s , and such machinery i s o f course a r e q u i r e ment f o r e l e g a n t mathematical t h e o r y .

However f o r purposes o f seeing how

MODULI SPACE

271

matters f i t together one does n o t need what some would consider excessive pedantry; t h u s we omit some a1 gebraic "thoroughness" unless i t r e a l l y con-

tributes to say f i r s t level understanding o f the picture.

Thus we do not

make omissions of d e t a i l e t c . t o spit i n t h e temple of mathematical orthodoxy b u t only t o make t h e s u b j e c t palatable and a c c e s s i b l e t o readers more i n t e r e s t e d in o t h e r matters and w i t h l e s s t r a i n i n g i n "algebraic t h i n k i n g " . We will show how t h i n g s f i t together and leave the deluge of modules, sheaves, schemes, e t c . t o the p u r i s t s (except when the concepts a r e essent i a l ) . On t h e o t h e r hand i t i s a f t e r a l l easy enough to t a l k about ringed spaces, schemes, sheaves, e t c . and o f t e n very productive so we will not hesit a t e t o use these ideas when useful or necessary.

In t h i s d i r e c t i o n we re-

f e r to [ HSl;MU4,51 ( s e e a l s o Appendix B and 5 1 2 ) .

I t should be noted t h a t

a l o t of material i n this s e c t i o n will be seen t o f i t together simply by writing i t out.

-

Thus l e t S be a RS of genus g 0 and K(S) t h e f i e l d of meromorphic functions (S,(aj, on S. We have the standard canonical homology marking ( S , ( a , B ) ) 6 . ) ) and t h e T o r e l l i space Q Q N in Remark 88 ( S i s i d e n t i f i e d w i t h JacS o r J 9 9 with t h e corresponding curve a s appropriate i n context - note t h a t this ide n t i f i c a t i o n i s dependent on t h e marking a s indicated i n LAG41 and Appendix 9 = N / S P ( 2 g , Z ) i s t h e coarse moduli space ( c f . Remark 88) B ) . Recall a l s o I 9 9 and Pico(S) i s the set o f divisor c l a s s e s on S o f degree 0 ( c f . 54,5 and 1 Appendix 6 ) ; a l s o H (0*) PicS and Pic's 2 JacS. Let now Q E S and 8 be

Q

Q ,

the r i n g of meromorphic functions on S which a r e holomorphic a t Q; l e t m t h e u n i q u e maximal ideal of 0 c o n s i s t i n g of functions vanishing a t Q t o

Q

be

Q

f i r s t order. Let v ( f ) = order of t h e zero o f f a t Q = - order o f t h e pole Q a t Q, so one o b t a i n s a f i l t r a t i o n {FnKK3 i n K. Evidently 0 = I f E K; v Q ( f ) Q 9 A lim K/m: 10Q ( c f . [ HA1 I a n d Appendix B ) . > 0) a n d one s e t s (*) KQ =

For 1; a sheaf of germs o f holomorphic s e c t i o n s o f a l i n e bundle L on S l e t Let Ho(L(*Q)) be meromorphic s e c t i o n s of I; holomorphic except a t Q. and B(S,Q) = Ho(S,O(*Q)) = meromorphic vector f i e l d s , holomorphic o u t s i d e of Q ( 0 = holomorphic vector f f i e l d s on S ) . The s e t PicS of holomorphic l i n e bundles L on S i s parametrized by t h e degree d ( d = c l ( L ) ) or by x = d + 1 - g (Euler c h a r a c t e r i s t i c ) and we r e c a l l Pic's = J ( S ) . The s e t of isomorphism c l a s s e s ( S , ( u , B ) ) = E 9

L Q - meromorphic s e c t i o n s of L, holomorphic a t Q.

-

272

ROBERT CARROLL

T o r e l l i space ( ( a , B ) i s a homology b a s i s and f: S = 8; f o r f,:

H1 ( S )

-+

-+

= a;, f,(Bi)

S ' , f,(ai)

H1 ( S ' ) c o n s t i t u t e s a homomorphism).

The coarse moduli

space i s ill = C /My M = SP(2g,Z) ( c f . Appendix B). Over C 9 9 9 o f RS,Sg u S, u n i o n o v e r [ S,(a,B)I E E ( i . e . 'TI: (S,(cc,B)) 9 Sg + C g ) . The u n i o n o f PicS f o r Sl E C forms a f a m i l y P 9 9 t The f i b r e p r o d u c t H = S X P is and we l e t p: P 9 g' 9 g t 9 l e s {((S,(a,@)),Q,C), w i t h Q E S and C E P i c 9 and t i l e r e i s -+

one has a f a m i l y

*

[ S,(a,@)l:

PicS SIEE the set o f t r i p = U

a commutative

diagram ( Q i s g r a t u i t o u s h e r e )

P

A

9

A A lim n+l One d e f i n e s 1; = CQ/kwQ C j as i n (*) and H = u q u i n t u p l e t s {(S,(~,B), Q 4 h A g Q,i:,u,t); Q E S, C E PicS, $ 0, t : C % 01. Here GQ = 0 /mntl with A lim ntl QQA Q C = CQ/mQ CQ and a s p e c i f i c a t i o n o f an isomorphism u: $ * 0 = C[[sll

'2

+

Q

Q

+

C algeA r i n g o f convergent power s e r i e s 0 c 0

formal power s e r i e s ) i s e q u i v a l e n t t o g i v i n g a v a l u a t i o n p r e s e r v i n g b r a homomorphism 0

Q

n

-+

0. I f u(0 )

Q

C

t h e n t h e s p e c i f i c a t i o n o f u corresponds t o choosing a holomorphic c o o r d i n a t e The map t i s a u l i n e a r module isomorphism ( i . e .

a t Q.

for f E

A

t(fs)

:iQ'

u ( f ) t ( s ) ) and i s c a l l e d a

f i x i n g o f i: a t Q.

The map

A

(S,(a,B),u,Q)

A

with

E

2

q:

S

+

A

S maps u

Q;

+

A

Q

G: s*9 -+

= t-'(l)

4

denotes by S t h e s e t o f isomorphisms u: 0 + 0 w i t h Q

t,

s

formal t r i v i a l i z a t i o n o r formal gauge

T h i s t i s determined by t h e formal s e c t i o n

which i s a formal g e n e r a t o r o f i: a t Q.

u

5Q and

E

S, and S

hereone *I

9

=

u S

%

Now c o n s i d e r P ( i ) w i t h C* a c t i o n ( C * C $*) and l o o k a t = P(ig)/SP(2g,Z). 9 9 T h i s i s t h e m o d u l i space o f framed and gauged RS ( c a l l e d t h e W e i e r s t r a s s system). t)

-+

There i s a ( K r i c e v e r ) map ( A )

t(H'(S,i:(*Q)))

C

?.

r:

P(ig)

-+

UGM(?): ((S,(a,~)),Q,i:,u,

The image i s t h e s e t o f Laurent s e r i e s expansions

o f meromorphic s e c t i o n s o f 1; holomorphic o u t s i d e o f Q by means o f a chosen k formal c o o r d i n a t e and gauge. Note i s f i l t e r e d by Fni? = I a k s 1 = Y-'

?

([n,=))

( v ( i n ) = n ) and Fo? =

$,

F'K =

1;

i, FnK = Gn

(n > 0 ) .

Hence UGM(c),

Maya and Young diagrams, etc.,

can be c o n s t r u c t e d i n t h e same s p i r i t a s be-

f o r e f o r UGM(V) ( w i t h say ent'

%

cn).

Then

r: P(?)

%

9

-+

UGM where

i'9

lies

FERMION OPERATORS

273

over Px (note e.g. Px = UPx,x = d+l-g) and r induces a map ( 0 ) F: $ g g A 9 9, P(2g)/SP(2g,Z) -+ U G M ( K ) , which i s injective f o r g > 1 ( f o r g 2 1 use P(H ) / SP(2g.Z) X A u t ( S ) ) . needed 1a t e r ,

9

Further information about t h i s structure will a r i s e as

(FERIIIZIDN IDPERAEBW).

RETlIARI( 18.4

One wants t o define End(V) action on the

diagram ry

UGM

(18.3)

1_)

UGM

F‘ P(F)

Thus with some r e p i t i t i o n from other sections ( c f . §14,20,21) P E Zh. with [$,,,JIv1+ = [ ? v . ~ v l + l = 0 and [6,.?,1+ = 6,,+” we consider +,,$, The associative algebra w i t h 1 generated by the JI,,,~,, will be called A. One defines action o f 9, a n d j = v-’ o n F = Q = II CeM, and = Q C$M ( c f . P’ FP !k Remark 18.2) via ( 4 ) $, = a/ae’, 3, = e’ A ( l e f t action on F) a n d JI, = AS,,, j’ = a / a 6 II ( r i g h t action on 3 ) . Let W (resp. W*) be the vector space generated by {JI 1 (resp. {$,I) for IJ E Z h a n d s e t V = W Q Recall from Remark ( c f . (18.1)).

L

G,

N

1.1

...

18.2, I p ) = eP-+ A ep-3/2 A E F a n d ( pI = one checks e a s i l y t h a t for ( uI E f a n d l v ) E F A

(ul(;,,v)) >

-ply

and

w

(18.4) A

w-

= 0 (I.!

JI,IP) = 0 (11 > P I , Set now < -p).

w-; w+

= B,,~CJ~,;

= (( U I $ , ) l V ) .

(PI; , =

w+ A

= 5,<tp,; u

@

Y

V+ = W

,Q W+ = I $

w-

A

E

(

... ep-3,2 n

( + I ( u l ($,,lv))

B,<~CJI,;

7; $10) =

w

A

=

w+

?.

= ((

A

5

&

w-; w,

=

Then

ulJI,)lv),

jlllP)

PI$,, = 0 ( P < P I . A

=

n

A ep,+E

= 0 (lJ

el.l,c$,,;

4

Ly

03; V- = W- B WU

Y

where V - = 111, E V; ( O I $ = 0 1 ) . This gives annihilation (V,) and creation w ( V - ) operators a s usual. The n o t a t i o n corresponds t o Remark 20.3 where .I.,, i s used, corresponding t o $,, a n d here,but we do not bother with s p e l l ing o u t a l l such identifications l a t e r .

I);

Gn

We note again the a l t e r n a t i v e representation of F via ( m ) a E A/Av+ -+ a10 F with dense image (note = 0 ) a n d isomorphically a E r - A \ A -+ ( O l a . . u u E F. The normal ordering map i s defined via (*+) : : : AV AV- PP A V + + A t o be a l e f t AT- a n d right AY+ bimodule isomorphism with : 1 : = 1 ( c f . E

Ar+lO)

-f

ROBERT CARROLL

2 74

(A*)l4 where t h e notation i s d i f f e r e n t b u t t h e concepts i d e n t i c a l ) . One deA 3 ( z ) = IvEZh$,,z f i n e s now (*A) $ ( z ) = &EZh$,z-’-’; (fermion f i e l d operat o r s - c f . here (21.3)) and f o r ( z [ > IwI t h e vacuum expectation values ( V E V ) a r e defined via ( c f . ( * A ) 2 1 e t c . )

m

1/(z w);

=

-

One extends to (C*)’

=

$(Z)$(W)

= 0

diagonal e t c . by a n a l y t i c continuation ( c f . [ T S 2 , 3 ] )

and Wick’s theorem has t h e form ( $ = (18.6)

azww)

$ ( z l ) . * * $ ( z N )=

I(*)

$

or

j)

$(<)$(gj):

rest

pairs

(2 =

@(zk): A

s i g n a t u r e o f t h e permutation).

Now f o r U E UGM define U C

as UL=

{ f E Hom(V,C); f l ,”, = 01 and l e t [Ul = image o f U i n P(F) ( c f . ( 1 8 . 3 ) ) . DeA f i n e f i l t r a t i o n s on W a nd W via FmW = QJ; = C$, and FmW^ = A C$ 4u = C$’. This gives a topology on W,W and one writes W,N f o r the and ^m = completions ( c f . ( * ) ) w i t h ( * A ) W = W + @ W- = (gp,oC$’) @ (Bcl
I,,

~,,,-,

@p
nd V

Iu<-,,

I,,>,,,

There a r e f i l t r a t i o n preserving isomor

i: e’

5,

-

A

and one defines U,(!)

W and W, Then i t follows G r u ) = 0; t h a t (**) W+(U) = I$E U);$[u) = 0; l u ) E [ U l l ; i + ( U ) = 1; E l u ) E [ U I I ; and [ U l = t l u ) E F; $ 1 ~ ) = j l u ) = 0; f o r a l l q~ E W,(U) and E

it

(U) C

A

A

+

’

+

-t

f o r U C V and U C V a s images under these maps.

i;

A

Thus W+(U) l3 W+(U)

i,(U)I. with U

E

UGM.

RrmARK 18.5

(CURRENC BPERACBW, Srf32ZNCER CERFR, SrllCAWARA COWCatRLICCZ6N)- One

considers D, C End? f o r

D,

=

i s a space o f a n n i h i l a t i o n operators associated

tP(z,a,)

=

^K

%

C ( ( 2 - l ) ) now ( c f . Remark 18.3) defined by (*dl This corresponds to a E C[[azlll.

na n ( a z ) ; a n ( a z )

& < , w ~

Fourier transform o f microlocal d i f f e r e n t i a l operators E = C [ [ t l](a-’ ) v i a

z

+

a

and

a,

+

t.

Order a n d valuation will be discussed when needed a n d we

simply s t a t e formal r e s u l t s , mostly d i r e c t l y v e r i f i a b l e (some analogous c a l c u l a t i o n s appear i n 5 1 7 ) . For P E D,, P* i s defined via z* = z, a: = -az, A a n d ( P Q ) * = Q*P*. The P E D, a c t on f i e l d operators $ ( z ) , $ ( z ) and w r i t i n g

$

f o r a clockwise contour integral around Q

Q

m,

one has (*+)

$

dz

CURRENT OPERATORS :(Pv(z))$(z): =

6

275

I t follows t h a t , f o r operators $ ( P ) =

dz:$(z)(P*?(z):.

6

( 1 / 2 a i ) d z : ( P $ ( z ) ) ? ( z ) : (*.) I $(P),Jl(z)I = P$(z), [ + ( P $ ( z ) I = - P * j ( z ) , and [ $ ( P ) , $ ( Q ) l = + ( [ Q , P J ) t c(P,Q)Id. where the Schwinger term c i s deter-

mined, in an obvious notation ( c f . §17),via

Formally t h i s i s proved by writing ( c f . § 1 7 )

The l a s t term provides c(P,Q). The d e t a i l s of c a l c u l a t i o n a r e l e f t a s an e x e r c i s e ( c f . (18.5)-(18.6)). The following p r o p e r t i e s of c a r e useful (note c: DZ X Dz

-+

C determines a cohomology c l a s s t c l

E

H 2 (D,,C))

(1) c(P,

Q ) = - c ( Q , P ) ( 2 ) c ( P , [ Q , R l ) + c ( Q , t R , P l ) + c(RA P , Q l ) = 0 (3) f o r t h e basis 0, n E Z ) , c ( u kn , uPm ) = ( - l I k n k (n+j)amtn/(k+p+l)!. -P One a l s o defines now c u r r e n t operators ( c f . §17,20,21)

unk = z k+n ( a Z )k / k ! ( k

(18.10)

Jn = -$(zn) = - ( l / Z n i ) /

a

J(z) = :$(z)$(z):

dtzn:q(z)j(z):; =

1

Jnz-"'

Z

and Virasoro operators ( f o r spin j ) (18.11)

L:

=

$(zn(zaz+j(n+l))= ( l / Z n i ) /

dzznt+':(ll-j)az$(z)$(z)

-j+( z ) a z$ ( z 1 :

276

ROBERT CARROLL

One a l s o expresses t h e Sugawara form o f T v i a

:

where

p u t s J,

(n

1. 0 ) t o t h e r i g h t and Jn (n

<

0) t o the l e f t .

RElRARK 18.6 ( 3 0 S 0 N FERm10N C0RRESP0NDEMCE, VERCEX 0PERAC0W, HIR0CA EQUAtZ0lusc). One makes t h e boson fermion correspondence as f o l l o w s . L e t H = C [[tl,t2

,... ]I B C[eto,e-tol

=

1 Hm = 1

Elements o f H

~[t,,t2,...11emt0.

a r e boson s t a t e v e c t o r s and one d e f i n e s degree ti = i, degree eto = 0, charge ti = 0 ( i = BI$) =

1

O), charge ento

(n < n One uses

-+

H i s defined v i a

(A*)

W r i t i n g an = a / a t n = an ( n 2 0) and a-n =

O), q

0' ."

= to, one has ) A . ( [am,anl = and l a m , e x p ( q ) l = 6 eq. m y0 t o p u t c r e a t i o n o p e r a t o r s (an, n < 0, eq) t o t h e l e f t o f a n n i -

h i l a t i o n o p e r a t o r s (any n 2 0 ) . aologz

A map B: F

e x p ( n t o ) ( n l e x p ( l y Jmtm)l$). Then B i s a t o p o l o g i c a l isomorphism

p r e s e r v i n g charge and degree.

nt

= n.

D e f i n e t h e f i e l d o p e r a t o r (A@)

Q(z)

= q t

- 1

(18.14)

anz-'/n ( c f . §20,21) and s e t ( v e r t e x o p e r a t o r o f charge k ) n+O V k ( z ) = o' ,kQ(zIo" = ,kly t n z n e k t e k l o g z a - k l ; zenan/n on o f @ one has ( c f . (*A) f o r $,$)

z)

~ ( z )= a Z q ( z ) =

1 anzqn-'

9

(A&)

TB(z) =

and energy-momentum opera t o r s r e s p e c t i v e l y ) . A&

Now f o r t h e t a u f u n c t i o n d e f i n i t i o n t a k e €;

) ;1

E

F;.

Here UGM'

UGM'

( c f . ( 1 8 . 1 ) ) w i t h image

i s t h e " b i g c e l l " i n UGMo and t h i s i s discussed i n 511,

12,13 a l o n g w i t h t h e B r u h a t decomposition ( c f . [ B P l ; P R l

I).

Let

T

Q

(tl,t2,

4,20,21,

277

TAU FUNCTIONS

13).

We l i s t now a number o f formulas d e r i v e d i n o t h e r s e c t i o n s by v a r i o u s

techniques and f o r which some a l t e r n a t e d e r i v a t i o n s a r e i n d i c a t e d i n [ KM1 1. Thus f o r c(T,z) = tnzn and I z J = (l/z,7/2z2 ,...)

1:

w

$(z,T,U)

(18.16) A

+(z,T,U)

= e - S ( T . Z 1 ~ ( T + [ z l , ~ ) / T ( T , U ) = e- S ( T y z ) ( l

= e S ( T ' z ) ~ ( T - [ z ], ~ ) / T ( T , C ) = e S ( T y z ) ( l

where w k Y i k

C"T1I.

E

+

1;

+

1;

W,(T)Z-~);

;,(T)z-~)

The H i r o t a b i l i n e a r i d e n t i t y i s expressed as

Resm +(z,T,U)j(z,T',U)

(A+)

= 0 and consequently m

1;

(18.17)

L)

Sj(-2y)Sj+l

(D,)eZ1

'palatp

T(T).T(T) = 0

.-

where t h e S . a r e Shur polynomials, DT = J Now one c a l l s

N

$(z,T)

(Am)

(al,+a2,...),

= 1 N 4 J N ( ~ ) T and j ( z , T )

etc.

=

1 GN(z)TN E

C((z-'))

@

C [ [ T l I a c o n j u g a t e p a i r o f g e n e r i c wave f u n c t i o n s when t h e y s a t i s f y (18.16) for

wk,tk

(no t a u f u n c t i o n ) and

(18.18)

(TN

Resm+(z,T)j(z,T' n

= tl

n t2

..., N

denoted by WF'.

n ., n.1 -> 0, ni E Z ) . The s e t o f such p a i r s i s 1 ' 2" i s ca l e d t h e g e n e r i c t a u f u n c t i o n i f T s a t i s f i e s

= (n

Then

T

(18.17) and ~ ( 0 #) 0; t h e s e t o f such [ KM1

I

$,(z)

t h a t f o r any p a i r $,? E WF' E U and $,(z)

E

A U

=

T

i s denoted by TF'.

t h e r e e x i s t s a unique U

-

I U ( f u r t h e r t h i s map WF'

?

One t h i n k s here o f UGM(K) f o r

1 = C( ( z - ) )

Thus one expands wn(T) = cn + 2 O(T ). D e f i n e elements i n C(

@,(z)

(18.19)

n

U =

=

1;

UGM

9

A

JI,;]

V i n the earlier construction

N*

and t,(T)

(z) = 1 +

=

where No = (0,O,...) and N

Cqn(z)

w i t h $,(z)

= $

C

? and

E U and j,(z)

=

E

1;

C;,(z)

6 (cf.

= (0,

j, C K.

such t h a t

9 UGM i s b i j e c t i v e ) .

1 and

The c o n s t r u c t i o n o f U i s w o r t h i n d i c a t i n g ( c f . [ S E ? ; P R 1

o f UGM.

(oj

-+

One shows i n E

+

=

Cn

c1

m~

+

* ( z ) = -zJ(1 +

NJ

...,l j,... ) .

511 ).

+

cnjtj

lm cnz-? 1

Then d e f i n e (**) A

l

A

One checks t h a t U = U and U = U

[ KM1 1 f o r t h e d e t a i l s and t h e r e m a i n i n g

1

278

ROBERT CARROLL

assertions). Next t h e map TF'/C*

WF@ can be determined v i a (.A)

-+

[ z I ) / . r ( T ) and $(T,z)

= ~XP(F(Z,T))T(T-~ZI)/T(T).

+(T,z)

= e-S(zyT),(T

t

T h i s makes t h e f o l l o w i n g

diagram o f i n j e c t i o n s commutative (18.20)

One shows a l s o t h a t f

E

only i f there exists U

REmARK 18.7

:H s a t i s f i e s t h e H i r o t a equations (18.17) i f and

E

UGMo such t h a t f ( T ) = T(T,U).

(SPZN StWCCUREg AND A CAN0NZCAt CNl FLINCtZ0N).

One p u t s t o g e t h -

A

e r now t h e K r i c e v e r maps T, T (based o n (A)) from Weierstrass data t o Grassmannians w i t h t h e P1 i c k e r maps Grassmannians

+ Fock

space t a subsequent

b o s o n i z a t i o n t o g e t t h e diagram

NI\

The l e f t l o w e r e n t r y i n v o l v e s s p i n s t r u c t u r e and i s discussed below; P(Hg) V

N

and S ( 3 ) can be regarded as p u l l b a c k bundles. R e c a l l t h a t t h e p u l l b a c k 9 f* : f * E -+ B ' o f a:E B under f : B ' -+ B i s d e f i n e d v i a f * E = { ( e y b ' ) ; T ( e ) = -f

f ( b ' ) l (f*s(e,b')

= b').

For s p i n we r e f e r t o §19-21,5,

more d e t a i l and here proceed f o r m a l l y f o l l o w i n g [ K M l

I.

and Appendix B f o r Given a RS S l e t KS

be t h e c a n o n i c a l sheaf ( r e c a l l from Appendix B and 15 e t c . t h a t t h e canonic a l l i n e bundle K clr meromorphic d i f f e r e n t i a l s has degree 2 ( 9 - 1 ) and K 'L sheaf o f s e c t i o n s ) . L e t u . U P * [ S l + (1 SI,Ks@ ) jbe a s e c t i o n ( n o t e j* 9' degree K ': i s 2 ( g - l ) j and c f . Q18.2) e t c . ) . This s e c t i o n l i f t s t o $j : A A A H s i n c e u: 6 + 0 induces (du)': f ( d u - l ( t ) ) j -+ u ( f ) . Using t h e 9 Q Riemann c o n s t a n t A f o r (S,(a,B)) E U one determines c a n o n i c a l l y a l i n e bun2 = 9KS = K ( c f . Appendix B y 15, e t c . d l e LA(Sy(a,B)) = LA such t h a t LA we -f

ig +

t!;,

-+

g:

-

w i l l f r e q u e n t l y use t t o r e f e r t o a bundle o r t h e sheaf o f s e c t i o n s ) .

Then

SPIN STRUCTURES

279

for j E +Z define u * t -+ P ( 2 j - 1 ) ( g - 1 ) a s u j ( ( S , ( a . B ) ) = ( s , ( ~ , B ) , c ~ ~ ) (2j j’ g A -1 = 2 ( 2 j - l ) / 2 ) . One uses a formal t r i v i a l i z a t i o n (du)’: h 2 A h 9 with = K -+ 0 which i s determined u p t o +1. (Jdu)‘ = d u : (C Then for j E 3-,Z w i t h K j = C2”define uj: 9 Zig 6 + P(Hg ) ( g - l ) ) by ( 0 . ) :j((S,(a,B),Q,u) = -+

(S,(a,B),Q,u,C2jy(du)J).

8

This yields a commutative diagram

A

a n d the modular embedding of spin j i s denoted by ( 0 6 )

TOGj:

lg

-+

UGMP, p =

(23-1 1(g-1). n

P(;O) by and write J(S) = Cg/L, L = (1g,n)Z2g via ? = Now denote $ * ” T H0m(Ho(S,K),C) 2 Cg (thus for c = (c,, c g ) E Cg, the corresponding c E cu J s a t i s f i e s c(wi) = ci - recall Ho(S,K) % holomorphic d i f f e r e n t i a l s with P S J; P + (& wi)/modL. For basis w i ) . One has the standard embedding I Q: c E J l e t Cc = l i n e bundle o f degree 0 % c modL in J . For sections of Cc one looks a t multiplicative functions on S (@+) i f : ?-+C; f ( r + tma + t n B ) t = exp(2Ti n c ) f ( p ) , m , n E Zg where ? i s the maximal abelian cover of S, a n d the f in ( a + ) multivalued meromorphic function on S ( c f . GZ1 I ) . Such a A n h section sc determines a t r i v i a l i z a t i o n s c : ( t c ) Q + for Q E S, c E J.

%

-+

....

-

-+

N

Q

Let now ? = family of universal coverings o f J(S)lS considered as a VB over 9 t of rank g. The dual basis ( w i ) above gives a V B t r i v i a l i z a t i o n . Thus 9 one has (18.23)

U set 4

s9

s”,~?

A

=us

{(S,(a,B

(1 8.24 )

9

280

ROBERT CARROLL

Given a l l t h i s s t r u c t u r e t h e t a u f u n c t i o n can be d e s c r i b e d a s a s o l u t i o n o f t h e problem o f c o n s t r u c t i n g a nonzero holomorphic f u n c t i o n

T:

dg(s")

-+

Hi

such t h a t t h e f o l l o w i n g diagram i s commutative

A

N

i s t h e c o m p o s i t i o n o f maps i n t h e bottom row o f (18.21), A+: S g ( J ) Here -+ 2 0 UGM + P(Fo) P(Ho). Such a l i f t i n g T can be c o n s t r u c t e d u s i n g t h e roo t h e o r y o f KP equations (see below). G e o m e t r i c a l l y Z i s a p e r i o d i c map o f -f

t h e m o d u l i space

ti^9 (?)

and t h e r e i s a c e r t a i n l a c k o f uniqueness t o be d i s -

cussed be1 ow.

RB!tARK 18.8 context.

(CHECA AND CAAU fUNCtL0Nk). L e t Xc = ( S , ( a , ~ ) , Q , u , t ~ ) E

(18.26)

A(Xc) = {$(z,T)

( 6 = E(z,T).

=

1N

We t o now t o t h e BA f u n c t i o n i n t h i s A

s(

h

9

.

( 3 ) be g i v e n data and d e f i n e

$ (z)Tn};

$,(z)

N

E

U(X,)

= (du)+Ho(S,CA

A

Gk

= Gk(T) E C"T11).

Elements o f A ( X c )

( r e s p A(Xc)) a r e BA

f u n c t i o n s ( r e s p . c o n j u g a t e BA f u n c t i o n s ) a s s o c i a t e d t o Xc. §8,13,14,

B

... f o r

g e n e r i c data ( i . e .

t.

o(cjn)

By r e s u l t s from

0 ) one knows t h a t A(Xc) i s a

f r e e C [ [ T l I module o f rank 1 generated b y n (18.27)

$ = f(z)e-l:

tn'

(Z)a(IT)+I( [zl)+cln)/o(I(T)

i.c l Q )

( t h e p r o o f i n [ KM1 I i s s i m i l a r t o [ DU1 1 and i s sketched i n 15). ?(Xc)

i s generated b y

(08)

i=

f(z)exp(I;

Here I ( T ) = ( I j ( T ) ) ,

o(I(T) + c w .

tn$n(z))n(I(T)

Ij(T) =

1;

-

Similarly

I [ z l ) + cIn)/

I:tny I [ z I = ( I j [ z ] ) ,

and 1'

1;

I:zn/n where 1; i s d e f i n e d v i a w j ( z ) d z = -d(l: Iiz-n/n). For t h e n l e t wn = wndz be d i f f e r e n t i a l s o f t h e second k i n d w i t h wn = 0, 1 w k Qn -m c Q bx Q = Z n i I n , and w dz = d ( z n qnmz /m). Then d e f i n e meromorphic f u n c t i o n s [z] =

/a.

Q

- 1:

w i t h p o l e s a t Q v i a ( b * ) $'(z)

= lZ wn = zn

Q

- 1;

qnmz-"/m.

Note t h a t t h e

TAU FUNCTIONS

281

A

normalizations gSve w o ( T ) = G o ( T ) = 1 so @,$ can be i d e n t i f i e d a s t h e wave ,\ 4 functions associated with U ( X c ) E UGM . Now define a tau function f o r $ (J) 9 T: -+ H: via nl

ig(?)

T(T,X,) = e 'q(T)o(I(T)

(18.28)

t

cln)

where q ( T ) = :1 qnmtntm ( c f . (&*)I. One checks t h a t (18.25) i s t h e n commutative and r(O,Xc) = o(cln) depends only o n 7 %F X C g ( c f . ( 1 8 . 2 3 ) ) . 9 There a r e various modular transformation properties of T . In p a r t i c u l a r l e t y = :) E S P ( 2 g , Z ) = M e f f e c t a change of homology basis via y ( i ) = ( BD AC ) a T T ( b ) . Let MA = I y E M; diagC D = diagA B = 0, mod 2 ) . Then M, preserves the Riemnn constant and ( c f . [ KM1 1 f o r proof) 4 Tic T (CntD)-1C(21(T)+c) T , y ( X C ) ) = E(y)det(Cn + D ) e T(T,X,)

(t

There is s t i l l a l i f t i n g ambiguity f o r

T

which plays no r o l e i n 2-D CFT

since the c o r r e l a t i o n functions depend only on r a t i o s of T o r i t s derivat i v e s . B u t i n string theory the l i f t i n g i s important since i t provides t h e integrand o f t h e string amplitude. This will be discussed below.

REmARK 18.9 (C0RREtACZ0N FllNCCIONk AND CHE FRZ$ECANtJ ZDENCZCg).

Given T a s i n Remark 18.9 we record here t h e following formulas ( s p e l l e d o u t more i n The vertex operators o f (18.15) a c t via [ KM1 1). U

(18.31)

V

(z )...V k, 1

n

1< i < j N

kN

( z ) T ( T , X ~ ) = n: f ( z i ) k i K etoK el;

E(zi,z ) j

If

k*J esiq(T) O(I(T) t

1;

tnl,

kign(zi)

kiI[zi] + cln)

c1

where K =- N ki ( t o t a l charge o f Vk, ( 2 )...V (2,) and E ( P , Q ) is the prime 1; qnmz-n-1 w -m-1 k).N Here one uses (6.) Vk(z) ," and (1; k i i i ) ' = -1 k . k . ( x i - xj)' t K C k i x i2. 1 J

For N = 1 and ki = & l one obtains $I and $ as i n (18.27) and

(0.).

Similarly

ROBERT CARROLL

282 (cf.

PO)

(18.32)

J ~ ( Z ) T ( T , X=~ )

1

t wn(z) + n Q

1;

wi(z)a/acih(T,Xc)

There is a l s o a formula f o r TE(z)Z(T,Xc) which we omit ( c f . [KMl I ) . Now f i x X, w i t h ~ ( 0 . x ~= )e(cln) = 0 and w r i t e I X , ) w r i t e s the Szegb' kernel as

=

B-lr(T,X,)

E

Sc(z,w) = @(Irzl-I[wl+cl~)/o(c~~)E(~,w)= l / ( z - w ) + IC

(18.33)

!J"

F.

One

z-'-%-"-'

( u , ~> 0 ) and N - p o i n t functions of operators O i ( z i ) o n I X c ) a r e defined via (18.34)

(01 (21 )

..-0N(ZN))xc

= (

-

0101 (21 1. ' ~ N ( Z NI )xc)/(

olxc )

while i n bosonic form this is (18.35)

( O 1 ( ~ l ) . . . O N ( ~ N ) ) =X O , I B ( ~ l) . . . O N B

I T=O

.

An i n t e r e s t i n g consequence o f c a l c u l a t i n g ( J I dJ (wl ) x in t h e s e two d i f f e r e n t ways i s a version of Fay's t r i s e c a n t formula ( c f . a l s o [ SAT1-3;E4;RN1 1)

First i t is given by (18.38) s a t i s f i e s t h e Hirota equations (18.18) ( T ( T ) =

REmARK 18-10 (LINXqLIE CHARACCERZZACZ0N OF CXHE CALI FLINCCZ0N). shown t h a t

T

.r(T,Xc)) and two o t h e r equations. is explained below) (18.37)

The equation o f "motion" is ( t h e notation

C O ( D ) + a(D,Xc)lT(T,Xc) = $ E ( D ) ~ ( T , X c )

where a(D,Xc) = -(l/lZ)Resm D(z)S(z,Xc) ( D

1;

(18.38) for $(z)

E

E

C ) and t h e r e i s a gauge formula

(1/2ai).f d$(a/aci)r(T,Xc) = $B($)T(T,Xc) bi

x(Xc).

A

A

For t h e notation here one defines C = DerK = Kd/dr; as t h e 4

Lie algebra of derivations o n K .

S(z,Xc) i s a p r o j e c t i v e connection term,

TAU FUNCTIONS A

S(z,Xc): S

+

(2-w)), $B!D) %(df/dc)),

C((z-')),

r(z)

defined by (6.) S(z,X,)(dz)* = -61im w-+z dzdwlog(E(z , w ) /

$,(XD)= B$(soD)B-l t2 =

10

283

(where s f(c)d/dc + f(c)d/dc + t2' C$'(z) where $n is defined i n ( 4 * ) , and f i n a l l y 0: C = a

Der? Ho(k^ 0- ) ( 0 'L sheaf of holomorphic vector f i e l d s ) is a Lie algebra g' $3 antihomomorphism whose a c t i o n on T can be expressed v i a -+

The main theme on t h e T function now i s t h a t f o r a holomorHX the following equations determine f uniquely u p t o a constant a s f(T,Xc) = c^r(T,Xc) (2 E C ) . The equations a r e (18.30), (18.37), and (18.38) w i t h f i n place of T. Another r e s u l t o f i n t e r e s t here i s t h a t i f f is a l i f t i n g of A& in (18.35) s a t i s f y i n g ( y E M A ) (18.29)-(18.30) w i t h f i n place o f T and f(;,X ) depending only on C X Cg then ( 6 6 ) f(T,Xc) = 29 n: S(3) + t and 2: C C* is holomorphic and MA i n ~ ( n ( X C ) ) T ( T y X where C) variant.

(wi = w i ( z ) ) . phic map f :

ti"9 (?)

-+

-f

19, fl0RE 0%KRZCDJER DACA, CIIRUES, CRWSIIIANNZANS, ECC,

In t h i s s e c t i o n we

will t r y t o gather together some threads i n t o a theme and sketch various extensions. Some o f relevant references a r e LAC4;BWl ;CY1-4;DLl; JR1 ;GN1 ;GHl; ML1-4 ;FR1 ,2 ;NO5 ;KR1-14 ,1 6 ,1 8; PE3; PR1; PM1 ;Ql ;S El ;W5,lO; W T l 1 a n d o t h e r r e f e r ences a r e c i t e d a s we go along. Let us r e c a l l f i r s t the BA function from Thus given a compact RS S o f genus g , and a point Q % m w i t h local v a r i a b l e l / k near Q ( k ( Q ) = -) t h e r e i s a unique ( u p t o a c o n s t a n t ) BA funct i o n JI on S characterized by t h e p r o p e r t i e s ( 1 ) J, i s meromorphic on S except a t 9 where JI(P)exp(-q(k)) i s a n a l y t i c ( 2 ) On S/Q,J, has poles o n D (a non2 3 special divisor). Typically q ( k ) has t h e form q ( k ) = kx + k y + k t t ... and J, can be represented via 0 functions as in (5.2) f o r example. We r e c a l l t h a t J, % e x p ( q ( k ) ) ( l + 1 ; c i / k i ) near Q and some asymptotic a n a l y s i s w i l l y i e l d d i f f e r e n t i a l operators L,A, e t c . such t h a t a J, = LJI, at$ = A$, e t c . Y ( c f . §5). We a l s o saw i n 14 f o r h y p e r e l l i p t i c s i t u a t i o n s how t h e BA func2 t i o n J, (based on LJI = A$, L = D + q ( x ) f i n i t e gap, B$ = !.I$, w i t h p = (X,u) REWRK 19-1 (ef[E 3A FUNCCZ0N)-

4,5,etc.

284 E

S

R O B E R T CARROLL 'L

u2

-

Jl(0,p) = 1 , and x

P ( A ) = 0, q ( k ) = kx, k = (A)',

-+

$ ( x , p ) holo-

morphic i n a N B H of 0 f o r p E S - m - 0 ) i s expressed i n terms of s e c t i o n s o f a l i n e bundle L X = L6 B 1;;. Here 1;& i s defined v i a t r a n s i t i o n functions =

f / f a on Ua n U

where fa has zeros only a t those P 1 , . . . , P

gaD B B U a ' while Uco 0 {Pi} = 0 with f

p"

= 1.

i:; i s defined v i a g '

aB

g

lying in

+

= 1 i f a , ~m

A section 0 of (f; = 1 ) and g;,(p) = exp(-(A)'x) i n Ua n Um (f; = g,"f;). C x s a t i s f i e s oa/fa = 0 / f = F ( d e f i n i t i o n ) w i t h F holomorphic function D B exp((,)'x) near so F 2. $. We saw how such y could be used t o describe the

-

Q

C . Neumann problem associated with L and a l s o indicated some r e l a t i o n s be-

tween

$

and t h e functions U , V , W

describing t h e a f f i n e coordinates of t h e

curve associated with S. REmARK 19.2

Krizever data came u p in various places ( c f .

(KRICWER DACA).

a l s o e.g. 9 2 1 ) .

For example one considers ( C , P , z , L , o 0 ) where

C

i s a compact

E C w i t h local coordinate z ( P ) = m, L i s a holomorphic l i n e bundle o n C, and (I i s a local t r i v i a l i z a t i o n over Urn 3 P. We a s s o c i a t e a point W 0 RS, P

E

'L m

Gr t o such data via W

Q

Ho(C-P,L).

Similarly in 518 we considered data

or such data augmented by local coordinThen one had a map I': P ( p g ) + U G M ( ? ) : data a t e s u , t t o form elements i n H g' t ( H o ( S , C ( * Q ) ) ) C K where t h e image here is in t h e s e t of Laurent s e r i e s expansions of sections of 1; holomorphic away from Q. Thus t h e s e construct i o n s a r e b a s i c a l l y the same ( t h a t of 918 i s simply more s o p h i s t i c a t e d and elegant - i n addition t o including more information). { ( ( S , ( a , @ ) ) , Q , i : ) ;Q E S; i: E PicSl 4

h

-f

REmARK 19-3 (BURCHNAU tXAUND&J i?BE8R&J)- Let us note in passing another

The tau functions and t h e KP hierarchy a r e r e l a t e d via Lw = zw, a n w = Bn w , L = a t co u n +1 a - n , E n = L,,n a n L = [Bn.L], e t c . where w = V ( Z ) T ( X ) / T ( X ) a n d L = Pap-' e t c . I f one has dependence on only a f i n i t e basic connection.

l1

number of xn then [ B n , L I = 0 f o r n N and such d i f f e r e n t i a l operators Bn commuting w i t h L generate a commutative ring ( c f . 512) a n d determine a n a l gebraic curve.

We r e c a l l a basic idea o f Burchnall Chaundy theory ( c f . [W5,

l o ] ) . Let L , Q of order n,m be 2 commuting (monic) d i f f e r e n t i a l operators (generating such a r i n g ) . Let V, be t h e n-dimensional eigenspace of L so Q i s a l i n e a r operator Q, on V,. Similarly l e t W be t h e p e i g e n s p a c e of Q

v

of dimension m with L a l i n e a r operator L o n W . By eigenspace we think of i ' !J s o l u t i o n s fixed by i n i t i a l conditions 0 f . ( x ) = 6 j j a t x = a , and one can J

BURCHNALL CHAUNDY THEORY w r i t e say L = D n t C;-lui(x)Di

-

det((Q, 2

285

with Q = Dm t ~ ~ - l v i ( x ) O i Define . f,(X,p) =

P I , ) ) and f2(X,v) = d e t ( ( L P - XIm)) with f3(A,v) = detM where M i s

There a r e (mtn) operators Q - u , D ( Q Dm-l 1.1), ..., (Q-1.11, L - A , D(L-A),..., (L-A). Write t h e ith o f these operators i n t h e form mtn-lm ( x ) D j ( i = 0 , . . .,m+n-1) and s e t M = ( ( m . . ) ) . Then ij 1J one can show t h a t (*) f l = ( - l I m n f 2 = f 3 ( s e e [WlOl - we omit t h e proof a

(mth)

matrix defined as follows.

on-l

lo

s i n c e t h e matter is discussed more generally in 512).

In p a r t i c u l a r f l i s

obviously a polynomial i n 1.1 and t h e r e s u l t (*) shows i t i s a l s o a polynomial

i n h ( s i n c e f 2 i s a polynomial in A ) . This can a l s o be seen d i r e c t l y from looking a t Q, a c t i n g on a standard basis f o r V, ( e x e r c i s e ) . Now s e t f = f , = + f 2 = f 3 a n d one shows t h a t f(L,Q) = 0. To see t h i s look a t f(L,Q) res t r i c t e d t o v, which is f(h,Q,) = 0 (Cayley-Hamilton theorem). Thus f(L,Q) a n n i h i l a t e s a l l eigenfunctions of L and has i n f i n i t e dimensional kernel (EF” d i f f e r e n t EV” a r e l i n e a r l y independent). B u t f(L,Q) commutes with

L and hence has constant leading term and t h i s gives a contradiction unless f = 0 ( e x e r c i s e ) . One can a l s o show d i r e c t l y by simpler arguments on order t h a t f(L,Q) = 0 f o r some polynomial f ( c f . [ WlOl). Finally f o r A l p such t h a t L$ = A$ and Q$ = PIJJ have a s o l u t i o n we see t h a t f ( $ , v ) = 0 and one can show this defines an i r r e d u c i b l e a l g e b r a i c curve u ( L , Q ) = SpecmR where Specm R % maximal ideal space o f the algebra o f operators generated by L a n d Q ( c f . §12 where t h e matter is t r e a t e d more generally and more completely).

(KRZCEUER DACA AND GRAktilllANNZAW). Let us make a few remarks here following [ SE1 ;PR1 ;W10] connecting KriEever data to Grassmannians. T h u s

REmARK 19.4

f i r s t l e t G be t h e i d e n t i t y component f o r real a n a l y t i c maps g: S’ + G L ( n , C ) . Let P C G be the s u b g r o u p of maps extendible t o holomorphic maps Do GL(n, -+

C ) (Do = I z ; IzI 5 1 1 ) . The Grassmannian Grn 2 6, P i n t h e following con2 1 Crete r e a l i z a t i o n . Thus l e t Hn = L ( S , C ) , o n which G a c t s i n an obvious 2 1 f ( z ) = fo manner and i d e n t i f y Hn w i t h H = L ( S , C ) via (A) ( f o , . . . f n - l ) -+

.. . +

z n-1 fn-l ( z n ) ( t h u s f k ( z ) = ( l / n ) C f ( r ; ) s - k , s r u n n i n g over the nth roots of z ) . This i s an isometry Hn > H and we will w r i t e H, zn)

t

Zfl

(Zn)

+

C H f o r functions which a r e boundary values o f holomorphic functions on

Do

Grn i s defined to be t h e s e t of a l l closed subspaces W C H obtained by a c t i n g o n Ht w i t h elements o f G ; i . e . (.) Grn = Here t h e a c t i o n o f g on igH,; g E GI 2 G/P ( P i s t h e isotropy group of H,) ( s i m i l a r l y f o r H: C H n ) .

286

ROBERT CARROLL

H, and mu1 t i p 1 i c a t i o n by z on Hn % mu1 t i p l i c a t i o n by H commutes w i t h g a c t i o n ( n o t e W E G r n has t h e p r o p e r t y znW C W ) . The k bases isiz , E~ a b a s i s f o r Cn, 1 5 i f n, k E Z1 f o r Hn correspond l e x i c o k nk+i-1, E.zo % zi-l g r a p h i c a l l y t o t h e b a s i s zk f o r H ( i . e . E ~ Z% z , E 1. 2 % 1 Hn i s t r a n s f e r r e d t o

zn on

Z

'+'-',...,

%

z 5 z n , slzz % zZn, ...). Note a l s o ( 4 ) G r n 1 W l where G r = G r ( H ) i s t h e s e t o f c l o s e d subspaces W c H

o r e.g.

IW

G r ; znW

E

C

such t h a t p r : W

elz

0

%

1,

E

H, i s Fredholm and p r : W

-+

+

H- i s compact ( H i l b e r t - S c h m i d t

= HS i s u s u a l l y used here i n p l a c e o f compact and we r e c a l l t h a t HS i m p l i e s

This corresponds t o G r n = { W

compact).

E

Gr(Hn);

zW C W) w i t h a s i m i l a r

d e f i n i t i o n o f Gr(Hn). For W

E

G r n t h e r e i s a unique (BA) f u n c t i o n

z

x E C,

$J~(X,Z),

t e r i z e d by t h e p r o p e r t i e s (*) q ~ ~ ( x , * E ) W (when d e f i n e d ) and

(1

+

lyai(x)z-')

i n g t h a t e-"JIW

( c f . here a l s o 111).

t h e spacesV

E

-+

H,

charac-

$,(X,Z)

= ex'

The d e f i n i t i o n i s c l a r i f i e d b y say-

i s t h e (unique) element o f e-"W

map p r : exp(-xz)W

E S,

p r o j e c t i n g t o 1 under t h e

( n o t e i n f a c t exp(-xz)W E Gr').

G r n p r o j e c t i n g i s o m o r p h i c a l l y o n t o H,

One notes here t h a t form a dense open sub-

Under t h e i d e n t i f i c a t i o n G r n 2- G / P t h e

set o f Grn called the "big cell".

b i g c e l l i s t h e s e t o f p o i n t s gP such t h a t t h e RH problem f o r g has a s o l u tion, i.e.

g = 9-9,

where g+

Ez

GL(n,C) where Dw = = { x E C;

4

exp(-xz)W

E

C U

E

P and g extends t o a holomorphic map Dm ( z ( 1. 11. Then one shows t h a t t h e s e t

Em);

t h e t a u f u n c t i o n , which i s t h e d e t -

zero s e t o f a n a n a l y t i c f u n c t i o n -iw(x), -+

H,).

We o m i t d e t a i l s here ( c f . [ S E l ; W

t h e f u n c t i o n JIw i n (*) i s t h e r e f o r e d e f i n e d f o r x

a r e e s s e n t i a l l y a l s o covered i n §11,21,22

These m a t t e r s

E A.

w i t h more d e t a i l s p r o v i d e d .

L e t now A denote r e a l a n a l y t i c f u n c t i o n s on S 1 o f f i n i t e o r d e r ( i . e . F o u r i e r s e r i e s has o n l y a f i n i t e number o f p o s i t i v e powers o f W a l g = elements i n W o f f i n i t e o r d e r . and n o t e t h a t zn

E

A

b i g c e l l } i s a d i s c r e t e subset o f C ( i n f a c t i t i s t h e

erminant o f t h e p r o j e c t i o n e-"W

lo]);

-+

Aw so C [zn

I

C

AW.

Define (m)

%=

Then f o r each f

If

E

E

the

z) and l e t

A; f W a l g

C

Walg)

AW t h e r e e x i s t s a

unique OD0 ( i n x ) L ( f ) such t h a t (**) L(f)qJW(xyz) = f(z)qJ,(x,z)

and i f t h e

F o u r i e r s e r i e s o f f has l e a d i n g t e r m a z N t h e n t h e o p e r a t o r L ( f ) has l e a d i n g t e r m aa

N

( a = a/ax).

To see t h i s s i m p l y equate c o e f f i c i e n t s i n t h e power

s e r i e s expansion i n (*)

t o f i n d a unique L ( f ) such t h a t L ( f ) $ J W- f$JW = ex'

GRASSMANNI A NS

28 7

But qW E W f o r each x, so do i t s x d e r i v a t i v e s , and so does f$wby

O(2-l).

d e f i n i t i o n o f AW.

Thus L ( f ) q W

-

f q W E W b u t e-"W

and hence c o n t a i n s no f u n c t i o n o f t h e form O ( l / z ) .

E b i g c e l l f o r generic x

-

This implies L(f)QW

fqlw= 0. Thus f

-f

L ( f ) embeds AW i n t h e a l g e b r a o f d i f f e r e n t i a l o p e r a t o r s and W

-f

AW

determines a map G r n .+ a commutative a l g e b r a o f d i f f e r e n t i a l o p e r a t o r s i s o -

%.

morphic t o

Now i n what f o l l o w s we w i l l o c c a s i o n a l l y make some unsupport-

ed statements whose p r o o f w i l l e i t h e r be p r o v i d e d below,or can be found i n [SEl;WlOI, E Grn,

o r i s a l r e a d y i n §11,12.

SpecAW i s a c u r v e ( c f . 512).

I t t u r n s o u t n a t u r a l l y t h a t f o r any W

Now W a l g i s a t o r s i o n f r e e AW module o f

r a n k 5 n ( r e c a l l a module A o v e r a r i n g R i s t o r s i o n f r e e means A A

= {a E

A;

Oa # 0) w i t h Oa = C r E R; r a = 0 3 ) .

t s i o n f r e e sheaf o v e r SpecAW ( c f . [ HA1 ;MU21,

o t h e r hand any W = KricYever d a t a .

E

= 0 where t Hence $lg defines a t o r -

912, and Appendix B).

On t h e

G r n can be c o n s t r u c t e d from {a curve, a sheaf, o t h e r datal

Indeed l e t X be a complete a l g e b r a i c c u r v e o v e r C ( c f .

Appendix B) and xm a n o n s i n g u l a r p o i n t . Assume t h e r e e x i s t s a r a t i o n a l func2 t i o n f : X -t S whose o n l y s i n g u l a r i t y i s a p o l e o f o r d e r n a t xm (always t r u e i n t h e p r e s e n t c o n t e x t f o r some n ) .

F i x a l o c a l parameter z - l near xm

so t h a t f ( z ) = zn l o c a l l y , and suppose z d e f i n e s an a n a l y t i c isomorphism be2 tween a NBH o f xm E X and an open NBH o f Dm = {z; I z I 2 1) C S (always pos1 s i b l e by r e s c a l i n g ) . Then z i d e n t i f i e s S C C w i t h a small c i r c l e around xm (and Dm w i t h a s e t Xm i n X ) .

L e t 1; be a holomorphic l i n e bundle on X o f de-

gree g = genus X ( o r i f X i s s i n g u l a r a rank one t o r s i o n f r e e coherent sheaf

-

c f [ HA1 1) and l e t $ be a t r i v i a l i z a t i o n o f t o v e r D C X. To t h i s data 2 T (X,C,xm,z,$) = K r i z e v e r data one assigns W C H = L ( S ,C) as W = Cclosure o f a n a l y t i c f u n c t i o n s on S1 t h a t extend t o s e c t i o n s o f C o v e r t h e e x t e r i o r o f

0-1.

Then W

- {xml. I n 1: Pi, xm #

E

G r n and

$will

be i d e n t i f i e d w i t h t h e c o o r d i n a t e r i n g o f X

e.g. [KR8] one has n o n s i n g u l a r X and a nonspecial d i v i s o r Pi;

a holomorphic s e c t i o n o f t h e l i n e bundle 1;

D

=

0, v a n i s h i n g

a t P., g i v e s a t r i v i a l i z a t i o n $ o v e r a NBH o f xm. One notes f o r KdV, W E 2 ' 2 G r so z i s meromorphic on X w i t h second o r d e r p o l e a t x ; t h i s means X i s h y p e r e l l i p t i c and xm i s a W e i e r s t r a s s p o i n t . These c o n s t r u c t i o n s , when a n a l y z e d i n more d e t a i l , i n v o l v e c e r t a i n t e c h n i c a l

288

ROBERT CARROLL

c o m p l i c a t i o n s t o deal w i t h s i n g u l a r i t i e s ( c f . [ SE1;WlOl

and R e m r k

B11 f o r

One can a l s o do t h i s f o r rank r 5 n v e c t o r bundles

some expansion on t h i s ) .

t ( c f . I W l O l ) b u t we o m i t t h i s e n t i r e l y and mention o n l y t h a t i t d i f f e r s from t h e c o n s t r u c t i o n s i n [ MUZ;KR8]. Now l e t

r + be

morphic y : Do

t h e group o f r e a l a n a l y t i c maps y : S’ -f

Cx w i t h y ( 0 ) = 1 .

y ( z ) = exp(xz +

(19.1)

w i t h ( s u i t a b l e ) x s ti

E

C.

One w r i t e s f o r y

E

Cx e x t e n d i n g t o h o l o -

E

r,

1”2 tiz’) W

As above f o r

E

( y , z ) d e f i n e d f o r y i n a dense open subset of (*A) $,(y,.)

-+

W and $,(y,z)

= y(z)(l

+

1;

G r n t h e r e e x i s t s a unique $w

r + and

ai(y)z-’).

z

E

S

1 c h a r a c t e r i z e d by

One w r i t e s QW(x,t,z)

f o r J, (y,z) and ~ , ~ ( x , O , z ) $ w ( x y z ) o f (+I. Also ( p r o o f a s b e f o r e ) ( * a ) f o r n‘ r W E G r and each r 2 1 t h e r e e x i s t s a unique d i f f e r e n t i a l o p e r a t o r Pr = a Q

+ 1y-2pi(xyt)a i such t h a t n + u an-*

t h e n L = Pn = a

n- 2

From L J , ~= zn$” and (19.2)

+

= PrW J,

... +

(ak

=

a/atk),

and P n ~ w = z n$w. L e t

uo w i t h c o e f f i c i e n t s depending on ( x , t ) .

= Pr$w we g e t immediately

arL = [ Pr,LI

I n t h i s c o n t e x t t h e t a u f u n c t i o n a r i s e s as f o l l o w s . age W E Grn.

For y

E

r+,

y : H+

L e t w: H,

H+, t h i s determines (yw): H+

-+

+

H have i m -

-f

H via the

commutativity o f W

(19.3)

>

One can choose w such t h a t H+%

H

H

pr, H,

d i f f e r s from t h e i d e n t i t y by a

t r a c e c l a s s o p e r a t o r ( c f . [ DF1-4;PRl;SEl;WlOl

and 511) and e.g.

(y-’w)

will

have a s i m i l a r p r o p e r t y so t h a t d e t can be d e f i n e d i n -1 (19.4)

T,(Y)

= det(H+

( I ! ! )

H

H+)

REmARK 19.5 (FREDH0tm 0PERAC0W AND SHEAF C0H0fit0t0cg). An i n t e r e s t i n g a r g u ment t o e s t a b l i s h t h e Fredholm c h a r a c t e r o f p r : W

-+

H+ when W i s c o n s t r u c t e d

FREDHOLM OPERATORS

289

v i a (X,L,xm,z,$) i s g i v e n i n [ S E l I ( c f . a l s o [ PR1 I ) . Thus l o o k a t W E G r w i t h n u n s p e c i f i e d . The E u l e r c h a r a c t e r i s t i c ~ ( t )i s x ( C ) = dimHo(X,t) 1 dimH (X,C) ( c f . Appendix B) and t h e v i r t u a l dimension o f W i s x(C) - 1. x(C) = 1 f o r l i n e bundles

i:

Q

d e g ( t ) = g by Riemann Roch ( c f . Appendix

B).

Now

,D a s before (Iz] 1. 1 ) and Xo Iz; I z I 5 1). L e t Um and Uo be op1 en NBHs w i t h S C U, n Uo and c a l c u l a t e t h e s h e a f cohomology v i a Mayer-Vie-

l e t ,X

Q

t o r i s ( c f . (12.7) and Appendix B) (19.5)

0

-+

Ho(X,t)

+

Ho(Uo,C)

@ Ho(U,,l:)

-f

Ho(Uo n U m , t )

-+

H1(X,t)

1

Taking d i r e c t l i m i t s as Uo

(19.6)

0

(where C ( S1 )

-+

Ho(X,C)

+

Ho(Xo,C)

-+

Xo and Urn

8 Ho(Xm,C)

0

1 = H (Um,t)

Here C i s a t o r s i o n f r e e coherent sheaf f o r example w i t h H (Uo,l:) = 0.

-+

* Xm g i v e s ( c f . a l s o 512) 1

-+

C(S )

-+

1 H (X,C)

*o

r e a l a n a l y t i c f u n c t i o n s o n S1 ) .

Since i: i s t o r s i o n f r e e i t s 1 s e c t i o n s o v e r Xo o r Xm a r e determined by r e s t r i c t i o n t o S (so one make i d 1 e n t i f i c a t i o n s i n t(S ) ) and t h e two m i d d l e terms i n (19.6) become (*4) Wan

fII zHtn

-+

Q

Han ( i n c l u s i o n o n t h e f i r s t f a c t o r and

Van = a n a l y t i c f u n c t i o n s i n V C H),

-

i n c l u s i o n on t h e second

o b v i o u s l y t h e same a s t h o s e o f t h e p r o j e c t i o n Wan

*

Z H ? ~ and one checks t h a t

these do n o t change on passing t o t h e c o m p l e t i o n ( e x e r c i s e n o t e t h a t W * H, has c l o s e d range).

-

But t h e k e r n e l and cokernel i n (*4) a r e

-

c f . [ SE1

I

and

The same argument shows ( c f . IKR5,8])

a l s o t h a t t h e k e r n e l and cokernel o f t h e p r o j e c t i o n W -+ H, can be i d e n t i f i e d 1 where C = 1: B [ - x m l i s t h e s h e a f ( o f degree g-1) w i t h H o ( X , t ) and H ( X , t ) whose s e c t i o n s a r e s e c t i o n s o f C v a n i s h i n g a t xm ( e x e r c i s e SE1;VDll). = 0.

-

c f . [ MU2;PRl; 1 I n p a r t i c u l a r W i s t r a n s v e r s e i f and o n l y i f Ho(X,L) = H ( X , t )

T h i s should a l l be f a i r l y c l e a r g i v e n t h e exposure i n §l2,21 and Ap-

pendix B. L e t us n o t e how t h e s t r a i g h t l i n e f l o w i n J(Z) corresponds t o t h e l i n e bund l e approach w i t h Kric'ever data ( c f . a l s o 121). For g E r+ l e t I: be t h e

g

l i n e bundle ( o f degree 0) o b t a i n e d by g l u e i n g t o g e t h e r t r i v i a l bundles o v e r 1 Thus t comes Xo and X, v i a t r a n s i t i o n f u n c t i o n s g o n a n open NBH o f S

.

(X,t,x,,z,+)

+

9

o v e r Xm and t h e a c t i o n o f r, o n G r i s g i v e n by (*+) 9 g(X,t,xm,z,$) = (X,C Il:g,xm,z,$ DD 4g). Then t + 1; B 1: i n 9

with a trivialization 4

290

ROBERT CARROLL

p a r t i c u l a r and t h e g e n e r a l i z e d Jacobian o f X corresponds t o holomorphic 1 i n e bundles o f degree 0 (which can a l l be c o n s t r u c t e d v i a g

E

r+ -

c f . [ SE1 I).

REmARK 19.6 (CCM$ZFZCAEZ@N OF CQ)mEAEZUE AtGE3W OF 0RDZNARg DZFFERENEZAL 0PERAE0W). We go a g a i n t o [MS51 and make a few comments. The r e s u l t s t h e r e g i v e a g e n e r a l i z a t i o n o f t h e K r i z e v e r map and p r o v i d e a c l a s s i f i c a -

-

t i o n o f a l l commutative a l g e b r a s o f OD0 ( o r d i n a r y d i f f e r e n t i a l o p e r a t o r s

T h i s a l s o shows t h a t KP f l o w s produce a l l g e n e r i c VB on

c f . a l s o [AC4,51).

We w i l l o n l y g i v e a s k e t c h o f some

a r b i t r a r y a l g e b r a i c curves o f genus > 1.

o f t h i s ( c f . a l s o 512 f o r r e l a t e d techniques and some d e t a i l s ) .

M a i n l y we

w i l l e x h i b i t some o f t h e o b j e c t s and m p s , w i t h o u t much p r o o f o f a n y t h i n g . L e t K = C as i n 112 ( t h e f i e l d K i s more general i n [MS51)

and d e f i n e Vn =

C [ [ Z ] ] Z - ~ a s i n 512 w i t h C ( ( z ) ) = V = U Vn, (0) = n Vn, and Vn+' 2 Vn. One says v E V has o r d e r n ifv E V n/ V n-1 and f o r W C V, y(V), i s d e f i n e d v i a

I

v-v

(19.7)

For p , v

E

u and l e v e l v i s (*.) G(u,w) = { c l o s e d ~ ( v i s) Fredholm ~ o f i n d e x u l . The b i g c e l l o f

Z t h e Grassmannian o f i n d e x

v e c t o r subspaces W such t h a t index 0 i s

(A*)

Although G(u,v)

+

G (0,v) = (W E G(0,v);

k e r n e l ~ ( v =) cokernel ~ y(w),

i s isomorphic t o G(p,v+v')

= 0).

it i s important t o note t h a t

t h e r e i s no canonical isomorphism between them. For r

E

Z, r 2 0, p , w

E

Z, a p a i r (A,W)

i s c a l l e d Shur o f r a n k r, i n d e x

and l e v e l v i f (u) W E G(u,v) and A C V i s a K subalgebra such t h a t K K # A, AW C W,

and r = r a n k A = GCD(order a; a E A ) .

s e t o f such Shur p a i r s .

Let

xw = { v

E

V;

Sr(u,v) Then if K

VW C W I . N

i s a (maximal) Shur p a i r , b u t f o r g e n e r i c W, AW = K.

u, A,

C

denotes t h e

+ iw, (KM,W)

The s e t o f f i n i t e r a n k

N

p o i n t s o f G(p,v)

is

(Aa)

Gfin(pyv)

= I W E G(u,v);

AW # K I and one has a canN

o n i c a l i n j e c t i o n (A&) s : Gfin(p,v)

-f

U

br(u,v)

= S(u,v)

v i a s(W) = (AW,W).

The geometrical data i s c o m p l i c a t e d because o f t h e g e n e r a l i t y t r e a t e d i n [MS5] so we w i l l o n l y s p e c i f y t h i s f o r K = C and n o n s i n g u l a r c u r v e s C.

Thus

one c o n s i d e r s q u i n t e t s ( C y p , F y r y ~ =) Q a s i n d i c a t e d below i n o u r s p e c i a l s i t u a t i o n (general q u i n t e t s

-

u n s p e c i f i e d here

-

w i l l be r e f e r r e d t o as QG).

COMMUTATIVE ALGEBRAS

291

The definitions are (A+) ( 1 ) C i s a reduced irreducible complete (nonsingul a r ) algebraic curve over K ( K = C here) ( 2 ) p E C i s a smooth k rational point ( 3 ) F C of rank r 1 dim H (C,F) around p i n

i s a torsion f r e e sheaf of 0c modules (here a vector bundle on a n d degree 1-1 + r(g-1) for g = genus C ) satisfying dim Ho(C,F) = 1-1 ( = deg F - r(g-1) here) ( 4 ) Let U be a small open s e t P C and Uo a small open disc in C around 0. Let n : U -+ U be a O

P

holomorphic covering of U ramified a t p and z a local coordinate a t 0 ( 5 ) I$: P F -+ n*O ( v ) a n isomorphism of t r i v i a l holomorphic VB on U ( 4 i s a local UP UC P t r i v i a l i z a t i o n of F over U and 0",(v) i s a so called twisted structure P sheaf of Uo a s a formal scheme - c f . [ HA1;SERl 1 and Appendix B ) . For 0 ( n ) roughly one thinks of rational functions P/Q with P a n d Q polynomials s a t i s fying degP - degQ = n. Then f o r r = 1 , n: Uo -+ U i s a n isomorphism a n d gives a local coordinate y P This on U while 4 i s a t r i v i a l i z a t i o n of the l i n e bundle i; over U P P' i s consequently the same Krizever d a t a covered before a n d aside from singul a r i t i e s a n d general f i e l d s K the extension of [MS51 involves the local covering IT and the special kind of local t r i v i a l i z a t i o n 4 . The Burchnall= n(z)

Chaundy theory ( c f . [ BN1 ;KR5,8;MU2;SE1 ;W5,10 I ) establishes a canonical bijection between B1 = {commutative C algebras with identity, with a m n i c P of order n regular a t some point, and of rank 11 and the moduli space 1111 = { t r i p l e s ( C , p , t ) ; C a n algebraic curve of a r b i t r a r y genus g, with a smooth point p; i; a l i n e bundle on C of degree g-1 having no nontrivial global holomorphic sections}. Thus i; 2 i n Remark 1 9 . 5 a n d (C,p,i;) % Krizever data/ Q

expl i c i t ( z , 4 ) . The extension i n [ MS5 ] now establishes a 1-1 correspondence between Shur pairs (A,W) a n d quintets

4,.

For W E G(O,-l) there e x i s t s a unique PSDO S

of degree 0 determined by W, and, identifying z-l with a d / d x , A becomes a ring o f PSDO with constant coefficients; the condition AW C W i s equival e n t to B = SAS-' consisting only of differential operators ( c f . 512). In t h i s s i t u a t i o n 1-1 = 0 and (C,p,F,n,$) a semistable VB F on C o f rank r a n d degree d = r(g-1) having no nontrivial holomorphic global sections. We ref e r t o [MS5] for d e t a i l s a n d discussion. Q

REmARK 19.7 (B090N 0PFRAvDR REALZZACZ0N BF CHE KRZCWER C0WCWCCZON). There

292

ROBERT CARROLL

1 concerning a

i s a n i c e c o l l e c t i o n o f i n f o r m a t i o n i n [CKl;FG1,3,6;SW1,25;Vl v a r i e t y of t o p i c s but y e t related.

We do n o t g i v e any d e t a i l s b u t e x t r a c t

a few general comments (some i n t e r s e c t i o n w i t h m a t e r i a l i n o t h e r s e c t i o n s o c c u r s and t h i s should augment t h e understanding o f t h a t m a t e r i a l and enhance i t s i n t e r e s t ) .

There i s a l s o some i n t e r s e c t i o n w i t h o t h e r work, most-

l y r e f e r e n c e d a l r e a d y ; we make no a t t e m p t t o s o r t o u t h i s t o r i c a l f a c t s here.

In [ V l

1 one s t u d i e s t h e b o s o n i z a t i o n o f c h i r a l f e r m i o n t h e o r i e s on an a r b i -

t r a r y compact RS.

The fermion and boson c o r r e l a t i o n f u n c t i o n s a r e expressed

i n terms o f t h e t a f u n c t i o n s and proved equal; v a r i o u s c h i r a l determinants a r e a7so analysed and a p p l i e d t o t h e p a r t i t i o n f u n c t i o n o f t h e c l o s e d bosoni c string.

I n p a r t i c u l a r one l o o k s a t two c o n j u g a t e c h i r a l f e r m i o n f i e l d s

b and c o f conformal s p i n A and 1-A.

There i s a l o t o f good d i s c u s s i o n o f

s p i n s t r u c t u r e , prime forms, d e t e r m i n a n t c o n s t r u c t i o n s , e t c . I n [ CK1 ;FG1,3,6 nian.

I one f o r m u l a t e s CFT on t h e i n f i n i t e dimensional Grassman-

One r e c o v e r s known formulas f o r e.g.

c o r r e l a t i o n f u n c t i o n s o f b-c

systems and produces c o n s i d e r a b l e general i z a t i o n .

The Grassmannian p r o v i d e s

a g e o m e t r i c a l i n t e r p e r t a t i o n o f i n f i n i t e dimensional K-M and V i r a s o r o a1 gebras l e a d i n g t o computations o f c e n t r a l charges and conformal dimension One a l s o can compute c o r r e l a t i o n f u n c t i o n s

( w e i g h t s ) f o r v a r i o u s models.

using vertex operators o f the type a r i s i n g i n s o l i t o n theory.

Relations t o

t a u f u n c t i o n s and h i e r a r c h i e s a r e i n d i c a t e d . I n [ SW1,2,5]

one d e a l s w i t h o p e r a t o r valued o b j e c t s and c o n s t r u c t s t h e a l -

gebro-geometric t a u f u n c t i o n s i n terms o f a bosonic CFT o n Riemann surfaces. This l e a d s t o t h e o p e r a t o r b o s o n i z a t i o n f o r m u l a t i o n f o r f e r m i o n i c b-c systems o n RS.

We e x t r a c t a few comments from [ SW21 which has t h e t i t l e

"A

bosonic o p e r a t o r r e a l i z a t i o n o f t h e K r i c e v e r c o n s t r u c t i o n and b-c systems on Riemann surfaces".

As i n Remark 19.2 f o r example, t o K r i c e v e r data

(C,P,z,

0

L,oo) one a s s o c i a t e s W E G r v i a H ( c - P , L ) and t h e passage from G r t o

been discussed e x t e n s i v e l y .

More g e n e r a l l y (C,w

J

,Q,u,o)

(w

J

T~

has

a bundle o f

s p i n J d i f f e r e n t i a l s ) d e f i n e s a r a y clW> i n a f e r m i o n i c H i l b e r t space H , r e l a t e d t o t h e Grassmannian ( c f .

TI^,

§18,21 and [ AG1,6;SE1 ]) and t h e passage t o

= (Olexp(H(t))lW) i s similar.

On t h e o t h e r hand f o r a s p i n J b-c sys-

tem one can c o n s t r u c t d i r e c t l y v i a t h e t a f u n c t i o n s an a l g e b r o - g e o m e t r i c

T

KRICEVER CONSTRUCTION

293

where ( 1 ) t = ( t r ) , r 2 1 , i s a collection of KP terms ( 2 ) L = (25-1) ( g - l ) , g = genus C , the Pa a r e a rbit ra ry generic points where L insertions of b operators occur in accordance with ghost number counting, and P = P + 1 ... + P L i s the corresponding divisor ( 3 ) z E Z i s arbitr ar y a n d identified with i t s coordinate z ( 4 ) 0 i s the appropriate theta function with charact e r i s t i c determined by the spin struc t ur e ( 5 ) A i s the Riemann ”class’and the Qm n a r e defined below. These ideas have a l l been developed elsewhere in the book (see e.g. Appendix B, §4,5,20,21), except for the notion of counting ghosts a n d we will n o t go into t h a t (note Jb 1; , b ( b i ) ) . Then event u a l l y one can conclude via unique bosonization that T~ = ~ 1 Once ~ ) this i s established then acting on T by vertex operators (see below) one IN) creates b-c correlation functions. A

Q

For vertex operators one uses ( ( ( t , u - ’ ) (19.10)

~ ( u ) exp(-tologu v*(v)

=

t

=

Q

1tru-r)

S ( t , u - l ) ) exp(- 1u r ar/r + ao);

exp(tologv

-

c ( t , v - l ) exp( l v r a r / r

t

ao)

acting on T to obtain ( k n o w n ) expressions for correlation functions. In J which reparticular one gets a complicated formula for V ( u ) V * ( v ) T J ( t ) / r J ( t ) duces a t t = 0 t o the correct re sul t for the b-c twooagator on a RS a s in [Vl 1. The bosonic C F T of [ SW21 on C i s the theory of a current I(u) = a n operator valued 1-form o n C , normalized t o 0 a periods, with propagator (19.11 ) ( 0 1 I ( u ) I ( v ) ~ O=) auavlogE(u,v)dudv where 10) i s the boson vacuum a n d E i s the prime form (radial % time orm n dering i s assumed). Using (a*) logE(u,v) = log(v-u) + l m , n , lv(/rnn)Qmn ~ one obtains (19.12) ( O ( I ( u ) I ( v ) l O )= ( l z ( n + l ) un / v n+2 + l;Qmnu m-1 v n-1 )dudv

294

ROBERT CARROLL

One d e f i n e s H a m i l t o n i a n s = (l/Zni)l:

t r % dvl:

Qrnv

around a base p o i n t Q). [H(t),I(x)l

=

Hl(t)

(0.1

n-1

f

\r

= ( 1 / 2 7 r i ) l 7 rtr& du/ur+lfu

I w i t h H = H1 + H2 ( 9 i s a c o n t o u r i n t e g r a l ( 0 0 ) t H1 ( t ) , H 2 ( t ) I = - 1Qrstrts and

There r e s u l t s

1 rtrwr(x).

Then one forms a bosonic o p e r a t o r BJ(P1,

Q) r e p r e s e n t i n g b - o p e r a t o r i n s e r t i o n s b(P1 1.. .b(PL); (19.13)

BJ(Po,,Q) = e x p ( ( 2 5 - 1 ) / Z n i ) ( $

‘j.1)

I and H 2 ( t )

exp(-jlPz

I)B(P -(ZJ-l)A

&;

I

Wi(u)Lu

..., PL,

t h i s has t h e form

- Ib I)/O(P -

- 4,

duaulogE(Q,u)-

(25-1)A)

Then one develops a formula ( T J ( t ) = exp

w i t h normal o r d e r i n g understood.

(Hi ( t ) ) B J ) (19.14)

= (OITJ(t)lO)

T,(t)

= ( OleH(t)BJe-H(t)lO)

( w i t h s u i t a b l e normal o r d e r i n g ) .

i n terms o f 8 etc..

which we o m i t .

e’[H2(t)yH1

(t)l

An e x p l i c i t formula i s a l s o a i v e n f o r TJy One can t h e n c a l c u l a t e V(u)V*(v)TJ(t)

e x p l i c i t l y and a l l h i g h e r c o r r e l a t i o n f u n c t i o n s have t h e form ( 0 6 )

(

O/TIV(ui)

Thus t h e t a u f u n c t i o n T J ( t ) i s produced b y an o p e r a t o r TJ

V*(vi)TJ(t)/O).

and t h e “ o p e r a t o r b o s o n i z a t i o n ” achieved v i a ( 0 6 ) e t c . under1 i e s t h e c o r r e s pondence between boson and f e r m i o n c o r r e l a t i o n f u n c t i o n s ( c f . [ AG3,4;SWlY2, 5;Vl

1).

RRllARK 19.8

(KRICEVER N0VZK04 (KN) A L G W W ) .

We had o r i g i n a l l y i n t e n d e d t o

g i v e a s k e t c h o f r e s u l t s and ideas r e l a t e d t o KN a l g e b r a s b u t t h i s w i l l be l i m i t e d due t o l a c k o f space.

KR10,14,16,18;MATl;N05;SL2]

L e t us r e f e r t o

[APl;BH1-3;BWl:DLl;DXl;JRl;

and we s i m p l y g i v e a few h e u r i s t i c ideas here

Thus l e t C = compact RS o f genus g, P+- two p o i n t s o n There e x i s t s a

f o l l o w i n g [ KR16,18].

w i t h holomorphic c o o r d i n a t e s z*(Q), s a t i s f y i n s z+(P+) = 0.

(me) ( l ) - i t has s i m p l e p o l e s a t w i t h r e s i d u e +1 and i s holomorphic on c - {P+ u P-1 ( 2 ) Rek(z) i s s i n g l e

unique d i f f e r e n t i a l dk w i t h t h e p r o p e r t i e s P+ valued on c ( i . e .

Set

a l l p e r i o d s o f dk on C t i m e ) and CT = {z;

{P+

U

T(Z) =

as T

C;,C’!)

J

-f

imY

with

( A ,

(ern)

~ ( z )=

P-1 a r e p u r e l y i m a g i n a r y ) .

TI

small c i r c l e s around This z- near P-. i i corresponds t o a one s t r i n g s i t u a t i o n ; f o r more s t r i n g s one uses (C,P+,P-, P,

Rek(z)

-

dk = dz+/z+ near P,

f o r s u i t a b l e Ci,C’!. J

%

and dk = -dz-/

Consider t e n s o r s o f w e i g h t X on Z ( f = f ( z ) d z

x

KRICEVER NOVIKOV ALGEBRAS

295

l o c a l l y ) w i t h transformation law (&*) f ( z ) +. f(z(w))(dz/dw)'. Let S(A,g) = 4g - A(g-1) and then except f o r c e r t a i n special cases, f o r any A and integral n t 4g t h e r e e x i s t s ( u p t o c o n s t a n t s ) a unique f i such t h a t ( 1 ) f i i s holomorphic on C except a t P where i t has possible poles of f i n i t e order ( 2 ) near ,P,- f,,A = cz;+ n - S ( A y g ) (tl O(z+))dz:. Let M, be t h e space of meromorphic tensors on C w i t h poles only a t P+ and weight A . There is a s c a l a r A 1-A A 1-A and a f t e r s u i t a b l e choice o f conproduct (&A) ( f , g ) = (1/2ri)q f g x 1-A Z s t a n t s (fn,f-, ) = 1 5 ~ ~Further .

(Fourier-Laurent expansion). A f n to BA functions.

The proof can be based on connections of the

There i s a l s o an almost graded s t r u c t u r e r e l a t i v e to the mu1 tip1 i c a t i o n

and (except f o r special c a s e s ) t h e bracket (go h

(19.17)

[en.fml

=

1I k llgo

=

2g/2)

-1 n t m - k ; en = f n

RA,k f A

nm

Here an almost N graded algebra L o r module M has t h e form L

1 Mi

=

1 Li

or M

=

with

(19.18)

'

L.L. c

1I k l 5 N

Litj-k;

'

L.M. c

1IklLN

Mj+i-k

There a r e natural analogues o f Heisenberg and Virasoro algebras f o r example. One can now do a l o t w i t h this machinery i n terms o f s t r i n g s a n d CFT; we r e f e r t o the references c i t e d f o r d e t a i l s . The s u b j e c t o f strings and s u p e r s t r i n g s i s s t i l l i n a s t a g e o f development and we make no attempt to cover a l l o f the mathemat i c s o r physics (some of which we a r e s t i l l l e a r n i n g ) . However i t i s poss i b l e t o give a few basic ideas and i n p a r t i c u l a r to connect some o f the mathematical technique w i t h " s o l i t o n mathematics" (meaning mainly tau funct i o n s , vertex o p e r a t o r s , Grassmnnians, Hirota equations, e t c . ) . For d i s 20. R€RIARK$ 8N 8CRZNG8.

296

ROBERT CARROLL

general d i s c u s s i o n o f s t r i n g s l e t us mention GAD1 ;KA1 ;ML2;LSl ; O r 1 ;PK1; PV1 ;RJ1 ;SIE1

RrmARK 20.1.

BAG1 ;BAR1 ;BAR1 ;BK1 ;GF1,2;BO1;

1.

BACKGROWD FOR F3E 8090NZC .HRRZNC,

I

GF1,2;BK1;KA1;ML1;PKl;PV1;SIEl

f o r “ c l a s s i c a l ” ideas, and will update mat-

t e r s l a t e r i n several d i r e c t i o n s .

We c o n s i d e r 1 i t e r a l l y s t r i n g s , sweeping

W,

o u t w o r l d sheets W, w i t h c o o r d i n a t e s ( u , T ) o n

(IJ = 1,.

..,d)

a r e maps W

+

%

U)

point

a/aU,

%

-

%

a / a ~ ; X’(U,~)

d-dimensional Minkowski space ( w i t h m e t r i c II

r aB

d i a g ( ( - l y l y . . . y l ) ) y and T i s a s t r i n g t e n s i o n .

u’

We e x t r a c t here from [ LS1;

W

i s t h e induced m e t r i c o n

r

with det

=

aax’aBxv~pv

=

’V

(ao

%

7,

< 0 ( s i g n i f y i n g t h a t a t each

W has 1 t i m e l i k e and 1 space l i k e t a n g e n t v e c t o r ) .

The Nambu-Goto

(NG) a c t i o n i s t h e n

One chooses 0 5 strings.

0;,

u

5 : where 2 =

IT

(resp.

=

271)

one o b t a i n s t h e n ( L

(20.2)

aTaL/a?

%

+ aUa~/ax” =

o

= 0 at u = 0 , ~ f o r open s t r i n g s (no momentum f l o w s o f f t h e

w i t h aL/aX’’

ends) and X’(U+~IT)

= X’(U)

f o r closed strings.

Another approach i s through t h e Polyakov a c t i o n . ha,(o,T)

f o r open ( r e s p . c l o s e d )

w i t h GX(o,r) a r b i t r a r y a t t h e ends u = -T((i-X’)‘ - i2X I 2 ) )

Keeping ~ X ’ ( T ~ ) = 6X’(.cl)

on

W

with h = -det((h

aB

One i n t r o d u c e s a m e t r i c

) ) and c o n s i d e r s an a c t i o n

T h i s g e n e r a l i z e s a l s o t o a curved background m e t r i c g ’V

The energy momentum t e n s o r TaB = -(1/Th4)6S/6haB tem t o changes i n t h e m e t r i c and u s i n g 6 h = -h

aB

(X) i n p l a c e o f

rl ’V.

i s t h e response o f t h e sys(6haB)h one o b t a i n s ( i n d e x

l o w e r i n g and r a i s i n g conventions a r e i n d i c a t e d i n Appendix A ( c f . a l s o [ CS11 ) and we w i l l n o t d w e l l o n t h i s )

The e q u a t i o n s o f m o t i o n a r e t h e n (*) TaB = 0 ( a l o n g w i t h (1/hL’)aa(h4h“%,X’)

STR I NGS

297

= 0 which corresponds t o a minimal area e q u a t i o n

-

c f . below).

Indeed one

checks ( e x e r c i s e ) from t h i s t h a t (A) VaTaB = 0 ( c o n s e r v a t i o n o f energy-momentum) and det(aaXIJaBXIJ) = %h(h%

a X')*

X

l a s t e q u a t i o n i n t o Sp one sees t h a t S p

aB

fix'

= SaaaX',

aa(c[Yh4) (5' stants).

= 2AhaB, fix'

6haB = SYayhaB

+

Putting the

SNG. One asks here t h a t Sp be i n -

5

= atX" + b'

v a r i a n t under Poincare/ motions: fix' under Weyl r e s c a l i n g : fih

( f r o m TaB = 0 ) .

Y'fi

(apv = -a

VlJ

) w i t h 6haB = 0;

= 0; and under r e p a r a m e t r i z a t i o n :

+ vBc,,

aaEYhyB + aBSYhav = vacB

and &h'

=

and A a r e a r b i t r a r y f u n c t i o n s o f ( a , ~ ) and apv, b' a r e con-

The Weyl i n v a r i a n c e i m p l i e s t h a t T i s t r a c e l e s s , i . e . haBT = aB aB use (20.4) d i r e c t l y , Tab = 0 i s n o t i n v o l v e d ) .

-

0 (exercise

Going now t o conformal c o o r d i n a t e s as i n 517 we use a d i f f e r e n t n o t a t i o n ,

I);

common i n s t r i n g t h e o r y ( c f . [ G f l ,2;LS1

t h e i d e n t f i c a t i o n s can be e a s i l y

o r g a n i z e d b u t i t w i l l be w o r t h w h i l e t o be exposed t o n o t a t i o n i n common use. 2 2 2 Thus one can t a k e a' = T?U (conformal gauge), ds = -d.r t da ) = -do+da-, haB = naB,

%(a

?

n+-

= -%,

= rl-+

aa), e t c .

rl

+-

=

n-+ - -2,

= Q-- =

T,++

++

= TI

--

= 0,

a -+

=

Then Sp t a k e s t h e form

Sp = -%T/d2a~aBaaX'agX" = $ T / d 2 u ( i 2

(20.5)

n

-

X ' 2 ) = 2 T l d 2 a 8 + X 3-X

F o l l o w i n g s t a n d a r d v a r i a t i o n a l procedures ( e x e r c i s e ) one o b t a i n s t h e n

( a T2 - ac)X 2 l J = 4a+ a X'

(20.6)

with

(0)

X'(O+~T) = X'(IJ)

t i o n i s (6) X'(U,T)

= 0

'I

( c l o s e d ) and X '

= X:(o-)

+

X[(u+)

a=o,71

= 0 (open).

A general s o l u -

( r i g h t and l e f t moving modes).

Fur-

t h e r f o r v a n i s h i n g o f t h e EM t e n s o r : To, = Tol = L , ( X - X ' ) = 0 and Too = Tll '2 2 = %(X + X ' ) = 0 i m p l i e s %(i? X ' ) ' = 0 and i n l i g h t cone c o o r d i n a t e s one has Tt+ = %a+Xa+X = 0 = T-- = %a-Xa-X

--

and T

= %(Too

-

To,)

-

vaTaB = 0 (EM c o n s e r v a t i o n ) becomes a-T+, s i m p l y (+) a-T++ = a+T--

= 0).

!>+

= %(Too + To,) -2 T h i s i m p l i e s XR = XL = 0 and

w i t h T+- = T-+ = 0

c f . Remark 17.1).

+

a+TqS = a+T-- + a-T+- = 0 ( o r

--

Thus T++ = T++(a ) and T

= T-- (u-) and

(+) i m p l i e s t h e r e e x i s t s an i n f i n i t e number o f conserved charges. f o r any

+ f(a )

served ( n o t e

one has a-(fT++) = 0 so ( =n.) Qf = 2 T f t d a f ( a

aTQf = 2 T I d a a T ( f T + + )

= 2TI:

0 u s i n g s u i t a b l e boundary c o n d i t i o n s ) .

+

+ )T++(o )

Indeed i s con-

ir

( 2 3 - + aa)(fT++)do = 2T(fT++)lo =

298

ROBERT CARROLL

The H a m i l t o n i a n i n t h e conformal gauge w i l l be ._

(20.7)

H =

lo do(h-L) 0

w

= %Ti‘

0

where t h e canonical momentum i s (20.8)

{X’(~,T),X~(~’,T)}

d. ,

do(?

n’

t

X I 2 ) = Tl‘

dO((a+X)’

0

= aL/ai’

+

2

(a-x) )

The Poisson b r a c k e t s a r e

= Ti’.

= I ~ ’ ( o , ~ ) , ~ ~,T)} ( d = 0;

{X’(o,-r),i’(o;

t)> = (l/T)rl”S(o-o’)

T h i s should be regarded as a s t i p u l a t i o n here; f o r c a l c u l a t i o n s see 517 and see 86 f o r r e l a t e d formulas. t

,.

kets {f,g}

=

Jf

((6f/6X)(63/6II)

N

= 2TJf

Q

f f+a+X.

As a consequence one has e.g.

2 do

f++(a,X)

= TJ:

Then from (**),,

-

(6f/6II)(6g/6X))du

+ X I ) ) 2do, and

f,(+(i

f o r 6Qf/611 = (1/T)6Q/6i, 6Qf/611 = % f + ( i

f

X’) =

SX(o)/SX(ol) = 6 ( o - o ’ ) ( i m p l i c i t i n ( 2 0 . 8 ) ) and we

+

A,

have { Q f , X ( o ) I = -J:

(**) CQfyX(o)>

t h i s i s c o n s t r u c t e d w i t h o p e r a t o r Poisson brac-

Note e.g.

= - f ( o )a+x(o).

ftatX6(o-o’)do’

( f t = f ( o ) and sX/6n = 0).

= -f+a,X(o)

P o i n c a r 6 i n v a r i a n c e g i v e s two conserved

c u r r e n t s , t h e EM c u r r e n t Pa = - T lJ

a$

b a$ a X and t h e a n g u l a r momentum c u r r e n t Ja = -Th2h (X a X - XvagXv) = 8lJ n4 u Vr-4 ’ B V X Pa - X Pa. Thus P = :1 doa X and J do(X aTX, - X a X ) a r e con’ V v u T ’ EV = TJf 2 5’ 2 V T ’ ? served. To see t h i s n o t e e.g. a P = J , doaTXu = 10 dua X = a X 1 0 = 0.

h2h

’

o!J

T ’

(0SCZttAE0R EXPAWZ0NS AND VZRASi0R0 0PERAC0Rd)-

REMARK 20.2

o s c i l l a t o r expansions and f o r t h e c l o s e d s t r i n g X ’ ( ~ , T )

0 ’

One goes now t o

= X R ( ~ - o )+ XL(’+u)

with b

X R ( ~ - u )= $xu + ( 1 / 4 1 ~ T ) p ’ ( ~ - o ) + ( i / ( 4 1 ~ T ) ~ )ane l n=O

(20.9)

Here x’ x’, =

’L

c e n t e r o f mass p o s i t i o n f o r

p’ a r e r e a l ;

Cr’

0

,a!

= (1/41~T)+p’.

m

$l-,(a-nag

t

= exp(imo-).

C1-nCLn)

T

= 0; p’

= (a:)+ and Gyn = ( E : ) + ;

‘L

-in(?-o)

/n;

c e n t e r o f mass momentum;

and g e n e r a l l y one d e f i n e s a :

The H a m i l t o n i a n i n terms o f o s c i l l a t o r s i s

(*A)

H =

f o r m a l l y and f o r conserved q u a n t i t i e s one chooses f,,,(o’)

The V i r a s o r o o p e r a t o r s a r e d e f i n e d t h e n v i a ( c f . (*@)17)

OSCILLATOR EXPANSIONS

Cm =

(20.10)

2TfndueimUT++

= TfndoeimU(a+X)2

=

-

t

299

$1Um-nan -

-t

I t f o l l o w s immediately t h a t Ln = L-n and Ln = L-n.

These o p e r a t o r s s a t i s f y

and we g i v e t h e q u a n t i z e d v e r s i o n below ( c f . (17.13)).

Note f o r m a l l y (as

i n d i c a t e d a f t e r (20.8)), 6Lm/6n = exp(-imo)a-X, w h i l e f o r 6Lm/6X c o n s i d e r Xl)(-aU6X) = -Tf"duexp(-imo)a-X(au6X) = T/o2 'do T1,2"daexp(-imo)(2)4(~

-

(exp(-irno)a-X)GX. Then f o r m a l l y {L,,L,> = /o do(Ta&exp(-imo)a X ) e x p ( - i n o ) a - X - Texp(-ima)a-Xau(exp(-ino)a-X)) = T(f'(-im + in)exp(-i(m+&r)(a-X) 2 do 2T

= i(n-m)Lnm

= -{LnyLm} which agrees w i t h (20.11).

-- (u),T -- ( u ' ) I

(20.11')

= -[l/2T)(T

IT

T-- ( a ' ) > = 0; IT++(o),T++(a')> For t h e open s t r i n g see e.g.

-- (u) + T-- (O'))aU6(U-o');

{T++(u),

+ T++(U~))~~~(CJ-CI~)

= (1/2T)(T++(u)

[GF1,2;LS1

One checks a l s o t h a t

I.

For t h e " f i r s t q u a n t i z a t i o n " o f t h e ( c l o s e d ) bosonic s t r i n g one r e p l a c e s

, I so

C , 1 by ( l / i ) [ [X'(U,T),X"(U',T)

(20.12)

t

1 =

X',~'I

Rescal i n g w i t h a:

(20.8)

-+

(*@I [ X p " ( a Y ~ ) , X " ( o ' , ~1) =

= 0 = [ X ~ ( ~ , T ) , X " ( ~ ~ ,I. T ) S i m i l a r l y t h e am,am s a t i s f y v I -p" a m y a nI

in'"';

= aI/m4,

+a:

v = 0; [a;.a,

= aYm/m'

- p " v -

I =[am!zn I -

s p e c i f y t h i s one expresses i t as an e i g e n s t a t e

)

"p"

10, p 5

p'110, pp")

(^pp"

J =n,,,,6,

p"V

.

(m > 0 ) and i n o r d e r t o

o f p',

w r i t t e n 10,p').

means p' as an o p e r a t o r ) and a;lO,p') 0

'

I-lV

m6 m+n

(m > 0 ) one has [ a i , a i

One d e f i n e s a ground s t a t e as one a n n i h i l a t e d by a: (*6

(i/T)npv6(o-o'),

Thus

= 0 (m > 0 )

0

I n t h i s s i t u a t i o n s i n c e no' = -1 one has I a m , a - m l = Cai,a'J = -m and s t a t e s ao IO),m > 0, s a t i s f y (*+I (0la:a~,,10) = - n i 0 1 0 ) ( i . e . ( a a'+) = -m; c f . -m m -m here Remarks20.1, 14.1 and 517 f o r d i s c u s s i o n o f normal o r d e r ) . These a r e

al

n e g a t i v e norm s t a t e s c a l l e d ghosts and w i l l o n l y be discussed below b r i e f l y i n v a r i o u s places.

Now :am@-,,: = a-,,am (m > 0 ) which corresponds t o p u t t i n g

c r e a t i o n o p e r a t o r s t o t h e l e f t o f a n n i h i l a t i o n o p e r a t o r s ( n o t e here t h e r u l e

:w1w2:

= w1w2

-

( w w ) g i v e s :ao a o * = 1 2 -m m' corresponds t o while :a a . = m -m* a-mam

ao

:ai;!,m

a'

0

0

w i t h (a-mam)= 0 f o r m > 0 = a0ao m -m

- ( a oma -mo )

= aoao

m -m

+

m

300

ROBERT CARROLL

= ao a o * t h u s no minus s i g n a r i s e s a s i n t h e case o f fermion operators

-m m y

-

cf.

Going back t o " c l a s s i c a l " QFT ( c f . iBM1;LDl;RRl I ) one defines

Remark 20.1).

propagators f o r f i e l d s X'(U,T)

via

(20.13) 4 X ' ( U , T ) X ~ ( C I ' , T ' )=) T ( X p ( ~ , ~ ) X " ( ~ ' ,- ~N' )( X ) '(U,T)X"(U'~T')) where T (resp. N ) r e f e r s t o time ( r e s p . normal) ordering. Hence ordering of this type p u t s e a r l i e r terms to t h e r i g h t . There i s a whole philosophy here of Green's functions Feynman propagators and graphs, Dyson-Wick expans i o n s , path i n t e g r a l s , generating functional s, vacuum expectation values of time ordered products, e t c . which we will n o t t r y t o reproduce here, b u t recommend a s important background. As a c l a s s i c a l example of (20.13) the 4 4 2 2 Feynman propagator DF(x) = - ( i / ( h ) )J exp(ik-x)d k / ( k + (m-iE) ) w i t h k - x * a 2 ' 2 2 4 3 = k-r - k t, k4 = i k k = k - ko, and d k = d kdko i s defined by D F ( x ) = 0

0'

$ ( x x O ) = T ( @ ( x ) @ ( O ) )- : @ ( x ) @ ( O ) : a(x) =

1 cakexp(ik-r -

now :p"xp:

=

x'pv

( @ ( x ) @ ( O )where ) $ ( x ) = a ( x ) + .+(XI, I = 6' k , Ak l , e t c . ( c f . [ L D Q ) . One defines

Q

-1

2

iwt), [ a ; , a i l w i t h ~ ' 1 0 ) = 0 and ( z , z )

'L

(e

ei(T+u)

i(T-0) 3

).

For

T >

T I

i t follows t h a t (20.14)

(

X ~ ( C I , T ) X ~ ( U ' ,=T ~' )a )' q ' " 1 0 g Z - +a'q'"log(Z-z');

TI)) = %al''"lOgZ (

-

&'l)'"lOg(Z-Z');

(

(

X~(U,T)X~(U~,

x ~ ( U , T ) x ~ ( C I ' , T ' =) ) -~C"'~'"lOgZ;

X'(r3,T)X"(U',T')~ = -+a'rl'v(log(z-z')

a "slope parameter" and

+ log(z-2'))

2 f o r t h e closed string i n a natural choice of u n i t s . The quantized L, a r e defined via (*.) Lm = $Im, 2 is replaced bv Lo - a with a t o be d e t e r :a ; m-n a n '* and Lo = +a, + 1 mined. The algebra VIR i s then determined by ( e x e r c i s e - c f . [ LS1 ] and (17.13) - note a s i m i l a r expression holds f o r In) Here

a'

= (2rT)-l i s

(20.15) REMARK 20.3

a' =

[ L m , L n l = (m-n)Lmtn + (d/l2)m(m 2 - ~ ) C Y ~ + ~ (FREE FERml0N 0PERAC0W).

We begin w i t h [ GB1 ;KClY2;SAT1-6;S01 I

and will t r a n s l a t e some of t h e f r e e fermion formulas o f 114 i n t o s t r i n g l a n guage ( f u r t h e r string background will be g i v e n a s we go along to make t h e language meaningful and a p p r o p r i a t e ) .

One defines $n,$: a s in Remark 14.3

FREE FERMION OPERATORS

1;

with H(t) =

tn

$Glvac) = (vacl$, (-H(X)), $(z) =

301

1 $m$i+n = 1 t n A n and $ [ v a c ) = (vacI$; n = 0 (n

= 0 (n

0) with

0). D e f i n e now as i n 114,$n(x) = exp(H(x))$,,exp $,,zn, $*(z) = +;z -n , T ( x ) = ( g ( x ) ) = ( O l e H ( X ) g / O ) , e t c . 9 so t h a t (14.49) h o l d s and a l l t h e o t h e r formulas i n Remark 14.3. The Fubi-

1

1

ni-Veneziano c o o r d i n a t e s o f an open bosonic s t r i n g a r e g i v e n b y a c o l l e c t i o n o f harmonic o s c i l l a t o r s and t h e c e n t e r o f mass c o o r d i n a t e ( a K l 0 ) = ~ ’ 1 0 ) = 0) (20.16)

X’(z)

= xu-ip’logz = ig’”’

where [x’,pvl

+

1”1

+

and [a:,aif7

attzn)/n’

= 6mng’v

( w i t h o t h e r commutators = 0 ) .

T h i s a r i s e s from (20.9) q u a n t i z e d ( c f . a l s o (20.19) below).

A c t u a l l y we can

t a k e here ( c f . [ SAT1 I ) (20.17)

a:

= (i/n’)l:m$m$i+n;

with H(t) =

-ily

index, e.g.

1-1

’a:

= -(i/n’)lrw$m$i-n

L

tnan/n2 (dropping the u index f o r s i m p l i c i t y

5 26).

-

LI i s a w o r l d

Note here (20.9) i n v o l v e s a c l o s e d s t r i n g and t h e c o r -

responding formula f o r an open s t r i n g i s i n f a c t X’(O,T)

(20.18)

Q’”’.

+ (l/aT)p”.r

+ (i/(rT)’)l

cine - i n T Cosna/n nSO

Again {cx$cxil = -imn*”G m+n and {x’,pVl T h i s does n o t appear t o be o f t h e same form as (20.16) b u t s h o u l d be

(satisfying XI’ =

= x’

= 0 a t cr = 0 and IT).

-

convertible (exercise

c f . [ KCZ])

Note i n (20.9) w i t h z,T as i n (20.14),

4 r T = 1 a g a i n f o r convenience, and 1-1 o m i t t e d (20.19)

x,(z)

= +x

-

iplogz

+

il

cinz - n /n;

x,(z)

= 4x-iplogT+i1;

Z-”/n

n*O

niO Since from (20.12) l a ~ , [ x ~ , , J= n6,--,, and -iayn/nL2

%

a’:

we can i n any e v e n t r e g a r d ici’/n4 % a’” n n and (20.16) as ( w i t h +x % x ) t h e holomorphic p a r t f o r a

closed s t r i n g . Now i n 514, f o l l o w i n g [ 011, i f one extends g(V,V*) 18) one can extend t h e a c t i o n o f

t o a l a r g e r a l g e b r a (A i n

a s f o l l o w s (; ; i s used here t o

a v o i d an i n f i n i t y i n t h e vacuum e x p e c t a t i o n v a l u e s i n First recall

‘1 CmnEmn;

(A*)

cmn

=

xw e t c . -

c f . 58).

1 = 1 $jJjm6k_? ( = $m 6 kn ) and one d e f i n e s 91” = 0 f o r (in-nl > > O ) ( i . e . g l m x, i n 18). Hence g l m = { l c m n [ :$, ,lm~l~:,$~

-

N

302

ROBERT CARROLL cmn = 0 f o r Im-nl > > O } ( n o t e Emn = ( ( c S ~ , , , ~ ~ , , and ) ) r e c a l l Emnvk =

:$,(I;:

6

$k and ad:$,,$;:

nk v m i n 18 so t h i n k o f vk

S e t t i n g Y+(n) = 1 ( n 2 0 ) and = 0

t ( $ $*)).

$:;

-

Emn

m n

n o t e a l s o $+ ,;

= :$,,,

f o r n < 0, one has then

( c f . 58) (20.20)

[:$

$*:,:I)

m n

n ( Y t = Y,(n)).

(A,

=

X,

rn

= 6nm':$

n

--

6mn':$

One w r i t e s x(p,q)

so t h a t X(p)X*(q)

( 0 ) one f i n d s ( e x e r c i s e

-

$*:

m n

+ 6nm'6mn'(Yt

= exp(c(x,p)

= (l/l-q/p)X(p,q)

m Y+)

-

-

c(x,q))

exp(-c(T,p-') + For a E A

( c f . Remark 14.3).

-

and ( B * ) ~ ~r e c a l l i: a l v a c )

cf.

n

8 C-1 i s d e f i n e d a s i n 18

The c e n t r a l e x t e n s i o n g l ( m ) =

@ cc).

<($,q-'))

$*

m n"

4.1( m l

eH(x)a lvac )zm)

Hence s i n c e ( $ ( p ) $ * ( q ) ) = q / ( p - q ) < 0 ) t h e a c t i o n o f :$(p)$*(q):

(exercise

-

r e c a l l ( $ $*)

= 6

V I J

PV

f o r P,V

on A ( 0 ) l v a c ) i s g i v e n by ( c f . (8.23) w i t h

s u i t a b l e adjustments and i n t e r p e r t a t i o n ) ( b )

-

(q/(p-q))(X(p,q)

1 ).

To com-

p l e t e t h e correspondence w i t h 18 one d e f i n e s d i f f e r e n t i a l o p e r a t o r s Zmn v i a (20.22)

-

(q/(p-q))(X(p,q)

When cmn = 0 f o r Im-nl>>O,

1) =

1 ZmnPmq-n

1 cmnZmn a c t s

on C [ x l and i d e n t i f y i n g :$Im$;:

with

Zmn one o b t a i n s a r e p r e s e n t a t i o n o f g l W o n C [ x 1 i n terms o f d i f f e r e n t i a l operators.

RRnARK 20.4 (REmARKSI. BN N0RIIIAI; 0RDERING). t h e : : n o t a t i o n and normal o r d e r .

A A(V)

%

A (W ) = A(V*)-A(V),

term o f a

A(W).

E

w1w2 = :w1w2:

w i t h $,,$:

satisfying

v v

O), $;lvac) (

O I $ n$*lo) m

A -+ :A:,

:1: = 1, and < a > i s t h e zero degree

Here : : i s t h o u g h t o f as t h e map A ( W )

w1 = :wl:, (JI*J, ) = 6

L e t us comment here f u r t h e r on

Thus i n (14.9) one works i n A ( W ) = A(V*)

t(w1w2),

[$,,,,$;It

=

e t c . (cf.

(14.9)).

[$m$~y$ml$:l

1

-+

A(W) above w i t h

I n 514 we a l s o d e a l t = 6nml$m$~l

-

6mnl$ml$:,

2 0), (~l,$;) = 6 ( P , V < 01, $,lvac) = 0 = (vacI+; ( n PV (n L O ) , ( ( I ~ , I J I ; ) =. , ,6 e t c . Note from (14.41) = 0 for n 2 0 or m 0 and (OIIJI;$,~O) = 0 f o r n < 0 o r m < 0.

PV

(U,V

= 0 = (vacl$In

<

NORMAL ORDERING

303

Also f o r $ $* = :$ $*: + ( $ $*) one has ad:$ $*:(qk) = $m6nk so ad:$,$;: m n inn m n m n Basis vectors f o r Alvac) c o n s i s t of ...$* $ ...$ Ivac) f o r ml

. Em n

... < ... >

$1,

...

nK

%

<

nl

0 5 nk < < n l ; s i m i l a r l y ( v a c l $ * ...$* ...$ with ml > ml mp nh n, m > 0 > n k > ... > n1 would be a basis of (vaclA. Then e.g. formally P( ( vacl$;,$,lvac)) = (vacl$*$ Ivac) = 6mn f o r m,n 2 O and some normalizing m n constant w i t h ( 8 . 1 2 ) f o r example would be appropriate. Alternatively take basis vectors $ ...$v,$t-, . .$;-"I 0 ) ( v ~ -> ~ ... > 0 > v - ~> ... > v -m ) Vh-I and (019 . . . V J ~ - , ~ I ~ * . . . $ ; ~ -( ,v - ~< < v - ~5 0 < ... < v n - , ) . One V-m w r i t e s W = V fil V* (with Qn generating V and $; generating V*) or W = Wcr fil W a n where U* = W t &20C$n ( c r e a t i o n o p e r a t o r s ) and U = Wan = c r = &
P

<

...

Cn
n

and F* = F

= WcrA(W)\ A(W).

I t i s perhaps worth noting here t h a t s p i n module f o r A(W) based on W = Wcr Wan; note t h e C l i f f o r d algebra

A(W) can be t h o u g h t o f a s a r i s i n g from W = V 8 V* or W = Wcr

fil Wan ( s e e be-

low). Let us give some embellishment to (14.9). Thus ( 1 ) = 1 , ($.I)%) = 6 i j 1 J (I)?$.)= 1 i f i = j < 0 and = 0 otherwise. T h i s says ( $ . $ * ) = 1 f o r i = j J 1 1 J < 0 and($*$.) = 1 f o r i = j 2 0 a s s t a t e d e a r l i e r . Generally u s i n g Wick's J 1

theorem (20.23)

(

w l . . . w r ) = 0 ( r odd),

( r even; o ( 1 ) < a ( 3 ) Now t h e s p e c i f i c a t i o n

<

... <

=

1 s g n d w u ( 1 )wo(2)).*.(

a(r-I), a(1)

-

<

'o(r-1 ) w a ( r ) )

u ( 2 ) ,....a ( r - 1 ) < a ( r ) )

m n causes some concern s i n c e normal order i n physics p u t s c r e a t i o n o p e r a t o r s t o t h e l e f t of a n n i h i l a t i o n seems seno p e r a t o r s . T h u s f o r A ( W ) = A ( U * ) - A ( U ) t h e map A + : A : of s i b l e b u t not perhaps f o r A(W) = A(V*).A(V). From ( @ ) we deduce f o r m = n , : q ~ ~ $ ; : = qm$;, b u t f o r m < 0 a n d n < 0 this has t h e wrong order s i n c e $, E U and q ~ ; E U*. However then of course from $$,; = -$*$ we simply p u t i t i n n m the c o r r e c t o r d e r . Note a l s o f o r m = n < 0, :$,$;: A = $$ ,*, 1 = -$,*$,,,. l i t t l e thought shows t h a t due t o C , I + r e l a t i o n s i t r e a l l y doesn't matter whether we construct A ( W ) from (V,V*), (V*,V), ( U , U * ) , o r ( U * , U ) ; i f we use U , U * we would have the c l a s s i c a l normal order : : o f physics and t h a t is what we will t h i n k o f a s defining : : ( c f . a l s o [GCl;KZ;LOl I f o r information on Clifford algebras and normal o r d e r i n g ) . (A@) :$,$;:

= $$ ,;

( $ $*)

-

3 04

ROBERT CARRCLL

RRRARK 20.5 (V€RC€X OP€RA&ORS( AND C0illl!tllCA&0RS(). Going back to strings the vertex o p e r a t o r of a bosonic string w i t h momentum k ( o r kA) is given a s (A&) V ( k , z ) = :exp(ikX(z)): where X(z) appears i n (20.16) ( t h i s is discussed below). Also h i s t o r i c a l l y t h e r e is some i n t e r e s t i n expressing X ( z ) a s an i n t e g r a l ( s e e a l s o (**) below) (20.24 1

X(z

1

=

$

(1 / 2 ~ ) (dw/w )$(w)$*(w)l og( (w-pz )/ (z-pw)

Ip=l

( c f . [ KCly2;SAT1-41). In o r d e r t o get c o r r e c t answers here f o r various commutators one makes a change o f v a r i a b l e s and removes an i n f i n i t e term ( c f . [SATl,31); this i s r a t h e r ad hoc mathematically, b u t customary i n physics, and could presumably be avoided by r e c a s t i n g t h e algebra a s i n 58 w i t h cent r a l extensions e t c . , b u t we include i t f o r i l l u s t r a t i o n of t h i s procedure (and f o r convenience). Thus write (A+) $n = 5; ( n L 0 ) ; q n = q - n ( n < 0 ) ; $; = 5, ( n 2 0); a n d $; = q r n ( n < 0). Then [5,,5,1+ = [nm,q;l+ = A,, with o t h e r commutators = 0. Also one notes t h a t (Am) 5,IO) = 0 ( n 2 0 ) ; <01q_*, = o ( n < 0) 5 < O l q * = O ( n > 0); q 10) = O ( n < 0 ) E n n l O ) = O ( n > 0); n -n ; s;zn + - I q- zn = (015; = 0 ( n L 0 ) . Note a l s o e.g. $ ( z ) = 1 $,zn = 1

1; cizn -+n 1; T-I,z-~ = F*(z) = 1 ; snz + 1; n;z” = i(z) + T(z)$ (z) + ~(Z);;*(Z) -

?r

+ { ( z ) while $ * ( z ) = 1 $ ; I ” =1 ; cnz-’+ q?,z + $;*(z). I t follows t h a t $(z)$*(z) = E * ( z ) ~ ( z )

=

-h

?(z);(z) + 1 ; 1 . The l a t t e r i n f i n i t e term corresponds to t h e energy o f t h e f i l l e d Fermi-Dirac sea and will be c a v a l i e r l y omitted. We don’t r e a l l y l i k e t o do this b u t i t i s customary i n physics and i t is worth while seeing i t a t l e a s t once (and only again a s indicated b r i e f l y a f t e r (20.28)). W i t h these provisos one o b t a i n s formally ( c f . [ S A T 1 I t h e 6 function s(w/z) = l / ( l - z / w ) + (w/z)/(l-w/z) = lIm(w/z)n a r i s e s i n t h i s computation ( c f . [ SAT1 I and ( 7 . 2 0 ) )

Now expand t h e log term i n (20.24) i n a Taylor s e r i e s to get f i r s t (cf.(20.2

(20.2))

COMMUTATORS

305

“Naive” c a l c u l a t i o n o f commutators i n (20.26) ( u s i n g t h e $,$* e x p r e s s i o n s ) y i e l d s zero f o r a l l b r a c k e t s and t h i s i s why t h e more n a t u r a l p h y s i c a l fields a r e i n t r o d u c e d . Using and (20.25) i n (20.26) one o b t a i n s e.g.

c,;

(20.27)

tyc

[ am,an

1

= -(1/2n2(mn)’)$

fp/z)/(l-w/z)2)wmz-n

(dz/z)$

= -(1/4n2(mn)’)#

-(i/Zn)(m/n)$

-

(dw/w)((z/w)/(l-z/w)‘

dz$ dwwmz-n/(w-z)2 =

- &mn

dzzm-n-l

S i m i l a r l y ( c f . [SAT1 I)[ x , p l = i, e t c .

as desired.

REmARK 20.6 (K0BA N L E U E N QARIABLU, DACllllIR EWPECCAEL0M, AND C i U FllNCCX0N$) Now go t o t h e v e r t e x o p e r a t o r V i n

q;:

(All.

By t h e d i s c u s s i o n above a b o u t :qm

we see t h a t V o p e r a t e s o n Ivac> s u i t a b l y so t h a t Vlvac) w i l l be r e p r e -

sented by a p o i n t i n UGM ( c f . t h e d i s c u s s i o n a f t e r (14.42) and [ KJl;SAT1,3,4 41). $,

We n o t e now t h a t t h e n o t a t i o n i n IKCl;SAT3,41

f o r some v a r i a t i o n s i n n o t a t i o n ) . so we i n t e r c h a n g e 6, =

ly

involves interchanging

p l u s a few o t h e r v a r i a t i o n s o n [ SATl,21 and on [ D1 1 ( c f . a l s c SO11

and $;

t n l Z$,,,$J;:+~).

with ?(t,z)

=

and $:

We want t o r e t a i n t h e p r e s e n t n o t a t i o n

in[SAT3,41 t o have t h e same H ( t ) as above ( H ( t )

Then w r i t e (**) ?(z)

-I1 tnz-n.

It f o l l o w s t h a t

=

1 $*r‘ n

(@A)

and

T(z)

=

1 $nz-n-l

exp(H(t))~(z)exp(-H(t)) =

7

Further using = p e x p ( ? ) p ( z ) and e x p ( H ( t ) ) 7 ( z ) e x p ( - H ( t ) ) = exp(-?)y(z). i n (20.16) ( s i n c e a z -+ l / z change i s i n v o l v e d i m p l i c i t l y ) and = -a+, m m n = -a one o b t a i n s n’

zt

(20.28)

1 $$,-;,.

-

an = -(i/n’)l

= (1/2nn4)g

$$ ,+ :,

= -(1/2m’)$

dzy(z)p(z)zn; u

p = -(1/2n)$

( c f . (20.26)

-

d z T ( z ) p ( z ) z - n ; =: ;

x = (1/2a)+

L

(i/nz)

dz$(z)?(z)logz;

dz$(z)p(z)

t h e power l / z i s absent here i n t h e i n t e g r a l s ) .

With care

r e g a r d i n g t h e i n f i n i t e t e r m as above one o b t a i n s t h e c o r r e c t commutation

3 06

ROBERT CARROLL

r e l a t i o n s (20.27) (as ,:'I iplogz t (gnzn +:nfz-')/>

=

1;

T(w)p(w)dw.

I n t h i s context u i k X ( z ) . = eikXt(z) eikX -(z); N

N

(20.29)

Y

[ x , ' j j l = i. Thus ( 0 0 ) X -,X = x t = (1/2s)P dw$plog((w-pz)/(z-pw))lp=l % i fZ

$,,,,Iand

V(k,z)

N

= :e

;r_

=

1- gnznIn%;

.+

1 N

X,

= i'i;logz

11 a n m,t

t

z-'/nJi

T h i s i s i n t h e standard s p i r i t o f p u t t i n g c r e a t i o n o p e r a t o r s l e f t o f a n n i h i l a t i o n operators ( 0 1 ~ ' = 0.

the

r e c a l l ~ ' 1 0 ) = 0 and we d i s c u s s below

I n any event t h e N - p o i n t t r e e a m p l i t u d e d e n s i t y ( 0 6 ) ( O l e x p ( H ( t ) )

...V(kN,zN)IO)

N

(a",);

(xi) t o

A0

V(kl,zl)

i s a t a u f u n c t i o n ( c f . below f o r p r o o f and comments

-

we w i l l d i s c u s s more s t r i n g t e r m i n o l o g y l a t e r and see [ BM1;LOl ;RR1 ] f o r c l a s s i c a l v e r s i o n s o f such formulas).

t ( A .J)

Note here from (8.15) and (8.19),

ciFT+j

%,aj and r ( A .) % j x w i t h L a px 3 -J I'h+ j j,'P 2 8 ) ) t h i n k o f i;i,"' % an and - i n % % nxn so [a,,~$I !L =

Thus ( c f . (20.

= pSpj.

= 6 % [ a n , ~ x P i= [ i r ~ % ~ , - i p % + ] = n6 Then i n i k X + ( z ) we w i l l have i k lE,ln. anz -n,n+ % - k l y P np' These a r e o f course zmnxn and i n i k X - ( z ) we f i n d ikl; znzn/n4 % k17 znan!n.

-

and r*(u) = oX*(u)u-' where X*(u) = y-;j, r ( u ) = um t l z e x p ( 1 Jl;l -1 u j x j ) exp(-l u-'a./j), and r*(u) = u z e x p ( - l uJxj) e x p ( 1 u - j a . / j ) . Note J J i n (14.3), (20.21 ), e t c . r,r* w i t h s u i t a b l e a d j u s t m e n t a r e s i m p l y r e f e r r e d

t h e terms i n v e r t e x o p e r a t o r s T(u) = uX(u)cr-' ( c f . (8.17) and (8.20)) X(u) =

1 u%

1

t o as X,X* e t c . and i n d e a l i n g w i t h vacuum e x p e c t a t i o n v a l u e s a r e s i d u e adj u s t m e n t i s made a s i n d i c a t e d i n [ AG1 I. Note here from ( 2 0 . 2 2 ) 1 Zmnpmq - n

-

= (q/p-q)(X(p,q)

1 ) (Zmn

%

:$,$;:)

i n v o l v e s a normal o r d e r i n g a d j u s t m e n t

and corresponds t o formulas i n [ AG1 from ( 2 0 . 2 2 ) and from (8.23), action).

and

I

(where however t h e mu1 t i p 1 i e r d i f f e r s

1 u-'v-jtij

i s used

-

A

denotes o p e r a t o r

I n any event a n adjustment i s needed t o c o u n t f e r m i o n i c charge

(which corresponds t o e n l a r g i n g t h e admissable g a c t i o n s ; g and t h i s i s accomplished v i a a d d i n g a t e r m xou operators (cf. Now H ( t ) =

i1;

0

E

rt

-

c f . §ll)

and -loguao i n t h e v e r t e x

(66) to follow).

tnan/n'

and one d e f i n e s new v a r i a b l e s (Koba-Nielsen v a r i a -

p.

b l e s ) z . v i a (a+) tn = ( l / n ) C i F.zn where t h e correspond t o momenta o f J J J J s t r i n g s i n t h e ground s t a t e ( s i m i l a r v a r i a b l e s were a l s o used i n [ D8;MWZI). D e f i n e t h e n (ern)

F(p") =

H(t)

+

i t o x where to =

1;

pj i s set = 0 a t f i r s t .

KOBA NIELSEN VARIABLES

307

u

p. p.y(zj)):. t icy FjlT

R e c a l l t h a t (vacl;: = 0, a n l v a c ) = 0, and w r i t e = ix1; zn/n4 = = *p j-X - p j ) . Hence (&*I
ic0

-a

iFjl

But

f i r m (4*).

(Olp

= 0 and one i d e n t i f i e s a o - z o

s t r i n g ( c f . (20.9) and comments t h e r e a f t e r ) .

%

(z ) = J + j = 0 t o con-

i n t h e bosonic

Here one can w r i t e e.g.

1

1.

J,

iF.X

$*$ and t h e n aolO) = ( O l a o = 0. a. = $.I)* JLO J J jovivg

N

For p

'L

-

liov;vi

a

0

'L

t h e n (&*)

a, V

A

i s used

i n a different context).

(C0NE€tC10N$ %€efDE€N .50I;1&0W A M , %IRING.5).

REIRARK 20.7

i n [ KC1,2;SAT1-4] [ SO1

Now one e s t a b l i s h e s

a c o n n e c t i o n between s o l i t o n ideas and s t r i n g s ( c f . a l s o

1) i n t h e f o l l o w i n g s p i r i t .

F i r s t r e f e r t o Remark 14.3 where we had a

d e r i v a t i o n o f the H i r o t a b i l i n e a r i d e n t i t y v i a f r e e fermion operators ( c f . (14.49)-(14.52)).

We r e c a l l n o t a t i o n here i n a capsule m

(20.30)

+(k) =

eH(x)q,(k)e-H(x)

Q; gV

= Vg;

$*(k) =

= e'(xSk)$(k);

gV* = V*g;

( n o t e eH(')

1 $;k-n;

H(x) =

eH(x)q,*(k)e-H(x)

V = BC$

.

jy

V* = QC$;l;

UGM

1;

:exp(ikX(z)):

= exp(iky+(z)) exp(ik? ( 2 ) ) .

(00)

x p l VJ,+,*~;

5

= e-SfxYk)q*(k);

%

G(V,V*)/GL(l);

a c t s t h r o u g h adH i n t h e a l g e b r a g(V,V*)).

...

'3;*(w)] v i a

1 Qkkn;

7 €G(V,V*).

l

xnkn;

G(V,V*)

v = g10)

-

By checking [ adT,S(w)] and [ad?, d i r e c t l y as

Hence i t determines a t a u f u n c t i o n

A+

v i a T ( x , ~ )= (exp(H(x))V(k,z)) tau function. :exp(-i?(z)):

and t h i s procedure c o n f i r m s t h a t

We n o t e i n p a s s i n g t h a t

=

Now c o n s i d e r V(k,z)

(compute a s i n (20.27) e t c . ) o r v i a ad(:$,,,$;:)

i n (20.20) one f i n d s t h a t

=

p(z)

= :exp(iz(z)):

(*(I)

is a

and T ( z ) =

i n t h e s p i r i t o f t h e boson ferrnion correspondence ( c f . [ K2;

=

308

ROBERT CARROLL

KCly2;SAT3,41,

take (&*) and (20.29))

c

-

(INi = Jl v(Fi,zi)

1 p.J

= 0, e t c . ) .

Now w i t h to = 0, );(;

I).

c f . [KClY2;SAT2,4;S01

t i n u o u s l y d i s t r i b u t e d momentum ( 0 5

CI

One notes t h a t i f

5 2 ) between z

V ( p, z ) = :exp( iEf( z ) ) : becomes (&* ) :exp( ( ;/2r)/,2’da?(

-

d

= H ( t ) and (see

u

roi s a con-

= fo and z = fZnt h e n

f,

)Fu) : which

becomes

z = exp(ia)

for

which l e a d s t o v a r i o u s general v e r t e x expressions ( c f . [ SAT1 I ) .

I n any ev-

e n t one sees t h a t s t r i n g a m p l i t u e d e n s i t i e s l i k e ( 0 6 ) e v o l v e a s t a u functions.

zl)

More p r e c i s e l y one l o o k s a t I = ( O ~ i l ~ ( ~ i , z i ) ~ O )and t h i n k s o f

f o r example a s exp(H(S)) where H ( E )

( r e c a l l H(S) =

1;

E n l $,$J;+~

=

1;

iF,y-(z,)

%

Thus En

Sninkgn).

:(Fly

= i q ( x t lyTnzy/nSi) %

rlzy/n

and I can be

c

t h o u g h t o f as a t a u f u n c t i o n ‘r(tS,g),

g = niilV(ri,zi).

c u s s i o n o f t h e p h y s i c s i n [ SATl-6;SOl ;KC1,2

I

There i s more d i s -

and one can p u t a l l t h i s on a

RS (which we w i l l d i s c u s s i n 521 i n a d i f f e r e n t f o r m a t ) .

One p o i n t o f n o t e

here i s t h a t t h e s t r i n g v e r t e x o p e r a t o r s r ( k , z ) e t c . correspond t o elements

o f G(V,V*)

UGM a s a parameter space.

and thus one can use

Interpertation

o f (20.30) i n v o l v e s more p h y s i c s t h a n we c a r e t o d i s c u s s here ( c f . [ SAT1-6;

SO1

I).

The c o n n e c t i o n o f Koba N i e l s e n v a r i a b l e s t o H i r o t a b i l i n e a r d i f f e r -

e n t i a l equations can be discussed i n t h e RS c o n t e x t ( c f . [ D8;MWZl f o r t h i s t o p i c , where many formulas i n v o l v i n g f r e e f e r m i o n o p e r a t o r s a r e e x p l i c a t e d ) . The p r e s e n t a t i o n i n [ SO1

1 uses s l i g h t l y d i f f e r e n t n o t a t i o n b u t p r o v i d e s a

v a l u a b l e c o n n e c t i o n t o t h e development i n §14.

RBIkWS 20.8

(XNCR0DllCC10E CO R Z M SllRFACB AND SCR1NW).

RS techniques

l a [EG1,2;IHl

MP1 ;OG1 ;SW1 ;V1 ,P;VAl;ZBl

;SS1

] (cf. also

I).

[AG1,3,5,6;CKl;Fgl;BH1-4;KMl;MTl;

T h i s s e c t i o n i s p r i m a r i l y m o t i v a t i o n a l and

t h e m a t e r i a l w i l l be expanded and c l a r i f i e d i n 521. (20.33) with

T(z) =

{Gn,gl,=,,,,,6,

1

-

Jlnz -n-+

;*(z)

=

1

n€ZtSi

{Tn,?,1

We go now t o

Thus w r i t e ( c f . [ I H 1

1)

wn-%

n€Zt% =

{?,?In m

= 0,

TnlO)

.u

=

$:lo)

=

0 ( n > 0), e t c .

RIEMANN SURFACES AND STRINGS

309

T h u s ( r e c a l l { $ ,$*I 6mn before w i t h $,lo) = 0 f o r n < 0 and $:lo) = 0 for m-n-4 % 1 $,zn-% ( n = 0 being n 0 ) we have say $, $-, and ?(z) % 1 $-nz excluded here). We will now drop t h e % notation in (20.33) and simply use $ n , ~ : e t c . The elements g E Clifford g r o u p G can be written i n t h e form g = e x p ( k dz& dwf(z,w)A(z,w); A = 1 m, n€Z++

(20.34)

A

mn

z-n-b$-m-4

=

:QZqJ;:

($, = $ ( z ) , $; = $*(w)) where f ( z , w ) i s an a p p r o p r i a t e 4 order d i f f e r e n t i a l in z,w ( c f . here 514,5,17,19 and Appendix B - note expecially Zmn % :$,,,I)::

i n (20.22)). The c u r r e n t operator i s defined via ( b e ) J, = & E Z J n ~ - n - ’= diagonal element o f A. One t h i n k s o f UGM a s t h e G o r b i t of 10) and from g10) = g’10) f o r g = g ’ h , h = : e x p ( l amnAmn: ( n > 0 o r m < 0 ) one w r i t e s G/g % gh f o r t h e G o r b i t space, Each point i n g10) i s c a l l e d a g vacuum.

n

One defines ( 6 6 ) H(x) = l n , O ~ n J nand V(x) = exp(&,oxnz ) exp(aologz z-“an/n) (Q X ( z ) i n (14.3) modulo t h e a. term; h e r e a, i s needed t o a d j u s t

-

residues

c f . [ AG1 ] and see [ CH1 ] f o r o t h e r motivation); one s e t s a l s o V*

( z ) = V ( Z ) ~ ~ + -Then ~.

e n t i t y is ((*)

tors on

T

T(X)

= (Olexp(H(x))g10) and t h e Hirota b i l i n e a r i d -

? d z ( V ( z ) T ( x ) ) ( v * ( z ) T ( y ) )=

0.

The a c t i o n of vertex opera-

is equivalent t o t h e i n s e r t i o n of f r e e fermions so t h a t defining

( c f . (14.50)). One can a l s o recover g10) from T ( X )= (Olexp(H(x))g(O) a s follows. Writing a form o f Wick’s theorem a s ( c f . (14.42)) (20.37)

(

O l e H ( X ) + ( z).l

..$ ( z n ) i * ( w n ) . . . i * (wl ) g I o ) / . r ( x )

=

d e t ( ( O ( $ ( z i) $ * ( W j ) g l O ) / T ( X ) )

-

one has ( e x e r c i s e (20.38)

g/O)

=

c f . [ IH1;ZBl I ) T(O):exp((l/ZTi)

2

dz+-

dwf(z,w)A(z,w)): 10)

f o r f ( z , w ) = (1/T(o))v(z)v*(w)T(X) evaluated a t x = 0.

31 0

ROBERT CARROLL

One can i l l u s t r a t e a v e r s i o n o f (20.38) w i t h an example based on [ ZB1 I. To 1 10) corresponds say $: A A (so $:lo) = 0, n 2 0 ) o r I z o , z 1 =

$7 ...

H,

i n an ,H,

,...

H- decomposition f o r t h e Grassmannian GrH, H = H, B H- ( c f .

§ll). The replacement o f zm i n H, by z - ~i n H- i s t h e c r e a t i o n o f a p a r -

* $:n$mlO).

H such t h a t pr,:

Recall G r H

%

W C

i s Fredholm (assume i n d e x 0 h e r e ) and p r - : W

-+

H- i s compact

t i c l e - h o l e p a i r and

ments o f t h e C l i f f o r d group have t h e form g = :exp(c

*+

e a r l i e r and we t h i n k o f g i n t h e f o r m g = ('-+

p+-

HS).

H,

+

Ele-

as indicated

pmn$,~:):

) r e l a t i v e t o H, B H-.

p--

P

(or

W

Then i n g10) one has o n l y p-+ v i a

Now one makes a correspondence o f (20.38) d i r e c t l y t o a Grassmann o b j e c t W a s i n 111 v i a T,(x) = d e t ( 1 + a - l b A ) where A = w-w: -1 a b and y, = ( o d ) r e p r e s e n t s a f l o w group a c t i o n (% gl0)

%

W; g: H,

product o f W

%

-+

W).

Here we t h i n k o f

*

P-'

g ( 0 ) ( c f . here a l s o (20.31)).

H,

-+

H- has graph W

exp(H(x)) E

r+ -

note

A and t h e maximal e x t e r i o r

We w i l l g i v e more d i s c u s s i o n

o f t h i s p o i n t o f v i e w l a t e r , once t h e RS background i s i n p l a c e ( c f . 121). Before g o i n g t o a more s y s t e m a t i c t r e a t m e n t i n 121 l e t us remark t h a t i n a t t a c h i n g a l l t h i s t o RS one encounters a v a r i e t y o f n o t a t i o n s and p o i n t s o f view ( c f . [ AG1,6;EG2;IHl;KCl;SAT2-4;SW1,2;Vl

] f o r example).

Moreover t h e Con-

n o t a t i o n i s sometimes d i f f e r e n t from t h a t used i n §4,5 and Appendix B. s e q u e n t l y we w i l l o c c a s i o n a l l y w r i t e formulas i n d i f f e r e n t n o t a t i o n s and

make t h e a p p r o p r i a t e connections o f n o t a t i o n when t h e m a t t e r i s e s p e c i a l l y i m p o r t a n t o r t o be pursued l a t e r .

Thus one knows t h a t (see below f o r exten-

s i v e discussion) *

(20.40)

r ( x ) = ec Qmnxnxm O (

AnxnlQ)

i s t h e t a u f u n c t i o n f o r a f i n i t e dimensional ( q u a s i p e r i o d i c ) KP s o l u t i o n ( w i t h s u i t a b l e A,

Qmn d e f i n e d below) i f and o n l y i f

i s the period matrix

f o r some genus p RS S. P i c k a p o i n t q E S w i t h l o c a l c o o r d i n a t e z such t h a t -1 z ( 9 ) = 0 and t h e n wi(c) wid< % A b e l i a n d i f f e r e n t i a l s o n S n o r m a l i z e d v i a Q ,

Li w j = 6ij (1; Ainsn).

and J6: w j =

Qij

w h i l e t h e An a r i s e from l o c a l expansions wi

To d e s c r i b e t h e Qmn l e t nn

%

=

d i f f e r e n t i a l s o f t h e second k i n d

STRINGS AND RIEMANN SURFACES

31 1

w i t h zero a periods and a s i n g l e pole a t q o f order n+l ( c f . §4,5 and Appen-n d i x 8 ) ; then normalize nn a t q via (dm) Jq q n = z - 21; Qnmzm/m (2,; will be discussed l a t e r ) . Now ~ ( x given ) by (20.40) must have t h e form (Olexp ( H ( x ) ) g l O ) where g l 0 ) 1~ vacuum s t a t e f o r t h e RS S w i t h p handles. Then from t h e formula

where E is the prime form. T h i s theme will be picked up again i n §21 a f t e r some f u r t h e r ground work has been developed. Let us remark i n passing t h a t t h e magic number 26 f o r strings a r i s e s i n various ways (see e.g. [BKl;GF1,2; LS1;PKl;SIEl I). In p a r t i c u l a r i f one includes ghost f i e l d s w i t h t h e Xu and computes the Virasoro commutator [ LmyLnl= (m-n)Lm+n + A(m)%+n f o r t h e comb i n e d system, then one requirement f o r A ( m ) = 0 is d = 26. The A ( m ) i s ref e r r e d t o a s an anomaly. 21. IRORE 01 NRINGS, RlBllANN $URFACEk, AND t A l l FLINCU110N$-

We will expand now on t h e material i n Remark 20.8 q u i t e extensively i n order to c l a r i f y i t and s e t the s t a g e f o r f u r t h e r s t u d i e s . We follow here mainly [ AG6 1 a t f i r s t w i t h some modification, and r e f e r t o [AG5,61 f o r much more d e t a i l and discussion ( c f . a l s o [SWl,Z;ZBl I). We will review some notation a s we go along ( c f . §4,5 and Appendix B ) a n d r e f e r t o $12 f o r connections of algeb r a i c curves t o KP equations. Take a (compact) RS c o f genus g, with canonical homology b a s i s a , b : a i y bi ( i = 1 , . . . , g ) with ( a i , a J. ) = ( b i ' bJ. ) = 0 and in an obvious n o t a t i o n . Take w . (1 5 j 1 9 ) a s (aiyb.) = -(b a . ) = 6 3 i' J ij J holomorphic d i f f e r e n t i a l s normalized such t h a t /a.t w j = 6 i j and Jb,. w j - 'ij. R = ( ( a . .)I is symmetric w i t h p o s i t i v e imaginary p a r t . One w r i t e s ds = 1.l 2gz-,dzdZ. The holomorphic one forms determine a canonical l i n e bundle K and i r r e d u c i b l e tensors t = t ( z , z ) d z q a r e s e c t i o n s of Kq. The t r a n s i t i o n func9 t i o n s f o r Kq, q i n t e g r a l , a r e g = (df,$z6)-q ( f a 6 ( z B ) = za) and they s a t aB i s f y cocycle conditions gaBgBa = 1 and g g g = 1 . For spinors we r e c a l l a6 BY Ya 1 (of order 22g). t h a t the set of s p i n s t r u c t u r e s i s parametrized by H (C,Z,)

2

312

ROBERT CARROLL

P i c k some r e f e r e n c e s p i n s t r u c t u r e saB s t r u c t u r e has t r a n s i t i o n f u n c t i o n s The t r a n s i t i o n f u n c t i o n s s

Q

La w i t h La2 = K.

x

Any o t h e r s p i n

x

i s a Z2 valued 1 c o c y c l e . aBSaB -1 with a r e square r o o t s o f kaB = ( d f a B / d z B )

aB

where

c h o i c e o f square r o o t c o n s i s t e n t w i t h t h e c o c y c l e c o n d i t i o n . Now r e c a l l a n o t h e r p o i n t o f view v i a d i v i s o r s where d i v ( f ) = nipi

has degree

1 nii’

p h i c l i n e bundle o v e r C, Appendix

B).

T,

Zg

W r i t i n g L:,

1 nipi,

given D =

5

+

1

(8divisor -

c f . below) i n J(C)

between p o s i t i v e d i v i s o r s o f degree g and p o i n t s i n J ( C ) .

o(;)(zln)

=

1 exp(ia(f;+;)n(i&)

t 2ai(fi&)-(;+B’))

t h e v e c t o r symbol) o ( i ) ( z + fin + m ) = e x p ( - i n n - n

-

e(z

theorem says t h a t f ( P ) =

P

E

t

I n t h e l a t t e r case t h e r e e x i s t s a vec-

I

+ I ( 1 Pi)

( v e c t o r o f Riemann c o n s t a n t s ) such t h a t (*) z E

J(C) where

o

vanishes i s t h e

o

d i v i s o r and @(el,)

f o r a nice proof o f this).

= A.

= 0 i f and

-

I ( P 1 + ...+ P ) (see .g-1 Such f a c t s have been discussed elsewhere

o n l y i f t h e r e e x i s t g-1 p o i n t s Pi E C such t h a t e = A [ MU1

Riemann’s v a n i s h i n g

I(P)ln) e i t h e r vanishes i d e n t i c a l l y f o r a l l

c o r i t has e x a c t l y g zeros Pi.

The s e t o f z

and ( o m i t t i n g

2an(z+B) t 21riam)@(zln).

T h i s p r o p e r t y i n f a c t c h a r a c t e r i z e s 0 up t o a c o n s t a n t .

t o r A(Po,a,b)

1

Zg f o r t h e p e r i o d l a t t i c e , J(C) = Cg/L, and PI(D) = niJp0‘w d e f i n e s t h e Abel-Jacobi map.

Outside o f a s e t o f complex codimension 1 One d e f i n e s

D =

e q u i v a l e n c e c l a s s o f d i v i s o r s 0 (as i n §4,5 and

w i t h Po f i x e d ,

t h e r e i s a l-l*map

1 nnPi,

For 5 a holomor-

d i v ( f ) = 0 f o r f meromorphic, e t c .

i n t h e book; r e p e t i t i o n i s designed t o f i x i d e a s . n

K has degree 2g-2 and s p i n o r s come from L w i t h LL = K (degree L I n J ( c ) t h e image o f K (Q image o f t h e d i v i s o r o f any holomorphic

The bundle = 9-1).

I f L12 = L2 2 = K then I(2D1)

o r meromorphic d i f f e r e n t i a l ) i s a s i n g l e p o i n t . = I(2D2)

I ( K ) so L1 IL’;

w i t h d i v i s o r D1-D2

i s f l a t w i t h 21(D1-D2) = 0.

Thus t h e d i f f e r e n c e between any 2 s p i n s t r u c t u r e s i s r e p r e s e n t e d b y a p o i n t o f o r d e r 2 i n J(C).

These c o n s i s t o f p o i n t s ,el

Moreover g i v e n t h e marking chosen ( i . e . Do such t h a t A = I ( D o

-

t

e2 where el,e2

E

(Z/2Z)g.

$,$) t h e r e e x i s t s a s p i n s t r u c t u r e

( g - l ) P o ) (Do depending on

i,;

b u t n o t on Po).

(and some o f t h e f o l l o w i n g ) i s discussed f u r t h e r i n §19,22.

This

The s p i n s t r u c -

t u r e s a r e i n 1-1 correspondence w i t h f i r s t o r d e r 0 f u n c t i o n s where a,B have components 0 o r %.

They a r e even o r odd a c c o r d i n g t o whether t h e y a r e sym-

m e t r i c o r n o t under z -+

-2.

Note o(:)(-zln)

= exp(4nia

~ ) o ( i ) ( z l n so ) 4aB

SPIN STRUCTURES

31 3

even spin structures. Now f o r the prime form E pick a n o n s i n g u Q, ( a , B ) on C a n d write J = O ( T f ) ( c w l Q ) for arbitrary z,w. For w fixed J will vanish as a function of z a t g-1 points P1,...,Pg-l for fixed z; J vanishes a t the same points as a function of w a n d for z-w small a n d near P i , J (z-w)(z-Pi)(w-Pi) (exercise - note one i s dealing w i t h f i r s t order theta functions when a , @ have components 0 or %). Differentiate i n z a n d s e t z = w t o obtain a holomorphic 1-form (A) h ( z ) * = 1 wi ( z ) o ' ( " ) ( D l Q ) where 0 ' means the derivative o f 0 with respect t o i t s arguB 2 ment. Evidently h ( z ) has double zeros a t the Pi a n d does not vanish or become i n f i n i t e elsewhere. Hence i t s square r o o t i s well defined on C (no cuts are necessary) a n d the prime form E can be defined via ( c f . 55) =

0 mod 2

Q

l a r o d d s p i n structure

Q,

(21.1 )

E(z,w) = -E(w,z)

=

O(a)(z-wlQ)/h(z)h(w) B

(z-w % image o f z-w E C under the Abel map I ) . I t can be easily seen t h a t E i s independent o f the odd spin structure, has a single zero a t z = w, behaves l i k e a (-%,-%) differential, i s single valued around a i cycles, and E ( z t b i , w ) = exp(-ianii - EniI(z-w))E(z,w). An e x p l i c i t formula for a mero1; Qi i s then morphic function on C w i t h divisor D = $ P i

-

(21.2)

f ( z ) = II E ( z , P i ) / I I

E(z,Qi)

Meromorphic differentials o f the f i r s t , second, and t h i r d kinds are defined i n 55 a n d we rephrase t h i s by saying t h a t abelian differentials o f f i r s t k i n d % holomorphic 1-forms as above; differentials o f the third k i n d have f i r s t order poles a t P,Q for example a n d zero a i periods. They can be written i n the form (.) w ( z , P , Q ) = dlog(E(z,P)/E(z,Q)). Other differentials of third k i n d can be b u i l t up from ( 0 ) . Differentials of the second kind a r e holomorphic everywhere on C except for a pole o f order n a t P . If one requires t h a t the a i periods vanish these are uniquely determined u p t o the coefficient o f the leading singularity a t P. Setting w 2 ( z , P ) = a 2logE(z,w)/ azaw (second order pole a t P, w being a local coordinate vanishing a t P ) differentials o f the second k i n d w i t h higher order poles a t P a r i s e from n- 1w2(z,w)/n!aw n-1 , n = 2 , 3 ,.... W ~ + ~ ( Z . P=) a We recall also the Riemann Roch theorem.

Given a holomorphic l i n e bundle 5

ROBERT CARROLL

31 4

1 over C , H o ( Z , 5 ) = ker(3) acting on sections a n d H ( C , s ) = coker(5). One has 1 (6) dimHo - dimH = deg(5) -1 + g (recall deg(5) % deg(D) f o r D a divisor 1 determining 5 c f . 15 and Appendix B ) . Serre duality implies ( 4 ) dimH ( c , C ) = dimHo(C,K IBI 5 - l ) where K i s the canonical l i n e bundle. Note how t h i s implies deg(K) = 29-2 since Ho(C,P.) 2, holomorphic sections of K = abelian d i f f e r e n t i a l s with dimension g; b u t K B K-’ = t r i v i a l l i n e bundle h a v i n g

-

only the constant holomorphic section so REllARK 21.1

(6)

-

(+) implies deg(K) = 29-2.

(BSCSLZU30V &RANSFSZU!lA&ZQ)W, CRMS~ANNZAN.5, AND FREE FERmZ0NS).

Consider C as a Wick rotated ( t * k i t ) version of S1 X R with z = 0 (resp. z = -) % t = -- (resp. t = m ) ; see here 117 for pictures. Express the free fermion operators via expansions ( n E Z+g) (21.3)

c(z)dz’ =

1 cnz-”’dz+;

b(z)dz’ =

1 b n z-n-+dz4

with cnlO)= b n l O ) = 0 for n > 0 (annihilation operators) a n d ( m ) (cn,bml = with {cn,cm}= {bn,bm} = 0 ( c f . here (20.32) where 7 c and 2, b n ; Q,

&n+m t h u s c ( z ) Q , ‘ ~ ; ( z )and b ( z ) % p ( z ) ) . One thinks of l,z,z2y ... a”s negative -1 -2 L energy s t a t e s a n d z , z ,... as positive energy s t a t e s . The d z 2 terms a r e inserted since e.g. c transforms as a spinor. Define now the s t a t e s t o be f i l l e d in constructing the Dirac sea via (**) wn = z n-1 + 1 ; B,,Z-~ (assume The new vacuum i s the highest exterior proconvergence in some NBH of -). duct of s t a t e s (**) which can be represented in standard notation via g ( B ) = exp(-l Bnmc-.,++ b -,,++I m,n>O -p -k d z k dwB(z,w)c(z)b(w)); B(z,w) = 1 B z w kP k ,P>O

(21.4)

IB) =

=

exp(-

Some comparisons a r e already possible with (20.33) e t c . b u t e t us defer this ( t h e calculations i n (21.4) a r e evident). The fermion 2 point function i s given by (21.5)

(Olc(z)b(w)g(B)I0)dz’dw’

= dz’ddw4(

(z-w)-’

-

B(z,w))

= S(z,w,B)

The Bogoliubov transformation dCfined by B transforms c ( z ) in the form (wn as in (**I)

BOGOLIUBOV TRANSFORMATION

31 5

For m u l t i p l e i n s e r t i o n s i n (21.5) Wick's theorem i m p l i e s (21.7)

(

Olnc(zi)nb(wj)glO)

( c f . (20.36), (20.23),

etc.).

= d e t S(zi,wj,B)

Since g(B) preserves f e r m i o n number t h e # o f

b and c f i e l d s i n (27.7) must be t h e same.

lB>

Another p o i n t o f view here t o c h a r a c t e r i z e g(B) t o c n l O ) = bnlO) = 0 ( n > 0 ) t o g e t T h i s says t h a t t h e charges ( n = 1,2,

= gbng-l I B ) = 0.

...)

t

N

Qn = t b ( z ) w n d z ;

Qn = $ c ( z ) w i ( z ) d z ;

IB)

gcng-l

(*A)

The Q can a l s o be w r i t t e n

a n n i h i l a t e t h e vacuum ( e x e r c i s e ) . (21.9)

involves t h i n k i n g o f applying

W:(Z)

= zn-l

f &,,>OBnm~-m

This idea i s t o work o u t a f r e e f e r m i o n f i e l d t h e o r y s t a r t i n g from a n a r b i t r a r y Bogol i u b o v t r a n s f o r m o f 10 ). One notes a l s o t h a t w i t h H

'L

{znl

'L

n e g a t i v e energy s t a t e s

el A

( e n (zm)

...

one can w r i t e t h e s t a n d a r d vacuum as (**) 10) = O0 A

= ),6 ,

Then t h e c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s o f b

( c f . here s8).

and c a d m i t a r e p r e s e n t a t i o n bn+L W1

A W2 A

(Z-n-l

t h e space o f

H* generated by a dual

s o l u t i o n s o f t h e f r e e D i r a c e q u a t i o n ( a f = 0 ) and basis8 n

%

%

... =

e n A (so 8 10) = 0, n > 0 ) and

n

)wl

A

... A

wk-l

A wk+l

A

...

(*&I cn+% ( n o t e 8k

10) = 0); here t h e w . a r e l i n e a r forms on H ) = 0 here, n > 0, so c + + n$% J w i t h w- as i n (21.9) (see (8.14)). One can check and I B ) = wo A w1 A

...

t h a t t h i s I B ) agrees w i t h (21.4)

(exercise).

Thus one c r e a t e s w i t h t h e Bogoliubov t r a n s f o r m a new c h o i c e wn o f n e g a t i v e energy s t a t e s and a new D i r a c sea I B ) .

Now i n (*)

t h e graph o f a HS o p e r a t o r B: H+ = tzn; n > 01 -+ -m Bnmz and graph(B) = {z"' + B(znml)I B(zn-') =

H-

1

t

1 Bnmzm-'

w i t h H+

%

{z-';n

2 01

-

wn(z) can be w r i t t e n a s = {z-*;

n > 01 where

( n o t e i n [AG51,wn = z

c f . Remark 21.7).

The Grassmannian G r

( H ) i s d e f i n e d as i n 811 as t h e s e t o f subspaces W C H such t h a t pr+: W projection

-n

-

-+

r e c a l l T: Hl + H2 i s 2 The HS i f f o r any complete orthonormal sequence en i n H1, IITenll < -). i s Fredholm and p r - : W

-+

H- i s HS ( p r

1

H+

31 6

ROBERT CARROLL

group GLres i s (as i n 511) composed o f g = (: HS.

k)

w i t h a,d Fredholm and b,c

Given W one c o n s t r u c t s a u n i t a r y U E GLres such t h a t W = UH,

and w r i t e s

I

(*+) U = ('+ 'i ) ( c f . [ AG6;HEl;PRl;SEl I). L e t r, = I g = exp(1; xnzn)); and w- w+, r- = Cg = e x p ( l l Y , z - ~ ) } . These a r e considered a s f u n c t i o n s holomorphic i n Do = Cz;IzI < 1 1 and ,D = C z ; ( z / > l } ( w i t h v a n i s h i n g values a t 0 and m r e s p e c t i v e l y ) ; t h e y g i v e r i s e t o boundary v a l u e elements g E GLres d e f i n e d as 1 The r e p r e s e n t a t i o n o f maps S1 -+ C. We w r i t e r f o r f u n c t i o n s S +C;r+Cr. -

r -+ on

these groups

t h e fermonic Fock space i s given by t h e f frequency modes

o f t h e c u r r e n t j ( z ) = : c ( z ) b ( z ) : where : : r e f e r s t o normal o r d e r r e l a t i v e t o 10).

One o b t a i n s t h e n (*.)

n > 0, j n l O ) = 0 ) .

Note

j(z) =

cz

jnz-n-';

jn =

lz

:ckb,,-k:

(for

I jn,jml = n6mtn.

REFIARK 21,2 (KRZCEUER DAFA, CALlCIPJ RZERANN QPERAeOR.5, AND DECERFIZNANF BUN-

DCS).

b!ow h e u r i s t i c a l l y ( d e t a i l s a r e p r o v i d e d i n p a r t i n 519,221 i n o r d e r

t o work w i t h CFT on a RS t h i n k o f a f r e e f i e l d t h e o r y . w i t h l o c a l c o o r d i n a t e z(P) =

m;

e,

Pick P E

P

n,

m,

then p i c k a holomorphic l i n e bundle C , and

a l o c a l t r i v i a l i z a t i o n s e c t i o n uQ o v e r a p a t c h Urn where z i s d e f i n e d .

This

i s necessary i n o r d e r t o expand f i e l d s i n terms o f i frequency modes.

Let

Uo = c-P w i t h Uo n Urn % a n annulus.

a p o i n t W E GI-:

W

= K(C,P,z,C,uo)

To K r i i e v e r data ( Z , P , z , C , u o )

= s e t o f meromorphic s e c t i o n s o f C which

a r e holomorphic o f f P (Q H0(c-P,L)).

5

o f f i n i t e dimension ( i i s i n c l u s i o n and j g i v e s t h e t i o n s o f those w

-

To see t h i s r e p r e s e n t s a p o i n t W

one checks pr, i s Fredholm v i a t h e e x a c t sequence 0 + H o ( C , C ) C) H, A - H1 (c,C) + 0 w i t h k e r ( p r t ) = k e r ( 5 L ) and coker(pr,)

o v e r Uo and Urn).

associate

t

5

W

E

Gr

0

H (1-P,

= coker(TL)

Laurent t a i l s o f sec-

W which cannot be w r i t t e n as a sum o f holomorphic s e c t i o n s

E

T h i s i s a l i t t l e clumsy and we r e f e r t o (19.6) f o r a b e t -

t e r d i s c u s s i o n ( c f . a l s o 522).

Note t h e r e i s a n a t u r a l M a y e r - V i e t o r i s se-

quence as i n (12.7) o r (19.5)-(19.6) o f t h e form 0 -+ Ho(z,L) Ho(Uo,C) @ o( A 1 1 Urn). NOW Ho(c, Ho(Uo n Um,C) -+ H ( Z , L ) + 0 (Uo C-P, S C Uo

Ho(Um,L)

-f

Q ,

C) corresponds t o $;

C ) = Ker(a)

'L

=

0, JI a g l o b a l holomorphic s e c t i o n o f

ker(a) (but

lomorphic s e c t i o n s o v e r U

0

5

L, so Ima9eHo(c,

i s n o t a p r i o r i d e f i n e d on domain(a)).

n Urn which cannot be w r i t t e n a s $,

+

$m,

The ho-

JIo

E

0

H

1 Now i n genE Ho(Urn,L), determine nonzero elements o f H (Z,C). (Uo,C), e r a l t h e r e w i l l be elements o f W 2, H o ( U o , C ) n o t coming from Ho(C,C) and s i m i l a r l y t h e r e w i l l be elements o f Ho(Uo r~ Urn,L) L(S 1 ) = r e a l a n a l y t i c func-f

31 7

DETERMINANT BUNDLES 1 t i o n s o n S ( c f . (19.6)) n o t coming from q0 + qm.

Now i f one l o o k s a t pr,

i n s t e a d o f a we r e s t r i c t a t o W = H o ( U o , t )

( n o t e elements o f W can have ne2 1 I f we p r o j e c t t o H, = L ( S ) we chop o f f such n e g a t i v e powers ( c f . here (19.6) and n o t e t h e image o f a a f t e r p r o j e c t i o n i s i n t(S1 ,) 1 c H,); t h e n kerg, a c t i n g on i;(S ), % p o s i t i v e powers o f elements o f W n o t g a t i v e powers o f z ) .

( t h e s e a r e + Laurent t a i l s a l l u d e d t o

corresponding t o s e c t i o n s qo t

The s i t u a t i o n i s e n t i r e l y analogous t o (19.6) and comments a b o u t

above).

coker(ZL) w i l l be c l a r i f i e d i n § 2 2 . Now i n G r c o n s i d e r determinants d e f i n e d v i a an admissable b a s i s f o r W ext r a c t e d from image(w, TRC) and w- i s HS.

+ w-):

H,

Use e.g.

orthogonal b a s i s here ( c f .

+

W

H where w+ = 1

C

+

+

Trace c l a s s ( = 1

t h e Gram Schmidt o r t h o g o n a l i z a t i o n t o g e t an

(*+)I.

Then wt has a d e t e r m i n a n t ( c f . [ S I M l

I).

A

B = space o f admissable bases and t h i n k o f i t a s a bundle o v e r G r

Now l e t

w i t h group T = { i n v e r t i b l e elements o f GL(H+) o f t h e form 1 t TRCI.

The de-

A

B

t e r m i n a n t bundle i s d e f i n e d as DET = 4

($t-l,,Adet(t)).

DET*:

(;,A)

%

fore

-

%

( a l s o c a l l e d 0ET-l) has a g l o b a l h o l o m r p h i c secOne c o n s t r u c t s an e x t e n s i o n E o f GL

(g,q)

E

GLres X T such t h a t aq-’

-

by T t o a c t reSa b ) 1 E TRC ( g = (c as be-

E i s a p r i n c i p a l bundle E-> GLres w i t h f i b r e T).

(g,q)$

= g$q-’

($,,A)

DET does n o t have any g l o b a l holomorphic s e c t i o n s b u t

(it,Adet(t))

t i o n u(W) = (;,det(w,)). on DET v i a E

XT C w i t h equivalence r e l a t i o n

and t h e new w+ w i l l be o f t h e form 1

+

Then f o r $

€

8,

The elements o f

TRC.

T have a d e t e r m i n a n t by d e f i n i t i o n and one forms now a c e n t r a l e x t e n s i o n o f GLres,

GL- = E/T,

where T1

C

T corresponds t o o p e r a t o r s w i t h d e t = 1 .

Then

GL“ i s a l i n e bundle o v e r GLres, which however has a n o n t r i v i a l f i r s t Chern c l a s s and i s d i f f i c u l t t o d e s c r i b e . exists. g2)g;

For gi E U w i t h g g2 = g

-3) = det(ala2a3

where c(g,,g2)

o v e r U i s d e f i n e d by (g,a). t i o n u(W), (21.10)

with (g,,a,)$

Hence one l o o k s a t U one shows t h a t i n GLh (a3

To check

%

(A*)

a b a s i s f o r W; then e.g. =

C l

bl)(:r)a;l d,

=

C

GLres where a - l

(A*)

gig;

= c(gl,

g3) and t h e c r o s s s e c t i o n o f E one l o o k s a t t h e canonical secg2 l i f t e d t o

[ a2Wt

+

w*

becomes

bZW-1.

dpc p t L e t U(g) be t h e l i f t t o DET* o f g and one o b t a i n s ( A A ) U(g1)U(g2)u(W) = U(g1g2)u(W)det(ala2a:) ( t h e l a s t term t o balance a -1 2 al-1 and a;’). E also +

31 8

ROBERT CARROLL

a c t s on DET via ( g , q ) ( w , A ) = (gwg-',X) and s i n c e T1 = { ( l , q ) , d e t ( q ) E a c t s t r i v i a l l y o n DET, GLh l i f t s t h e a c t i o n of GLres t o DET.

= 1)

REmARK 21-3 (KAHCER %ERllCeLIRE ON CR), Consider next Diff S1 4 GLres.

A t the 1 Lie algebra level t h e Diff S a c t i o n is given by Lie d e r i v a t i v e s w i t h r e s n +1 pect to generators L n = z d / d z a c t i n g on elements o f W. If elements o f W transform a s tensors o f s p i n j then (A*) Lnf = (zn+'d/dz + j ( n + l ) z n ) f . One k computes e a s i l y (A)) ( z p I L n l z ) = ( k + ( n + l ) j ) 6 p , k + n . Now t h e Lie algebra cocycle % c ( g b , g 2 ) i n (A*) i s w ( a l y a 2 ) = T r ( [ a l y a 2 1 - a 3 ) = Tr(clb2 - b l c 2 ) where g = (: d l is thought of a s an element o f t h e Lie algebra o f GLres ( c f . here [ PR1 I ) . Noting t h a t b: H +. H+ and c : H+ +. H one o b t a i n s ( W ) w ( L n , 3 Lm) = ( n - n ) ( ( 6 j 2 - 6 j + l)/6)tin,-,, ( c f . equations l i k e (17.13) f o r j = 0 t h e f a c t o r o f 2 is not s i g n i f i c a n t here). If now f , g 6 Lie algebra r then

w(f,g) = ( 1 / 2 n i ) 5 1 f ' g d z = 1 nfngn which i s equivalent t o j n y j m=l n The cocycle w ( a l y a 2 ) above can be i n t e r p e r t e d a s a t 6 h l e r 6n+m a f t e r (*a). form on Gr. Indeed s i n c e the u n i t a r y subgroup Ures C 61res a c t s t r a n s i t i vely on Gr one defines a Hermitian metric on Gr by giving a Hermitian form on i t s tangent space a t H, E Gr, i n v a r i a n t under t h e isotropy subgroup U ( H + ) X U(H-). The tangent space a t H+ = (HS operators A: H+ H-1 and t h e unique i n v a r i a n t inner product is ( A , B ) = 2Tr(A*B) giving r i s e t o an i n v a r i a n t two form w ( A , B ) = -iTr(A*B - BAA). To see t h a t this agrees w i t h w above note -:*IE Lie algebra Ures. t h a t any tangent vector A a t H+ can be mapped t o One checks a l s o t h a t w is closed via d w ( X , Y , Z ) = Lxw(Y,Z) + L y w ( X , Z ) + Lzw (X,Y) + w([ X , Y ] , Z ) t w([ Y,Zl,X) + w([ Z , X l , Y ) ( L x = Lie d e r i v a t i v e - s e e Appendix A) and Lxw = 0 s i n c e w is i n v a r i a n t . (Am)

-f

(i

Next one introduces t h e tau function a s a measure of t h e lack o f equivariance r e l a t i v e to r+ o f the canonical s e c t i o n o(W) o f DET* (note this i s b a s i c a l l y t h e d e f i n i t i o n i n First one notes t h a t o(W) = 0 unless W is t r a n s v e r s e to H-. For [ SE1 I ) . t r a n s v e r s e W t h e tau function is defined f o r g E r+ by RRN\RI( 21-4 (Eft€ EAll FUNCEZON AND EQll1OARlIWCE)-

(21.11)

Tw(g)g-llJ(w) = u(g-lW)

a b where g-l l i f t t o DET* of r+ a c t i o n . For any g E F+, g-l = ( o d ) y w i t h a i n v e r t i b l e , so g E U w i t h l i f t to E o f t h e form (g-',a). Thus f o r an admissible basis w o f W

EQUIVARIANCE AND TAU Tw(g)

(21.12)

=

31 9

det(1 + a-lbB)

where B = w-w+-1 : H+ * H- w i t h graph W. Note here i n DET* (GlYul) 5 (G2,u2) i f and only i f p1 = p2det(;2Gl) so (gQa-',A) (gi,Adet(a)) ( i . e . Adet(a) = A-1 -1 A Adet(aw g g w ) ) . Hence w r i t i n g o u t (21.11 ) one has ( c f . (21 . l o ) ) (21 . 1 2 ' )

T,(g)(g$,det(a)det(w+))

=

(gG,det(aw+

+

Bw-))

( g t , d e t ( a ) d e t ( w + ) d e t ( l + a-l bw-w;'

=

))

One can a l s o w r i t e (21.12') i s a d i f f e r e n t (equivalent) way a s (21.13)

Tw(g) = ( O l e H ( X ) I B ) ;g

=

exp(ln,oxnzn); I B )

=

g10); ~ ( x =) l x n j n

( c f . (21.4), (*.), e t c . and note t h a t t h e operator representation o f r+ i s + + via H(x) - exp(H(x)) l i f t s r+ t o DET*). From (21.9) and I B ) wo A w, A we can t h i n k o f l B > a s a s e c t i o n o f DET* ( t h e i n f i n i t e wedge a determina n t ) . I f l B ' ) i s another s e c t i o n one w r i t e s ( B I B ' ) = deK wi lw!) where wi J ( r e s p . w!) represent admissable bases f o r ( B ) ( r e s p . I B ' ) ) . Via l i f t i n g

...

Q,

Q,

3

exp(H(x))lB) = l B g ) i s t h e Bogoliubov transform corresponding t o t h e admiss i b l e b a s i s g^wa"l and (21.13) % (21.12). The space o f sections of DET* i s t h e f r e e Fermion Fock space F. One embeds ( p r o j e c t i v e l y ) the Grassmannian i n F via W E Gr -+ J14 E r(DET*) = F (^w an admissible basis f o r W ) where $k($') = det($lG' ). A canonical basis o f F comes from "Dirac" p a r t i t i o n s S % p a r t i c l e s and a n t i p a r t i c l e s . T h u s S C Z s a t i s f i e s S-N = ( n l , n 2 2 2 . . . )and N-S = (ml ,...) f i n i t e a n d one w r i t e s (.*) $ ( S ) = c c ...b b ...I 0 ) w i t h nt n 2 m t m, $(W) = n,(W)$(S) where n,(W) P l k k e r coordinates o f W ( c f . 513 e t c . ) Further s i n c e t h e $ ( S ) will be orthogonal one has ($(W)l$(W)) = 1 n,(W)*

Is

Q,

ns(w). (C0NNECCl0W OF CAAn AND CHECA FU"CCZ0NZ).

Next we r e c a l l t h a t ( w i t h obvious notational changes) one goes to the f r e e fermion Fock space F with o p e r a t o r s :cn+L2bm+4:generating an algebra A ( c f . (21.3) e t c . ) Write F = 1 F(m) and r e c a l l ( v i a 18 e t c . ) t h a t t h e Heisenberg algebra generated by the j n of (*=) can be canonically represented i n C[x, , x 2 , . .] by j-., nxn R?3ARK 21.5

.

Q,

ROBERT CARROLL j,, =

-

etc.

.. I

The i n n e r p r o d u c t from F i s t r a n s p o r t e d v i a ( X ~ I . > 0). nni!/l n12nz... i n t h e form (fig) = f(an)g(x)lx,o ( c f . (8.12)-(8.

an ( n

..)

f(an)

one sometimes wants

i s ( O l c ( z ) b ( w ) I O ) = (z-w)-’

w i t h ( m l c ( z ) b ( w ) l m ) = ( w / ~ ) ~ ( z - w ) - ’ (see d i s c u s -

This l e a d s t o (@.)

s i o n below).

(mlnc(zi)b(wi)lB,m?

exp(H(x))lB,m)lx=O where IB,m)

X ( Z ) = q-iplogz+iC j n z n+O (20.28),

( c f . (21.3), N

%

$(Z), b ( z )

: ~ l - ~ $ ; - ~ :%

The f e r m i o n i c 2 p o i n t f u n c t i o n

here).

-n

-

i; jn,jml = n6,,.,

etc.

-

n o t e from (20.32) and ( * m )

$ibn 1 :$k$itk:y H(x) = 1 x n j n y e t c . ) . We ‘L

T*(Z),

$-,

%

Qn 2,

-4

Cn,

%

jn =

$is

%

t h e n o t a t i o n a l correspondences a t any one t i m e . s i g n s w i t c h e s e t c . i n 120 o c c u r i n g i n (20.27), X(z) i n (21.14)

(m(

i s any s t a t e w i t h charge m and

/n; [ q , p ~

(20.32),

= nV(zi,x)V*(wi,x)

etc. c(z)

-1

l z :Ckbn,k:

do n o t t r y t o p i n down a l l We do r e c a l l however t h e

(20.16)

(@@),

-f

etc.;

thus

i s n o t a t i o n a l l y more r e l a t e d t o X(z) i n (20.16) t h a n t o

y,

Thus i n view o f (21.13) we need o n l y know r W ( g ) i n o r d e r t o determine t h e c o r r e l a t i o n functions.

D i r e c t computation o f

T

from say (21.12) i s d i f f i -

c u l t , b u t one can e x p l o i t t h e f o l l o w i n g h e u r i s t i c argument based on [ SE1 I ( c f . [AG61 f o r more d e t a i l s ) . Urn c o v e r i n g as b e f o r e (Uo

%

L e t us f i r s t s k e t c h t h e i d e a s .

Use t h e Uo,

f o r K r i z e v e r data (C,P,z,C,uo).

Let V =

C-P)

space o f boundary values o f holomorphic f u n c t i o n s d e f i n e d on Do = I I z l < 1 1

r+

(i.e.

= exp(V)

-

note generally

rt

%

exp(1;

xnzn) w i t h no 0 t e r m ) .

Any

holomorphic l i n e bundle C can be represented v i a t r a n s i t i o n f u n c t i o n s g ( z )

i n Uo n Urn and g can be t h o u g h t o f as an element o f f i r s t Chern c l a s s o f 1; = w i n d i n g # o f g. (or

-

r-1;

= 0.

i.e. cl(t)

-f

3

r-

U

r+

Thus f l a t l i n e bundles

where t h e %

g

E

r+

I f g = exp(ko)exp(km) t h e n 1; i s t r i v i a l ( c f . [ FT11

t h i s i s a general r e s u l t i f c o c y c l e s gij

m: V

r

s p l i t as g.g.). 1 J

One g e t s a map

J ( Z ) o f t h e form A

(21.15)

f

-+

f = (1/21~)/, f w

S For f

E

K = ker(V

-+

J(C)) one has

?=

nn t m (n,m

E

Zg).

Thus elements o f

K have t h e form e x p ( k ) = $(k)exp(krn) where l o g $ ( k ) has s h i f t s h i p around

TAU AND THETA FUNCTIONS

321

t h e homology cycles o f C. The idea here i s to describe the geometric a c t i o n o f r+ say on W % ( C , P , z , t , u o ) ( W % meromorphic s e c t i o n s of 1; holomorphic o f f Let e.g. 1; be the l i n e bundle based on t h e t r a n g Then think of g E r+ a s a c t i n g o n (c,P,z,i:,u0) by tensoring (t,uo) with (1; ,u ) (us i n Urn w i t h uo I u t r i v i a l i z a t i o n i n U_ 9 9 9 f o r 1; Ip t g ) . Thus r+ a c t i o n moves 1; i n t h e moduli space of l i n e bundles of the same degree over C ( i . e . t moves on J(C)). This motivates t o some ex9 t e n t looking a t t h e map V J(C) above ( r + = exp(V)). Now one wants t o desP = Ho(C-P,C) a s before).

s i t i o n function g above.

+.

c r i b e t h e t a u function i n J(C),

via the t h e t a function a s i n ( 2 0 . 3 9 ) .

t h e moment we r e f e r t o [AG5,6;SE1

For

I f o r more d e t a i l and simply r e s t a t e (20.

39) i n a somewhat d i f f e r e n t way without motivating discussion. F i r s t change z + l / z so rf 'L exp(V) % exp(1; x n t q n ) . One examines the properties o f T~ ( f ) , a(W) E DET*, K = ker(V +. J ( C ) ) , e t c . For f = 1 x n t - n E V s e t Q ( f , f ) = 1 Qnmxnxm where Qnm i s defined a s i n (+*Izo w i t h z t . Let An = d n w / d t n / n ! evaluated a t t = 0 as i n 120. The function y ( f ) = T ( f ) e x p ( - Q ( f , f ) ) ( T ( f ) % TW(g), g = e x p ( f ) ) is found to have c e r t a i n p r o p e r t i e s and F ( f ) = T ( f ) e x p ( - a ( f ) ) can be i d e n t i f i e d with a t h e t a function (here a ( f ) is a l i n e a r funct i o n a l on V coinciding w i t h l o g 3 f ) when f ko f km E KO - t h i s defines KO; Q,

ko extends t o Uo and km to Urn).

Then one determines e and

CL

Hence

via information from W

E

Gr.

Now l e t us give more d e t a i l following [AG5,6]. First note f o r f = ko + km A or f E KO (corresponding t o a t r i v i a l i: ) f = 0 i n (21.15). Recall t h e a b e l 9 ian d i f f e r e n t i a l s w . a r e holomorphic globally and one can deform t h e cont o u r i n (21.15)

(%

J 9ip ) towards P o r to t h e back o f

1

c t o get 0.

clenerally 1 and 0 -+ H ( c , S ) /

r e c a l l t h a t isomorphism c l a s s e s of l i n e bundles h H (c,O*) 1 1 H (C,Z) + H (C,0*) Z + 0 i s e x a c t , t h e l a s t n a p c1 t o Z determining the 1 Thus c l ( t ) = 0 % H (C,O*)O. Chern c l a s s . Note a l s o t h a t f l a t bundles

-

+.

4

f f : V -+ Cg, V / K *II J(C), V / K o % H1 (c,S), H1 ( c , O ) / H 1 (c,Z) (V/Ko)/(K/Ko) = V / K , and K/Ko 1 H 1 (c,Z) = Z 29 . Write now r+ via g % exp(1 x n t - " ) a s above

in:

+.

and look a t some p r o p e r t i e s of

defined via (21.11) (we follow [AG6! here a somewhat d i f f e r e n t point of view is used i n [AG5]). F i r s t ~ ( 0 =) 1 and T

-

322

ROBERT CARROLL

f o r W transverse e x p ( f ) here).

T

-

(exercise

has (.*)

Next t h e canonical s e c t i o n o(W) o f DET* i s e q u i v a r i a n t under a 0 use g E -'l o f t h e f o r m ( c d ) ) . Next i f E r- and g E rt one

.r5W(f)= exp(w(?,f)).rw(f) Finally for f

c f (AH)).

E

for

V and k

K

E

-

( a ( k ) = k-).

To see t h i s one

and s i n c e rb(k)W C W t h e e q u i v a r i a n c e

y i e l d s r ( k ) = exp(-a(k)

l i t t l e c a l c u l a t i o n (note e-a(k)e-f (21.17)

and g = e x p ( f ) ( e x e r c i s e

w r i t e e x p ( k ) = $(k)exp(km) and one

computes T ( k ) = u(exp(-k)W)/exp(-k)o(W)

r-

6

r = eXp(?)

o b t a i n s ( 0 6 ) .c(f+k) = T ( f ) T ( k ) e x p ( w ( a ( k ) , f ) ) o f DET* under

where I B ) = g10) ( g =

= ( O l e x p ( H ( x ) ) l B ) a s i n (21.13)

T(f)

.(e-f-kW)

e

=

= .r(f)e-a(k)e-fo(W)

which i m p l i e s ( 0 6 ) ( c f . (21.11),

(AH),

= T ( f ) . r ( k ) e - w ( a ( k ) y f ' e - f - k o(w)

etc.).

Now one can i n t r o d u c e Qmn and

An a s i n d i c a t e d b e f o r e (21.16) w i t h

( c f . 15 and see Remark 21.3 f o r a more thorough development).

Then one de-

However t h e p r o o f i n [AG5] i s r e a l l y much

velops (21.16) e t c . as i n [AG6].

n e a t e r here, and u s i n g t h e above f o r m o t i v a t i o n and background we t u r n t o [ AG51 now i n Renark 21.7

RmARK 21.6

( c f . a l s o [ SE1 I)

(SCRZNC FIELD$, SPZN0R BUNDtEk, AND 0SCZLLAC0R EWPAWZ0W). Go b ( z ) and c ( z ) a r e a s

t o LAG51 and make a few n o t a t i o n a l changes as f o l l o w s . b e f o r e i n (21.3) b u t w i t h z-"'

-f

z"';

zn, n

0,

'L

t

energy s t a t e s , as be-

f o r e ; < O l c ( z ) b ( w ) l O > = - l / ( z - w ) now (a change i n s i g n ) ; and j(z) = : c ( z ) n-1 ( Z - n - l n-1 z Iz-',n 2 0 ) i n s t e a d o f {zn; n ). H, now b(z): = jRz n m (*); Bn(z-m) = ;,,,6,, t z Iz ) = B zm-' i n s t e a d o f > 01; wn z- + m > O nm k = ;,,6, bnt+ = en A and cnY wo A w1 A = &,,o(-l ) ( 1 / 2 a i ) * z-"'dz/z ntl wk(z )wo A A w ~ A- wktl ~ A ( c f . (**)-(*&)): B ( Z , W ) = ~ n , m > O B n m ~ n w m

Iz

Q

-f

1

...

...

( c f . (21.4));

On =

...

en +

1 Bnme-m+l;

I B ( w ) ) = On,

A o~~ A

... where

W i s the

span o f an,.; (21.4) holds w i t h t h e r e v i s e d B; t h e r e a r e forms l i k e (21.5) and ( 2 1 . 7 ) ' b u t w r i t t e n now as ( a * ) G(z,w,B) = t O l b ( w ) c ( z ) l B ( W ) ) / t OlB(W)) = B(z,w))dz %dw%w i t h det((G(zi,wj,P))) = ( OlnlN b(wi)c(zj)lB(W))/ (l/(z-w)

-

tO)B(W) ); (21.8)-(21.9)

a r e g i v e n a s l i g h t l y d i f f e r e n t form v i a

THETA FUNCTIONS (21.1 9)

Qn = g(B)bn-+g(B)-l

=

323

(1/2ai )ppb(z)wn(z)dz;

QA = gCn-+ g-l = (1/2ni)$p c(z)w,!,(z)dz

+,,

where “,I = z-n t ~ m , o B m n ~ - m y wn(z)w,l,(z)dz = 0, [Qn,Q;l+ = [ Q n , Q m l += Qi,Q;]+ = 0, and one thinks of lB(W)) as t h e boundary of a d i s c w i t h ( 0 1 puncture a t t h e c e n t e r ( i n f i n i t e p a r t ) and o t h e r punctures zi,wi i n s e r t e d

[

w i t h i n the disc.

%

A l t e r n a t i v e l y one can t h i n k of bn =a/ac-, and c n = a / a b - n .

The Q o p e r a t o r s s a t i s f y Q n l B ( W ) ) = Q,’,lB(W)) = 0 and may be thought o f a s ann i h i l a t i o n operators f o r l B ( W ) ) a s before. Note a l s o 10) = e0 A el A a s before i n our notational correspondences.

...

Now take a RS c w i t h a s i n g l e puncture a t P and l e t W = meromorphic s e c t i o n s of a spinor bundle 1; having poles only a t P (assume t h e r e i s no holomorphic s e c t i o n ) . As a b a s i s o f W one c o n s t r u c t s meromorphic spinor f i e l d s w i t h a r b i t r a r y order poles a t P and extended holomorphically to the r e s t o f c . The Szeg6 kernel f o r s p i n 4 is ( c f . 95) (21.20)

G(t,y) = 0‘;”’:

wlT)/O(”,(ob)E(t.Y)

where (”) c h a r a c t e r i z e s the d i f f e r e n t choices f o r t (we a r e taking t ( P ) = B t y(P) = 0, E is t h e prime form, T i s t h e period m a t r i x ‘L R , I w .n. Abe’l map). Y

One gets the b a s i s indicated f o r W by w r i t i n g ( c f . (21.8))

wn

(21.21)

=

(l/(n-l)!)an~’G(t.y)~y=O Y

( I / ( n - I )!(m-1

)!)a

m-1 an-l t Y (G(t,Y)

-

=

t-n +

1;

Bnmtm-’;

Bnm =

l/(t-Y))It=y=o

S t r i n g f i e l d s now a r i s e a s follows ( c f . § 2 0 ) . For a b e l i a n d i f f e r e n t i a l s w i , s . define n n ( t ) = -An (Im)-’(w-G) = 5, where ( c f . (+*)20, (21.18), e t c . )

wi

wi(t)

(21.22)

(w

=

(w.), i = 1 1

=

1”1 A i n t n - ’ d t ;

,... ,g,

An

Q ,

nn(t)

(Al,n

=

- ( l / ( n - 1 )!)a t aYn l o g E ( t , y ) l y = O

,... h,A gYn ) ) .

Then X ( t ) = J’ 5, i s s i n g l e

; (2Qnm valued and harmonic on C-P w i t h X n = X + X: whet-e (am) X: = t - n - 1 a a n, -1 m t m / m t nAn-(Im-r)-’imtm/m)and X n = ~ Y T A , ( I M T ) A m t /m. Now one w r i t e s an o s c i l l a t o r expansion a s i n §20

--

324

ROBERT CARROLL X(t) = q + iplogt + iplogt +

(21.23)

il

t n j n / n + conjugate n=O

% a n n i h i l a t i o n (resp. i , [ jn’ j m 1 = n6n+m, and jn ( r e s p . j - n ) q,p 1 c r e a t i o n ) o p e r a t o r s f o r n > 0. Note a l s o t h a t ( 6 * ) jn % - i a n , j - ” % i n x n’ j, % - i a / a T n , and j-n% in; n ‘ The e q u a t i o n o f m o t i o n f o r X i s a a X = 0.

where

-

d

(CALCUACZON O f CAU OZA CHECA).

R€i!tARK 21.7

ground we go t o t h e t a u f u n c t i o n a g a i n .

W i t h t h e preceeding as back-

The map ((21.15) f

J(C) i s w r i t t e n as f = (1/2ni)9,,f(z)w(Z)

+

V -+ C g o r

;:

now and V i s t h o u g h t o f as

r

=

A

For k E K, k = Ta + b where a,b a r e g dimensional v e c t o r s w i t h it

exp(V).

teger entries.

Here I?+

exp(&,Oxnzn)

exp(f),

=

r-

%

exp(ln,Oznz-n)

?J

ef , N

a . .

j(z)f(z), = &>Oxnj-n = ( 1 / 2 n i ) P v j ( z ) f ( z ) , H ( x ) = l n > O x n j n = (1/2ni)*,, a nd (&A ) ex p ( H ( x ) ) exp ( H (2) ) exp ( H ( x ) ) exp ( H ( x ) ) = e xp ( - S ( f ) where S ( f“,f ) =

-

(l/2ni)9p d r f =

-1 nxn,;

the cocyclew(fu,f)

in

(exercise). (Am)

-

7,

T h i s g i v e s perhaps a b e t t e r p i c t u r e o f

( n o t e t h e s i g n changes).

We work now w i t h o u t con

j u g a t e terms i n (21.23) e t c . and ( c f . (21.14) e t c . ) (60)( m l c ( z ) b ( w ) l m ) = ( W / Z ) ~ / ( Z - W ) ;i f one t h i n k s o f I m ) a s 1 one can w r i t e (21.14) e x a c t l y as i t i s , and ( o h ) h o l d s .

The t a u f u n c t i o n i s now d e f i n e d as ( c f . ( 2 1 . 1 3 ) ) ( 6 0 ) ( w i t h some m o d i f i c a t i o n due t o l m ) ) .

~ , ( f )= (mleH(X)IB,m)/(mlB,m) A

now k = Ta

+ b as above ( k

= ln21ynt-n E K ) w r i t e (66) $ ( k ) =

Given nn(s)yn +

= e $ ( k ) e k 1. S i m i l a r l y one can c o n s t r u c t f u n c t i o n s i n H O ( ~ - P , ( 6 + ) g n ( t ) = f t nn(s) 2niAAni. The II ( t ) have zero ai periods,

Eaiw(t)-a (e

0) v i a

J‘C

-

n w i t h bi p e r i o d s f . n ( t ) = - 2 n i i n ( c f . (am)). Then l B > i s a n n i h i l a t e d by b, n t h e charges (6m) Q(gn) = 9rP j ( t ) g n ( t ) and f o r f u n c t i o n s k E K, !Q($l),Q($2):

-

a - b ) ( e x e r c i s e ) . This i m p l i e s (+*) e x p ( Q ( $ l ) ) e x p ( Q ( $ 2 ) ) = 2 1 exp(Q($2))exp(Q($l)) and t o compute t h e change o f T under K c o n s i d e r ( c f . = 2ri(al-b2

n n-1 Cn,oxnz , j ( z ) % jnz = 0 (n > 0), e t c . and t h i n k o f

f =

j ( z ) = :c(z)b(z):,

j n l O ) = 0 ( n > 0), ( O l j - n

k i n (21 .24) as coming from

r+

C

so t h e o p e r a t o r s i n (21.24) commute and r e p -

r e s e n t e x p ( h j ( k + f ) as i n d i c a t e d i n (+A) below). Now e x p ( k ) = e x p ( $ ( k ) ) exp(km) k = k0 + km ( b u t ko i s m u l t i v a l u e d - c f . (66)). Then ( 9 % (1/2 Q

Ti)&

) one has ( c f . a l s o [ SE1

I)

TAU AND THETA T(f+k) =

(21.25)

(ole*

j(t)f(x,t) T( f ) T (

(see ( 0 4 )

- we

325

e P j k o ( B ) etz[j(ko),j(kJ1 k)e-S(k-s

-S(km,f)

f,

=

use -S here i n s t e a d o f S i n [AG51 and t h i s s h o u l d agree w i t h

(04)). The argument b e f o r e v i a (21.17) was i n f a c t e a s i e r . Here we r e f e r t o [AG5] f o r d e t a i l s b u t n o t e f o r ko 5 &>Ownz n , k, % F,,z-~, Ko(a) = (1/

1

N

2 d ) e j k o = &>ocinjn,

,K

I

= ( l l 2 n i ) S t jk,

= ~ n > O (~ OIexp(K,! njnj=n (01, , etc.

t h e Baker-Campbell-Hausdorff formula ( c f . 1 RRl;VR21) can be a p p l i e d t o exp since [ j(ko),j(km)l

(K,tt)

Thus one knows g e n e r a l l y (+A) l o g

...

= A t B t %[A,Bl + where t h e h i g h e r o r d e r commutators A, v a n i s h here. Consequently exp(A+B)exp(+[A,BI) = eA eB

(exp(A)exp(B)) [A [A,Bl,

,...,

i s a number.

...I a l l

.

Then from This accounts f o r t h e z = e x ~ ( S [ j ( k ~ ) , j ( k , ) l ) t e r m i n (21.25). ((A) one has ( W ) eHeK = e He l be Ko e x p ( r i [ j ( k o ) , j ( k m ) l ) = e e He -S(km,f)EeKo: Now (Olexp(Km) = (01 and we o b t a i n t h e second l i n e i n (21.25). l i n e involves f o r f =

-1 -. S(km,f) = &,n>OQnmynym 2 A i a (An % D e f i n e t h e n Q(f,f)= Qnmxnxm (so f o r k E KO, S(km,f) =

-

Now g i v e n (21.25) we can w r i t e f

i

r

\

k = ratb).

(An),

The l a s t

K 0, T ( k ) = Z : ( O l e * I B ) .

1

i n Remark 21.2). ( F ‘L One g e t s immediately F(kl+k2) = F(kl )F(k2) f o r ki E KO so 1og.r i s l i n e a r on Extend b y l i n e a r i t y , f + a ( f ) , t o V and t h e n ? ( f ) = F ( f ) e x p ( - a ( f ) ) i s KO. -. d e f i n e d o n V. Using ( + 4 ) one g e t s ( + m ) ?(ff+k) = ?(f)?(k)exp(-2Ki(Anxn)-a) ( c f . (21.25)). By d e f i n i t i o n ~ ( 0 =) 1 and one can w r i t e ( r e c a l l V / K 2 J(C)) M A (m*) :(f;ratb) = F ( f ) F ( r a + b ) e x p ( - 2 a i - a ) o r ?(;) = @(;)(?)/@(;)(O~T). To de2 Q ( k , f ) ) and d e f i n e F v i a (M)r ( f ) = e x p ( Q ( f , f ) ) F ( f )

t e r m i n e a ( f ) one l o o k s a t t h e 2 p o i n t f u n c t i o n f o r fermions when a ( f ) = 0 so one o b t a i n s f o r s p i n S fermions ( a , B

%

a choice o f spin s t r u c t u r e

5

c h o i c e o f l i n e bundle) (21.26)

T(X)

( c f . (21.16)).

= e l Qnmxnxm

o(”,(l

A,X~IT)/@(;~)(O~T)

For h i g h e r s p i n j t h e 2 p o i n t f u n c t i o n i s d i s c u s s e d i n [ A G

5,6] and one a r r i v e s a t (21.16) expressed v i a O(01.c) i n t h e denominator and -L

o ( 1 Anxn and

c1

and e

+ e1.r) i n t h e numerator.

We r e f e r t o [AG5,6]

f o r d e t a i l s about e

and remark here o n l y t h a t a,B a r e absorbed here i n t h e c o n s t r u c t i o n i n v o l v e s t h e canonical bundle K, t h e Riemann c o n s t a n t A , and t h e

prime form.

We r e f e r here a g a i n t o 518 f o r r e l a t e d i n f o r m a t i o n .

326 22-

ROBERT CARROLL

REmARKs

Dl;E$L

ON CAll fXINCCZONS, C€WE~-RZEmA!N 0PERAC0R5, ANI) DEEERmZGABC PUN-

I n 521 we encountered some n i c e i n t e r a c t i o n between d e t e r m i n a n t bun-

dles, t a u f u n c t i o n s , Grassmannians, e t c . and we w i l l develop such themes a 1i t t l e f u r t h e r i n t h i s section.

For r e f e r e n c e s see e s p e c i a l l y

BOS1;FR1,2;LOl;ML2;GH1;PM1;Q1;PR1;WTl

1.

AA2;BISl;

PM1

F T r s t we e x t r a c t from

I

on det-

e r m i n a n t s o f Cauchy Riemann (CR) o p e r a t o r s as t a u f u n c t i o n s t o make connect i o n s between m a t e r i a l i n 514 and 21 i n p a r t i c u l a r . i n t o a s e r i e s o f remarks a s u s u a l .

We break t h e d i s c u s s i o n

The main p o i n t i s t o show t h a t t h e t a u The a n a l y s i s i n [ PM1 ] i s

f u n c t i o n o f 514 i s a d e t e r m i n a n t o f a CR o p e r a t o r .

r e f r e s h i n g l y c a r e f u l and d e t a i l e d ; hence we w i l l o n l y s k e t c h i t and recommend r e a d i n g t h e paper.

R€I!WRK 22.1

(BPZN BLINDCElii AND CR 0PERACORs).

The c o n s t r u c t i o n s o f [ PM1

worked o u t i n d e t a i l t h e r e and we i n d i c a t e some main p o i n t s . “structural“,

1 are

These a r e

i.e. one e s t a b l i s h e s i m p o r t a n t correspondences between o b j e c t s That such correspondences a r e i m p o r t a n t w i l l be as

i n various contexts. sumed known by now.

-

That t h e development o f such connections i s e x c i t i n g i s

h a r d e r t o demonstrate ( u n l e s s immediately v i s i b l e ) b u t we have t r i e d t o convey some sense o f t h i s , w i t h o u t r e s o r t i n g t o too many cheap a d j e c t i v a l l a b e l 1 b e l s . Thus f i r s t c o n s i d e r t h e s p i n bundle o v e r P P i c k E , 0 < E < 1, and 1 P = ( z E C ; I z l < 1 t E ) and D L = { z E C ; I z l > 1 - E I U { m } . d e f i n e (*) DE

.

C u

{a}

as usual and D =

{IzI

5 1 1 w i t h D ’ = t I z l 1. 11.

X Cn be t r i v i a l bundles w i t h e ( P ) ( r e s p . em(P))

a C ),

(P,e.) E DE X Cn (resp, D L X J by t h e t r a n s i t i o n f u n c t i o n 2 - l P E DE n D.; h

I f fo:

1, eO J. ( P ) f OJ.(P) i n DE).

D

-+

%

L e t DE X Cn and D;

row v e c t o r s w i t h e n t r i e s

The s p i n bundle En o v e r P1 i s determined

(z = x t i y on D) v i a

(A)

e,(P)

Cn one d e f i n e s t h e l o c a l s e c t i o n

= z-’(P)eo(P), ( 0 )

eo(P)fo(P) =

and r e f e r s t o f o ( P ) as i t s l o c a l c o o r d i n a t e s . ( t h i n k

S i m i l a r maps fm: D i

-+

o f fo(z)

Cn determine l o c a l s e c t i o n s emf, and fm

Q

10-

c a l c o o r d i n a t e s (as f u n c t i o n s o f w = l / z ) . Now l e t Ap’q

1 denote ( p , q ) forms on P (dz

Q ,

p, dz

En i s a f i r s t o r d e r d i f f e r e n t i a l o p e r a t o r Cm(En) form

n, -f

A Cm(En I 9).

CR o p e r a t o r X on A0”)

with local

CAUCHY RIEMANN OPERATORS (22.1 )

Xeo ( z Ifo ( z )

327

eo (z)d?(dz+Ao ( z ) Ifo (z);

=

xeW(w)fm(w)= e_(w)di(Tw+ A,(w))f,(w)

zz = $(ax

- i a ) and Ao(z) (resp. A m ( w ) ) i s a smooth n X n matrix funcY I n order for t h i s t o be well defined one must have ( b ) d?Ao(z) = dG Am(w). A local section f i s holomorphic r elative t o the complex structure on E determined by a CR operator X i f a n d only i f Xf = 0. We note that X induces a CR operator X D on D as follows. Let H s ( E n ) be the Sobolev space of order s o f sections of En ( c f . " 3 , 2 0 1 for H s ) . Let H i ( D ) C H 1 (En ) be Here tion.

sections f such t ha t Xf(p) = 0 for p E D; (121 > 1 ) . Identify such f with C n valued functions o n 0 via eo and l e t X D = + Ao(z)lH;(D). Then ker a n d coker Xo ar e the same as ker and coker X . Indeed l e t f E H;((D) w i t h XDfo = 1 no 0. Then fo i s the eo coordinate of a section f E H ( E ) such that Xf(p) = 0 in D.; B u t then Xf = 0 since XDfo = 0, a n d kerXD C kerX. The other in2 clusion i s obvious so kerX = kerXD. Now l e t f o E L (D), R ( X ) = range X , a n d 2 coker XD = L ( D ) - R ( X D ) . Consider ( + ) m: fo d2eo(z)fo(z) t R ( X ) so t h a t m(fo) = 0 i f and only i f there e xist s g E H 1X ( D ) (c H ' ( E n ) momentarily) such t h a t Xg = dfeofo over D. This means fo E R ( X D ) so m: coker XD coker X (ker m = R ( X D ) so for fo E coker XD,m(f,) = dzeofo + R ( X ) puts dZe0 f 0 E coker X ) . To see tha t m i s surjective one need only show t h a t any section f 2 n E L ( E ) d i f f e r s from one supported on D by a n element in R ( X ) (exercise). Let 4 E Cm be 1 on D and 0 on G D; for some E . One can always solve Xg = $f for g locally defined on DL ( c f . [ FT1 I ) . Now l e t JI be smooth, JI = 1 on D ' , a n d $ = 0 in a NBH o f D; so pg E H 1 ( E n ) . Then f - X(Jlg) has support in D . 1 1 REClARK 22.2 (CRMSCM"NANS). Let now H%S ) be the Sobolev +space on S ( c f . [ SE1;PRl I ) and write H, for the subspace whose elements have analytic extensions into Do = i nte ri or D (also write H- for the subspace with analyt i c continuation into D a n d vanishing a t m ) . Let ( m ) f k = ( 1 / 2 n ) t n f ( e i e ) e-jked6 be the Fourier coefficients and use an inner product in H4 o f the form ( f , g ) = (1 t Ikl)fkgk. Then H t I H- under ( , ). Let J N ( X , D ) be b 1 the H Z ( S ) closure of the subspace obtained by r estriction of solutions f E 1 1 H ( D E ) o f Xf = 0 t o S (use the eo t r i v i a l i z a t i o n ) . Then via [ SE1 ] ( c f . § l 9 )

aZ

.+

.+

c

c

c

ROBERT CARROLL

328

a N ( X , D ) E Gro(H,) (see below). This means the same thing as in I SE1;PRl I; thus Gr(H,) ‘L Gr(H), WE Gr(H) means pr-: W + H i s HS and pr,: W -+ H, i s Fredholm, e t c . ( W i s said t o be close to H,). Here Gro(H+)= Gro(H) means L 2 ( E n PP4”’) index 0 which we will suppose momentarily ( C R operators H ’ ( E ) have index 0 c f. [FTl;R01 I ) . Similarly one defines a N ( X , D ’ ) a s the HL,(S’) closure of functions on S1 a ri sing from restriction of solutions f E H 1 (0;) t o S1 ; here however one uses the eo t r i v i a l i z a t i o n ( n o t e m ) . Then a N ( X , D ’ ) E Gro(H-) where W E Gr(H-) means now pr,: W -+ H, i s HS and p r - : W -+ H- i s Fredholm. In order t o see t h a t a N ( X , D ) E Gr(H) for example l e t A ( z ) be any +

-

smooth n X n matrix function recall also t h a t holomorphic al ( c f . [FTl I). Let $ l y . , . , $ erator a n d l e t + be a n n X n C ) i s smooth and (22.2)

on DE a n d consider d Z ( a z + A ( z ) ) on DE X C n ; bundles over open RS ar e holomorphically t r i v i n be a holomorphic t r i v i a l i z a t i o n for t h i s o p Then $ : DE GL(n, matrix with columns $ . ( z ) . J

-f

% z = $-’(z)(Zz + A ( z ) ) $ ( z )

Hence a N ( X , D ) = $ I S t H + so (roughly) W = a N ( X , D ) i s a smooth subspace of H close t o H, ( i . e . pr-W i s HS, e t c . ) . That aN E Gro(H) follows then since $ I s , extends continuously to a continuous map D -+ GL(n,C). RARARI( 22.3 (D€E€lZUNANE BUNDLE$). Now determinant bundles have been discussed already in §21 (c f. also[AA2;BIS1;B)S1;FRly2;ML2;Ql I ) . We follow here [ PM1 ] a n d note a n a lt e rna t ive way t o define the fiber s in Quillen’s l n 2 determinant bundle over X: H ( E ) -+ L ( E n I A o s l ) . X i s Fredholm o f index 1 0 so there e xi sts a n invertible q : H ( E ) L 2 ( E n PP A o S 1 ) such t h a t q-’x i s a compact perturbation of the identity (or trace class or even f i n i t e dimensional - c f . [SIM1,21). For q-’X a trace class perturbation of I one says q (or 9 - l ) i s an admissable parametrix for X (q E P ) . If q1 a n d q2 E P then -+

q i l q l i s a trace c la ss perturbation o f I (exercise). The fibre in the determinant bundle over X can be identified with the s e t of ordered pairs ( q , h ) , q E P, a E C*, with equivalence relation (**I (4, ,A,) I (q2,X2) i f and only 1 i f h l = a2det(q- q 2 ) . The determinant bundle i s designed so t h a t the M P (*A) a: X (q,det(q-’X)) i s a well defined (canonical) section. -+

The original definition of Quillen’s determinant bundle involves a fibre

DETERMINANT BUNDLES

329

isomorphic t o k e r ( X ) * B coker(X) ( o r t o C when X i s i n v e r t i b l e

-

see [ Q1 I )

and when kerX = kerXD, cokerX = cokerXD as i n d i c a t e d i n Remark 22.1 we can work o v e r X D i n s t e a d o f X.

XD a l l o w s one t o focus o n v a r i a t i o n s i n X which

o c c u r i n t h e e x t e r i o r o f D.

If

F i s a f a m i l y o f X a g r e e i n g o v e r D t h e n XD

changes v i a boundary c o n d i t i o n s on S1 d e t e r m i n i n g domains. v i a l i z a t i o n $ f o r X o v e r D and XD v a r i e s v i a aN(X,D'). a s a c o l l e c t i o n o f subspaces aN(X,D')

E Gr'(H-1.

Thus f i x a tri-

Think o f X D y X E

F,

The d e t e r m i n a n t bundle

o v e r XD i s t h e n t h e p u l l b a c k o f DET* o v e r Gro(H-).

More p r e c i s e l y g i v e n a

f a m i l y FA o f CR o p e r a t o r s o n En a g r e e i n g w i t h dZ(zz + A ) i n t h e eo t r i v i a l i z a t i o n o v e r D, t h e map X E FA +. a N ( X , D ' ) DET* o v e r Gr'(H-1

l i f t s t o a map from DET o v e r FA t o

and t h e l i f t i s an isomorphism o n f i b r e s .

Q

We do n o t g i v e

a l l t h e d e t a i l s o f p r o o f here ( c f . [ PM1 I ) b u t s t a t e t h e necessary c o n s t r u c tions.

1 ) = $H+ + $H- = aN(X,D) + $Hw r i t e (S' 1 L e t H+(D) % boundary values i n $H- and XD($)

Thus f o r $ a t r i v i a l i z a t i o n on D,

$ I s #when

appropriate). 1 = X r e s t r i c t e d t o H (D). F i r s t one shows XD($) i s i n v e r t i b l e and c o n s t r u c t s ($ =

4

1

Thus f E H (D) r e s t r i c t s t o

u n i f o r m l y an a d m i s s i b l e p a r a m e t r i x f o r XD. E

H4(S1) w i t h

flSl =

$g+ + $g-, g, E H.,

The map g+

+.

i s c o n t i n u o u s (a standard r e s u l t i n PDE 1 1 [ L1 1) and one d e f i n e s f o r f E H ( D ) a c o n t i n u o u s map (*.) P+: H ( D ) ( a g a i n w r i t t e n 9),

flSi

holomrphic extension

i n H'(D)

-f

cf. 1 H (D);

-

P $ f ( z ) = $(z)g+(z), z E D. Then f o r f E H1X ( D ) (*&) X D f = X D ( $ ) ( I P$)f 1 (note f P f E H ( D ) and v i a (22.2) XP f = 0 ) . One shows ( u s i n g harmonic $ 4 1 @ f u n c t i o n s ) t h a t (1 P+): HL(D) -+ H ( 0 ) i s Fredholm w i t h i n d e x 0, which i m -

-

-

4

= 0 ) so XD($) i s i n v e r t i b l e .

p l i e s XD($) has index 0 ( w i t h N ( X D ( + ) ) p a r a m e t r i x f o r XD = X,($)(l

-

-

Then a

P ) i s obtained v i a a uniform parametrix f o r $

This i s done by l o c a l l y t r i v i a l i z i n g t h e bundle o f a d m i s s i b l e P$). (1 frames ( c f . [PMl I ) . Elements i n t h e f i b r e o f t h e d e t e r m i n a n t bundle o v e r XD a r e t h e n p a i r s (*+) ( q , h ) , q = X D ( $ ) w - l ,

w a 4 a d m i s s i b l e frame f o r a N ( X , D ' )

( c f . (**) and r e c a l l t h a t a $ a d m i s s i b l e frame f o r W

E

Gr(H ) i s a c o n t i n u -

ous isomorphism w: +H- -+ W such t h a t t h e map prow: OH-

-f

class perturbation o f the identity).

w1 and q2

XD($)

F i n a l l y f o r q1

%

W 3 $H- i s a t r a c e %

w2,

since

does n o t change f o r X E FAYt h e e q u i v a l e n c e r e l a t i o n (**) becomes (*m)

X1 = x2det(q;'q2)

r e l a t i o n f o r (w,h)

= h2det(w,w;l) E

= h2det(w;'wl)

DET* o v e r G r ( H - )

and t h i s i s t h e e q u i v a l e n c e

( c f . remarks i n 121 b e f o r e (21.12')).

330

ROBERT CARROLL

The l i f t o f X

Gro(H-)

E

FA

+

aN(x,Dl)

t o a map between DET o v e r FA and DET* over

Q

i s t h e n c a r r i e d o u t v i a s u i t a b l e c h o i c e s o f parametrices ( c f . [ PM1

I).

The a c t u a l c o n s t r u c t i o n o f

Ql 1 i s bypassed h e r e when we use t h e correspon-

dences i n d i c a t e d (cf.2 a1 so

AA2 I).

REMARK 22.4

(RZBIIA” HICBERC PR0BtER5, m0N0DR0l&!,

AND CR 6PERAC0W).

We

s k e t c h now v e r y b r i e f l y some f u r t h e r developments i n [ PM1 I i n o r d e r t o make c o n t a c t w i t h §14 ( c f . a l s o [ M B l ; V D l I ) . Consider a c l a s s i c a l Riemann o r RH 1 problem on P v i a A ( z ) an n X n m a t r i x w i t h p o l e s a t {al, a 3 . The f u n -

...,

P

damental m a t r i x s o l u t i o n Y(z) t o dY/dz = AY i s g e n e r a l l y m u l t i v a l u e d w i t h branch p o i n t s a t t h e ai.

Choose a.

= ai

and assume t h e ai o c c u r i n o r d e r a s

one makes a c o u n t e r c l o c k w i s e c i r c u i t around a. j o i n i n g a.

c r o s s i n g t h e 1 i n e segment

t o ai.

L e t y . be a c l o s e d c u r v e based a t a. and e n c l o s i n g ai 1 1 {al,. ..,a 1 i s generated b u t no o t h e r a The fundamental group II, f o r P j’ P by equivalence c l a s s e s [yi I w i t h r e l a t i o n [y, 1.. [y 1 = i d e n t i t y . L e t P 1 be t h e s i m p l y connected c o v e r i n g space o f P where ill a c t s by deck t r a n s f o r -

-

.

mations and Y(z)

is t h e n holomorphic on

w i t h t r a n s f o r m s Y ( [ y . l p ) = Y(p1M-I J j c o r r e s p o n d i n g t o a l i n e a r r e p r e s e n t a t i o n IT( [ y . I ) = M . (monodromy group). The J J c l a s s i c a l RH problem here i s : S t a r t w i t h a r e p r e s e n t a t i o n [ y . ] = M . o f ill J J and f i n d Y(z) s a t i s f y i n g Y ( [ y j ]p) = Y(p)MI1 (see 914 f o r more d i s c u s s i o n o f J this). L e t us assume here t h a t t h e i s o l a t e d s i n g u l a r i t i e s o f t h e s i n g l e v a l u e d 1form A = dYY-’

( d e f i n i t i o n ) a t t h e a . a r e s i m p l e p o l e s . (% r e g u l a r s i n g u l a r J n o t e dY = AdzY). Then f o r s u i t a b l e c h o i c e o f a l o g a points f o r Y’ = AY

-

-L

j. Fixing r i t h m ( 2 a i L . ) f o r M . t h e l o c a l b e h a v i o r o f Y near a % ( z - a . ) J J j J t h e f u n c t i o n Y(z)(z-a . ) - Lwj i l l have branches o f Y(z) and (z-a . ) - Lnear j a J j’ J Now i n a s i n g l e valued a n a l y t i c c o n t i n u a t i o n i n t o a punctured NBH o f a j‘ t h e f o r m u l a t i o n o f [ M B l ] one d e f i n e s a c o n n e c t i o n vA on 7; X Cn such t h a t A

i s t h e c o n n e c t i o n 1-form f o r vA i n t h e s t a n d a r d t r i v i a l i z a t i o n e - ( p ) ( c f . th entry L e t e . ( p ) = (p,e.) and e ( p ) be a row v e c t o r w i t h j Appendix A ) . 3 J ej(p). L e t f ( p ) be a column v e c t o r o f f u n c t i o n s on ? ; and w r i t e (A*) e ( p ) f(p) =

5Y

11” f j ( p ) e j ( p ) .

= 0 i t follows that

Then vA i s d e f i n e d v i a (AA) VAef = e d f

+

eAf.

Since

vA i s f n t e g r a b l e ( i . e . has 0 c u r v a t u r e ) and i s holo-

morphic ( o n l y a dz t e r m appears).

When A ( z ) has s i m p l e poles one says VA

MONODROMY

3 31

has logarithmic poles. Then t h e RH problem can be rephrased a s t h e problem of constructing an i n t e g r a b l e holomorphic connection on P1 X Cn with logaa rithmic poles a t a and prescribed holonomy M-l on the y . (the holomony is j j J determined by p a r a l l e l t r a n s l a t i o n on t h e basis e ( a o ) around the curves y . ) .

.

J

For s i m p l i c i t y now assume t h e {al ,.. ,a 1 C D = open u n i t d i s c and i n s e r t P branch cuts r .-,, ( a j Y s j ) , s j = a j b e i n g s u i t a b l y chosen "sinks" ( s e e [PMl I). A simple choice i s s = 0 ( = ao) and r 'L ( a j , s ) = s t r a i g h t l i n e (assuming j s i s not c o l i n e a r w i t h any two a . ) . On t h e open d i s c DE one can solve t h e J RH problem w i t h ( z - a . ) - L j prescribed (any L.). Fixing a branch one gets a J P J s i n g l e valued Y(z) on DE - r . and any 2 such functions Y1 and Y2 d i f f e r by J an i n v e r t i b l e holomorphic function Y,(z)Y;'(z) on DE. Choose E small enough so t h a t DL $ a and l e t ( $ , l - $ ) be a p a r t i t i o n o f unity subordinate t o (DE, j D;) ( c f . Appendix A ) . Let M .-,, measurable s e c t i o n s o f En and define (A@) = ( f E M; Y$f E H 1 ( D E ) ; (1-$)f E H 1 (D;)}. Define a CR operator 5 Da,~ a,L a c t i n g on s e c t i o n s f E D by 2 a , L f = 5 a , L ~ f+ BaYL(1-$)f via a,L

(22.3)

i s well defined and does not depend o n (note Y$f E H1). One sees t h a t 5 a,L t h e choice of Y o r t h e p a r t i t i o n o f unity $. Now i n [ PM1 I t h e r e i s a careful a n a l y s i s o f ? (and many other t h i n g s ) so a,L one can s a f e l y summarize and r e f e r t o [ PM11 f o r d e t a i l s . The index o f 3 a ,L may not be zero b u t can be adjusted by choices o f t h e L . so one assumes i t 3 1 here t o be 0. The R H problem o n P will have a s o l u t i o n when some s o l u t i o n Y(z), holomorphic i n D, has an i n v e r t i b l e holomorphic extension t o D ' . Thus f o r fixed Y every s o l u t i o n has t h e form Y+Y f o r some i n v e r t i b l e holomorphic Y+ (on 0 ) and Y I S t will have a canonical f a c t o r i z a t i o n Y = Y;'Ywhere Yhas an i n v e r t i b l e holomorphic extension t o D ' ( r e c a l l Y1 = (YlY;')Y2 for 2 s o l u t i o n s Y1 and Y 2 with i n v e r t i b l e and holomorphic on D). This i s not always possible b u t when achieved the function on P1 which matches Yand Y+Y can be thought of a s a global gauge transformation intertwining 5a,L w i t h t h e standard CR operator on t h e spin bundle. The remaining

Y1Yi1

ROBERT CARROLL

332

a n a l y s i s i s q u i t e technical and we only provide a few guidelines. Let C . be 3 s u i t a b l e c i r c l e s around D . containingaoYj, where the R H problem % a o , j has J ) for a s o l u t i o n Yo(z) w i t h Y O ( m ) = I. Let a . E D . and Y.(z) = Yo(z-a.+a J J J J o,j s u i t a b l e z. One gets connections V l o c a l l y via A . ( z , a ) = Ao(z-a +a . ) and j J OsJ t h e r e will be gauge transformations S j ( z , a ) such t h a t (A&) S.V.S-' = Vo. To + J J ~ + solve the R H problem i t s u f f i c e s then t o f a c t o r S I J C j = SS. f o r SJ. holomorOne o b t a i n s a family a a , L f o r l a j - a o y j 1 < r phic and i n v e r t i b l e i n D j*

j*

RRllARK 22.5 (CR 0PERAC0W AND CAU FUNCEZ0W)- Now a s i n Remarks 22.2-22.3 the map taking 5a,L t o i t s n a t u r a l r e s t r i c t i o n on D ' l i f t s t o a map on determinant bundles. T h i n k then of such 5a,L r e s t r i c t e d to 0 ' and c l a s s i f i e d 1 via subspaces Y-lH, E Gro(H) where Y = S-'Yo near S (S-'VoS on DE - U D . is J t matched u p w i t h S.V.S+-l on D . ) . Thus reversing t h e viewpoint o f Remark 22. J J j J 3 we have a family agreeing over D ' and c l a s s i f i e d v i a aN(X,D) = Y - l H + E Gr defined by a (H). Then t o f i n d a determinant f o r the local family

sa,L

choice of branch c u t s r . ( a ) i t s u f f i c e s t o t r i v i a l i z e t h e DET* bundle over J subspaces YilS(.,a)H+. T h i s i s c a r r i e d o u t i n d e t a i l i n [ PM1 1. One f i n d s then ( c f . [ Dl;J5-8;MBl;MWl (A+) T = d e t ( 8 ). Thus a,L (22.4)

I ) formulas of t h e type (14.24), upon i d e n t i f y i n g

dalog(det($a,l)) =

c

$1j + k Tr(A.A )d(a .-a ) / ( a j - a k ) J k J k

t

0 0

Tr(A.A J k )dak/(ao, j-ao,k) j#k

where t h e A j ( a ) a r e t h e residues of the connection 1-form a t a J. ( n e a r a 0,J.) and A' = A.(a ). Thus

j

(22.5)

J

O

(dY/dz)Y-'

=

ElP

Aj(a)/(z-aj); d A

a j

=

-1j.+k

[ A . , A Id(aj-ak)/(aj-ak) J k

23, QUANCW I N V E W E SCACEERZNG, W e do not attempt any complete treatment o f this subject b u t will g i v e some introductory material based on [BOG1 ;DSl;F

5,6;KU6;KH3;GMl;SKl;THl 1. There a l s o i s some supplementary material i n 118 on t h e 1-D impenetrable Bose gas and NLS.

REmARK 23-1 (QUANCUR NG AND BECHE AWAC2)- We will emphasize t h e NLS equat i o n ( c f . 19) and f i r s t discuss t h e Bethe a n s a t z approach following [DSl; TH1 1. Consider t h e normal ordered quantum Hamil tonian

333

BETHE ANSATZ

(*I [ ? ( x , t ) , ? ' ( y , t ) l = % ~ ( x - Y ) and [ $ ( x , t ) , $ ( y , t ) I = [ ? ' ( ~ , t ) , ? ~ ( y , t ) l T h i n k o f k > 0 ( r e p u l s i v e potential which implies no bound s t a t e s ) and I$ means t h e o p e r a t o r CI, $; normal ordering here puts j operators to the r i g h t o f A$J t o p e r a t o r s . The Heisenberg equations o f motion a r e then (immediate

where =

0.

from (23.1 and ( * ) )

( s e e 16 f o r Hamiltonian ideas and r e c a l l t h a t Poisson brackets commutator brackets upon quantization. We do not however attempt to a d j u s t notation here to 16 o r § 9 ( b u t s e e Remark 23.2 f o r some comparisons). -f

One builds u p a Hilbert space o f s t a t e s via a vacuum 10) w i t h $ ( x , t ) ( O ) = 0 (so j t , c r e a t i o n o p e r a t o r s ) and t h e N p a r t i c l e s t a t e w i t h momenta k l , . . . , k l i has t h e form (take h = 1 ) (23.3)

( kl . . . k N )

=

b - N (l/(N!)')LIIldxi$(xl

...$ +(xN)IO); $ ( x , k )

=

,...,xN;kl ,..., kN)$

A t

t O / ? ( x l )...$(xN)]kl

( x , ) ...

...k N )

Thus $ is t h e N p a r t i c l e wave function. For a noninteracting system $ plane waves b u t f o r i n t e r a c t i o n s the Bethe a n s a t z is a way o f describing i t s s t r u c t u r e . One notes t h a t , I J

$ ( k & ) ) nNl

A t

(xi)lo)

(ai

Q

a/axi)

t o be an e i g e n s t a t e $ must be an eigenfunction o f the Laplacian w i t h 6 function i n t e r a c t i o n ; t h i s corresponds to a Bose gas w i t h point i n t e r a c t i o n s ( c f . I1 5 ) . T h u s f o r I kl . . . k N )

Consider the 2 p a r t i c l e wave function. (23.5)

(-A

+ 2k6(x1-x2))+

( $ = $(x,,x2;kl,k2)).

The eigenvalue equation is

= A$

The 6 function potential will lead t o a d i s c o n t i n u i t y

ROBERT CARROLL

334

i n t h e f i r s t d e r i v a t i v e o f t h e wave function a t the i n t e r a c t i o n points and one can check t h a t t h e s o l u t i o n g of (23.5) i s ( c f . STE21 f o r sample calcul a t i o n s i n this d i r e c t i o n )

4

(23.6)

e'l

=

-

(1 where

E(X)

(23.7)

k j X j(1

4sgnx and

4

=

P

( 2 i k / ( k 2 - k 1 ) ) ~ ( x 2 - x 1 ) )+ e i ( k 2 x 1 + k 1 x 2 )

1)

(2ik/(kl-k2))~(x2-xl

=

1

-

+ k 22 here.

A = k:

ei(kplxl

+

kp,X2) ( k

p,

-

Thus when x1 < x2 one has k

p,

-

ik)/(kp,- k

p,

))

where pi n, permutation of ( 1 , 2 ) and the sum is over permutations. a n s a t z generalizes this and a s s e r t s t h a t i n the region x1 < x2 < (23.8)

,...,xN;kl ,...,k N ) = 1P e i c p1 ,..., pN permutations o f 1 ,..., N . g(xl

The Bethe

... <

xN

k p j x j II !kpm-k - i k ) / ( k -kpj) m>J Pj Pm

The function where n, extended to t h e whole space and will be an eigenfunction w i t h eigenvalue 1 : k:. We do not discuss t h e anatomy of mark t h a t the combinatorics a r e very i n t e r e s t i n g and can t h i n g s (e.g. card s h u f f l i n g ) .

g can be n a t u r a l l y o f ;as

i n (23.4)

(23.8) here b u t rebe r e l a t e d t o many

(P0ZM0N %RACK€C$ AND R mACRZCEk). Now f o r quantum inverse s c a t t e r i n g (QIS) we r e c a l l t h e c l a s s i c a l p i c t u r e i n t h e form T, = TU from (9.2) and to connect notations w i t h [ DS1 ] we note t h a t E = k w i t h t h e NLS + 2 k ( $ l 2 $ and A = k L , ( < u + + $u-) - i c u 3 w i t h equation i n the form i$t = -$xx 01 0 0 The s i t u a t i o n E A = U = Uo + A U 1 ) . U+ = ( o o), u- = ( 1 0 ), and 5 n, @.(SO > 0 corresponds to no bound s t a t e s ( n o s o l i t o n s ) , and T here i s t h e t r a n s i t i o n matrix a s i n §9. Note a l s o g, = Ag i s the c l a s s i c a l eigenvalue probn lem which i n t h e quantized s i t u a t i o n becomes ( f o r k < 0, q = $, and A = -i?,

REmARK 23.2

u3 + ilkl'$'o+ (23.9)

ax$

+ ilklqu-) t*t

= : i ( x , c ) ? ( x ) : = -icu3;(x)

A

+ i l k l V (x)u++(x) +

I

+ i kl%-;(x)$(x) (by d e f i n i t i o n $(x) = : $ ( x ) : ) .

Now i n order t o deal w i t h various operator

R MATRICES

335

Poisson brackets (and eventually t h e corresponding commutators) t h e idea o f R matrices i s useful. Referring t o [ OSl;F2;RE6,10;SMl ,3,4;LY1 1 we w r i t e f o r say 2 X 2 matrices A,B

where j k , m n = 11,12,21,22. W r i t i n g f u r t h e r ( A ) P ( C @ q ) = TI I 5 (P a permuta2 2 2 t i o n matrix i n C IC ) with P = I ( 4 X 4) one has ( a ) P ( A I 8 ) = ( B I A ) P f o r A,B 2 X 2 matrices. Then e.g. (23.11)

{A

BCI = { A IB3(I IC ) + ( I B B ) I A I C )

BI = -P{B J AIP; {A

I

01

In terms of Pauli matrices ( 6 ) u (23.12)

P = %(I+

1

02 =

(0 -i

i

0 ' 9

= ('

o

'3

O) one has

-1

c13 aa Ia b )

( t h e l a s t expression i s i n the 1 ,12,21,22 b a s i s ) . In this notation f o r k < 0 i n NLS one can w r i t e ( e x e r c i s e (23.13)

(A(x,c)

bj

A(y,c')I = i k ( o + I u-

0 0 0 1 ( r e c a l l u + = ( o o) and u- = (l o ) ) .

(23.14)

-

a-

-

c f . [ DS1 I )

I o+)b(x-y)

Further an elementary c a l c u l a t i o n gives

[P,A(x,s) I I + I I A ( x , s ' ) l = 2 i ( ~ - s ' ) ( a +8 a-

-

a-

IBI a+)

One writes ( 4 ) r ( c - 5 ' ) = ( k / 2 ( 5 - s t ) ) P so t h a t (23.15)

tA(x,c)

A ( y , < ' ) I = b ( x - y ) C r ( ~ - s ' ) , A ( x , < ) II + I I A ( x s s ' ) l

Now P i s a c-number operator ( i . e . a complex m u l t i p l i e r o p e r a t o r ) SO (23.14) A s t i l l holds when A i s replaced by A . A l i t t l e elementary c a l c u l a t i o n a l s o gives f o r (23.16)

(m)

R(5-5') = I

-

ihr(c-5')

R(s-s')(i(x,s) I I + I I (i(x,s) B I + I I

i(x,s')

i ( x , c ' ) + nko-

+like+

B a+) =

I o-)R(s-s')

Again R is s t i l l a c-number matrix even though h i s present.

One s e e s t h a t

336

ROBERT CARROLL

(23.16)

I ,1

i s a q u a n t i z e d v e r s i o n o f (23.15) and as h (23.16)

0 with (l/ih) [

-+

, I

-f

(23.15).

+

(0PERACBR UACUED SPECQRAI; DAQA AND C0l!llTUCAC0W). Now one must r e w r i t e t h e v a r i o u s formulas i n v o l v i n g T i n Remark 9.1 f o r example t o g e t

REmARK 23.3

I

f o r m a l l y (we f o l l o w [ D S l A

(23.17)

where more d e t a i l appears) A

A

T(x,y,c)

= :T(x,y,s):

= : e x p ( c A(z,c)dz): A

w i t h (*)

f o r x > y > z ( n o t e exp(JX A(z,c)dz i s Y symbolic as i n 39 and 7-l i s n o t s p e c i f i e d here). The d i f f e r e n t i a l equa?(x,y,c)?(y,z,s)

= ?(x,z,s)

tions (9.2) become ( k < 0 ) (23.18)

A

axf(x,y.s)

i\kI%-?(x,y,c)$(x);

= -icu3? + i I k l

= :A(x,e)?(x,y,c):

ayi(xy~,s) = -:ffx,~,c)~(y,c):

L A f

F

A

(x)o+T(x,y,c)

+

= fCfo3r-

- iI k l % f ( y ) f ( x y y , c ) u + - iI kll'?(x,y,s)G(yl< and one sees t h a t these e q u a t i o n s can be w r i t t e n i n t h e e q u i v a l e n t i n t e g r a l form A

A

(23.19)

T(x,y,c)

= I +

dz:A(z.c);(z,y,c):

Y

I +

=

ly" dz:?(x,z,c)i(z,c):

F u r t h e r one checks e a s i l y t h a t ( k < 0 )

(23.20)

[ ~ ( x ) , i ~ x , y , s ) =~ l'ihIkl'u+?(x,y,c);

A

LA

4

%

6(x-Z) =

A t

Y

A

Next one d e f i n e s ?,(x,y,<) and w r i t i n g (*A) t,(x,c.c')

fi

ax(?,

=

A

=

A

B I and T = I B T. Using (23.18), A A A(x,s) B I + I IA(x,c') +%ku- B u+ w i t h L2,

= T(x,y,c)

K

shows d i r e c t l y ( k < 0 ) A

(23.21)

I

dzs(y-z).

JX

-c~,fl,a+,one

= -QinlklLZ

I n t h e s p i r i t o f (*&)6 one w r i t e s h e r e JX dz Y

-Jgitilkl ZT(x,y,c)u-).

4=

A

I$ (y),T(x,y,c)

~ ~ ~ ( x , y , e ) ; ~ ~ ~ ~ y ) ,I ~= (l'inlkl x , y ~ cT(x,y,c)u+; ) (aTo-

t

[ J I (x),T(x,y,c)I

n

A

A

( x , ~ , c ) T ~ ( x , ~ , c ) ) = :Ll (x,cyc')T1 ( x , ~ ~ c ) T ~ ( x , ~ ~ c ' ) : ; 1

A

A

h

ax(T2(xyy,c')T1 ( x , Y , ~ ) ) = :L2(xyc,s' )T2(x,~,s')T1 (x,Y,c):

AS Y MPTOT I CS

337 n

A

One sees a l s o be d i r e c t computation t h a t (*.)

R(c-c')L1 (x,c,c')

= L2(xyc,c')

R(c-c').

From t h i s and (23.21)

(23.22)

R(c-c')T1 ( x , ~ , c ) T ~ ( x , ~ , c ' ) = ?2(x,~,c')T1 (x,y,c)R(c-c')

results

A

A

A

T h i s r e l a t i o n w i l l l e a d t o v a r i o u s b r a c k e t r e l a t i o n s f o r t h e s c a t t e r i n g ope r a t o r s a,b,

REILIARI(

e t c ; when h + 0 i t reduces t o t h e c l a s s i c a l l i m i t

23+4 (M#!lmPC@CZCS)+ Next one checks a s y m p t o t i c s and we observe here Thus c l a s s i c a l l y i n [ DS1 ] one d e f i n e s (*&)

a n o t a t i o n a l change from 99.

T(c) = l i m exp(icu3y)T(y,z,c)exp(-icu3z) whereas i n 19 ( w i t h

-b

c

%

+A)

NOW, w r i t i n g (*+)

here.

one f i n d s t h a t

= (-;

,. (t

-b) %)

we have T(A) = lim = L1 (x,c,c') 1x1-

L' ;

( k < 0, y

-f

-,z

+

-m)

so b i n §9 corresponds t o lim = Ixl- L 2 ( x y c , c ' )

and

This leads t o

+iO))u-

A

R(c-c')( I + (ihk/Z(c-c'-iO))u-

(23.25)

I

IS+) =

(I- ( i k h / Z ( c - c ' + i O ) ) o +

8 o-+)T1

A

(c)T2(c')(I

-

(ikh/Z(c-c'

A

I

U - ) T ~ ( ~ ' ) ? ~ ( ~+ ) ( I

t (ikh/2(c-c1-iO))o+

T h i s i s t h e quantum analogue o f ( c f . [ DS1

191 u - ) R ( c - c ' )

I)

which c o n t a i n s a l l t h e Poisson b r a c k e t s i n v o l v i n g s c a t t e r i n g data. Now d e f i n e ( * m )

R-+ ( < - c ' ) = ( I

+

(iAk/Z(<-c'+iO))a+

Iu

)R(c-c')(I

+

3 38

ROBERT CARROLL

One notes t h a t a s 41 + (23.26), namely (23.29)

Ca(s),a(s'

( c f . [ DS1 I ) . Recall a l s o t h a t a c t i o n a n g l e v a r i a b l e s involve (AA) Q = argb and P = (2/ak)log1al ( c f . 59). In the present s i t u a t i o n we see t h a t (A*) [ log:(c),log$(<')] = 0 so log: will generate commuting q u a n t i t i e s (including t h e Hamiltonian 1). Consider next (23.30)

I

? ( 5 ) = ( f i k /^at( 5 )e"( c ) 1% '( 5 ) ;

it( 5 ) = (;

5 ) (ah 1 k I $t( 5 1; ( 5 ) )-'

I t follows t h a t (A&) [ $ ( c ) , $ ( c ' ) I = 0 = [ ? t ( e ) y $ t ( c ' ) l , 6 ( c - e ' ) = [;(c),$' ( < ' ) I , and [ l o g $ ( ~ ) , $ ' ( c ' ) ] = log(1 + ( i k R / 2 ( e - s ' + i O ) ) i t ( c ' ) ( e x e r c i s e ) . Thus one can t h i n k of$'and $ a s r a i s i n g and lowering operators i n t h e Fock A t A t -t space generated by vectors (A+) I k l . . . k N ) = 4 ( k l ) $ ( k 2 ) ...4 ( k N ) I O ) where A a ( s ) l O , = 0 (so a l l conserved quantum numbers of t h e vacuum will vanish). The Hamiltonian then takes the form (Am) ^H = d k k 2 z t ( k ) $ ( k ) w i t h i l kl . . . N 2 k N ) = (1, k i ) l k l . . . k N ) ( e x e r c i s e ) . One sees t h a t the s t a t e s in (A+) a r e t h e

iI

PIS

339

same a s the Bethe ansatz s t a t e s of Remark 23.1 ( u p t o constants of proportionality). The formulas above for R matrices lead naturally and directly to the YangBaxter equations and quantum groups b u t we will not discuss these topics here. For further information on the r matrix technique in classical s i tuations see [ DS1;F2;BOR1;GI1-6;CR5,6;LY1;RE6,?0;SMly3,41. There i s also much more material on QIS i n the references cited, including quantum Gelfand Levitan methods, e t c . (c f. [ BOG1 ;F5,6;KH3;TH1 I ) . See also BH6-91 for interesting uses of r matrices i n classical and quantum Toda f i e l d theory, with relations t o CFT, Drinfeld-Sokolov theory, monodromy, W algebras, etc.

This Page Intentionally Left Blank

34 1

APPENDIX A

DIFFERENTIAL GEOMETRY AND ELEMENTARY HAMILTONIAN THEORY There a r e many good books on d i f f e r e n t i a l geometry and we w i l l g i v e o n l y a C1

1 b u t c o n s i d e r a b l y expanded t o i n c l u d e

e.g. more a b o u t H a m i l t o n i a n systems.

We w i l l c o n c e n t r a t e on l o c a l c o o r d i n -

“snappy” i n t r o d u c t i o n modeled on

a t e f o r m u l a t i o n s and b a s i c a l l y i g n o r e many g l o b a l m a t t e r s o f t o p o l o g y u n t i l t h e y become r e l e v a n t .

I n p a r t i c u l a r we want t o e s t a b l i s h n o t a t i o n and w i l l 1 E C e t c . when such a p r o p e r t y i s c l e a r l y

f r e q u e n t l y o m i t hypotheses l i k e f indicated.

G e n e r a l l y we assume known b a s i c ideas and f a c t s from algebra,

p o i n t s e t topology, a n a l y s i s , e t c .

we d e f i n e a Cm (n-dimensional)

Thus e.g.

m a n i f o l d t o be a H a u s d o r f f space M = U Ua,

Ua

C

M open, w i t h

na

Ua

-+

Rn a

homeomorphism (1-1 b i c o n t i n u o u s map) o f U o n t o an open s e t C Rn, and i t “-1. i s supposed t h a t f o r each a , t~h e map I$ $a(Ua n UB) $B(Ua n U B ) i s B Cm ( i . e . t h e range c o o r d i n a t e s a r e Cm f u n c t i o n s o f t h e domain c o o r d i n a t e s ) .

.

-f

One can r e p l a c e Coo by Cm o r r e a l a n a l y t i c t o g e t o t h e r types o f m a n i f o l d . I n terms o f n o t a t i o n , i f f i s a f u n c t i o n o n M one says f E i f f o r every c h a r t (Ua,$a)

the functions

f“= fo$-’

cm,

a r e Cm, L’,

f E L~

etc. 1oc:, e t c . on U a .

“A.

Then e.g.

one speaks o f af/axi

w i t h c o o r d i n a t e s xi

i n Rn).

and W t

when t h i n k i n g o f af/axi

(here $,(Ua)CRn

I f M and N a r e Cm m a n i f o l d s w i t h a t l a s e s C ( U a ,

J, 1 t h e n a map f : M N i s Cm means t h a t whenever f ( U a ) C WB 6’ F1 t h e map +60foaa : I$“(Ua) - + +(W ) i s Cm ( t a k e maximal a t l a s e s c o n t a i n i n g 6 % maximal c o l l e c t i o n s o f c h a r t s ) . I f f : M N i s o n t o w i t h f and f-l Cm t h e n -+

-+

f i s c a l l e d a diffeomorphism.

RflllARK A l ,

1 F o l l o w i n g [ CE1 ] l e t C ( p ) be t h e s e t o f equivalence c l a s s e s o f

c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s i n some open neighborhood (NBH) o f p; t h e NBH Df depends on t h e f u n c t i o n f, and f i s i d e n t i f i e d w i t h g i f t h e y a r e equal i n some NBH o f p. We w r i t e f f o r t h e f u n c t i o n and i t s z c l a s s and r e 1 1 c a l l t h a t C ( p ) i s c a l l e d t h e s e t o f germs o f C f u n c t i o n s a t p ( t h u s c l a s s 1 f = germ f ) . We now d e f i n e an equivalence r e l a t i o n R i n C ( p ) ( c o n s i d e r e d

34 2

ROBERT CARROLL

a s a v e c t o r space o v e r R w i t h a f + Bg d e f i n e d and C _=

1

on Df n D

g i f a l l f i r s t p a r t i a l d e r i v a t i v e s o f f - g a r e 0 a t p.

by s a y i n g f 9 To s p e l l o u t t h e

‘Ga

C Rn, d e t a i l s , one means t h a t i f (Ua,$a) i s any c h a r t a t p w i t h ‘pa(Ua) = -1 E v i d e n t l y t h e d e r i v a t i v e s depend on t h e t h e n a/axi((f-g)oa ) = 0 a t $a(p). b u t t h e i r v a n i s h i n g i s independent o f c o o r d i n a t e systems because o f t h e

$a*

c h a i n r u l e ( c o n s i d e r e.g.

a/axi(((f-g)o$-’)o($~$~’))

etc.).

It i s e a s i l y

seen t h a t t h i s r e l a t i o n i s w e l l d e f i n e d on germs and i s an e q u i v a l e n c e com-

1

f” G ,

p a t i b l e w i t h t h e v e c t o r space s t r u c t u r e o f C ( p ) ( i . e . f _= g, 5 implies 1 a f + B i 5 ag+B;j). Thus C (p)/R i s a v e c t o r space which we c a l l T*M (cotangent l P 1 space) and we denote t h e canonical ( a l g e b r a i c ) homomorphism C ( p ) -+ C (p)/R

i

Next we w i l l show t h a t T*(M) i s an n-dimensional v e c t o r spaP space w i t h b a s i s dxl,...,dxn i n a c o o r d i n a t e system $a a t p a s above where by f

-t

df.

xi = prio$a

(pr denoting p r o j e c t i o n ) .

A b a s i s i s d e f i n e d a s usual t o be a

s e t o f l i n e a r l y i n d i p e n d e n t elements bi spanning T*M i n t h e sense t h a t any P Suppose t h a t Aidxi = 0. v e c t o r b E T*M has a unique expansion 1 a . b P 1 J j’ Then f = Xixi = 0 i n C ( p ) which means t h a t a t $,(p), X = a/ax.(fo’p;’) = j J 0 and consequently t h e dxi a r e l i n e a r l y independent ( i f f = x = pr.$% t h e n j J fo$,’ i s t h e f u n c t i o n x -t x . on $ a ( U a ) ) . On t h e o t h e r hand l e t f E C’(p) J ( i . e . Xi = a/axi(fo$a’) e v a l u a t e d a t $ a ( p ) ) . Then f E and xi = ( a f / a x . ) 1 P 1 hixi and d f = 1 Aidxi, which means t h a t t h e dxi span T*M and d f = 1 ( a f / P ax.) dxi. Thus T*M i s c o m p l e t e l y determined r e l a t i v e t o any c h o i c e o f co1 P P o r d i n a t e s + , a a t p, where t h e $a s i m p l y p r o v i d e a b a s i s i n which t o d e s c r i b e

1

1

t h e space T*M. P (* denotes a l g e b r a i c dual o v e r R and we w i l l have We now d e f i n e T M = T:(M)* P L e t a c o o r d i n a t e system $a a t p be selected, w i t h dxi t h e M T; = T (M)*). P a s s o c i a t e d b a s i s o f T*M and l e t v . ( a ) be a standard dual b a s i s f o r T M. Thus P J P = 6ij and any v E T M can be w r i t t e n t h e v . ( a ) a r e determined by (v.(a),dxi) P J J 1 v = 1 p.v.(a) r e l a t i v e t o t h i s b a s i s . Now i f f E C ( p ) we can d e f i n e an ac1 1

t i o n o f v E T M o n f by t h e c o m p o s i t i o n f d f -+ ( v,df) which we denote by P p . v . ( a ) , l ( a f / a x . ) dxi‘, = 1 p . ( a f / a x . ) which we repThus v ( f ) =
the d i r e c t i o n a l derivative. j = l,...,n, a/ax jy and we w r i t e v =

Consequently we can t h i n k now o f t h e symbols

as b e i n g a b a s i s f o r T M r e l a t i v e t o t h e $a c o o r d i n a t e s P

1 uJ. ( d / a x .J) .

TENSOR PRODUCT

34 3

( r e s p 4,) r e f e r to x ( r e s p . y ) coordinates. For any f E C 1 ( p ) we By t h e chain rule a / a x i ( f q , ’ ) = have df = 1 ( a f / a x . ) dxi = 1 (af/ayj)F:yj. Let now

13

a/axi((fe$i’)o($fio$, 1) = 1 a/ayj(fo$B )ayj/axi (note t h a t 0g0,~ maps x y ) Thus ( a f / a x . ) = 1 ( a f / a y . ) ( a y j / a x i ) . The matrix ( j ,X row, i % column) J 1 P J P Y ( x ) = ( ( a y . / a x i ) ) is c a l l e d t h e Jacobian o f t h e map qa6 = 480,1 and s i n c e J i s a diffeomorphism d e t J = 0 on $a(Ua A U 6 ) . To see this look a t t h e map yi = y i ( x ( y ) ) t o obtain 6ik = 1 ( a y i / a x j ) ( a x j / a y k ) o r J ( x ) = ~ ~ ( y 1 - l Y and det J ( x ) = 0. Hence ( a f / a y ) = 1 ( a x i l a y ) ( a f / a x . ) which a l s o f o l Y k P k P 1 P lows d i r e c t l y a s above. By d e f i n i t i o n o f df we have f u r t h e r dyj = 1 (ayj/a xk)pdxk. Therefore we can w r i t e ( e x e r c i s e ) a/ayk = 1 ( a x J a y ) a / a x . and i f J k P J 1 u i ( a / a x i ) = 1 A.(a/ay.) then X = 1 (ayj/axk)ppk. The l a s t two formulas j J J follow by d u a l i t y and a r e standard r e s u l t s of l i n e a r algebra. -f

va6

Let T(M) be the d i s j o i n t union U T M and T*M = U T*M. These P P s e t s a r e c a l l e d t h e tangent and cotangent bundles of M r e s p e c t i v e l y (they will be seen below t o be CObmanifolds w i t h a natural bundle s t r u c t u r e ) . The change of basis formulas a t a point p E M and corresponding c o e f f i c i e n t chan-

DEFZNZCZBN A2,

ges a r e indicated above. R€l!lARK A3.

Let now V be a f i n i t e dimensional vector space of dimension n

over C or R. To define t h e e x t e r i o r product V A V one can use several equiv a l e n t techniques. For example V A V = V 5 V/J where J is the vector space i n V IV c o n s i s t i n g of f i n i t e l i n e a r combinations with c o e f f i c i e n t s i n C o r R o f elements of t h e form x Iy t y B x. Recall here t h a t F IG i s defined a s follows (F,G vector spaces over C ) . Let B ( F , G : C ) = b i l i n e a r forms F X G -+ C and T(F,G,C) = T = formal l i n e a r combinations 1 a . ( f . , g . ) w i t h a E C j J J J and ( f . , g . ) E F X G. P u t an obvious vector space s t r u c t u r e o n T and l e t To J

=

J

subspace c o n s i s t i n g o f f i n i t e 1 inear combinations, w i t h c o e f f i c i e n t s in C,

of elements ( f l + f 2 , g ) - ( f l y g ) - ( f 2 , g ) , ( f , g l + g 2 ) - ( f , g l ) - ( f , g 2 ) , (af,g) - a ( f , g ) , and ( f , a g ) - a ( f , g ) . Then F IG = T/To and i f b E B(F,G,C), b extends to T and vanishes o n To so t h e r e e x i s t s a unique l i n e a r map 6 : F IG -+ C w i t h b ( f , g ) = 6 ( f I9 ) . This provides a 1-1 correspondence between B ( F , G,C) and L ( F 5 G,C). Returning t o V fl V we i n d i c a t e another point o f view.

Thus l e t P ( m )

mutation g r o u p on m symbols and f o r u 6 P ( m ) define o(xl I

=

... Ixm) =

per-

x

o(1 )

344

I

ROBERT CARROLL

... B

xo(,,,).

Then e x t e n d t h i s o p e r a t i o n i n t h e o b v i o u s way by l i n e a r i t y

t o BmV = V ffl

...

R V (n t i m e s ) .

P.n element T E B"V w i l l be c a l l e d symmetric

i f UT = T f o r a l l u ~5 P(m) and a l t e r n a t i n g i f oT = (sgnu)T f o r a l l u E P ( m ) .

D e f i n e Am = ( l / m ! ) l (sgna)a (sum o v e r u E P ( m ) ) as a map BmV

+

B"V.

Then

observe t h a t AmT i s always a l t e r n a t i n g and AmT = T i f T i s a l t e r n a t i n g (exercise).

One d e f i n e s AmV t h e n t o be t h e v e c t o r space o f a l t e r n a t i n g elem-

e n t s i n NmV ( D e f i n i t i o n 2 ) . One can a l s o d e f i n e AmV as fflmV/kerAm ( D e f i n i t i o n 3 ) b u t t h e n o t a t i o n w i l l be s i m p l e r i n v a r i o u s formulas i f we t h i n k o f AmV C iBlmV and use D e f i n i t i o n 2 . F u r t h e r l e t us say t h a t a l i n e a r map J , : BmV nating i f

= (sgno)J, f o r a l l u

$00

€

W a v e c t o r space,

W,

-+

is alter-

Then AmV as a q u o t i e n t ( D e f i n i -

P(m).

t i o n 3 ) i s c h a r a c t e r i z e d by t h e p r o p e r t y t h a t i f J, i s any a l t e r n a t i n g l i n e a r map lDmV -+ W,

t h e n t h e r e i s a u n i q u e l y determined l i n e a r map n: AmV

-+

W such

A

vimwhere Am: BmV + BmV/kerA i s t h e c a n o n i c a l map ( e x e r c i s e ) . I n m, t h i s f o r m u l a t i o n one w r i t e s c u s t o r a r i l y Am(v1 I . . . B vm) = v1 A . . . A v . m

t h a t J, =

V.

f o r a r b i t r a r y vi

However, we a r e g o i n g t o d e f i n e now ( u s i n g D e f i n i t i o n

3) u A v = (r+s)!/r!s!)Ar+s(u&Iv)

for u

and v

E ArV

duce a c o n v e n i e n t numerical f a c t o r i n t o t h e formulas.

... A

v1 A

vm = m!Am(vl

B

... Ivm).

I n any event,

t o r space o f a l t e r n a t i n g l i n e a r maps BmV

-+

W,

which w i l l i n t r o -

E AsV

Thus i n p a r t i c u l a r

i f La(BmV,W) i s t h e vec-

i t f o l l o w s immediately t h e n

t h a t La(BmV,W) i s isomorphic as a v e c t o r space t o L(AmV,W) under t h e c o r r e s pondence J, Now i f vi

-+

n i n d i c a t e d above.

(i= 1,

...% n k) i s

a basis f o r V then the n

c o n s t i t u t e a b a s i s form V ( e x e r c i s e ) . A v i K w i t h il < E

... c

k

elements v

it Indeed n o t e t h a t f o r T

k

ik span t h e v e c t o r space A V.

RkV, Ak(uT) = ( l / k ! ) l ( s g n r ) r o T = ( l / k ! ) l ( s g n r u ) ( s g n ( o ) r o T = (sgno)Ak(T)

which i m p l i e s i n p a r t i c u l a r t h a t an element vi

A I

... A

The (:)

EHEtXETI A4, k

elements vi,

b a s i s f o r A V whenever t h e vi Proof: il

A

A

...

A v. 'k

( i = l,...,n)

,

i, <

'U

... <

v.

lk

,..., ik) where

= 0 ( ( i ) = (il

ik, constitute a

a r e a basis f o r V.

Only l i n e a r independence remains t o be shown.

... A

i s m u l t i p l i e d by

v.

-1 i f two o f i t s terms a r e interchanged.

v

1...1 v i, A . ..

7, Hence elements o f t h e form v

i, <

... <

Thus suppose ik).

1a ( i )

Pick some par-

t i c u l a r ( i ) i n t h i s sum and l e t ik+l,...,in be t h e r e m a i n i n g i n t e g e r s i n t h e

345

EXTERIOR PRODUCT s e t (1

,..., n )

without r e p i t i t i o n .

a l l terms except v

... A

A il

v.

1K

M u l t i p l y by v .

A v.

A

1ktl

. .. A

1 k+l

v.

In

A

...

A v and then in’ w i l l i n v o l v e a repeated

element and hence must v a n i s h ( b y remarks above an i n t e r c h a n g e o f t h e repea-

...

t e d element would r e v e r s e s i g n ) .

Hence a(ilvi, A A v = 0. But vi A in I...&! v . ) i n v o l v e s t h e l i n e a r l y independent In terms u ( v i , I Iv.I n ). Hence a ( i 1 = 0 f o r any ( i ) . QED k The p r o o f a l s o shows t h a t A V = 0 f o r k > n s i n c e repeated elements would

. .. A

Vin

’

$ 0 s i n c e An(fi,

...

Now we can c o n s i d e r A k T*M w i t h AoT* = R and A 1 P P T* = T* P i c k i n g a b a s i s o f T* corresponds t o s e l e c t i n g l o c a l c o o r d i n a t e s P P’ P $a a t p and u s i n g t h e dxi ( i = 1, n ) a s a b a s i s . Then (modulo r e d u c t i o n

occur i n the basis vectors.

...,

... <

t o i n d i c e s j, < A dyin

j k )t h e change o f b a s i s formula f o r a t e r m dy

f o l l o w s d i r e c t l y v i a D e f i n i t i o n A2.

r e c a l l f i r s t (see e.g. [ B K l

1 a(i)dyi,

I..

.

A

...

i t

I)

As f o r c o e f f i c i e n t changes we

t h a t t h e standard t e n s o r l a w f o r a n e q u a l i t y

1 b(j)dxj,

... 1sI

i s a(i) = (axj,/ayi, I... (axjK/ayi,)b(j) w i t h summation on repeated i n d i c e s ( e x e r c i s e ) . Then e v i d e n t k l y under t h i s r u l e t h e ensuing i d e n t i t y i n B T* would map i n t o an i d e n t i t y P k i n A T* ( i t i s e a s i l y seen t h a t t h e necessary antisymmetry p r o p e r t i e s o f t h e P a(i) and b r e q u i r e d o f a l t e r n a t i n g forms would be p r e s e r v e d ) . F i n a l l y ( j1 k one c o u l d pass t o a b a s i s expansion i n A T*. T h i s can be summarized i n t h e P f o l l o w i n g way ( c f . [ B K l I ) . I f 1 a(i)dyi, A ... A dyik = 1 b ( j ) d x j , A . . . A

19 dyiK

=

...

8

dx

jk

1

...

with c < ik, j, < < j , then (exercise) a = det((ax./ j, k (i) J vi))b!j) sum o v e r j, < < jk);((axj/ayi)) = ((axj,/ayi,)) (1 5 p,m 5 k; m = row index, p = column i n d e x ) ; dyi A A dyik = d e t ( ( a y i / a x j ) )

dx

...

dx

j,

A

...

I

...

1

A dxjk.

k k D e f i n e A T*M a s t h e d i s j o i n t u n i o n U A T*M (a bundle s t r u c k p M be t h e p r o t u r e can be i n t r o d u c e d e a s i l y c f . below). L e t ak: A T*M k k j e c t i o n map o f A TfN p. A s e c t i o n o f A T*M i s d e f i n e d t o be a map s : M k P A T*M such t h a t n ~ = s i d e n t i t y - a n d w i l l be c a l l e d a d i f f e r e n t i a l form o f

DEFFINlEL0N A5.

-

-f

-f

-f

degree k (no c o n t i n u i t y o r d i f f e r e n t i a b i l i t y o f s i s r e q u i r e d a t t h e moment). Thus a d i f f e r e n t i a l form o f degree k i s a f u n c t i o n s on M such t h a t s ( p ) E A k T*M f o r each p and t h e change o f b a s i s and c o e f f i c i e n t formulas a r e g i v e n P above.

RERARK

A6.

Next i f gi

E

T* x. E T we d e f i n e , w i t h a s l i g h t abuse o f P’ 1 P’

34 6

ROBERT CARROLL

a

n o t a t i o n , (gl

... Igk)(xl ,. . .xk)

= ( Bgiy@xi

) = (

I.. . Igk),(xl

(gl

I. ..

Ix k ) ) = IItgi,xi).

S i m i l a r l y g(X l,...,xk) i s d e f i n e d f o r any g E I T * by exk P Thus IT* can be regarded as the space o f k - l i n e a r P maps ( o v e r R say) o f TI k T -+ R (where nk means T X X Tpy k times, and a P P Inmap i s c a l l e d m u l t i l i n e a r i f i t i s s e p a r a t e l y l i n e a r i n each argument). k k k deed,I T* i s t h e a l g e b r a i c dual o f IT ( t o see t h a t one gets a l l o f ( @ T )* P P P s i m p l y check t h e dimension). tending t h i s 'linearly.

...

k S i m i l a r l y A T; can be c h a r a c t e r i z e d as t h e v e c t o r space o f a l t e r n a t i n g kk

l i n e a r maps II T P a map ( x xk)

-+

,,...,

R.

Here an a l t e r n a t i n g m u l t i l i n e a r map i s d e f i n e d t o be

R which vanishes whenever two o f t h e xi c o i n c i d e ; such k a map determines an a l t e r n a t i n g 1 i n e a r map IT R and c o n v e r s e l y ( e x e r c i s e ' k Hence AkT* w i l l appear as t h e a l g e b r a i c dual o f A T ( c f . Remark A3) and t h e P P k d u a l i t y can be e x h i b i t e d by u s i n g t h e ( , ) p a i r i n g above between IT and k k k k P IT* ( r e c a l l t h a t we c o n s i d e r A T* C B T* and t h e a c t i o n o f w E A T* o n (xl, P P P k P x k ) i s t h u s e x a c t l y i t s a c t i o n as an element o f IT*). I n particular k P f o r w E IT* and xi E T P P -f

-f

...,

Akw(xl

(A.l)

,...,x k )

( l / k ! ) C sgno(w,o -1 (Elxi))

= ( A w.xl

k

... B

I

= (l/k!)1

xk) = ( l / k ! ) l

sgno<w,xo

I...Ix,)

sgna
=

( l / k ! ) C sgnuw(xa)

I r T * and v E BST*, A r + s ( ~ Iv)(x,, P P . . . ' x ~ + ~ )= ( r ! s ! / ( r + s ) ! ) l u(xi, x. ) v ( x j , xjs)sgno where t h e sum i s 'Il o v e r p a r t i t i o n s o f 1,. ..,r+s i n t o i n c r e a s i n g sequences il,. .,i r and j,, Then one can show ( e x e r c i s e ) t h a t f o r u

...,

E

...,

.

js. Consequently u A V ( X ~ , . . . , X ~ + ~ =)

. . .,

1 sgnau(xi, ,...,x i P )v(xjl ,..., x j s )

which d e f i n e s t h e wedge p r o d u c t u A v E ArfST* C IrtST* by i t s a c t i o n on P P t h e r u l e g, A We n o t e i n passing t h a t f o r gi E T* xi E T ( Xl,...,~r+s). P' PY A gk(xl x ) = k!Ak(gl I B gk) ( c f . Remark A3) y i e l d s g1 A A k gk(xlY".Sxk) = sgndgo(l) @ ID ga(k)YX1 8 xk) = sgndgo(l)YX1) ( g,(k),xk) = d e t ( ( ( giyx.))) which r e v e a l s t h e d e t e r m i n a n t i n i t s n a t u r a l J rs h a b i t a t . Note a l s o (u+v)Aw = uAw t VAW, uA(v+w) = UAW t UAW, uAw = ( - 1 )

...

,..., 1

...

VAU,

...

* * a

...

...

1

auAv = uAav = a(uAv), and (uAv)Aw = u A ( v A w ) .

Now t o d e f i n e t h e idea o f t a n g e n t v e c t o r t o a c u r v e we f i r s t con1 s i d e r two Cm m a n i f o l d s M and N and a Cm map f : M -+ N ( f E C here w i l l o f t e n

BEmARK

A7,

TANGENT VECTORS be used; i.e. the coordinates of f ( x ) a r e C

1

34 7 functions of t h e i r arguments

xi). J,

B

Let p E M belong t o a chart (Ua,+,) a n d f ( p ) E N belong t o a chart ( W B ) with f(Ua) C WB (such a s i t u a t i o n can obviously always be realized f o r

any (W ,I) ) given in advance). We will occasionally abuse notation by writB B ing p E (Ua,+,) instead of p E Ua in order t o simultaneously exhibit the lo-

T be an a r b i t r a r y tangent vector a t p ; then we P define i t s image f,v E T by the rule f,v(g) = v(g0f) a n d c l e a r l y f,v i s f(P) well defined by duality. In particular i f v = a / a x i we want t o express f, ( a / a x i ) in terms of the basis elements a/ay. a t f ( p ) . To do t h i s consider cal coordinates $a. Let v

€

G = ilBf+y

w a x i (go+; )+,f+; ( t h e l a t t e r evaluated a t $ a ( p ) ) .

write f,(a/axi)

=

!

)I+,(~)

= a/aYk(go$B -1 (p))(a;k/axi Hence using a previous abbreviation we

1 (aFk/axi)(a/ayk) a n d

the matrix ( ( a f k / a x i ) ) ,

1

k = row i n -

dex, i = column index, i s again called a Jacobian. The map f, i s a l s o sometimes denoted by df, which can cause confusion, and we will t r y t o avoid i t .

In particular consider a map q: J M where J C R i s say an open interval ( i . e . q i s a curve). Thus a generator of Tt ( J ) for to E J fixed can be denoted by d / d t and we consider q , ( d / d t ) = q ' ( t o ) E T Thus d / d t ( h ( q ( t ) ) ) q(t,)' = q ' ( t o ) ( h ) a t to a n d q ' ( t o ) i s called the tangent t o q ( - ) a t q ( t o ) . B u t i f q k are local coordinates a t q ( t o ) given by (Wy$) a n d d c k / d t = v k (where q^ = p q ) then ( d / d t ) h ( q ( t ) ) = 1 ( a h / a q k ) ( d 6 , / d t ) = 1 v k ( a h / a q k ) which shows t h a t q ' ( t o ) = 1 v k ( a / a q k ) . Conversely t h i s formula could be used t o define the -+

t a n g e n t vector t o a curve q ( t ) . If f : M + N i s a C' map then i t defines a map f,: T M -+ p 1 N, and g E C rule f,v(g) = v(g-f) where v E T M, f,v E T T f ( p ) N by the P f(P) ( f ( p ) ) i s a r b i t r a r y . The e f f e c t on basis vectors i s shown above a n d i f M = J i s an open interval, with t + f ( t ) a curve i n N, f,(d/dt) i s the tangent

1 vk(a/ayk) to

f ( t ) a t to (here v k = dyk/dt a t toyd / d t E Tt ( J ) , and the y k ( t ) a r e the coordinates of f ( t ) . One should note t h a t f, 0 does n o t generally provide a map of vector f i e l d s (= sections of the tangent

vector f ' ( t o ) =

bundles) since f ( p ) may equal f ( p ' ) for p

*

p'-

...

DEFZNZCZ0N A9. If u E A k T* i s given by u = 1 b ( . ) d x j , A A dxjK, j, < P ... A j,, then d u E A ~ + ' TP* i s defined by d u = 1 ( a b ( j ) / a x i ) d x i A d xj t A (sums over 1 2 i 5 n a n d j, < ... < j k ) . A form i s closed i f du A dx ju = 0 a n d exact i f u = d v f o r some form v .

...

$

348

ROBERT CARROLL d ( u A v ) = du A v + ( - l ) r s

The f o l l o w i n g f a c t s a r e e a s i l y proved ( e x e r c i s e ) :

Also i t i s c l e a r t h a t d(u+v) = du + dv. Next observe t h a t 1 N i s C as b e f o r e and p E (Ua,$a), f ( p ) E (WB,FB), f(Ua) C Wg,

u A dv, d2 = 0. if f: M

-f

t h e n one can d e f i n e a map f*: TT(p)N

+ T*M by t h e r u l e f * ( d g ) P Moreover f o r dg E T* M, v E T M, f(P) P ( d ( g e f ) , v ) = v ( g 0 f ) = f,v(g) = (dg,f,v) which e x h i b i t s f* and k maps. We can t h e n extend f* t o forms u E A T* a t f ( p ) by t h e

i s e v i d e n t l y well defined.

,... ¶ v k ) =

,..., f * vk )

= d(g-f).

This

(f*(dg),v)

=

f, as a d j o i n t

formula f * u

R e c a l l here t h a t P k f*u i s c o m p l e t e l y determined by i t s a c t i o n as a k - l i n e a r map on A T P' (v,

u(f,vl

EHE@REI A I O .

Iff : M

+

f o r vi

E

(see (A.1)).

T

N i s a C' map t h e n i t determines a map f*: TF(p)N

-f

T*M by t h e r u l e f * ( d g ) = d ( g 2 f ) f o r dg E TF(p)N. F u r t h e r f* i s a d j o i n t t o P k f, and i t extends t o a map f*: A TF(p)N -+ A'T*M ( k 2 0 ) determined as above P Further f o r k 2 1 and, when u i s a f u n c t i o n , by t h e r u l e f * u ( p ) = u ( f ( p ) ) . df* = f * d and f * ( u A v ) = f * u A f * v .

REmARK All.

The most n a t u r a l examples o f v e c t o r bundles a r e perhaps t h e

tangent and cotangent bundles o f a m a n i f o l d M.

To i n t r o d u c e t h e bundle

s t r u c t u r e we s h a l l r e d e f i n e them now i n a s l i g h t l y d i f f e r e n t way. here t h e Frechet d e r i v a t i v e d e f i n e d as f o l l o w s .

Thus i f f : U

+

We r e c a l l

F i s a con-

t i n u o u s map o f an open subset U o f a Banach space E i n t o a Banach space F then f i s F r e c h e t d i f f e r e n t i a b l e a t xo e a r map A: E x

0

.

-+

F such t h a t l i m ( l l f ( x )

-

E

U i f there e x i s t s a continuous l i n -

f(xo)

-

A(x-xo)lf)/llx-xoll

We w r i t e A = f ' ( x o ) = D f ( x o ) and f ' becomes a map U

-t

= 0 as x

L(E,F)

d i f f e r e n t i a b l e a t e v e r y p o i n t o f U; s i m i l a r l y , f " becomes a map U

-f

if f is -f

L(E,L(

L( Df(x): U E,F)) e t c . and t h e d e r i v a t i v e s a r e u n i q u e l y d e f i n e d . I f x 1 E,F) i s continuous we say f E C on U and s i m i l a r l y f o r h i g h e r d e r i v a t i v e s . -f

W r i t i n g L~(E,FI

f o r symmetric m - l i n e a r maps

I?"''+ F

-f

( i . e . m - l i n e a r maps un-

a f f e c t e d b y i n t e r c h a n g i n g two arguments) i t i s e a s i l y seen t h a t Dmf(xo)

L e t now U C El and V C E2 be open s e t s and suppose f : ous.

Consider ( x , y )

d e r i v a t i v e a t (x,y) exists).

E

U X V

E

+

F i s continu-

U X V and denote f o r example by Dlf(x,y)

t h e Frechet

o f f ( - , y ) w i t h r e s p e c t t o t h e f i r s t argument (when i t

Then ( e x e r c i s e ) Df(x,y) e x i s t s i f and o n l y i f b o t h D l f and 0 2 f

VECTOR BUNDLES

34 9

e x i s t a t (x,y) and a r e continuous; i n t h i s event one has Df(x,y)(vl,v2)

1 Dif(x,y)vi E.1

=

f o r vi E E.

=

The D i f a r e c a l l e d p a r t i a l d e r i v a t i v e s and f o r F =

R we o b t a i n t h e customary n o t i o n o f p a r t i a l d e r i v a t i v e .

The c o o r d i n a t e -

wise continuous d i f f e r e n t i a b i l i t y o f say $ $ i l on a m a n i f o l d M t h u s i s equiB v a l e n t t o t h e e x i s t e n c e and c o n t i n u i t y o f ( 4 4;’)’ which i n f a c t equals t h e B Jacobian J ( x ) ( e x e r c i s e ) . S i m i l a r l y i t means t h e same t h i n g t o say e.g. my $B$il C using e i t h e r E

wise v e r s i o n .

Frechet d i f f e r e n t i a b i l i t y

or

t h e usual c o o r d i n a t e -

Using t h e Frechet d e r i v a t i v e we can g i v e a n o t h e r d e f i n i t i o n

o f T(M) where t h e bundle s t r u c t u r e i s apparent ( c f . [ LA1 I ) . For x E M we c o n s i d e r t r i p l e s (U,@u) where (U,@) i s a c h a r t

DEfZNZCZdN A12.

a t x and u i s an element o f t h e v e c t o r space Rn i n which $(U) = U l i e s . say (U,$,u)

: (W,$,w)

i f ($$-’)‘u

= w (where ($$I-’)’

We

can o f course be de-

This i s c l e a r l y an e q u i v a l e n c e r e l a t i o n by t h e

f i n e d i n some NBH o f @ ( x ) ) .

c h a i n r u l e ( e x e r c i s e ) and i n p a r t i c u l a r one has t h e f a m i l i a r )-’ -1 $(X) The s e t o f e q u i v a l e n c e c l a s s e s o f such t r i p l e s a t x i s de= (@$ f i n e d t o be TxM and each c h a r t (U,$) a t x determines a b i j e c t i o n TxM Rn -f

( o n t o ) b y (U,$,u)-

-f

u.

One g i v e s TxM t h e t r a n s p o r t e d t o p o l o g i c a l v e c t o r

space s t r u c t u r e from Rn which i s e v i d e n t l y independent o f t h e c h a r t (U,$) selected t o e x h i b i t the b i j e c t i o n . t i o n TxM (U)

-f

-f

x.

I f (U,$)

Again s e t TM = U TxM w i t h

U X Rn determined by rUx(U,$,u)-

t e x t ) f o r x E U. t u r e o f U X R” t o

T

the projec-1

i s a c h a r t on M we have a n a t u r a l b i j e c t i o n = (x,u)

:T ,,

IT

( o r = u, depending on con-

We d e f i n e c h a r t s on TM by s i m p l y t r a n s p o r t i n g t h e Cm s t r u c -1 (U) b y a c h a r t (II (U), (9X 1 )oT”) and t h e n e x t e n d i n g

IT-’

t o a maximal f a m i l y on TM.

This c l e a r l y makes sense ( e x e r c i s e ) and TM w i l l

be a m a n i f o l d l o c a l l y isomorphic ( o v e r U ) t o U X Rn.

I f I? i s t h e c a t e g o r y

o f Cm manifolds where isomorphism means diffeomorphism one s t a t e s g e n e r a l l y

DEfZNZCZ6N A13.

Let E , M

E

M and

IT:E +

M be a morphism o n t o M.

Assume t h e

f i b r e I T - ~ ( x (complete ) i n v e r s e image) has t h e s t r u c t u r e o f an n-dimensional r e a l o r complex v e c t o r space f o r each x E M ( n <

i s t o be f i x e d f o r a l l x ) . -1 Assume t h e r e i s an open c o v e r i n g { V k 3 o f M and maps T,: IT (V,) V, X F m

-f

( t r i v i a l i z a t i o n ) where F i s a f i x e d copy o f Rn o r Cn c a l l e d the t y p i c a l -1 T, i s an isomorphism i n (B) T , ~ : IT ( x ) -+ F i s an i s o -

f i b r e such t h a t (A)

morphism o f v e c t o r spaces and

( C ) t h e f o l l o w i n g diagram commutes

ROBERT CARROLL

350

‘K

(A.2)

) V k X F

Evidently T~~ i s the r e s t r i c t i o n of T~ t o IT-’(x). Then extend ( V k , ~ k ) t o be a maximal family h a v i n g these properties and call the resulting object 71 = (E,M,F,*) a vector bundle ( V B ) . A section s of a VB i s a map s: M + E such t h a t IPS = identity. REIXARK A14, Let us make a few cursory remarks about singular homology a n d the deRham complex ( c f . [AOl;DU41). Thus very briefly k-cells on M a r e

t r i p l e s (D,f,O) = u where f : D M i s a smooth map of a k-dimensional polyk hedron D in R to M and 0 i s t o represent an orientation. If w E A k = kforms on M one defines Iu w = IDf*w. Chains a r e f i n i t e collections C = 1 k miui of c e l l s with multiplicity factors m i a n d given D C R as above with a D = 1 Di ( D i suitably oriented - we omit discussion of t h i s here) one defines for a c e l l 0,au = 1 u i where a i n, (Di,fi,Oi) ( d e t a i l s omitted - see [ A01 I). Similarly a C = 1 miaui. One says w E A k i s closed i f dw = 0 a n d via Stokes theorem J a C w = Jc dw we see t h a t JaC w = 0 i f w i s closed. A cycle on M i s a chain C w i t h ac = 0 so evidently Ic dw = 0 for any cycle. A form w i s called exact i f w = d v . If one considers the deRham complex 0 n k + R + A’ -t + . .. A -+ 0 (map i s d ) then H ( M , R ) = Kerdk/Imdk-l i s called the kth deRham cohomology group ( d k = d a t level k ) . A very beautiful s e t of theorems shows t h a t H k corresponds t o the real singular or Cech cok homology of M and moreover each cohomology class i n H will have a unique harmonic representative (Hodge theory); i n a d d i t i o n t h i s theory can be embedded i n the Atiyah-Singer complex of ideas ( c f . [ BS1; NS1 ;GY1; PA1 ;WR1 I ) . Locally over nice (topologically simple) regions the deRham complex i s however exact a n d t h i s i s the content o f the classical Poincar6 lemma (proof omitted - c f . [Cl;WRl I ) . k t?3KlA A15, Let U be a s t a r l i k e s e t in R n a n d l e t w be given with dwk = 0; then there e x i s t s vk-’ w i t h dvk-’ = w k -f

-f

.

We will also need the ideas o f i n t e r i o r product a n d Lie derivaThus the i n t e r i o r product i s determined by (ixwk )(51,...,5k-1) =

REmARK A16, tive.

L I E DERIVATIVE

351

1 k ( X , S ~ , . . . , S ~ - ~ 1 (ix = i ( X ) ) and t h e L i e d e r i v a t i v e o f a 1 - f o r m i s ( L ~ W) 1 t ( 5 ) = Dtw (g,5)lt,0 where gt i s t h e f l o w determined by X (see remarks below

ii w

-

e x p ( t X ) i n D e f i n i t i o n A24). gt % $,(e) v e c t o r f i e l d s S y n i s L5,nli = 1 Sjani/axj

Recall t h a t t h e L i e b r a c k e t o f

- n.ag./ax One can d e f i n e a L i e J 1 j' i n p a r t i c u l a r f o r a v e c t o r Y, ( L Y ) = Dt(gitY t ) X P 9 P e v a l u a t e d a t t = 0 ( = l i m ( g i t Y t - Y p ) / t as t + 0 ) . I t i s i n s t r u c t i v e t o 9 P draw a p i c t u r e here - c f . [ WR11- t h e g i t c o n t e x t i s a p o s i t i o n a l device. derivative of$nythin$and

One proves t h e n t h a t LxY = [X,Y]

( c f . [ C1;WRl

a r e one forms we w i l l l e t them a c t on Y. g )It=Otwhile

YDt(fog

Note n e x t t h a t L X d f = d L X f

To see t h i s n o t e t h a t s i n c e L X d f and d L X f

on f u n c t i o n s f (where L X f = X f ) . t

I).

Thus d ( L X f ) ( Y ) = Y ( L X f ) = YDt(f

t

t

0

t

L X ( d f ) ( Y ) = Dt
)ItZo as

Next one shows Lx = i ( x ) - d + d o i ( X ) .

To s e t t h e stage w r i t e i ( X ) w = X J w

f o r the i n t e r i o r product.

Also i t i s c o n v e n i e n t t o speak o f " a n t i d e r i v a k -+ A ~ + ' , e ( u A v ) = eu A v t ( - 1 ) u A ev ( k = t i o n " e i n AT*, namely e: k degree u; e i s c a l l e d a d e r i v a t i o n i f t h e r e i s no f a c t o r (-1) ). Thus d i s a n a n t i d e r i v a t i o n and so i s i ( X ) w h i l e L x i s a d e r i v a t i o n .

One shows e a s i l y

t h a t i f a ( s u p p o r t d i m i n i s h i n g ) map D s a t i s f i e s D ( f f ) = ( D f ) f + f, ( D f 2 ) 1 2 s and D ( f w ) = ( D f ) w t f D w ( f E A', w E A ) w i t h 0: Ao A and As+' (s

i1 1

-f

even) t h e n i t has a unique e x t e n s i o n t o a d e r i v a t i o n on AT* ( e x e r c i s e ) . But LX and i ( X ) o d + d o i ( X ) a r e d e r i v a t i o n s a g r e e i n g on flf2 and f w which i m p l i e s t h e y a r e equal on any form ( e x e r c i s e ) . LXf = (X,f)

Note here i f f € A',

= X ( f ) while i(X)df = (df,X) = X ( f ) .

i ( X ) f = 0 and

On b a s i s elements, LXdxi

=

dL x s i n c e LXd = dLX. On t h e o t h e r hand ( i ( X ) d + di(X))dxi = di(X)dxi = X i d(X,xi) = dL x so LX = i ( X ) d + d i ( X ) o n A T * . As examples l e t us c i t e X = X i Sia/axiy w = 1 aidxi. dw = 1 dai A dxiy X J dw = 1 ( X J dai) A dxi -

1

1

1 1 Ej(aai/ax.)dxi - 5.(aa./axi)dxi, X J J J 1 S.(aa./axi)dxi + 1 aj(ag./axi)dxi, Lxw = (i(X)od J J J (aa ./axi )dxi + 1 a . ( a S ./axi )dxi. J J J

dai A X J dxi =

w) =

RENARK A17,

1 aiSiy + doi(X))w =

-I w =

d(X -I

1 Sj

We g i v e here a few b a s i c ideas a b o u t L i e groups and a l g e b r a s

(cf. [TDl;WRl I);t h e r e w i l l be more o f t h i s a t v a r i o u s o t h e r p l a c e s i n t h e book.

Thus f i r s t some examples.

a l l n o n s i n g u l a r l i n e a r maps Rn

-f

GL(n,R)

(resp. GL(n,C))

Rn ( r e s p . Cn

-f

Cn);

i s t h e group o f

i t can be t h o u g h t o f

a l t e r n a t i v e l y as t h e group o f a l l i n v e r t i b l e n X n m a t r i c e s w i t h r e a l (resp.

352

ROBERT CARROLL

complex) c o e f f i c i e n t s . rices M

(i.e.

Fi = TM-’

L e t O(n) be t h e group o f r e a l orthogonal n X n matT M means t h e transpose o f M ) and l e t U(n) be

where

t h e group o f u n i t a r y n X n complex m a t r i c e s ( i . e . t h e complex c o n j u g a t e o f M I . group o f m a t r i c e s i n GL(n,R)

One w r i t e s SL(n,R) (resp. GL(n,C))

= ‘M-’

denotes f o r the

w i t h d e t e r m i n a n t equal 1 and

then we s e t SO(n) = O(n) n SL(n,R) w i t h SU(n) = U(n) SO(p,q) be t h e s e t o f m a t r i c e s i n SL(p+q,R)

where

(resp. SL(n,C)) SL(n,C).

Next l e t

l e a v i n g t h e form x12 +

...

+ x2

-

P i s t h e L o r e n t z group (we

2

i n v a r i a n t and n o t e t h a t SO(3,l) P+1 *-. p+q 2 2 2 2 use t h i s i n t h e form xo - x1 x2 x 3 u s u a l l y ) . We d e f i n e a l s o i n passing

- x

-

SP(n,R)

-

(resp. SP(n,C))

-

t o be t h e s e t o f m a t r i c e s i n GL(Zn,R)

C ) ) l e a v i n g i n v a r i a n t t h e form x1 A x

+

...

low). a6

-

( r e s p . GL(2n,

~ ++ ... ~ + xn A xZn ( r e s p .

‘1 ‘n+l A l l o f these groups a r e L i e groups.(see d e f i n i t i o n s be-

+ zn A z2,,). Consider e.g.

SL(2,CO now, c o n s i s t i n g o f 2 X

~y = 1 (hence SL(2,C) i s a 3

-

complex

-

2 m a t r i c e s (“ ’) w i t h y S

4

dimensional s u r f a c e i n C ) .

Since a,~,y,G cannot a l l v a n i s h t o g e t h e r one can express f o r example y as a f u n c t i o n o f a,’,&

when B

f

0 and thus t h r e e o f t h e q u a n t i t i e s a , B , y , 6

s u f f i c e as c o o r d i n a t e s i n t h e NBH o f any p o i n t o f SL(2,C). sing t h a t there i s a 2

-+

1 mapping o f SL(2,C) o n t o SO(3,l)

will

We n o t e i n pas( x and - x

+

y ) so

t h a t SL(2,C) i s l o c a l l y isomorphic t o SO(3,l ); t h e corresponding L i e a l g e b r a s s l (2,C) and so(3,l ) a r e isomorphic (see below).

I n f a c t the L i e alge-

c o n s i s t s o f 2 X 2 complex m a t r i c e s w i t h t r a c e 0 f o r which a ba0 1 0 0 ), H = (’ O). Note a l s o t h e expressions e x p ( t X ) - i s i s X = (o o)y Y = ( 1 0 0 -1 O t a l o n g w i t h t h e r e l a t e d expressions e x p ( t Y ) = and e x p ( t H ) = = (o (exp(t) ) which connect sl(2,C) and SL(2,C). 0 exp(-t) bra sl(2,C)

(i7)

DEflNlCI0N Af8.

A t o p o l o g i c a l group i s a group G w i t h a t o p o l o g y c o m p a t i b l e

w i t h t h e group s t r u c t u r e ( t h i s means t h a t (x,y) ous).

-+

xy-l: G X G

-f

G i s continu-

A L i e group i s a group G which i s a r e a l ( o r complex) a n a l y t i c mani-

f o l d w i t h t h e map (x,y)

+

xy-’

a n a l y t i c (when expressed i n l o c a l coordinates).

Here t h e p r o d u c t m a n i f o l d G X G i s d e f i n e d i n terms o f c h a r t s (Ua X UB,$a;;

4 ) i n t h e obvious way. B

REmARK Af9.

One can e a s i l y v e r i f y t h a t t h e groups above a r e L i e groups. One 2 uses here t h e t o p o l o g y and a n a l y t i c s t r u c t u r e determined by t h e n m a t r i x components i n GL(n,R) o r GL(n,C)

as l o c a l c o o r d i n a t e s ; t h i s i s t r a n s m i t t e d

LIE GROUPS

353

t o subgroups in t h e obvious way. Since the r e s t r i c t i o n MM* = I (M* = T-M or TM) f o r c e s the c o e f f i c i e n t s of M t o l i e i n a bounded region i n n2 dimensiona l real o r complex Euclidean space the groups O ( n ) , SO(n), U ( n ) , and S U ( n ) a r e seen to be closed compact subgroups o f G L ( n , R ) o r G L ( n , C ) ; S O ( n ) , S U ( n ) , SL(n,R), S L ( n , C ) , a n d U ( n ) a r e a l l connected while SO(n) i s the connected component o f the i d e n t i t y in O(n). We mention a l s o t h e Poincar; group c o n s i s t i n g o f t h e standard semidirect product of SO(3,l) w i t h t h e t r a n s l a t i o n s i n R 4 (inhomogeneous Lorentz group). T h u s G c o n s i s t s of p a i r s ( L , a ) w i t h L E SO(3,l) and a a real 4-vector where the product r u l e i s defined by ( L , a ) ( i , b ) = (L1,atLb). Then M = G / H ( H = SO(3,l)) is c a l l e d Minkowski space. We have discussed vector f i e l d s on a manifold e a r l i e r ; they form a module over the r i n g f ( M ) under t h e rules (X+Y)(f) = X(f) + Y(f) and (fX)(g) = X Y - YX i s a d e r i v a t i o n fX(g). Further i t i s e a s i l y v e r i f i e d t h a t [X,Yl i n COD(M)whenever X and Y a r e vector f i e l d s and t h a t t h e following Jacobi i d e n t i t y holds ( e x e r c i s e ) : [X,[ Y,Z11 + [ Y,[ Z , X l l t [ Z,[ X , Y l l = 0. Thus the s e t of vector f i e l d s on M forms a Lie algebra where this i s defined f o r f i i t e dimensional s i t u a t i o n s by DEFZNZCZ0N A20, A Lie algebra i s a ( f i n i t e dimensional) real o f complex vector space F w i t h a b i l i n e a r operation (X,Y) .+ [ X , Y l such t h a t t h e Jacob i d e n t i t y holds w i t h [ X , X l = 0. A morphism h : F -+ E o f real o r complex Lie algebras i s a l i n e a r map such t h a t [ h f , h f ' ] = h [ f , f ' ] f o r f , f '

E

F.

. g + pg: G + G the l e f t t r a n s P' l a t i o n by p. I t i s evident t h a t T i s an a n a l y t i c diffeomorphism ( e x e r c i s e ) P The map ( T ~ ) * = T T ~ :Tx(G) -+ T ( G ) i s denoted a l s o by d.r sometimes and i f PX P v deX i s a vector f i e l d , w r i t t e n a l s o g +. X * M TM, then g -+ ( T ~ ) * X ~= X PS g' termines another vector f i e l d y o n G . We say t h a t X is l e f t i n v a r i a n t i f x P9 -- x P g * Now given 5 E Te(G) ( e 'L i d e n t i t y ) one can c o n s t r u c t a unique l e f t i n v a r i a n t (C") vector f i e l d X on M such t h a t Xe = 5 . Indeed define X P = (TT ) 5 and noting t h a t t h e diagram p e REllARK A21,

Let now G be a Lie group w i t h

T

-f

ly

354

ROBERT CARROLL (TT ) (TT ) 5 = (TT ) 5 which i s l e f t SP P e SP e show uniqueness l e t Y be l e f t i n v a r i a n t w i t h Ye = 5 which

commutes we have (see [Cl;LAl;WRlD

To

invariance.

i m p l i e s (TT ) Y = 5; b u t (TT (TT - I ) = i d e n t i t y and hence Y = Xp. It P-' P P P e P P P remains t o check t h e smoothness, i . e . we want t o show I f E C"(G) when f E = v(g f ) ) ( X f ) ( p ) = ( X ,df) X (f) P P P and t h i s can be p u t i n t h e form ( X f ) ( p ) = Dtf(py

Now ( r e c a l l T f = f, and (f,v)(g)

C"(G).

= 5(foT ) P p e (t))ltzo where y i s a smooth curve, i . e . a Cm map J

=

(TT ) S ( f ) P e

+

G w i t h J an open i n -

t e r v a l c o n t a i n i n g t h e o r i g i n , such t h a t y ( 0 ) = e and y ' ( 0 ) = y * ( D t )

Indeed g i v e n such a

(smooth may a l s o mean a n a l y t i c , depending on c o n t e x t ) .

Y, one has

Dt(f0T

Po y I 0 = ( TPo Y ) * ( D t ) ( f ) p

5

=

= (TTp)(Y*Dt)(f)p

= 5(feTp)e-

Con-

s e q u e n t l y from above X i s e a s i l y seen t o be smooth as r e q u i r e d ( t h e e x i s t e n c e o f such curves y i s covered below).

We observe n e x t t h a t i f X and Y

a r e l e f t i n v a r i a n t v e c t o r f i e l d s t h e n so i s [ X,Y I. To see t h i s suppose N

!w

i n which case f o r f € G and w r i t e dg(X ) = X p I t f o l l o w s t h a t X(Yf)og = X(7f'fog) = X(Y(fo+)) one has (?f)o$ = X ( f o $ ) .

f i r s t t h a t $ i s a map G

C"(G)

+

U I V

and hence dg[X,Yl

= [X,Yl.

Therefore t a k i n g g =

T

P

we see t h a t [ X , Y l

is

l e f t i n v a r i a n t w i t h X and Y which g i v e s

mE0REF1 A22,

L e t G be a L i e group.

Then t h e l e f t i n v a r i a n t (C")

vector

f i e l d s o n G form a L i e a l g e b r a g which induces a L i e a l g e b r a s t r u c t u r e on Te(G) under t h e r u l e [ X , Y l e (C")

= [ 5,dwhere

v e c t o r f i e l d s a s s o c i a t e d w i t h 5,n

E

X,Y a r e t h e unique l e f t i n v a r i a n t

Te(G).

g i s called the Lie alge-

b r a o f G and we always i d e n t i f y i t w i t h Te(G) a s i n d i c a t e d .

REmARK

A23,

L e t J be an open i n t e r v a l c o n t a i n i n g 0 and l e t G be a L i e group

An a n a l y t i c homomorphism (when d e f i n e d ) g: J meter subgroup o f G ( n o t e t h a t sed.

I f X i s a (C")

$(O)

= e);

-+

G i s c a l l e d a l o c a l one para-

i f J = R t h e word l o c a l i s suppres-

v e c t o r f i e l d on G t h e n a l o c a l i n t e g r a l c u r v e f o r X a t

p i s a smooth map y: J

-+

To Y(t)' be a c h a r t a t p; t h e n i t i s

G w i t h y ( 0 ) = p and ~ ' ( 0 )= Y * ( D ~ ) =~ X

show t h a t l o c a l i n t e g r a l curves e x i s t l e t (U,g)

easy t o see t h a t a v e c t o r f i e l d X on G can be w r i t t e n l o c a l l y a s X = a/axi

w i t h Xi

tem o f ODE.

E

and t h e e q u a t i o n y ' ( t ) = X

Thus x i ( y ( t ) )

f = (Dt),f(v(t)).

(y(t)) =

C"(U)

1 Xi

= Xi(t)

can be w r i t t e n a s a sysY(t) and observe t h a t i f f E C"(G) t h e n y*(Dtlt

B u t now f ( y ( t ) ) = ( f o + - ' ) ( $ ( y ( t ) ) )

1 (a?/axi)(axi/at)

which one w r i t e s as

=

?(+h(t))) so Dtf

1 (af/axi)(dxi/dt)

according

L r E THEORY

N

=

-

Consequently t h e e q u a t i o n s a r i s i n g from ~ ' ( t= )X

t o o u r conventions.

can be expressed i n t h e form dxi/dt where Xi

355

,...,x n ( - ) ) ,

= Xi(xl(-)

0 - l and a n i n i t i a l c o n d i t i o n y ( 0 ) = p i s p r e s c r i b e d .

XiO

Y(t)

i = l,...,n,

Now by

s t a n d a r d theorems i n ODE t h i s system always has unique l o c a l s o l u t i o n s ; n o t e t h a t t h e Cm f u n c t i o n s

Yi

s a t i s f y a L i p s c h i t z c o n d i t i o n f o r example and t h u s I n particular, i n order t o

t h e so c a l l e d P i c a r d - L i n d e l o f f theorem a p p l i e s .

produce t h e c u r v e s y o f Remark A21, one c o u l d use any g l o b a l Cm v e c t o r f i e l d That such v e c t o r

X w i t h Xe = 5 and l e t y be a l o c a l i n t e g r a l c u r v e f o r X.

f i e l d s e x i s t i s e a s i l y seen. w i t h (U,$) define

a chart a t e =

for p

T;? '

P

compact, l e t J,

E

E

Indeed t a k e a t r i v i a l i z a t i o n

G, and w r i t e

Cm(G) be 1 on

vector f i e l d X =

T(U)

T":

?

+

U X Rn

Then

5 = ( r e c a l l 5 E Te(G)). Ue C U with U and, p i c k i n g open s e t s V c C W C

E

T

v w h i l e JI

= 0 outside o f W (exercise); the

t h e n f u l f i l l s o u r needs.

v e c t o r f i e l d on G w i t h Xe = 5 ( i . e . X

I?

w

Now l e t X be a l e f t i n v a r i a n t

= (TT ) <) and check a s a n e x e r c i s e

,.

P e t h a t X i s i n f a c t a n a l y t i c ( i . e . t h e Xi i n an e x p r e s s i o n X = 1 Xi(a/axi), Xi -1 L e t $,(PI be t h e unique l o c a l i n t e g r a l c u r v e a t , a r e a n a l y t i c ) . = Xio $ P ( i . e . $,(PI = P ) . BY uniqueness one has $ s ( $ t ( p ) ) = $s+t(P) w i t h $ - t ( $ t ( p ) ) = p, whenever e v e r y t h i n g makes sense, and a g a i n b y standard theorems

is i n some open NBH o f (0,e) f o r example on ODE ( c f . [cn 1) (t,p) -+ w h i l e p -+ $,(p) i s a diffeomorphism o f some open NBH o f say e o n t o i t s image. For s u i t a b l e t,p,g =

(TT ) X =

we can w r i t e i n an obvious n o t a t i o n ( $ t e ~ p ( g ) ) '

(T o

$t(g))'

and t h i s means t h a t

P g g P s i n c e t h e i n i t i a l c o n d i t i o n s a r e t h e same.

T

0 4 =~

P Hence g i v e n $,(el

some i n t e r v a l J we can d e f i n e $ t ( p ) f o r any p b y $,(p) means i n p a r t i c u l a r t h a t t

-+

.

one has t h e e q u a t i o n $s+t(e) g e t a n a l y t i c s o l u t i o n s o f Xi

$,(el

defined i n o$,(e).

f o r example t o a l l values

Hence i t makes t o d e f i n e

-+

exp(Xe):

W r i t i n g q s ( e ) = cpSt(e) one has q;(e) = exp(tXe).

T

be t h e (unique) one parameter subgroup o f G gen-

exp(Xe) ( o r expX) and t h e map Xe

s = 1, +,(e)

=

-

Then $,(e)

e r a t e d b y t h e l e f t i n v a r i a n t v e c t o r f i e l d X. map,

P

= $ ( $ ( e l ) = T (e)o$,(e) = $,(e)eS(e) (we s, t -@t = Xi when t h e Xi a r e a n a l y t i c c f . [CTl I ) .

of t € R u s i n g t h e group p r o p e r t y . L e t 4,(e)

XT ( 9 )

b y uniqueness,

This P i s a l o c a l a n a l y t i c homomorphism s i n c e

F u r t h e r we may now extend t h e d e f i n i t i o n o f $,(el

DEFZNZFZBN A24.

T~

;-+

i s d e f i n e d t o be

G i s c a l l e d the exponential

= t$;(e)

= t X e so t h a t ,

setting

ROBERT CARROLL

356

RrmARK A25.

We remark t h a t when G = G L ( n , R ) f o r example the exponential map i s t h e usual exponent o f a matrix ( r e c a l l ? = L ( R n ) ) . Similar situations

occur f o r Lie subgroups o f G L ( n , R ) .

One should observe t h a t t h e exponential map does n o t always map o n t o G even when G i s connected. A standard counterexample i s provided by G = SL(2,R) ( c f . [ TO1

I).

5

A few examples o f pairs (G,;) a r e as follows: ( A ) G = G L ( n , R ) and = L(Rn) = gl(n,R) = a l l l i n e a r maps R n + Rn ( B ) G = S L ( n , C ) and = sl(n,C) = a l l n X n matrices w i t h t r a c e 0 ( C ) G = U ( n ) and = u ( n ) = a l l skew hermitian ma= 0 ) ( 0 ) G = S O ( n ) and = s o ( n ) = a l l skew symmetric trices (i.e. M + T matrices ( 1 . e . M + M = 0 ) ( E ) G = S O ( p , q ) and s o ( p , q ) = matrices o f t h e form M = (5' 9' where X and X 3 a r e skew symmetric o f orders p and q xi x, 1 r e s p e c t i v e l y and X 2 i s a r b i t r a r y ( F ) G = S P ( n , R ) and =; sp(n,R) = matrices o f t h e form M =('I x, 'x,") where X 2 and X3 a r e symmetric n X n a n d X1 is a r b i t r a r y . To be more e x p l i c i t consider t h e r o t a t i o n group G = S O ( 3 ) w i t h 0 0 0 0 0 1 0 -1 0 ni E y = so(3) a1 = [ O 0 -1 ] ; a 2 = 0 0 0];a3= (A.4) 01 0 -1 0 0

t

[ b ;;]

[

Then

r o t a t i o n s of angle

rices

ak(e) =

c=

e a r o u n d t h e x k a x i s in R 3 a r e described by mat-

expake and e v i d e n t l y a k = ( d / d e ) a k ( e ) l e z o . T h u s the

.,(e)

are

one parameter s u b g r o u p s of G and we have f o r example Cose -Sine 0 a3(e) = Sine Cose 0 (A.5) [ o 0 1

I

We note i n passing a l s o t h a t t h e m u l t i p l i c a t i o n t a b l e f o r s o ( 3 ) i s determined by[a1,n21 = a 3 , [a2.n31 = n l y 1a3,n1 1 = a 2 . For t h e Lorentz g r o u p SO(3,l) we add a fourth row and fourth column of zeros to t h e a k o f (A.4) and denote them again by a k E S o ( 3 , l ) ; then elements Bk E s o ( 3 , l ) a r e defined by B~ = 1 i n t h e ( 1 , 4 ) and ( 4 , l ) positions, B~

=

1 in the (2,4) a n d

( 4 , 2 ) positions, B~ = 1 i n t h e ( 3 , 4 ) and ( 4 , 3 ) positions, and the o t h e r ent r i e s a r e 0. One s e t s .(,(e) = expake with B k ( e ) = expBke so t h a t the a r e a s before w i t h an additional 1 in t h e ( 4 , 4 ) position while e.g. f Coshe 0 0 Sinhe q(e) = 0 1 0 0 (A.6) o i o Sinhe 0 0 Coshe

lo

J

Cck(e)

R I EMANN I A N GEOMETRY

357

The m u l t i p l i c a t i o n t a b l e i n s o ( 3 , l ) i s t h e n determined b y [al,a21 = a 3’ [“2’ a ] = a1 3 [a3,”1 1 = a 2 , [al,B1 I = [a2,B21 = [a3,B31 = 0, [B1,B31 = - a 3 , tB2, B3 I = = -B3,

[B3sB1 1 =

[ ~ 1 , B 2 1 = 839 [ai,B31 = -B2,

1

[a2,B31 = B1”ay2’B1

[a3’B1

I

A26.

We w i l l make a few remarks here a b o u t Riemannian t y p e geometry;

= B2,

and [ c c ~ = -B1. ,B~I

f o r ideas o f connections and c u r v a t u r e we r e f e r t o t h e t e x t where t h i s i s developed as needed ( c f . [ CUl;DU4;NSl;SXl;TTl

] f o r general r e f e r e n c e s ) .

We

r e f e r t o e a r l i e r remarks about t e n s o r p r o d u c t s f o r background and t h i n k i n g i o f V = T M one d e f i n e s elements g = g. .e R eJ as c o v a r i a n t t e n s o r s o f r a n k P 1J i 2. Here e (resp. e l ) i s a b a s i s o f V ( r e s p . V*) so ei % a/axi and e % dxi.

i

E v i d e n t l y gij degenerate.

= g(ei,e.)

J

and g i s c a l l e d a m e t r i c i f i t i s symmetric and non-

We w i l l o n l y be concerned w i t h L o r e n t z m e t r i c s where gij

assumes always t h a t t h e r e i s a smooth dependence x

-f

= (1,

2 0.

-ly-ly-l) i n a s u i t a b l e b a s i s o r i n Riemannian m e t r i c s where gij

One

g(x) w i t h l o c a l i z a t i o n s

One The i n v e r s e o f t h e m a t r i x ( ( 9 . . ) ) i s dentoed by ( ( g i j ) ) . 1J has an i n d e x r a i s i n g and l o w e r i n g c o n v e n t i o n which goes as f o l l o w s . Given a

as i n d i c a t e d .

m e t r i c g on M a s above, f o r x E M, g ( x ) induces a map Gx: TxM

-+

M T;

v i a Gx

(v)w = gx (v,w) (gx % g ( x ) ) . I n p a r t i c u l a r ( u s i n g s u p e r s c r i p t n o t a t i o n on i i i i j i v a r i a b l e s x now) i f v = v a/ax t h e n Gx(v) = v l d x = g. .v dx (summation o n i k i k ) = gikv w h i l e gij repeated i n d i c e s ) s i.n c.e Gx(v)(a/ax ) = g(v a/axl,a/ax i j i k Thus i n terms o f components v , v dx (a/ax ) = g. .vJ6; = g v j = g j k v j . 1J kj vi = g. . v j and Gx l o w e r s i n d i c e s ( w h i l e Gi,’ r a i s e s i n d i c e s ) . Here t h e i n 1J ij i verse map ’G; i s determined v i a ( g ’ j ) ) = ((gij))-’ w i t h G;(w) = g w.a/ax J i i i i ij = w a/ax f o r w = widx (i.e. w = g w.). L e t us d e f i n e here a l s o t h e J Hodge * o p e r a t o r . Thus l e t t h e o r i e n t e d volume element be TI = ( l / n ! ) n ( i ) 1 dxil A A dxin = ( g I 4 d x A A dxn where l g l = l d e t ( ( g . . ) ) l and d e f i n e 1J = sgndet( (gi j ) ) g c i y j ) ~( j ) i n an obvious n o t a t i o n (e.g. (i,j) % ( i k y j k )

p)

...

...

-

) nj, ,..j,). Any k - f o r m can be w r i t w i t h g(iyj) % I I ~g’kjk; s i m i l a r l y Q ( ~ % t e n as a = ( l / k ! ) a ( i ) d x ( i ) where a(i) = a(ei, ,...,e ) f o r eij a/ax i i ). k ik Now *: A -+ An-‘ i s d e f i n e d b y * c t ( ~ ~ + ~ , . . . , vn )TI = a A Gx(~k+l).A A Gx(v,) J I .jn and t h e components a r e g i v e n by ( * a ) . = (l/k!)a,j, ...jk TI ik+, in i6+, j k+l I n p a r t i c u l a r i n R 3 w i t h s t a n d a r d m e t r i c g one has *1 ( c f . [TTl I ) . Igimjn2 2 3 2 3 = e A e A e *el = e A e , * ( e A e ) = e l , e t c . w i t h ** = 1. G e n e r a l l y ** = ( - l ) k ( n ) o nk k-forms where s i n t h e i n d e x o f g ( i . e . s i s t h e

...

...

...

+

-

ROBERT CARROLL

358

-

number of vectors i n a n orthonormal basis for which g ( e i , e i ) = -1 orthonormal means g ( e i y e i ) = tl a n d g ( e i , e . ) = 0 for i j ) . For Minkowski space J w i t h g % (1,-1,-1,-1) one has *1 = - n ( q = e0 A e 1 A e 2 A e 3 ), *eo = -e 1 A e 2 ~ e3, * e ' = - e O 2~ ~e e3, e t c . * ( e O1 ~ e )2 =Aee 3, * ( e 2 A e3 ) = - e0 A e l , etc., *(ep A e1 A e 2 ) = -e 3 , *(e1 A e 2 A e 3) = -eo, etc. and ** = (-lf+! We remark finally t h a t 6: A -+ Ak-' i s defined by 6a = ( - 1 ) n ( k + l )+s+l*d*a k for a E A and the Laplace-Beltrami operator is A = 6d + d 6 : A k -+ A k . In R 3 w i t h s = 0, 6a = (-1) 3(kt1 )"*d*a = (-1)3k*d*a while i n R4 w i t h s = 1 the -1 factor is 1 = (-1)4(kt1 It'. Using the index raising o r lowering convention i i i i 3 i c v . d x tf v a/ax i n the form viei 'L v ei one has i n R , 5 = v e . -+ 5 = v.ei 1 7 1 -+ *F E A' -+ S? = -*d*r E Ao and one writes divc -6?. Similarly one can define the curl a s curl6 = (*dt)".

+

RmARK A27. Symplectic geometry arises frequently i n the text and we give a sketch o f a systematic exposition (cf. [AOl;DU4;GLl;LMl 1 for a d d i t i o n a l information). F i r s t recall for mechanical situations w i t h coordinates q i and momenta pi = mivi ( i = l,...,n say) one considers kinetic energy T = 2 In the Lagrange formulapi/mi and potential energy U = U ( q i ) . +Cmiv: = t tion one s e t s L = T - U with @ ( q ) = ft' L ( t , q , q ) d t where q E A = admissable 1 0-r Classical varia(say C ) trajectories between q ( t o ) = qo a n d q ( t l ) = tional arguments give immediately t h a t a curve q ( t ) i s an extremal o f @ ( q )

:,.

provided Dt(aL/a{) = a L / a q along the curve q ( t ) . For q 'L (qj), p "u ( p j ) , etc. t h i s becomes a system D t ( a L / a + ) = a L / a q i ( p i = a L / a 4 i i s called a general ized momentum). These equations are called Lagrange's equations (or Euler-Lagrange equations) a n d the fact t h a t motions o f the system coincide with such extremals (which are often minimizing for @ ) i s referred t o as Hamilton's principle o f l e a s t action. We distinguish however @ ( q ) from the action integral S defined by S ( q , t ) = f L d t where y i s the extremal path Y connecting ( q o y t o ) t o ( q , t ) (see below for more on t h i s - again q % ( q i ) , etc. i s implicit). Evidently one recovers Newton's second l a w from the Lagrange equations when the force i s derived from a potential. Thus for L = Ign6' U ( q ) we have Dt(m{) = -U = F ot F = ma ( i f m i s constant). Consider 9 next e.g. planar motion in a central force f i e l d i n polar coordinates q1 = r a n d q2 = 0 . Thus U = U(r) a n d , l e t t i n g vr and v, denote u n i t vectors in the i radial a n d tangential directions respectively, r = Fv r + i r v , (exercise).

-

SYMPL ECTIC GEOMETRY

359

+ r2G2) so t h e generalized momenta a r e p1 = aL/acjl mr 208 . The Lagrange equations a r e then m r mre- 2 - U r and Dt(mr2.0 ) = 0. The coordinate 8 = q 2 is c a l l e d c y c l i c s i n c e a L / a q 2 = 0 2and p2 = mr e is then a constant (conservation of angular momentum). One can w r i t e now mF' = mM2/r3 - U = - V f o r V = U + ( M 2 m/2r 2 ) = e f f e c t i v e potr2 e n t i a l energy (setting 8 = M/r ). One notes t h a t f o r E = T + U (energy) = + V the conservation of energy follows from Et = mFr + FV = b(m? t V r ) 1-r = 0. Consequently from Mi2 = E-V we obtain F = ( ( 2 / m ) ( E - V ) ) ' and t h e o r 2 bits can be found via = M/r2 = ;(d@/dr) i n t h e form e = 1 ((M/r ) d r / ( ( Z / m ) (E-V))'. As an e x e r c i s e apply this t o an inverse square force (U = -k/r) and derive Kepler's laws o f planetary motion ( c f . [A01 I ) . Recall here t h a t conic s e c t i o n s have equations r = a/(l+eCose) e t c .

Then T

= +?n?'

m; and p2 =

=

+all(;'

aL/aG2

=

Thus l e t y = f ( v ) be a convex function (say f " > 0 f o r f d i f f e r e n t i a b l e ) a n d l e t v ( p ) be t h e point wher: pv - f ( v ) = F ( p , v ) is maximum; one defines then g ( p ) = F ( p , v ( p ) ) . Thus Fv For example an easy c a l c u l a t i o n = 0 a t v ( p ) and f ' ( v ) = p determines v ( p ) . 2 shows t h a t i f f ( v ) = +mv then F ( p , v ) = pv - +mv2, v ( p ) = p/m, and g ( p ) = 2 p /2m. To see t h a t t h e map f + g is involutive one considers G ( v , p ) = v p g ( p ) and i t follows t h a t max G ( v , p ) = f ( v ) ( v = vo f i x e d ) a n d p ( v o ) = f ' ( v o ) . We leave t h e d e t a i l s a s an exercise ( c f . A01 ;Cl;SCl I and draw some p i c t u r e s ) . Now one defines the Hamiltonian H = H ( p , q , t ) = p: L ( q , 4 , t ) a s the Legendre transform o f L ( q , ; , t ) (4 = v + p; H(p) = p{ - L ( q , G , t ) ; and here L i s a s sumed convex i n {). Then dH = ( a H / a p ) d p + ( a H / a q ) d q t (aH/at)dt i s equal to d ( p v - L ( q , v , t ) ) when p = Lv = aL/aG. Thus dH = v d p - ( a L / a q ) d q - ( a L / a t ) d t and one o b t a i n s ( r e c a l l L = and note pv d p - L v d p = 0 ) q P V P ZliWREm A28, The Lagrange equations a r e equivalent to the Hamilton equat i o n s qi = a H / a p i ; bi = - ( a H / a q i ) ; Ht = -Lt where H ( p , q , t ) = pq - L ( q , { , t ) i s the Legendre transform o f L ( 4 + p ) . We r e c a l l now the Legendre-Fenchel transform.

-

The t r a n s i t i o n t o Hamil t o n ' s equations is not just a n a r t i f i c e . A29, I t allows one to phrase the theory i n t h e cotangent bundle ( o r phase space) and u t i l i z e t h e techniques o f symplectic geometry. We will see l a t e r how this conceptual change is productive. Indeed placing ourselves now on a manifold M w i t h l o c a l coordinates q i l e t us sketch t h e theory. One can

RrmARK

360

ROBERT CARROLL

imagine t h e need f o r w o r k i n g o n a m a n i f o l d i f we t h i n k o f v a r i o u s c o n s t r a i h t s imposed on a dynamical system so t h a t t h e m o t i o n i s f o r c e d t o t a k e p l a c e on 3n say. We denote by TM t h e t a n g e n t bundle and some (smooth) subset M o f R cotangent bundle..

T*M

Observe t h a t a c u r v e q ( t ) on M corresponds t o a tan-

gent vector q(0) = v a t t = 0 i n the form v recall v ( f ) = (v,df)

= Dtf(q) a t t = 0).

%

1 ai(a/aqi)

-

w i t h ai = ;li(0)

One notes t h a t T*M = N i s even

dimensional and f o r s i m p l i c i t y we t a k e dimT*M = 2m (where 2m

%

6n).

A sym-

p l e c t i c s t r u c t u r e on N i s determined b y a c l o s e d nondegenerate 2-form w2 ( i . 2 e. dw2 = 0 and f o r a l l 5 4 0 t h e r e e x i s t s q such t h a t w (5,q) = 0, <,q E T*M has t h e n a t u r a l s y m p l e c t i c s t r u c t u r e determined b y w2 =

TxN). dqi.

To see t h i s ( f o l l o w i n g [A01

I)

l e t f : T*M

1 dpi

A

M be t h e p r o j e c t i o n T*M 3 9 TM t a k e s 5 t o a v e c t o r f& t a n -+

Then f,: T(T*M) + q and suppose 5 6 T (T*M). P 1 D e f i n e a 1 - f o r m e ( 5 ) = w ( 5 ) = p(f,.) ( i n l o c a l c o o r d i n a t e s 1 e v i d e n t l y w1 = 1 pidqi and 8 i s c a l l e d t h e L i o u v i l l e form). Then w2 = dw gent t o M a t q.

i s nondegenerate ( e x e r c i s e

-

n o t e here one can w r i t e p

% lBj(a/aqj), (a/apj) + aj(a/aqjl, e,f c o n s t r u c t i o n o f w' i s p e c u l i a r t o T*M

-

1

%

1 pidqi, 5 1 aj (1 pidqi)(c)). his %

p(f,c) % pjbj t h e r e i s no analogous form o n TM f o r

example. 1 Now t o each 5 E T(T*M) one a s s o c i a t e s a 1 - f o r m w1 E T*(T*M) by t h e r u l e w5 5 2 2 (TI) = w ( n , ~ ) , f o r a l l 11 E T(T*M) (w as above), and we denote by I t h e map

w51

T(T*M)). We w i l l s p e l l t h i s o u t i n l o c a l c o o r d i n a t e s as 2 1 ai(a/api) + Bi(a/aqi), 0 % f o l l o w s ( n o t e w (TI) = w (v,Iw 1). Set 5 % +.

5 (T*(T*M)

-+

1

+

yi(a/api)

while

1

si(a/aqi),

1 ajyjl+

=

we(q)

1

1

and w:

bjsj.

%

1 ajdpj

+ bjdqj.

Then

It f o l l o w s t h a t a

B . and b

= -a I n parJ j j' 1 T*M (any such f u n c t i o n ) t h e n t o dH % w =

j

t i c u l a r if H i s a C f u n c t i o n on on T*M one a s s i g n s a v e c t o r f i e l d 5 = IdH which i s c a l l e d k m i l t o n i a n . d e r t h e c a l c u l a t i o n s above f o r dH =

1 (aH/api)dpi

t h e H a m i l t o n i a n v e c t o r f i e l d (ai = -(aH/aqi) (-aH/aqi)(a/api) dqi so (dH,q) has

=

+ (aH/aqi)dqi

and B~ = (aH/api))

one o b t a i n s IdH =

1

1

+ (aH/api)(a/aqi). N o t e here dH = (aH/api)dpi + 2 = w (n,IdH); i n p a r t i c u l a r f o r q = Si(a/api) + qi(a/aqi)

1

1 ~ ~ ( a H / a p+~qi(aH/aqi), )

5

Un-

and

one

SYMPLECTIC GEOMETRY

ci ni (dqi,IdH)

= ‘I(dpi,IdH)

Therefore IdH =

1 (-(aH/aqi

361

I

)a/ap. + (aH/api )a/aqi)

#’

g i v e s agreement i n t h e

I % #: T* +. Tp). P Then i n l o c a l c o o r d i n a t e s t h e map I:dH +. IdH: ((aH/api)dpi + (aH/aqi)dqi) + (-(aH/aqi)a/api + (aH/api)a/aqi) has t h e form ( - 0I o1) ( w o r k i n g o n c o o r formulas and IdH i s c a l l e d (dH)

i n [ C S l l f o r example ( i . e .

1

1

0 1 O)(H

d i n a t e s i n p,q o r d e r t h i s corresponds t o an a c t i o n o f t h e form (

H ) q P The v e c t o r f i e l d

- H ), t h e l a t t e r two v e c t o r s b e i n g column v e c t o r s ) . P q IdH on T*M now g i v e s r i s e t o a f l o w which we assume t o be a one parameter group o f diffeomorphisms gt : T*M -+ T*M. One i s s i m p l y s o l v i n g t h e ODE as= (H

s o c i a t e d w i t h IdH

%

5 here ( i . e . i i ( t )

= Ci(t)

s t a r t i n g a t some p o i n t qo and

working i n l o c a l c o o r d i n a t e s ) . We r e f e r t o CAO1-3;Cl;Al;CTl;LM11 f o r a stut t dy o f such flows. Thus Otg q(t=O = IdH(q) and g i s c a l l e d t h e H a m i l t o n i a n t phase f l o w a s s o c i a t e d w i t h t h e ( H a m i l t o n i a n ) f u n c t i o n H. Now r e c a l l ( g )* t t w 2 ( ~ , n )= w 2 (g,s,g,rl ) by d e f i n i t i o n s and one can prove ( c f . [ A O l ; A l ; C l I )

A H a m i l t o n i a n phase f l o w preserves t h e s y m p l e c t i c s t r u c t u r e 2 2 gt*w2 = w where w i s g i v e n l o c a l l y by 1 dpi A dqi).

CHE0RZ111 A30. (i.e.

REIIIARK A31.

N

The theorem i s i m p o r t a n t ; f o r M = R w i t h

= T*M

%

R

2 this i s

t h e famous theorem o f L i o u v i l l e a s s e r t i n g t h a t t h e phase f l o w p r e s e r v e s area The theorem shows a l s o t h a t w t o n i a n phase f l o w f o r 5 = IdH (=

n)

i n the direction

-

2

i s a so c a l l e d i n t e g r a l i n v a r i a n t o f a Hamil-

2

.

2

F u r t h e r s i n c e dH(5) = w (S,Idh) w2 = f t w C g c we o b t a i n dH(n) = w x ( ~ , v ) = 0 so t h a t t h e d e r i v a t i v e o f H

n

i.e.

J

i s 0 (dH(n) = n(H) e t c . ) .

t e g r a l o f t h e phase f l o w determined by H ( i . e .

T h i s says t h a t H i s a f i r s t i n H = constant along the flow

and t h i s corresponds t o c o n s e r v a t i o n o f energy). Given now t h e s y m p l e c t i c m a n i f o l d T*M = N and wL d e f i n e a s above, t o any 1 t s u i t a b l e C f u n c t i o n H o n N we have a phase f l o w gH and t h e Poisson b r a c k e t o f two such f u n c t i o n s F and H i s d e f i n e d as t h e d e r i v a t i v e o f F i n t h e d i r t t 2 e c t i o n o f t h e phase f l o w gH, i . e . {F,HI(x) DtF(gH(x))lt,o = dF(1dH) = w ( I d H , I d f ) = -{H,F)(x). (aF/aqi)

-

Note IdH

(aH/aqi)(aF/api)

%

= {F,H)

-H D

2

+ H D and w ( I d H , I d f )

1

% (aH/api) P q so t h e canonical equations o f m o t i o n

q P

362

ROBERT CARROLL = {pk.Hl.

become Dtqk = (aH/apk) = { q k y H l and Dtpk = -(aH/aq

Evidently a

f u n c t i o n F i s a f i r s t i n t e g r a l o f t h e phase f l o w g i i f and o n l y i f {F,Hl

=

0.

+

One checks e a s i l y t h a t a Jacobi i d e n t i t y h o l d s i n t h e form I C A , B I , C I t {{C,AI,BI

IIB,C),AI

= 0 (exercise).

e.g. [ A01 1 whereas e.g.

We n o t e t h a t t h e r e i s a n o t a t i o n a l d i f -

Thus we a r e u s i n g w =

ference i n various references.

[ A1 ;AC1 ,3 ] use

1 dqi

A dpi

1 dpi =

A dqi f o l l o w i n g

-w which i n c e r t a i n The r e s u l -

r e s p e c t s i s more n a t u r a l i n d i s p l a y i n g t h e qi c o o r d i n a t e s f i r s t .

t i n g Poisson b r a c k e t has t h e same l o c a l form as above ( i n [A1;AC1,3;M0lY2]) b u t some formulas a c q u i r e a

-

sign.

Thus l e t us c o l l e c t a few formulas and

n o t a t i o n a l connections i n IdH =

(~.9) w

w(IdF,IdH)

1 (aH/api)a/aqi

G(XFyXH)=

[FZ;PL21)

(A.IO)

(aH/aqi)

a/api

dF.XH = XH(F) = -XF(H); = $(XF,s)

( r e c a l l here (i(XF);)(s) (e.g.

-

= dF-6).

xH;

=

xF

{F,H)=

w" =

w(IdH,IdF)

=

i ( X F ) c = dF

On t h e o t h e r hand some a u t h o r s

use a Poisson b r a c k e t (which we denote b y (F,G))

(F,G) = -IF,GI

=

1 (aF/apj)(aG/aqj) -

(aF/aqj)(aG/apj)

I t d o e s n ' t m a t t e r o f course b u t each s t y l e has many proponents and hence one must be c a r e f u l o f minus s i g n s i n r e a d i n g v a r i o u s sources. L e t us mention now a v e r s i o n o f t h e L i o u v i l l e - A r n o l d theorem a b o u t i n t e grability. {F,G}

= 0.

One says t h a t F,G ( f u n c t i o n s on say T*M) a r e i n i n v o l u t i o n i f B a s i c a l l y t h e theorem says t h a t , f o r dim M = n (dim T*M = Zn),

i f n independent f i r s t i n t e g r a l s i n i n v o l u t i o n a r e known, t h e n t h e system

i s i n t e g r a b l e by quadratures. tem

Gk

= IqkyHI,

pk

= {pk,HI

Here t h e system r e f e r s t o t h e H a m i l t o n i a n sys above.

For p r o o f see e.g. [Al;A01

c e p t o f i n t e g r a b i l i t y i s discussed f r e q u e n t l y i n t h e t e x t .

I.

The con-

Next one shows

t h a t under t h e hypotheses o f t h i s theorem one can f i n d a c t i o n a n g l e v a r i

-

I = (Ik), $ = ( $ k ) , Ik 'L a c t i o n , $k 1~ angle, such t h a t d I / d t = 2 0 and d $ / d t = w ( I ) , w i t h w =I dpk A dqk = 1 d I k A d+k. The p r o o f can be here; such v a r i a b l e s o c c u r natufound e.g. i n [A];AOl I and we o m i t d e t a i l s r a l l y i n t h e t e x t a t v a r i o u s places where d e t a i l s w i l l be e v i d e n t .

a b l e s (I,$),

REmARK A32,

We e x t r a c t now some m a t e r i a l based i n p a r t on n o t a t i o n i n

PO I SSO N STRUCTURES

363

OY1 ;PL2 1 and a t t e m p t t h e r e b y t o f a c i l i t a t e a l s o access t o v a r i o u s n o t a t i o n s used i n v a r i o u s i m p o r t a n t work ( c f . a l s o [AC1,3;AD1,2;F2;1Cl;RE1-9;RTl;SMl3;SY1,2;TK4;WSly21).

Recall CPL23 uses (F,G) as i n ( A . l O )

minus s i g n s as needed o r s i m p l y use (

,

).

so we w i l l i n s e r t

F i r s t a s suggested i m p l i c i t l y

x

b e f o r e Theorem A30 one can w r i t e , f o r x = (p,q) and V H = (aH,ap aH/aq.), jyJ = JVH where J = ( 0I - Io). Also as usual iJ ‘ = ( x j, H I o r = Cx,HI = XH. Now i n [ PL2 I

zH

i s r e f e r r e d t o as a v e c t o r f i e l d , i n s t e a d o f (more c o r r e c t l y ) XH There should be no problem i f we remember t h a t XH =

a s i n (A.9).

1 (aH/api)

Y

and xH = ({piyHIy{qiyH)) a/aqi - (aH/aqi)a/api I f one has now a Poisson b r a c k e t

= (-aH/aqj,aH/apj)

= (p.,q.):

J

J

(~.ii) (F,G) = wjka.Fa G J k (always sum o n repeated i n d i c e s ) w i t h w j k = - w k j and wjkakwPm t wPkakwmj + wmkakwjp = 0 ( f o r t h e Jacobi i d e n t i t y ) t h e n equations o f m o t i o n $ = (H,xj) * jk =Ixj,HI = become = -w akH and g e n e r a l l y one notes t h a t [ XF,XG] =

xi

-

xJ

x ~ = X(F,G). ~ , The~ space ~ F(M) o f smooth f u n c t i o n s on M w i t h ( , ) becomes a n i n f i n i t e dimensional L i e a l g e b r a ( n o t e t h a t t h e L e i b n i t z r u l e (F,GH) = (F,G)H + (F,H)G h o l d s a l s o ) . Now suppose w j k ( x ) = Cj k x P , t h e n t h e C j k must P P be t h e s t r u c t u r e c o n s t a n t s o f a L i e a l g e b r a and ( A . l l ) i s c a l l e d a L i e - P o i s son b r a c k e t . = adEi(E.)

J gebra and

S t r u c t u r e c o n s t a n t s o f a L i e a l g e b r a r a r e d e f i n e d v i a [ Ei,E.] J k = C .E f o r Ei a b a s i s o f g. I n p a r t i c u l a r l e t be a L i e a l iJ k t h e dual v e c t o r space. Then F(?j*) has a n a t u r a l Poisson s t r u c -

5

1

L e t ei be a b a s i s o f {, f

t u r e d e s c r i b e d as f o l l o w s .

k (so ( f ,ei)

k

t h e dual b a s i s o f

k = 6i),

;3*

1

Cp d e f i n e d v i a [e.,e I = 1 CP e , and x = x . f j E:*. jk J k Jk P J Then F(”g) c o n s i s t s o f smooth f u n c t i o n s F ( x ) = F(x.); one can d e f i n e (x.,x ) J J k =.I CPkxp and r e q u i r e a l s o t h a t t h e L e i b n i t z r u l e should h o l d . Then (F,G) =

a .Fa G p r o v i d e s a Lie-Poisson s t r u c t u r e on F(:*) ( t h e Jacobi i d e n t i t y f o r Jk J k ( , ) i s a consequence o f t h e Jacobi i d e n t i t y i n 5 ) . Next, i n general, i n

C?

standard notation, i f Ad o f G on E

5 and

?, adS(n)

=

comes from a L i e group G t h e r e i s an a d j o i n t a c t i o n

a c o a d j o i n t a c t i o n o f G on

p.

Thus f i r s t f o r S , n E

s”,

u

[ S , n l and ad;u(n) = - u ( [ E , n l ) = -u(adS(n)) = u(adn(5)). This = (u, h,c]>. Then d e f i n e ( c f . [ F2;FMl;SYl ,21) Ad*g 5

can be w r i t t e n (ad*u,n)

= (Adg-’)* f o r g E G where f i r s t Adg i s d e f i n e d v i a ( I )* = Adg w i t h I :G .+ 9 G: h ghg-’ ( t h u s Adg: $). C o a d j o i n t a c t i o n ;5* i s o f t e n defined -f

-f

2;.

ROBERT CARROLL

364 a s (Adg-l)* = Ad*g, - ( a d c ) * as above.

i . e . Ad*gu(S) = u(Adg-lS) ( u Further DtAdexpt5(n)lt=D

E

*;,

5 E

z)w i t h ad;

=

= [5,111 = a d e ( q ) and D t Ad*expt5

= ad;(@)

REIURK A33, For t h e discussion of momentum maps e t c . we r e f e r e s p e c i a l l y to [ TK41 where the matter is nicely organized, i n addition to t h e o t h e r r e f e r ences c i t e d in Remark A32. T h u s use t , } again and { f , g ) = q(X , X ) ( c f . f g X f ( g ) = { g , f } , and i ( X f ) G = d f . Let $ : G X M (A.9)) w i t h [X X 1 = X f' g {g,fI' M be a group a c t i o n of a Lie g r o u p G on M. Let 5 E a n d define t h e Kill i n g vector f i e l d o n M via S M ( p ) = Dt$(exp(tc) p ) a t t = 0. A g r o u p a c t i o n C"(M) such on M is c a l l e d Hamiltonian i f t h e r e e x i s t s a l i n e a r map J": For $g: M -+ M one gets an induced l e f t a ction $ = t h a t tM(p) = Xj(5)(p).

.+

-+

A g

T* M o n T*M where $ i s symplectic and Hamiltonian with J ( 5 ) = $g.X 9 e?~,,,) (6 is t h e Liouville form of Remark A29 and 5T*M i s t h e Killing vect o r f i e l d ) . To see this we r e c a l l f o r Xa E Ta(T*M): ea(Xa) = a(a,X,) ( a : T*M M ) . Note here by construction ($*O)(Xa) = 0 (a)(($g),Xa) = $,(a) $*-,:

TM ;

-+

-f

9

$9

(r,(qg)*Xa) = $g(a)(($g)*r*Xa) = a(a,X ) = 8 ( X ) so $*O = 6 which implies a a 9 a E T*M, and On t h e o t h e r hand f o r 5 E $*; = $ a n d J, i s symplectic. 9 9 ~ T * M= DtJ,expt5(a)lt = O one has LSTSrr0 = Dt($EXptt0) = 0 s i n c e 0 is preserved ( r e c a l l from Remark A16, LXf = ( d f , X ) = X ( f ) , Lx = i(X)d + d o i ( X ) , i ( X ) f =

0, i(X)df = X(f) = ( d f , X ) , LX(Y) = [X,Yl, L 0 = Dt(J,Expt50), e t c . ) . Hence 5,f o r 5 = cT*M, 0 = L 0 = i d o + d ( e ( 5 ) ) = -iw + d(<6,5)) and consequently 5 5 5 ( i ( X f ) i = d f ) ; thus t h e Killing f i e l d 5 * Hamiltonian f i e l d T*M = ( sTxM) = Next one dea n d s ~ i s* generated ~ by the Hamiltonian $ 5 ) . 8.5) A f i n e s the momentum map J: M -+? associated t o 0 by J ( p ) ( x ) = J ( x ) ( p ) (note ?: +.; C"(M) is l i n e a r so J ( p ) E ?). The s i t u a t i o n of i n t e r e s t involves $ :

5X

T*M a s above ( J , = I$:-,). Then i f H: T*M R is 9 a Hamiltonian i n v a r i a n t under J, ( i . e . HoJ, = H ) t h e associated momentum map I* J: T*M -+ g* is constant along t h e i n t e g r a l curves of X H . To s e e this l e t G X M

-+

M a n d J,:

T*M

-+

y ( t ) E T*M be a curve w i t h ; ( t ) = X H ( y ( t ) ) .

.+

Then f o r a l l 5

E

KIRILLOV BRACKET R

3

= E u c l i d e a n t r a n s l a t i o n group,

p.5. T X =

Thus J ( i , t )

-XI,

$x(G)

( 2 ) M = R3,

%

and f o r 5 E

y,

$

t.(Eq)

A

,;

t

G = S0(3),

J((,t)(5) =

365 A

4(4) =

= 5-(; A

J*(t)(G,c)=

2

and J(q,p)(S)

=

A,: T ={X E g1(3,R), t ) so J(<,$) = A $

( c o r r e s p o n d i n g t o a n g u l a r momentum).

RrmARK A34,

We go n e x t t o t h e Poisson s t r u c t u r e on

l o w [ T K 4 ] ( c f . a l s o [ADl;AOl;FLl;G3;SY1,2;WS21). o f can be d e f i n e d v i a d f l x = ( x , V f ( x ) ) E

T*M and a k f = a f / a x k

CY

E

-+

*:

so

E T,*?

'$((a)

c f . Remark A32).

%

F* =

a s i n Remark A33 so J:,

cand

TM

-t

T$

First recall for f

(i.e. Vf(x)

-

vF:

a

2

-t

v?(a) *;(

(A.13)

J * s ( v ) = 5(J,v)

Dt5(J(v(t)))lt=o

Next l e t f

= d,?(t).

E

(alf

,..., a n f )

=

E

C"(M),

where d f

t*, ? E C"(p),and t). Now t a k e J: M

-

-f

and J*: T * P -

one has J*5 E T*M and f o r v E TM, v = ;(O),

Hence J*S

%

L e t now M =

c z T*Z*

=

o f K i r i l l o v and f o l -

-+

T*M.

For 5 E

one has

Dt?(E)(v(t))lt=O

= d;(t)(v)

Cm(T*G) be l e f t i n v a r i a n t w i t h f t h e c o r r e s L

tion). f o r 1~. E

*:

(i.e.

f =

f o

P, P: T*G

N

g* = T*G b e i n g t h e p r o j e c e Then one can d e f i n e a Poisson b r a c k e t on via {f,gI(u) = {f,gl(v)

ponding f u n c t i o n on

+

-

(f,g

l e f t invariant).

N

This i s t h e K i r i l l o v b r a c k e t on

wants t o show i t makes sense and f i n d a n i c e e x p r e s s i o n f o r i t .

and one This i s d i s -

cussed b e a u t i f u l l y i n g r e a t d e t a i l i n [ TK41 and t h e r e s u l t i s

We i n d i c a t e a few o f t h e t e c h n i c a l c a l c u l a t i o n s .

Recall from Remark A32 t h e

Write I

d e f i n i t i o n s o f ad* e t c .

= L OR and n o t e t h a t P ( a ) = L* 9 9 g-' G (P i s t h e pullback t o the f i b r e over e). T* r(a) c Thus f o r f l e f t i n v a r i a n t and a E T* G, f ( a ) = F ( P ( a ) ) = ?(L:(a)a) = f R " n(a) R ( J ( a ) ) = ( J ) * f ( a ) where JR i s t h e momentum map % r i g h t G a c t i o n . To see

5

where n: T*G

.+

G and a

E

) =

and

366

(P

ROBERT CARROLL

g*,

E

OF(,,)

)!*GL

LI

(vf ( p

E

c-z T;R(,,)~)

and t h i s i m p l i e s Xf(u) = X J I R ( ~ F ( , , ) ) I ~=

I t=O'

= Dt( 'expv?( ,,) ('

This l a s t expression i s the K i l l i n g

f i e l d o f v f ( p ) generated by r i g h t a c t i o n R o f G

on T*G.

F i n a l l y t o estab-

l i s h (A.14) one w r i t e s ( c f . ITK41)

To f i n i s h t h i s remark we n o t e t h a t t h e H a m i l t o n i a n v e c t o r f i e l d s on *; a r e d .,

g i v e n i n t h i s K i r i l l o v s t r u c t u r e v i a X?(a)(F) = -{?,;>(a) = a( [ v f ( a ) , v F ( a ) l ) -ad;r(cc)a(Vg"(a)) comes X$a)

and i f a ( t )

= i ( t )=

E

*;

i s an i n t e g r a l c u r v e o f X$a)

The i n t e g r a l m a n i f o l d s o f H a m i l t o n i a n vec-

3

t o r f i e l d s are the coadjoint o r b i t s o f G i n

-

i s surjective an o r b i t

N

check w i t h f x ( a ) = a ( x ) ,

T, za(adp,ad*a)Y

(a)yv?-y(a)

I)=

REiURK A35.

-a( [X,Y

t h i s be-

x

( s i n c e ? E C"(?)

-t

vf"(a) €

c, so v f X ( u ) = x ) and o n such N

E

Iv

1 2 . -

= Ga(X<x(a),Xw

f -Y

I).

(a)) = If-x.f-yl(a)

= -a(

We go now t o t h e Adler-Kostant-Symes lemma (AKS lemma) and r e -

f e r t o [AD1;SYlY2I f o r background; we w i l l change a few minus s i g n s t o agree w i t h t h e n o t a t i o n o f Remarks A33,34. hr

h with

5yKL i e

subalgebras

o f the constructions). 'h'

II I, II~L

5

g

L e t t h e b a s i c L i e a l g e b r a be

=

13.

(rneed o n l y *L be a complementary subspace f o r some

Write

*:

n.1

= g

B h with

be t h e obvious p r o j e c t i o n s .

Let F E

h* and

C"(:*)

and

?*

- p; write v

;*'L

let

T19 -t F: g t h e g r a d i e n t as

f o r t h e g r a d i e n t o f F a s a f u n c t i o n on * ; w i t h OF: -+ -I a f u n c t i o n on K L C F. Then f o r a € h , VgF(a) = TI VF(a) ( e x e r c i s e cf. g [ SY1,2]). S i m i l a r l y i f ad* i s t h e r e p r e s e n t a t i o n i n g l ( g L ) induced b y t h e 9, c o a d j o i n t r e p r e s e n t a t i o n o f g on under t h e i d e n t i f i c a t i o n ;5* o,;'ne has

-

-1

-+ g l (p)(exer9 -L Note a l s o t h a t ad* induces a Poisson b r a c k e t on h v i a {F,G)(a) = 9 4 a , [ngvF(a),n VG(a) 1) and Hamil t o n i a n v e c t o r f i e l d s X F a r e determined v i a 9

for x

cise).

u

E

gy a E h

v

.u

c p*, ad*x(a) = n t a d * x ( a ) where ad*: p

AOLER KOSTANT SYMES

367

XF(a) = - ( a , s ~ a d * denote ad* i n v a r i a n t funch I T ~ V F ( ~ ) ~Now ) ' l e t S C C"(?) *: d e f i n e d by S = {F E C"(p);(VF(a),ad*x(a)) = 0 f o r a l l a E p,

t i o n s on x E

El.

(i.e.

The t h e AKS lemma s t a t e s t h a t ( 1 ) S ~ K L i s a system i n i n v o l u t i o n = 0 ) and ( 2 ) f o r F E

{F,GI

S l i ; ~ the

corresponding Hamiltonian v e c t o r

f i e l d X F i s determined by XF(a) = -(a,ad*(rgVF(a))(a))

i s that

The i m p o r t a n t p o i n t here equation.

= (a,ad*(shVF(a))(a)).

n h i i s m i s s i n g i n f r o n t o f ad* i n t h e l a s t

We s k e t c h a p r o o f f o l l o w i n g C S Y l l and n o t e f i r s t as a general

f a c t t h a t DtF(AdgxptXa)

= 0 i f and o n l y i f DtF(Ad;xpsXa)ls=O

= 0 i f and o n l y

i f a d p ( V F ( a ) ) = 0 i f and o n l y i n ad* a(X) = 0 ifand o n l y i f ad;F(a)a = vF(a) 0, so S c o n s i s t s o f Ad* i n v a r i a n t f u n c t i o n s . To show t h e i n v o l u t i o n w r i t e

then f o r a E -(ad*VF(a)a,x) (A.18) (

g',

x E

7,

and F E S, 0 = (ad*x(a),VF(a))

which i m p l i e s a d * ( r VF(a))a + ad*(ahvF(a))a = 0. 9

{F,G)(a)

= -( a , [n VF(a),r

9

-(

9

VG(a)

ad*(agVF(a))(a).rhVG(a))

= -( ad*(ngvG(a))(a),n

where t h e l a s t s t e p r e q u i r e s

1) =

-(

1)

=

Hence vF(a)) = 9

a , [nhVG(a),ngVF(a)

= ( ad*(ahoF(a))(a),nhVG(a))

( a , [ahVG(a),ahVF(a)

-I

1)

= ( a , [a VF(a),ahVG(a) 9

ad*(ahvG(a))(a),agVF(a))

=

= ( a , [vF(a),xl)

1)

=

= 0

t o be a subalgebra.

T h i s shows ( 1 ) and f o r

V F ( a ) ) ( a ) ) = (a,ad* -9-1 (ahVF(a))(a)) where n h i i s s u p e r f l u o u s because ad*h(h ) C Since F i s

(2) take F

E

S, a

E

h

SO t h a t XF(a) = -(a,ahsad*(n

ad* i n v a r i a n t ( i n t h e sense above) we have ad*(ahvF(a))(a)

( a ) and ( 2 ) f o l l o w s .

cL.

= -ad*(n VF(a))

9

This Page Intentionally Left Blank

369

APPENDIX B RIEMANN SURFACES AND ALGEBRAIC CURVES

There a r e many excellent sources of information and we mention especially [AE1~AT4~CE2~CM1~C01~FS1~FT1;FU1~FY1~GU1,2,4,6~DQ1~GR1;HA1;MA1;MJ1;MU1-6;RA1;

SL1 ;SP1 ;SHF1 1. ([ SL1 1 i s p a r t i c u l a r l y useful here a s a guide i n extracting items of i n t e r e s t i n s o l i t o n and s t r i n g theory). There i s obviously too much t o cover b u t fortunately one can limit concern t o c e r t a i n selected t o pics. Some r e s u l t s will simply be s t a t e d b u t some proofs will be sketched;

hopefully a l l the necessary d e f i n i t i o n s will be present and c l e a r l y s t a t e d . W e will t r y to capture the essential meanings by g i v i n g " i n t u i t i v e " descriptions a t times i n addition to a formal a b s t r a c t d e f i n i t i o n . We assume basic material about m a n i f o l d s ( c f . Appendix A ) and refer t o the t e x t f o r some i n troductory use of curve ideas and theta functions (these matters a r e usually duplicated here o r i n the t e x t ) . Occasionally i n the t e x t d i f f e r e n t notation is employed a t times. One should note t h a t many students and researche r s i n s o l i t o n theory know v i r t u a l l y nothing about algebraic geometry o r RS = Riemann surfaces f o r example, b u t these a r e essential ingredients f o r the theory. I t is impossible here t o write down and prove a l l of the basic mat e r i a l ; however i t is possible t o sketch the basic ideas and s t r u c t u r e s and t o show how they play a r o l e i n studying s o l i t o n topics. This leads to a perspective and conceptualization of the s t o r y which can be augmented l a t e r a s needed. Just knowing how some basic s t r u c t u r e s of one d i s c i p l i n e a r e rel a t e d t o the content of another can provide an enormous advance i n understanding. We especially d i s l i k e omitting d e t a i l s b u t i n the present s i t u a t i o n t h e r e is no choice. First we define a complex manifold a s a d i f f e r e n t i a b l e manifold M w i t h an open covering IU,) and coordinate maps $:, U, * caC C n such t h a t -1 $,o+~ is holomorphic. f holomrphic means a f / a Z i = + ( a f / a x i t i a f / a y i ) = 0 For several complex variables see e.g. [FZl;GU3;HMl;RGl 1 b u t (i = l,...,n). we will concentrate on n = 1 where M is c a l l e d a Riemann surface (assumed

REmARK 31.

370

ROBERT CARROLL

S).

a l s o connected and denoted by

R e c a l l a l s o Pn i s t h e s e t o f l i n e s

Pn = { [ z l P O E Cntll/[zl

through 0 i n Cntl,

Charts a r e g i v e n v i a Ui = I [ z l ; z i

...,zn/zi)

i;(

mann sphere C

U

1-1.

-+

I f P,Q

%

S

(S

=

(Z~/Z~,...,;~/Z~

1 Pn i s compact and P i s t h e Riea r e 2 curves on S from

= a R S ) and C1,C2

= Cl(t)

and h ( t , l )

(h(0,u)

= C2(t)

(any

P g i v e s t h e same

=

P and h(1,u)

=

P forms t h e funda-

The s e t o f e q u i v a l e n c e c l a s s e s o f c l o s e d curves a t

mental group IIl(S)

...:zn1).

C 2 (homotopic) i f t h e r e e x i s t s a c o n t i n u o u s map h: I X

S such t h a t h(t,O)

9).

E

([zl =[zo:

01 and $i([zo:...znl)

means t h i s t e r m i s m i s s i n g ) .

P t o Q one says C1 I

5

=[A21

One can d e f i n e homology groups

Itl).

v i a t r i a n g u l a t i o n (cover S w i t h curved t r i a n g l e s ) w i t h 0 s i m p l i c e s pk ( p o i n t s ) , 1 simpl i c e s (prps) angles).

(curves), and 2 simpl i c e s

(

prpspt)

O r i e n t a t i o n i s determined v i a permutations (e.g.

(

( c u r v e d tri-

p3p2p1 ) = 4 p1

p2p3)) and t h e group o f n-chains Cn i s t h e f r e e a b e l i a n group o v e r Z gener a t e d by the s e t o f p o s i t i v e l y o r i e n t e d n-simplices. map an: p p ) t r s an+l(Cntl)

(

Cn (

-+

C n- 1 d e f i n e d by

ao( p k )

= 0,

There i s a boundary

-

al( prps) = p

Pr.

a2( PrPsPt)

=

-

p p ) ( prpt) and e v i d e n t l y an-l an = 0. L e t Zn = keran and Bn = s t w i t h nth s i m p l i c i a 1 homology group Hn = Zn/Bn. One checks e a s i l y

t h a t Ho(S) = Z and H2(S) = Z o r 0 depending on whether S i s compact o r n o t . H1 ( S ) 2 n , ( S ) a b e l i a n i z e d ( i . e . d i v i d e o u t t h e commutator subgroup generated -1 -1 by aba b , a,b E nl). The r a n k o f H1(S) 2 ZZg i s t h e B e t t i number B1 ( i . e. B~ = 2g where g = genus

%

number o f h o l e s i n S ) .

Here t h e r e a r e v a r i o u s

ways o f r e p r e s e n t i n g a RS ( c f . 14) and we use whatever r e p r e s e n t a t i o n i s app r o p r i a t e i n a given context.

A compact s u r f a c e o f genus g i s t o p o l o g i c a l l y

a sphere w i t h g handles and one has an a s s o c i a t e d polygonal model a p

Think o f c u t t i n g and f l a t t e n i n g ( c f . [CNl;SLl I). Note i n ( B . l ) n,(S) i s 2 -1 -1 = 1. The E u l e r characgenerated b y aiyBi w i t h t h e r e l a t i o n nl ~ l pi ~ ~ ~ a ~ t e r i s t i c f o r a compact S i s x ( S ) = 1-2g+1 = 2-29 and i s a t o p o l o g i c a l i n v a r i ant.

For g e n e r a t o r s a i y ~ i as above one has an i n t e r s e c t i o n p r o d u c t H1(S) X

H1(S)

-+

Ho(S) = Z induced b y a i - a . = 0, B . e B . = 0, ai-B. J

1

J

J

= - 8 j - a 1. = ' i j .

RIEMANN SURFACES

1 niai

For c =

+ m . B . and c ' =

iyl

w r i t e s cat' = ( n m;

I "ai

(::I.

+

371

w i t h i n t e g e r c o e f f i c i e n t s one

I n t h i s f o r m u l a t i o n t h e ai,Bi

are called a

canonical homo1ogy b a s i s .

RRnARK 32.

L e t S be a (connected always) compact RS (compact i s a l s o always

assumed u n l e s s s t a t e d o t h e r w i s e ) .

A d i v i s o r i s an element o f t h e f r e e a b e l -

and D ' =

1 maa

i n t h e form

I (na + ma)a

1 naa

One adds D

i a n group DivS generated b y p o i n t s o f S ( f i n i t e sums).

and D 5 D ' i f and o n l y i f na 5 ma

For f a meromorphic f u n c t i o n on S t h e o r d e r o f f a t a ( o r d a f ) i s

f o r a l l a.

d e f i n e d as ( 1 ) 0 i f f i s a n a l y t i c a t a w i t h f ( a ) # 0 ( 2 ) k i f f has a zero o f m u l t i p l i c i t y k a t a ( 3 ) - k i f f has a p o l e o f m u l t i p l i c i t y k a t a ( 4 ) if f

5

(f) = (f)

+

0 near a.

For f meromorphic,

1ordaf.a.

@

f f 0, one d e f i n e s a d i v i s o r

These a r e c a l l e d p r i n c i p a l d i v i s o r s and e v i d e n t l y ( f . 9 )

(9) with (l/f) = - ( f ) .

map deg: DivS

f E M(S),

One says D E D ' i f D-D' = ( f ) . Next d e f i n e a

c npP c np.

by

+

=

+

DivoS

(%

degD = 0 ) c o n t a i n s a l l ( f )

s i n c e meromorphic f u n c t i o n s have t h e same number o f p o l e s and zeros w i t h multiplicity.

One d e f i n e s t h e P i c a r d groups D i v S / I ( f ) l = PicS and Pic's For an a r b i t r a r y d i v i s o r D w r i t e L(D) = { f

DivoS/{(f)l. Then e.g.

i f n - p i s a term i n D, f

E

E

M(S);

=

( f ) 2-Dl.

L(D) a r e a l l o w e d a p o l e a t p o f a t most

L e t Q ( D ) = dimL(D), so a ( 0 ) = 1, and t h e Riemann Roch theo-

m u l t i p l i c i t y n.

rem g i v e s a f o r m u l a f o r Q ( D ) (see Remark B3) Given w1 ,w2

E

C, l i n e a r l y independent o v e r R, w r i t e

r

+

= {nwl

mw2; n,mE Z}.

T h i s i s a two dimensional l a t t i c e and T = C/r i s a t o r u s ( w i t h induced s t r u c ture).

One can, f o r c l a s s i f i c a t i o n purposes, r e s t r i c t a t t e n t i o n t o l a t t i c e s

r

= { n t m(w /w ); n,m E Zl and s e t w2/w1 T. F u r t h e r one can choose T w i t h 2 1 I m T > 0 ( e x e r c i s e ) so t h e p e r i o d p a r a l l e l o g r a m has F = m a s a fundamen-

t a l region.

One c a l l s f o n C d o u b l y p e r i o d i c ( r e l a t i v e t o

= f ( z ) and f

%

(8.2)

P(z)

F € M(T).

= l/z

2

+

,I /

(El means 0 i s m i s s i n g ) .

r)

i f f(z+n+mr)

Define t h e Weierstrass P function as (l/(z-w)'

-

2 l/w )

T h i s i s holomorphic f o r z q!

r

and doubly p e r i o d i c ; 2 3 P'(z) = ( 2 / ( z - w ) ) and t h e r e i s an e q u a t i o n (*) (PI) = 4P - g,P - g3 4 1 4 6 where g2 = 601 l / w = 1 l/(n+mmt) and g3 = 140 c'1/w6 = l ' l / ( n + m z ) In 3 2 C ( x ) [ y l / ( y 2 , - 4x + g2x + g3) shows t h a t P d e t e r f a c t M(T) = C(P,P')/(*) mines a l l meromorphic f u n c t i o n s on T.

-1

3

'

.

ROBERT CARROLL

3 72

RRRARK 33.

For d i f f e r e n t i a l forms we r e f e r t o Appendix A a n d note here f o r complex manifolds one deals w i t h d z = d x + idy, d? = dx - idy, and forms w = f ( z ) d z t g ( z ) d ? ( a l s o a Z = %(ax - i a ) and a? = % ( a x t i a y ) ) . One speaks of Y (1,O) and ( 0 , l ) forms where g o r f = 0. A (1 ,O) form w is holomorphic i f w

w i t h f E O ( U a ) = holomorphic functions on Ua f o r a l l Ua in a covering o f S. A global meromorphic d i f f e r e n t i a l w is a holomorphic d i f f e r e n t i a l o n S/{pl,. . . , p k l with local representation around p g i v e n by a meromorphic mm i function. Thus around p,w = f ( z ) d z w i t h f ( z ) = 1, c i z , cm 0, and ord w P = m w i t h res w = c - ~ . One shows e a s i l y t h a t 1 res w = 0 and f o r a meromorP P phic d i f f e r e n t i a l w one defines a canonical d i v i s o r ( w ) = 1 ord w - p (with P c l a s s (w) = K - c f . a l s o Remark 8 6 ) . Note any two meromorphic d i f f e r e n t i a l s w1,w2 determine t h e same equivalence c l a s s ( w ) s i n c e ( l o c a l l y ) w, = hw2 w i t h h holomorphic and ( h ) = 0. Recall L(D), Q ( D ) e t c . from Remark 82 = fdz

+

and t h e Riemann Roch theorem then says (B.3)

Q(D)

-

P(K-D)

= 1

-

g + degD

f o r an a r b i t r a r y divisor D. This i s an important theorem with connections t o many t h i n g s and we will give various comments l a t e r o r i n t h e t e x t . In p a r t i c u l a r f o r D = 0, 1 - P ( K ) = 1-gtdegO implies B (K) = g while f o r D = K, 2 B(K) - B(0) = 1-gtdegK implies degK = 29-2. Note f o r example o n T with co1 o r d i n a t e z t a (a % t r a n s l a t i o n ) ( d z ) = 0 w i t h deg(dz) = 2g(T)-2 and o n P 2 w i t h coordinates z and w = l / z ( d z = -dw/w ), ( d z ) = -2[m] and deg(dz) = - 2 1 = 2 g ( P ) - 2. Generally equivalent d i v i s o r s D have isomorphic L ( D ) and f o r

w a meromorphic d i f f e r e n t i a l L ( ( w ) ) 2 n(S) = global holomorphic d i f f e r e n f-w (note ( f ) 1. -(w) implies (fww) = ( f ) + ( w ) 0). t i a l s via f -f

Next one writes 2-forms l o c a l l y

via w

= f ( z ) d z A dT

and s e t s df = af + z f

For a 1-form w = fdz + gd? we have dw = (g,-f:)dz A d z . A form w is closed i f dw = 0 and exact i f w = df. The deRham cohomo1 logy group i s defined via H ( S ) = {closed l-formsl/{exact 1 forms} = kerdl/ Imdo and this will agree w i t h s i n g u l a r o r Cech cohomology ( c f . [ FT1;GMl 1 we will discuss t h e deRham cohomology i n t h e t e x t when needed). One speaks (af = f dz, 5f = f z d r ) . Z

of i n t e g r a t i n g k-forms over k-simplices on k-chains and Stoke's formula

JACOB I A N S

373

JC dw = JaCw f o r C a ( k + l ) L h a i n and w a k - f o r m i s c r u c i a l . Ifylyy2 a r e paths from P t o Q and w i s a c l o s e d 1 - f o r m g e n e r a l l y J w C J w. Rather, YI YZ f o r a k y B k a b a s i s o f H1 (S,Z) ( k = 1 ,g) one has (modulo boundaries) ylyil =

1 nkak

' mkBki,

l y Hk(S,C)*

y

,...

2

1

mkynk E Z, so J w - J w = nkJ w t m J w. Generalk yc Y1 ak k Bk deRham cohomology. Another i m p o r t a n t f a c t

H ( S ) where H

i s t h a t t h e r e e x i s t s a unique s e t o f holomorphic d i f f e r e n t i a l s wkY k = ly..., = 6. and these wk form a b a s i s o f t h e C v e c t o r space o f @J 1j holomorphic d i f f e r e n t i a l s ( d i f f e r e n t i a l s o f t h e f i r s t k i n d DFK d i f f e r -

g, such t h a t J . wi

-

-

e n t i a l s o f t h e second k i n d a r e meromorphic 1-forms w i t h no r e s i d u e s and d i f f e r e n t i a l s o f t h e t h i r d k i n d a r e meromorphic forms w i t h o n l y s i m p l e poles; these a r e discussed l a t e r i n more d e t a i l ) .

R?JURK 34. L =

1 Zwi

L e t now

W~,...,W~~

E C2n be l i n e a r l y independent o v e r R and l e t

be t h e corresponding l a t t i c e .

Cn/L = Tn i s a t o r u s and i t i s s a i d

t o have a p o l a r i z a t i o n H i f ( 1 ) H i s a p o s i t i v e d e f i n i t e h e r m i t i a n form o n

Cn ( 2 ) E = I m H i s i n t e g e r valued and a n t i s y m m e t r i c on L.

I f detE = 1 on a

l a t t i c e b a s i s T i s c a l l e d p r i n c i p a l l y p o l a r i z e d and one must remark t h a t n o t e v e r y t o r u s a d m i t s a p o l a r i z a t i o n (T,H). i f t h e r e e x i s t s an a n a l y t i c isomorphism $:

(TlYH1)

T,

-+

i s isomorphic t o (T2,H2) T2 ( $ ( O )

= 0 ) such t h a t $*

Cn/L2 can be expressed as a 1 i n e a r isomor( H 2 ) = H1. Note here 4: Cn/L1 phism $ # : Cn + Cn w i t h $ ( L ) = L2; $* i s d e f i n e d v i a $$(H2)(x,y) = H2($# # 1 be t h e ( x ) ¶ $ # ( y ) ) . For p r i n c i p a l l y p o l a r i z e d t o r i now l e t A = (wl,...,wn) -f

n X 2n m a t r i x whose columns a r e t h e c o o r d i n a t e v e c t o r s o f t h e l a t t i c e b a s i s . By s u i t a b l e bases changes i n Cn and L one can assume A = ( I P) where I i s an

n X n i d e n t i t y and P i s symmetric w i t h p o s i t i v e d e f i n i t e i m a g i n a r y p a r t . A t t h e same t i m e one can a r r a n g e t h e l a t t i c e b a s i s so t h a t E has t h e form J =

(-:

i)

on L and such bases a r e c a l l e d s y m p l e c t i c . The s y m p l e c t i c group SP A B T (2n,Z) = {M = ( c D) E Mat(2n,Z); M JM = J1 and t h e isomorphism c l a s s e s o f p r i n c i p a l l y p o l a r i z e d t o r i a r e g i v e n v i a A = ( I P) as i n d i c a t e d w i t h ( I P) E

( I P I ) for PI = (AP t B) (CP t L e t now aiyBi

w

~1-l.

(1 5 i 5 g ) be a canonical homology b a s i s ( g

1 ) w i t h wlY.

..)

Define the (unique) holomorphic d i f f e r e n t i a l s such t h a t J wi = 6ij. 9 a; This i s a symmetric g X g m a t r i x w i t h p o s i t i v e p e r i o d m a t r i x TI. = J wi. ij Bj d e f i n i t e i m a g i n a r y p a r t . Then L = Zg (3 l I Z g i s a l a t t i c e generated by t h e

374

ROBERT CARROLL

The Jacobian of S is defined a s JacS = Cg/L. Another d e f i n i t i o n , independent of bases, uses R(S) = global holomorphic d i f f e r e n t i a l s , which has dimension g ( w l . . . . , w a r e a b a s i s ) . Write Hom(R(S),C) = R* and 9 via w + I w one embeds H1(S,Z) i n t o a*. Then i n f a c t JacS = Hom(n(S),C)/ Y H1(S,Z). JacS comes with an a l t e r n a t i n g b i l i n e a r form from the i n t e r s e c t i o n columns o f TI.

pairing o n homology ( c f . [ GR1 ;SL1 I) so JacS a r i s e s a s a p r i n c i p a l l y p o l a r ized t o r u s .

To embed S i n t o JacS one has the Abel ( o r Jacobi) map p

+ cp(p) A change o f base point involves only a global PO Po 9 t r a n s l a t i o n o f t h e image. A theorem of T o r e l l i says t h a t S 2 S ' i f and only i f JacS 2 JacS' w i t h the corresponding principal p o l a r i z a t i o n s . Another r e : pi - ly q i + s u l t of Abel and Jacobi looks a t $: DivoS + JacS where D = 1 1; ( J ~ ~ w l , . . . q,, j ~ 9,w) mdL. Then ker $ = principal d i v i s o r s a n d $ i s s u r j e c t i v e so Pic's = Div S / { ( f ) > JacS.

= (Ipw1,...,fp

REElARK B5.

w

modL.

Generally speaking one i d e n t i f i e s compact RS and a l g e b r a i c cur-

ves although these a r e r e a l l y d i f f e r e n t s u b j e c t s . This involves prescribing a l i n e bundle ( o r s u i t a b l e sheaf - see below) on S plus some o t h e r informa-

t i o n ( c f . [ CM1;GRl;GUl l ) .

One notes t h a t h y p e r e l l i p t i c RS ( a s in 54,5) a r e

a special c l a s s having q u i t e special properties a t times.

We do n o t t r y to

give here a l l the d e t a i l s about anything b u t mention t h e f a c t s and d e f i n i t i o n s needed i n t h e t e x t . Generally ( c f . here [Gull ) a compact RS can be represented a s a branched covering S + P 1 o f P1 ( h y p e r e l l i p t i c 2-sheeted covering w i t h g > 1 + some technical c o n d i t i o n s ) . The f i e l d M S of meromorphic functions on S is a f i n i t e a l g e b r a i c extension of a simple transcenden-

P i c k i n g any two f , g E M S which generate MS (Ms = C(f,g)) l e t P be t h e polynomial ( i r r e d u c i b l e ) such t h a t P ( f , g ) = 0. Then P describes

t a l extension o f C .

MSy and hence S, i n a sense l e f t imprecise f o r the moment. Such a P (say o f degree n ) determines a homogeneous polynomial P o ( t o y t l, t 2 ) = t:P(tl/toyt2/ t o ) ( x = t / t , y = t2/to)with P(x,y) = P o ( l , x , y ) . Although Po i s not well defined i n P 2 O i t s 0 locus (locPo) i s well defined a n d t h i s determines an a l 2 In t h e coordinate NBH to # 0 gebraic plane curve P o ( t o y t l y t 2 )= 0 i n P 2 i n P t h e curve i s determined by P(x,y) = 0 . One shows by a r g u i n g v i a Po ( e x e r c i s e ) t h a t t o a given plane a l g e b r a i c curve P(x,y) = 0 ( P i r r e d u c i b l e ) one has a canonically associated RS, S p ( t h i s RS can d i f f e r from locPo a n d t h e i r a n a l y t i c functions can d i f f e r ) . However given S w i t h f , g generating

.

THETA FUNCTIONS

375

MS and P ( f , g ) = 0 t h e r e i s a canonical a n a l y t i c homeomorphism from S onto A

F u r t h e r one can show t h a t S and S a r e a n a l y t i c a l l y e q u i v a l e n t i f and SP. (as f i e l d s ) . o n l y i f MS "M; T Mat(g,C); P = P , I m P > 01 and one T T = &EZsexp(ain Pn t Z n i n z ) where z E Cg and

The Siegal upper h a l f space i s H d e f i n e s e f u n c t i o n s v i a e(z,P) P

E

H

= {P

E

9

8 i s a w e l l d e f i n e d holomorphic f u n c t i o n on Cg X

g' Cg/L, w i t h

L = Zg @ PZg one has e ( z + m,P) T T P ) = exp(-aim hn h i m z ) B ( z , P ) (m E Zg).

-

= e(z,P)

H

and w r i t i n g T = 9 (m E Zg) and e ( z + Pm,

A s l i g h t l y expanded d e f i n i t i o n

uses a,b E Qg ( Q = r a t i o n a l numbers) and d e f i n e s

(8.4)

e(i)(z,P)

so e(z,P)

= e

T a i a Pa

0 = e(,)(z,P).

+

T 2 n i a (z+b)

e ( z + Pz + b,P)

One can use such e f u n c t i o n s t o embed t o r i i n t o p r o -

j e c t i v e space and t h e r e s u l t i s t h a t a t o r u s i s embeddable i n t o some Pn i f and o n l y i f i t a d m i t s a p o l a r i z a t i o n ( t h e embedding depends on t h e p o l a r i z a t i o n chosen).

REmARK

86.

Such embedded t o r i a r e c a l l e d a b e l i a n v a r i e t i e s .

We want t o g i v e now some d e f i n i t i o n s and r e s u l t s concerning

sheaves, l i n e bundles, e t c .

This s h o u l d a l l be s t a n d a r d knowledge f o r mathe-

m t i c s s t u d e n t s (and today f o r p h y s i c i s t s ) b u t i s perhaps n o t as f a m i l i a r t o engineers and a p p l i e d mathematicians.

F i b r e bundles and v e c t o r bundles a r e

developed i n Appendix A and here we make o n l y a few a d d i t i o n a l remarks. Thus we t h i n k m a i n l y now o f complex m a n i f o l d s M and VB n : E M with trivializing -1 maps T ~ :n (V,) Vk X F ( F 2 C'). A t a f i x e d z t h e maps ( T . 0 ~ ; : ) : Cn + -f

-f

JZ

Cn a r e l i n e a r isomorphisms and can be r e p r e s e n t e d by a n X n m a t r i x C . . ( z ) J1

E GL(n,C)

depending h o l o m o r p h i c a l l y on z.

c o c y c l e c o n d i t i o n s : Cii domains o f z ) .

= 1, Cij

= (Cji)-';

One checks t h a t such Cji

Cki

= Ckj-Cji

satisfy

(on a p p r o p r i a t e

The isomorphy c l a s s e s o f r a n k n bundles over a g i v e n t r i -

v i a l i z i n g c o v e r i n g (V,)

i s determined by t h e cohomology c l a s s o f t h e Cji.

For l i n e bundles ( c f . §4,5) n = 1 and we work m a i n l y on a compact R S . L be a holomorphic l i n e bundle o v e r S w i t h t r i v i a l i z i n g f u n c t i o n s L e t U C S be open and s a holomorphic s e c t i o n o v e r U ( s : U

Ui. s i s given v i a t h e

{aly...,am1

T~

as holomorphic f u n c t i o n s si o v e r Ui n U.

and s a s e c t i o n o v e r U A.

i f s i s g i v e n by meromorphic si

fi

+

T~

E).

Let over Then

Now l e t A =

Then s i s c a l l e d meromorphic o v e r U

E

M(Ui)

( w i t h poles a t t h e a k ) . D e f i n e

ROBERT CARROLL

376

o r d s = ord f . and a l o o k a t t h e t r a n s i t i o n cocycle function C i j shows t h i s P P 1 is well defined. Hence t o such neromorphic s e c t i o n s s of L one has a d i v i s o r ( s ) = 1 o r d s - p . Since every holomorphic l i n e bundle has a t l e a s t one P nonzero meromorphic s e c t i o n ( c f . [ F T l 1 f o r proof) we have a t l e a s t one d i v i s o r a s indicated. Given two such d i v i s o r s one has ( s l ) - ( s 2 ) = ( h ) where h is a meromorphic function a n d one writes c , ( L ) (Chern c l a s s - more below) f o r t h e z c l a s s o f such ( s ) . This c l e a r l y depends only on the isomorphy Given L and M l i n e bundles, s a n d t meromorphic s e c t i o n s , and f i , g i t h e i r respective t r i v i a l i z i n g functions over U i , one has s It a sec-

class o f L.

t i o n o f L (81 M w i t h t r i v i a l i z i n g function f i g i over U i . Since ord ( s (81 t ) = P o r d p ( f i g i ) = o r d ( f . ) t ord ( 9 . ) we get immediately ( s B t ) = ( s ) + ( t ) and P ' P i c l ( L IM ) = c l ( L ) t cl(M). Further i f ( s ) L i s a principal d i v i s o r ( i . e . ( h ) ) then r = ( l / h ) s i s a s e c t i o n ( s ) = comes from a meromorphic function h , of L a n d ( r ) = ( s ) ( h ) = 0. Hence h %nowhere v a n i s h i n g holomorphic funct i o n and L can be defined by C i j = 1 so L i s t r i v i a l . Hence t h e map c1 i s

-

-

a n i n j e c t i v e homomorphism of a b e l i a n groups. Moreover c1 (L*) = -cl ( L ) and one can show e a s i l y ( c f . [SLl I t a k e p E U0 ) p U i , f o = z, f i = 1 f o r i > 1 , c i o = l / z , coi = z, coo= 1 , and c i j = 1 f o r i , j 2 1 ) t h a t f o r any p e S t h e r e e x i s t s L such t h a t c ( L ) = [ P I = d i v i s o r c l a s s of p . Hence any d i v i P 1 P p generates a corresponding c (ILMnp)= [ D l (where LBnp means 1 P P :p (L;)'InP = i f n < 0 ) . Consequently t h e group of isomorphy c l a s s e s of l i n e P bundles over S is isomorphic via c1 t o t h e equivalence c l a s s e s of d i v i s o r s which is equivalent t o PicS.

-

F

Now i f K = ( w ) is the canonical d i v i s o r c l a s s o f S a s in Remark 83 we can think of w a s a meromorphic s e c t i o n of the cotangent bundle L = T*S (T*S can be taken a s t h e canonical l i n e bundle). One reformulates Riemann Roch now a s follows. Let L be a l i n e bundle a n d c , ( L ) = [ D l where say D (s) = 1 np.p. Recall L ( D ) = { f E M(S); ( f ) 2 -DI and Ho(S,L) = holomorphic s e c t i o n s of L. As mentioned e a r l i e r i n Remark 83 L ( D ) 2 Ho(S,L) v i a f + f . s (look a t ord ( f a s ) and o r d (s'/s)) a n d one defines degL = degcl(L) = 1 n From ReP P P' mark B3, Riemann Roch says dimL([ D l ) - dimL(K [ 01) deg([ D l - g t 1 and

-

apply this t o D = ( s ) with deg[Dl = degL a n d cl(w BI L*) = c,(w) - c , ( L ) = K - [ 01. One o b t a i n s dimHo(S,L) - dimHo(S,w IL*) = degL - g t 1 and dim Ho(S,L) - d i m H1 (S,L) = degL - g t 1 ( s i n c e H 1 (S,L) 2 Ho(S,w 19 L*)* by Serre d u a l i t y - c f . [ FT1 ; G R 1 I ) .

377

SHEAVES

For sheaves we refer to [C2O;CEZ;FTl;GN1;GRl;GU1,2;HA1 I. For open s e t s U C M l e t there be assigned a n abelian group G ( U ) (resp. ring, module, ...) a n d for V C U open l e t there be homomorphisms p vU : G ( U ) 3 G ( V ) satisfying P { = I a n d P: P: = P: for V C U C W . One writes also p UV ( f ) f l V for f E G ( U ) . This i s presheaf data a n d we a r e only concerned with s i tuations where G ( U ) i s some class of functions over U , e.g. 0(U) = holomorphic functions over U , or sections o f a VB over U . If in addition (1) U = U U i , s , s ' E G ( U ) , a n d s = s ' on U i implies s = s ' ( 2 ) s i E G ( U i ) , si = s . J in U i n U. implies there exists s E G ( U ) such t h a t s = s i in U i , then the J presheaf i s called a sheaf. This i s purely algebraic a n d we d o n ' t deal here with the topology of the associated espace i t a l e ' (EE) associated with a presheaf (see [ C2O;GMl ] for d e t a i l s ) . We do note however t h a t the EE i s d e fined via direct l i m i t s . One orders open subsets containing a point p by saying V > U i f V C U. The collection ( p vU , G ( U ) ) form a directed system and 1 im one forms G = -+ G ( U ) . This means one considers in the d i s j o i n t union P U G ( U ) the equivalence relation which identifies a E G ( U ) a n d B E G ( V ) i f U V there e x i s t s W C U n V such that p w a = p W p . The s e t of equivalence classes i s G a n d there i s a canonical map p u * G ( U ) G p based on the p vU . G i s P P' P e will generally use fancy l e t t e r s C instead called the s t a l k of G a t p . W o f G for sheaves. In particular we write 0 t o refer t o the sheaf of germs of holomorphic functions on a compact RS,S,defined via natural d a t a 0(U) on open s e t s U . A sheaf C i s called a sheaf o f 0 modules i f C ( U ) i s a module over 0(U) for a l l open U a n d the r e s t r i c t i o n maps a r e compatible with module structure. A sheaf homomorphism $J: F -t P; i s a collection $J,, of homomorphisms of appropriate type compatible w i t h r e s t r i c t i o n s . A sequence F G H i s exact a t 6 i f q,o$, = 0 for a l l U a n d i f $J,(s) = 0 for s E C ( U ) then for x E U there e x i s t s an open NBH V o f x a n d r E F ( V ) such t h a t +,,(r) = s I v . A short exact sequence i s an exact sequence 0 + f + C + H -+ 0, and an importa n t example i s 0 Z $ 0 0 0* + 0 where Z = sheaf of locally constant functions with values i n Z, 0*(U) = {f E 0(U); f ( z ) P 0 for z E U l , i n, injection, a n d e ( f ) = exp(2nif). The exactness i s straightforward (noting t h a t locally a branch o f the log function i s well defined). One says a sheaf F of 0 modules i s locally free of r a n k k i f every point has a n open NBH U such REEIARK B7.

-+

-+

3 78

ROBERT CARROLL

k t h a t 0u ? F l u .

For F t h e s h e a f o f s e c t i o n s o f a VB F one sees by l o c a l t r i -

v i a l i z a t i o n t h a t F i s l o c a l l y f r e e o f r a n k dimF.

I n t h i s s p i r i t one o f t e n

F w i t h t h e i r sheaves o f s e c t i o n s F, based on F(U)

i d e n t i f i e s VB

o f F o v e r U (Coy

CODy

= sections

o r whatever i s a p p r o p r i a t e ) .

Now sheaf cohomology can be d e f i n e d i n v a r i o u s ways ( v i a coverings, d e r i v e d f u n c t o r s , e t c . ) and these w i l l be e q u i v a l e n t i n s i t u a t i o n s o f i n t e r e s t i n t h i s book. ( c f . [ F l l ;GR1 ;GM1 ;GU1 ;HA1 based on c o v e r i n g s .

I).

We w i l l d e s c r i b e t h e Cech t h e o r y

B a s i c a l l y l e t F be a sheaf on M ( s u i t a b l e ) and

M (each z

a l o c a l l y f i n i t e open c o v e r i n g o f

= $ f o r a l l b u t a f i n i t e number o f Ua).

= (Ua)

E M has a NBH V such t h a t V n Ua

One d e f i n e s a s i m p l i c i a 1 complex

c a l l e d t h e nerve o f t h e c o v e r N ( l j ) a s f o l l o w s : V e r t i c e s Uo,

...,Uq

span a q

+.

...

s i m p l e x o = ( U o y ...,U ) i f and o n l y i f 1 0 1 = suppo = U n nU # Then 9 0 9 a q-cochain w i t h values i n F i s a f u n c t i o n f : IS f ( o ) E i - ( l o l y F ) ( r denotes -f

c o n t i n u o u s s e c t i o n s f ( z ) E FZ and, u s i n g t h e t o p o l o g y o f t h e EE, one can define

F v i a s e c t i o n data f ( U )

-

which w i l l be sheaf data

-

I).

c f . [C20;GMl

The s e t o f q cochains i s Cq(V_,F) w i t h f + g d e f i n e d " p o i n t w i s e " o v e r u as above

The coboundary o p e r a t o r 6 : Cq

t o g e t an a b e l i a n group.

-+

Cq+l

i s de-

f i n e d by

= r e s t r i c t i o n o f the section f E r(Uo n

where p 101

) t o 1 ~ 1 1= uq+4 and Z (U_,F) = i f

uo E

n

... I, uq+l

Cq(U_,F);

6f =

... n

01 i s a subgroup ( c o c y c l e s ) .

8

uUa ( =

n

...

6Cq-'(g,F) C

(Ho = Zo) denotes co-

It i s easy t o see t h a t Ho(UyF) = r(M,F).

= ( V ) i s a refinement o f

vering

n Ui+,

This i s a group homomorphism w i t h 6 6 = 0

*

Cq(U,F) i s t h e group o f coboundaries and Hq = Zq/ C q - l homology groups f o r U.

Ui-l

= (Ua)

i f t h e r e e x i s t s a map

Now a coy:

1 .+ ,U

f o r some 8 ) . There may o f course be many such mappings. B E v i d e n t l y LI induces a map y: Cq(!,F) -+ Cq(_V,F) v i a : f E Cq(UyF), o = (Vo,...y

w i t h Va C

U

...

n pv 3 f ( y V o ,..., uv ) ( n o t e uVo n implies (uf)(u) = p 101 q q V n n Vq # $ ) . Since 1.16 = 6v one o b t a i n s a homomrphism y*: Hq(LJ,F) -+ lim q H (V,F) which can be passed t o d i r e c t l i m i t s Hq(M,F) = + H (_U,F) ( e x e r c i s e

vq)

E

N(!),

.. .

\

cise).

Given a s h o r t e x a c t sequence 0

-f

E

9F 2 G

-f

0 (i.e.

I m $ = Ker$) one

gets t h e n ( f o r a l l s i t u a t i o n s d e a l t w i t h i n t h i s book) an e x a c t cohomology sequence (see comments below o n v a r i o u s cohomology t h e o r i e s )

379

COHOMOLOGY x

t

(B.6)

0 * Ho(E)

Ho(F) $*Ho(C)

9 H1 (E)

-+

H1 (F)

-+

. ..

Here 6* i s obtained via maps, a t some stage of refinement, constructed as follows: Given f E C q ( G ) such t h a t 6f = 0 pick g E Cq(F) such that Q g = f; b u t $69 = 6$g = 0 implies there i s an h E Cq+'(E) such that $g = 6g; define 6 * [ f ] = [ h ] and note 6 h = 0. Applied t o 0 Z 0 0* 0 one gets a sequence 0 Ho(Z) 2 2 H'(0) 2 C (since M a compact RS) HO(O*) C/{O) 1 H ' ( Z ) * H'(0) + H (0*)* 2 -+ 0 (c f. [ GR1 ;GU1 ;SL1 I ) . Recalling from Remark 86 t h a t l i n e bundles a re determined by (nonvanishing) cocycles C . . ( z ) J1 1 locally we can identify H (0*)with {isomorphy classes o f l i n e bundles} ( = 1 PicS) a n d the map H (0*)* 2 above i s the equivalence classes of divisors One notes also that sections r ( F ) o f Chern number = degL = degcl(L) = 1 n P' a sheaf F a r e isomorphic t o Ho(F) (exercise). We note also in passing t h a t locally the map e: f e xp(2ri f) i s o n t o 0" b u t globally t h i s i s n o t true. For example in a n annulus M: 1 < I z I < 2, z E r(M,0*) b u t z C exp( 2r if ) since necessarily f ( z ) = (1/2ri)logz, which will n o t be holomorphic single valued. -f

-+

-+

Q

-+

-+

-+

-+

-+

-+

-+

The Cech theory (described above) i s the easiest one t o describe b u t i t does n o t always agree with other cohomology theories (moreover (8.6) may n o t be exact beyond H1 ) . Generally i f e.g. M i s a suitable space or the sheaf has certa-in properties then various cohomology theories are the same ( a n d there a r e spectral sequence relating e.g. Cech a n d derived functor cohomology i n general). We will n o t give any real discussion of t h i s b u t point o u t some cases of agreement. Thus e.g. Cech cohomology (C,) f or coherent sheaves on a1 gebraic v a ri e t ie s (with Zariski topology) agrees with the derived functor theory (C,) as does Cech theory of coherent analytic sheaves on a complex analytic space. Over paracornpact Hausdorff spaces many theories coincide b u t t h i s i s n o t applicable over schemes ( c f . Remark B9 for schemes). When M i s a (Noetherian or n o t ) a f f i n e scheme ( M = specA) a n d F i s quasicoherent 0 a n d C F E Cc ( t h u s (definitions l a t e r ) then ( c f . [ HA1 I ) Hi(M,F) = 0 for i Hp = H F ) . This will cover a l l situations of inter est here. In any event F the structure sheaf 0, of any scheme i s coherent a n d quasicoherent. Regard-

380

ROBERT CARROLL

i n g t e r m i n o l o g y here Noetherian r i n g s a r e d e f i n e d i n Remark B9 and a scheme X i s Noetherian i f i t can be covered by a f i n i t e number o f open a f f i n e

schemes SpecAi w i t h Ai

For coher-

Noetherian (such SpecAi a r e Noetherian).

ence we a l s o s i m p l y s t a t e d e f i n i t i o n s .

A coherent sheaf w i l l be quasicoher-

e n t and we d e f i n e quasicoherent l a t e r .

A sheaf F (F an 0 module o v e r t h e

s t r u c t u r e sheaf 0 ) i s coherent means F i s o f f i n i t e t y p e and, f o r each open U C X and each hommorphicm 4 : 0“U)

* F(U), k e r 4 i s o f f i n i t e type; F o f

f i n i t e t y p e means each x has a NBH V such t h a t F ( V ) i s generated by a f i n i t e number o f s e c t i o n s o f

F o v e r V ( i . e . SP(V)

-f

F ( V ) i s o n t o f o r some p ) .

Simply s t a t e d F coherent means t h e r e e x i s t s an e x a c t sequence 0r One f u r t h e r comment here ( f o r s u i t a b l e s i t u a t i o n s where HC

0.

Let M = U

U

-+

=

0’

-t

f

-f

HF e t c . ) .

V be e.g. a d i f f e r e n t i a b l e m a n i f o l d w i t h U,V open; t h e n t h e r e

i s a c l a s s i c a l e x a c t M a y e r - V i e t o r i s cohomology sequence

...

(8.7)

-+

( c f . [ KJ1 I ) .

Hk(U) @ Hk(V)

* Hk(U n V)

-+

Hktl(M)

-+

Hktl(U)

B Hk+l(V)

-+

...

This can be extended t o s u i t a b l e sheaves o v e r s u i t a b l e schemes

as needed i n 112 f o r example ( c f . [ B X 1 ; I N l ; K J l ; K W Z l ) .

RENARK B8- We go n e x t t o t h e moduli space o f curves f o l l o w i n g [Bl;FEl;NKl; MU3;SLl

1.

T h i s p l a y s an i m p o r t a n t r o l e b o t h i n v e r y t h e o r e t i c a l q u e s t i o n s

and i n v e r y p r a c t i c a l problems o f s o l i t o n c l a s s i f i c a t i o n ( c f . [ BL31).

Thus

M i s t h e s e t o f a n a l y t i c o r a l g e b r a i c isomorphism c l a s s e s o f RS o r curves 9 o f genus g. For g = 1 i t i s known t h a t any RS can be r e a l i z e d as a nonsin2 2 g u l a r c u b i c c u r v e C i n P ( e l l i p t i c c u r v e ) of t h e form y2 = 4x g2x g3

-

-

-

c f . Remark B2) and j ( C ) = ( t h e e q u a t i o n o f t h e Weierstrass P f u n c t i o n 3 3 2 1728g2/(g2 - 27g3) depends o n l y on t h e isomorphism c l a s s ill o f C. Thus j : ill

-+

C i s 1-1,

39

-

3.

space H Q ,

(I

g i v i n g l!ll a complex s t r u c t u r e as w e l l .

Now r e c a l l t h a t ( c f . Remarks5.1, 9

7)

For g > 1 dimm = g B4, and 86) t h a t t h e Siege1 h a l f -

( o r H ) o f dimension g(g+1)/2 parametrizes t h e p a i r s (A,(ai,Pi)) 9 where A i s a p r i n c i p a l l y p o l a r i z e d a b e l i a n v a r i e t y (complex t o r u s

T E H ) and (ai,Pi) i s a canonical s y m p l e c t i c b a s i s . Those p a i r s com9 i n g from Jacobians o f c u r v e s C form a subset N C H ( t h e p e r i o d m a t r i x % T) 9 9 Thus A = H /SP(2g,Z) i s t h e coarse moduli space o f ( p r i n c i p a l l y p o l a r i z e d ) 9 9 a b e l i a n v a r i e t i e s and M = N /SP(2g,Z) i s t h e coarse moduli space o f curves 9 9

plus

MODULI SPACE

381

o f genus g ( t h e notation i s s l i g h t l y d i f f e r e n t in Remark 5.1 and H i s used 9 t h e r e ) . One r e f e r s t o data { ( A , ( a . , B . ) ) ) = N as t h e T o r e l l i space ( o r betJ J 9 t e r use F c l a s s e s o f data under isomorphisms f : A -+ A'; f,: a + a ! and 8i i i -+ 6 :). Ill i s a complex o r b i f o l d ( i . e . i t i s a smooth a n a l y t i c manifold ex4 c e p t f o r " o r b i f o l d " p o i n t s where i t looks l i k e a complex V B modulo a f i n i t e A

i s Teichmul9 g l e r space. e* is a topologically t r i v i a l complex a n a l y t i c manifold a n d I3 = 9 9 ?9/ r 9 where r 9 i s the d i s c r e t e mapping c l a s s group ( o r modular group). We a s needed i n the t e x t ( c f . 118 where N % cg). will say more about g 9 RElMRK 39. We go now to some topics in commutative algebra and a l g e b r a i c geometry which a r e needed in ii12,18 f o r example. The sources here a r e [AT4; DQl;F~l;MA1;MQ1;JU1-6;HA1;MJ1;KEN1;GRO1;SHFl 1. Again a c e r t a i n amount of introductory material appears a l s o in t h e t e x t and t h e r e will be some duplic a t i o n . We consider commutative r i n g s R w i t h i d e n t i t y 1 , a typical example being K[xl = polynomials i n x over a f i e l d K ( K = C always b u t we r e t a i n the K n o t a t i o n ) . An ideal i s prime i f R/I i s an i n t e g r a l domain and maximal i f R/I i s a f i e l d . The multiples a x of x E R form a principal ideal ( x ) and x is a u n i t i f and only i f ( x ) = ( 1 ) = R . If a C R i s an i d e a l , a # ( l ) , then t h e r e e x i s t s a maximal ideal containing a a n d maximal i d e a l s a r e prime. Every ring has a t l e a s t one maximal ideal ( r i n g always means commutative ring with i d e n t i t y ) . We note t h a t i n K t x l i f f i s an i r r e d u c i b l e polynomial then ( f ) is prime (by unique f a c t o r i z a t i o n ) . Further in K[x] ( b u t not in K[xl , . . . , ~ n ] , n > 1 ) a l l i d e a l s have t h i s form ( f ) f o r f a n i r r e d u c i b l e polynomial. T h u s K[x] i s a principal ideal domain (PID - a l l i d e a l s a r e p r i n c i p a l ) and every nonzero prime ideal i s maximal ( c f . [AT41). The s e t N of a l l n i l p o t e n t elements i n R i s an ideal c a l l e d t h e n i l r a d i c a l and N = n p, p prime. Generally f o r a an ideal rada = Ex E R ; xn E a f o r some n} a n d rada = i n p, p prime, p 3 a). We r e c a l l a l s o t h a t a ring i s Noetherian i f every ideal i s f i n i t e l y generated ( f i e l d s and PID's a r e Noetherian). If R i s Noetherian so i s R[x l , . . . , ~ n ] and in p a r t i c u l a r K[x, ,... , x n ] i s Noetherian f o r any f i e l d K ( c f . [ FU1 I ) . group action).

The universal a n a l y t i c covering space C of GI

The s e t o f prime i d e a l s i s c a l l e d SpecR.

-

For p a prime ideal the s e t S =

p i s a m u l t i p l i c a t i v e subset ( a , b E R-p implies ab E R-p since i f ab E p e i t h e r a o r b E p - a l s o 1 E R-p). Then t h e l o c a l i z a t i o n R i s the P R

38 2

ROBERT CARROLL

commutative r i n g d e f i n e d by R = S - l R = RS = {a/s, a E R, s E S) (where a/sl P = b / s 2 ifand o n l y i f s3(s2a - slb) = 0 f o r some s 3 E S ) . Note here S - l R

=

r ) when s(slr2 - s2rl) = 0 f o r 2’ 2 s-lr. This can be d e f i n e d v i a a u n i v e r s a l mapping S - l R i s a r i n g homomorphism and any r i n g homomorphism

= S X R modulo t h e E r e l a t i o n (sl,rl)

some s E S and ( s , r ) *

Q

p r o p e r t y , namely v: R

.+

(s

$: R + T, w i t h a l l s E S i n v e r t i b l e i n T,

i t has o n l y one maximal i d e a l

note t h a t i f b / t

4m

4

then b

m

p so b

Note v ( r ) =

$ 0 ~ .

I). R i s a local ring P a E p 1 = pR ). To see c h i s P R-p and b / t i s a u n i t i n R Hence

l - l r = ( l , r ) * and ( v s 1 - l = s - l - 1 = ( s , l ) * (i.e.

f a c t o r s as J, = ( c f . [RWl

= {a/s, E

.

One i f a i s an i d e a l i n R and a $ m t h e n a c o n t a i n s a u n i t and hence = R P P’ n o t e s t h a t t h e prime i d e a l s o f S - l R = R a r e i n 1-1 correspondence q -+ S - l q P w i t h prime i d e a l s o f R which d o n ’ t meet S = R-p ( R -+ R c u t s o u t a l l prime P i d e a l s e x c e p t those c o n t a i n e d i n p). For S = Ifm; f E R; m 2 0) RS i s w r i t t e n a s Rf.

Now l e t A be a commutative r i n g w i t h 1 ( A i s a common n o t a t i o n here s i n c e For a n i d e a l a C A l e t V(a) =

commutative a l g e b r a s a r e r i n g s ) .

e a l s o f A c o n t a i n g a ( n o t e a C h i m p l i e s V(h) C V ( a ) ) .

Z a r i s k i t o p o l o g y on SpecA v i a a b a s i s o f open s e t s D ( f ) = SpecA {p

SpecA; f $ p } ( f o r f

E

E

A,

( f ) i s t h e i d e a l generated b y f

Indeed t h e f a m i l y F = {V(a); a

C

I f t h e n V(a) i s c l o s e d and p

(ai). =

U

C

p

E

q and s ( q ) = a / f i n A

from A ) . V

C

U.

V(1

a,)

$ V(a)

=

= n V(a,)

(3)

V(qai)

=

uy V

i t f o l l o w s t h a t a $ p and t h e r e

X = SpecA open one d e f i n e s a r i n g ( o r a l g e b r a ) 0,(U)

t h e r e e x i s t s a NBH W o f p, W

4

V((f))

( f ) = Af).

a, f # p; hence p E D ( f ) and D ( f ) n V(a) = Q. Hence SpecA - V(a) f # a . One notes t h a t t h e Z a r i s k i t o p o l o g y i s never H a u s d o r f f .

t i n g o f maps s : U - + U A p , W, f

-

E

D(f),

For U

-

A an i d e a l ) s a t i s f i e s t h e axioms f o r c l o s e d

s e t s ( 1 ) V(0) = SpecA, V(A) = Q ( 2 ) exists f

prime i d -

One d e f i n e s t h e

C

4

U, w i t h s ( p )

E

A

P’ U, and elements a,f

(i.e.

o r BA(U) c o n s i s -

such t h a t f o r each p E

A satisfying, f o r 4

U, E

l o c a l l y s i s a q u o t i e n t a / f o f elements

There i s a n a t u r a l r e s t r i c t i o n homomorphism rVU: OA(U) This gives a sheaf o f r i n g s on X ( s t r u c t u r e sheaf

and ( x y 0 A )

E

-

-+

OA(V) f o r

c f . Remark 87)

i s c a l l e d a r i n g e d space o r a f f i n e scheme; a c t u a l l y here a l o c a l

r i n g e d space s i n c e t h e s t a l k 0 A i s a l o c a l r i n g ( i . e . 0 has a unique P - P Pn maximal i d e a l ) . Note a l s o f o r U = D ( f ) , O A ( D ( f ) ) = Af ( a / f 1 ( c f . [HA1 I Q

Q

SC H EM ES

383

f o r p r o o f ) and f o r V = D ( f g ) C U = D ( f ) one has rVU: a/fn

':'

(note roughly A Afg V ( ( f ) ) , and

$

(X,Ox)

-+

1,

Oy(U)

+

agn/(fg)n:

Af -+

f u n c t i o n s w i t h poles on t h e s e t where f = 0, i . e . on

Afy f

4p -

cf. [DQll).

A morphism o f r i n g e d spaces 4 :

(Y,Oy) i s a c o l l e c t i o n o f a c o n t i n u o u s map $: X

morphisms $,:

+

open U

0,(+-'U)for

C

F i n a l l y a scheme (X,Ox)

homomorphisms o f sheaves.

such t h a t t h e r e e x i s t s an open c o v e r i n g X

Y and r i n g homo-

-+

Y, which commute w i t h r e s t r i c t i o n U

i s a l o c a l l y r i n g e d space

U, w i t h (U,,Ox(U,))

an a f f i n e

scheme. L e t us add a few more d e t a i l s and o b s e r v a t i o n s a b o u t (SpecA,BA) and r e f e r t o [ DQl ;GRO1 ;HA1 ;MA1 ;SHF1

I

f o r much more.

There i s a c e r t a i n amount o f "path-

o l o g y " connected w i t h schemes and t h e i r s t r u c t u r e sheaves due i n p a r t t o t h e We w i l l i n d i c a t e

Z a r i s k i t o p o l o g y (and t h e n a t u r a l sheaf t o p o l o g y o f EE). j u s t a l i t t l e o f t h i s and s o r t o u t a few i t e m s .

{pl

E

F i r s t one notes t h a t p =

SpecA i s c l o s e d i f and o n l y i f p i s maximal,

i f and o n l y p = n V(E),

C

4 (p,4 a r e prime i d e a l s ) .

p E V(E) (i.e. p

roughly the closed points A has o t h e r p o i n t s ideals.

Q,

E so V(p)

3 Q,

C

IF3

= V(p), and 4 E) ;1

Note i n t h i s d i r e c t i o n t h a t

{TI

V ( E ) and t h u s n V(E) = V(p)). Thus

points o f the classical variety

1,

SpecA b u t Spec

a l l i r r e d u c i b l e s u b v a r i e t i e s and represented by p r i m e

One notes t h a t t h e prime i d e a l s o f A/p

c o n t a i n p so Spec(A/p) 2 V(p)

2 SpecA.

%

prime i d e a l s o f A which

T h i s means SpecA 2 Spec(A/N) where

N = nilradical.

A space X i s i r r e d u c i b l e i f e v e r y p a i r o f nonempty open s e t s i n X i n t e r s e c t (thus X i s h i g h l y nonkusdorff).

Equivalently X i s n o t the union o f 2 pro-

p e r c l o s e d subsets or e v e r y nonempty open s e t i s dense i n X. d u c i b l e i f i t i s i r r e d u c i b l e i n t h e induced t o p o l o g y .

Y

C

X i s irre-

One sees e a s i l y t h a t

{XI and {yl a r e i r r e d u c i b l e and i f Y C X i s i r r e d u c i b l e and Y = ):1 x

E

X then x i s c a l l e d a generic p o i n t o f Y (points y

i a l i z a t i o n s o f x). i n t e g r a l domain.

E

I:{

f o r some

a r e c a l l e d spec-

Now X = SpecA i s i r r e d u c i b l e i f and o n l y i f A/N i s an I n t h i s d i r e c t i o n we n o t e t h a t i f A has no d i v i s o r s o f 0

t h e n ( 0 ) i s prime and i s c o n t a i n e d i n e v e r y prime i d e a l ; hence and ( 0 ) i s a g e n e r i c p o i n t .

(T) =

SpecA

We see a l s o t h a t SpecA has a (unique) g e n e r i c

p o i n t i f and o n l y i f t h e n i l r a d i c a l N i s prime (and t h e n N i s g e n e r i c ) . Gene r a l l y i f X i s a scheme, e v e r y i r r e d u c i b l e c l o s e d subset has a unique generic point.

This gives a l i t t l e o f the flavor.

384

ROBERT CARROLL

k on

Now g i v e n A a r i n g and M a n A module one d e f i n e s a s h e a f lows.

s

=

Let M

P

~ - one p forms S - ~ M = M v i a an e q u i v a l e n c e r e l a t i o n on M

P

i f and o n l y ift h e r e i s a t

(m',s') %

(m,s)',

SpecA as f o l -

be t h e l o c a l i z a t i o n o f M a t p = prime i d e a l i n A.

etc.

E

such t h a t s ( p ) E M

P

-

S such t h a t t ( s m '

E(U) =

For U C SpecA open now d e f i n e

x

Thus f o r

S: (m,s)

s'm)

0; then m/s

+UMP

functions s: U

and l o c a l l y , f o r each p E U t h e r e e x i s t s an open NBH V

o f p, V C U , such t h a t f o r 4

E

V s ( 4 ) = m/f,

m

E

M y and f E A.

One sees

t h a t t h i s determines a s h e a f and such sheaves a r e i n f a c t t h e models f o r One says a sheaf F o f modules o v e r a scheme (X,OX)

quasicoherent sheaves.

i s q u a s i c o h e r e n t i f X can be covered by open a f f i n e schemes Ui

= SpecAi such

Y

FlUi? Mi.

t h a t f o r each i t k r e i s an Ai module Mi w i t h Mi

i s a f i n i t e l y generated Ai

module t h e n

F i s coherent.

t h e s t r u c t u r e sheaf 0x i s q u a s i c o h e r e n t and coherent. LI

M(D(f))

zMf

with

P

= M

P

I f i n a d d i t i o n each On any scheme X

One notes a l s o t h a t

( c f . [HA1 I).

L e t us add a few comments now a b o u t ProjB d e f i n e d i n Remark 12.5.

Intuitiv-

e l y t h e c o n s t r u c t i o n o f t h e s t r u c t u r e sheaf on P r o j B i s modeled on p r o j e c t i v e space.

Thus one wants an a f f i n e scheme s t r u c t u r e on each D + ( f ) .

t h i n k o f Pn(K) = ProjK[xo,

...,xnl

and open s e t s

x ~ - ~ / x ~ , x ~ + ~ / x ~ , . . . , ixn~ Euclidean /x~) space. D+(xi)

and a l s o

%

SpecK[xo,...,?

,... ,x n )

i a l Q(xo,... , x ~ - , , x ~ + ~

,...,

X ~ - ~ / X ~ , X ~ + ~ / Xxn/xi) ~

a1 f u n c t i o n P/xi

%

i,. . .,xn].

%

xi # 0

%

Now

...,

p o i n t s (xo/xi,

0

Consider t h e s e t xi

%

Here t o a (homogeneous) polynom-

,...,

r a t i o n a l f u n c t i o n o f degree 0, Q(xo/xi k ( k = degree P) and t o a r a t i o n -

P(xo,...,xn)/xi

k o f degree 0 v i a xi

= 1 one g e t s a homogeneous polynomial Q

= 0 degree component o f t h e grad1 n (xi) A ed r i n g o f f r a c t i o n s P/xr, corresponds t o K [ x O y . . . y ~ i,...,xnl, and D+(xi) %

(xoy..

SpecK[ this.

.,;li,...,xn).

Thus K [ x o y . ..,x

1 for further "intuitive" explication o f a l l I n general ( c f . [DQl;GROl;HAl I)one emphasizes t h a t f o r B = @Bdy d 2 We r e f e r t o [ D Q l

0, ProjB i s made up o f homogeneous i d e a l s p

#J

B,

= $Bd,

eous i d e a l s can be generated by homogeneous elements bn

d > 0 ( n o t e homogenE

Bn and a homogen-

eous i d e a l p i s prime i f and o n l y i f f o r any 2 homogeneous elements a,b; E

p implies a o r b

E

p).

F u r t h e r i n d e f i n i n g D + ( f ) = (p

c a l l f i s t o be a homogeneous element, i . e . f

€

E

ab

ProjB; f q p ) r e -

Bd f o r some d.

For B

P

one

38 5

SPEC AND PROJ

looks (by definition) a t 0 degree elements in T-lB where T i s the multiplicative system of a l l homogeneous elements f E By f 4 p, a n d locally 0(U) has the form { a / f l , a , f homogeneous of t h e same degree, f 4 q, 4 E U ( s ( 4 ) = a/f E Bq). Recall also in defining SpecA, 0 ( D ( f ) ) ? A f = { a / f n l . Analogously here with Proj, D+(f) 2 SpecBYf) with corresponding structure sheaves, and % t h i s i s spelled o u t in [DQl;HAl I. Note again f i s homogeneous a n d Bo (f) { b / f n l ) , degb = degfn. Evidently open s e t s D+(f) for homogeneous f E B+ c o ver ProjB (note D+(f) = $ i f a n d only i f f i s nilpotent). I t i s often the case t h a t Bo i s a f i e l d and B i s generated by B1 over Bo in which case Proj B = {p # B+I a n d for f E B1, B Y f ) = { g / f n y g E B n l . Note also ( c f . [ SZPl I ) i n this case Bf = B ~ f ) [ T , T ' l l ( T f ) . In general take f E B+ a n d consider the identification 0 ( D + ( f ) )?O(SpecBqf)). F i r s t look a t $ : a -t (aBf) n 1 (a C B a homogeneous i d e a l ) . Recall for a suitable (nongraded) r i n g R ideals R S-lR = Rf map via 4: q S-lq, q n S = 4 and here one i s concerned w i t h S % {f",n 01. I t i s easily seen t h a t $*: SpecRf SpecR has image -+

BYf

+

-+

-+

D(f). Now go back t o J, a n d one sees t h a t i f a E D+(f) ( i . e . f ( a ) E SpecBYf) and we refer t o [ HA1 I for the r e s t ,

$

a ) then J,

If M i s a graded B module one associates t o M a sheaf K o v e r ProjB as folwhere lows (cf. Remark 12.14). Over D+(f) one defines the sheaf via Mo (f) M:f) = degree 0 elements of M f y defined as above for BYf); M Y f ) i s evidently More specifically for f E Bd, d > 0, and x E M one has a module over Bo (f)' P fx E M so consider ( M f ) n = { x / f n ; x E M n k k d } with ( M f ) o = M Y f ) % x E M k d . (a E B k d ) a n d x/fr E C1earl!+iO i s a Bo module since for a / f E Bo (f) (f) One defines then a M Y f ) ( x E M r d ) one has a x E M ( k + r ) d so a x / f k + r E M ry (f)& sheaf M o n ProjB in the obvious way with G(D+(f)) z M ( ~ ) ; M i s a quasicohere recall also ( c f . [ DQ1;HAl I ) t h a t 0 ( n ) of Reent SX module ( X = ProjB). W mark 1 2 . 6 can be p u t in a more general context ( c f . Remark Bll ). Let B =

cfd

-

= $Bd ( d 2 0 ) , where we assume f o r convenience t h a t ( A ) B i s generated by B1 a s a Bo algebra. Set B ( n ) k = Bn+k t o define a graded B module B ( n ) = ... I?l B ( n ) - n 63 fR B ( n ) o @ The associated sheaf B(n)w i s called 0 (n) ( X = ProjB) a n d O x ( l ) i s the twisting sheaf. Evidently ( B ( n ) Y f ) % { z / f k f z E Bn+kdy degf = d1 since B ( n ) o % B n . I n the classical picture B = K [ x O , . . . , ~ r xy], f = x i , B;f) % r a t i o n a l fractions P ( xo , . . . , x r ) / x i k with degP = n + k . Given the assumption (A) one knows O x ( n ) = O X ( l ) B na n d for a quasicoherent

...

....

ROBERT CARROLL

386

,xtl

module F one defines F ( n ) = F @Ix 0,(n); i t follows t h a t f o r

as above

M(n) = M(n)-. Let us i n d i c a t e a few examples of Spec and Proj.

Take K t o be a n a l g e b r a i -

c a l l y closed f i e l d ( K = C is f i n e ) . For K = A SpecK = 1 point and 0 = K. 1 The a f f i n e l i n e over K i s AK = SpecKixI; i t has a generic point 5 'L 0 ideal

1

2

with = AK and the o t h e r points ( a l l c l o s e d ) % maximal i d e a l s o f K[x] (maximal = prime here and maximal i d e a l s % i d e a l s generated by monic i r r e d u c i b l e polynomials) so f o r K a l g e b r a i c a l l y closed t h e closed p o i n t s % points o f K. 2 The a f f i n e plane AK = SpecK[x,yl whose closed points a r e i n 1-1 correspondence w i t h points ( k l y k 2 ) . There i s a l s o a generic point 5 'L 0 ideal with 2 5 = A K and f o r each i r r e d u c i b l e polynomial p(x,y) there i s a point TI whose closure =

rl

+ a l l closed points ( a , b ) such t h a t p ( a , b ) = 0

(0

is c a l l e d gen-

e r i c f o r p(x,y) = 0 ) . I f A i s a r i n g P i = ProjA[xo , . . . , x n ] ( B = A [ I w i t h natural grading) i s p r o j e c t i v e n space over A and f o r A = K a f i e l d t h i s i s

the standard PF. For A a r i n g ProjA[x] = SpecA which reduces t o a point f o r A = K. In p a r t i c u l a r w i t h B = A[xo , . . . , ~ n ] , Bo = K, B1 = A, l e t I be an Then B ' = B/I is a ideal generated by homogeneous polynomials ( f l , . . . y f r ) . graded r i n g a n d X = ProjB' i s t h e closed subvariety of P; = ProjB defined by the ( f,

,. ..,f r ) .

One r e c a l l s a l s o the important r o l e of n i l p o t e n t s (a E R, a ( x ) = 0 f o r a l l x E SpecR i f and only a E np i f and only i f a is n i l p o t e n t ) . Thus f o r ex2 1 ample ( c f . [MU4,51) Y = SpecK[x]/(x ) i s a s i n g l e point, say 0 E A K , support2 i n g functions O( E K and the function x which vanishes a t 0 ( O y = K [ x ] / ( x ) ) . T h u s t h e function x is not zero i n 0 y y e t i t vanishes on Y. Similarly Y, = SpecK[xl/(xn) is a s i n g l e point 0 b u t 0n involves functions and t h e i r f i r s t 1 C A . Since K[xI i s a PTD n-1 d e r i v a t i v e s a t 0; one w r i t e s Y1 C Y2 C

...

a l l nonzero i d e a l s have the form a = (nln ( x - a i ) r i ) and Y = SpecK[xI/a i n volves n p o i n t s a l ,

...

,a n w i t h sheaf 0y % ( r i - 1 ) order d e r i v a t i v e s a t a i . 2 One notes a l s o ( c f . [ KEN1 I f o r lovely p i c t u r e s ) t h a t "curves" such a s V(x + y2 - 1 ) i n P 2 (C) a r e spheres while V(x 2 +y2 ) i s two spheres touching a t one point.

REmARK 310.

We continue Remark B9 w i t h a few f u r t h e r comments ( c f . [AT4;

FU1;HAl;MAl;MJl;MQl ;MU1-6;SHFl

I).

W e go f i r s t to completion.

Localization

COMPLETION

387

preserves exactness and t h e N o e t h e r i a n p r o p e r t y and so w i l l c o m p l e t i o n f o r The idea o f c o m p l e t i o n goes b a s i c a l l y as f o l l o w s .

f i n i t e l y generated modules.

L e t A be a r i n g , Ia n i d e a l , and M a n A module.

The submodule InM d e t e r -

mines t h e I - a d i c t o p o l o g y o f M which d e f i n e s a fundamental system o f neighborhoods (FSN) o f x

M t o be { x + I'M}.

€

Note In 3 1" f o r m > n, M i s Haus-

d o r f f i f and o n l y ifnz InM = 0 (which we assume) and N C M i s c l o s e d means

n (N t I'M)

= N.

such t h a t n,m

A sequence xn i s Cauchy means f o r any

-

8 i m p l i e s xn

xm E I'M.

CL

there e x i s t s 8

M i s complete means t h a t e v e r y h

Cauchy sequence converges and t h e c o m p l e t i o n M can be d e f i n e d as t h e s e t o f equivalence c l a s s e s o f Cauchy sequences i n M. i n v e r s e ( p r o j e c t i v e ) l i m i t o f t h e M/InMM.

To c o n s t r u c t i t l e t M* be t h e

Then M* C Ern ( M / I n M ) 1

d u c t t o p o l o g y and one r e q u i r e s p o i n t s (ml,m2,...)

w i t h t h e pro-

o f M* t o s a t i s f y

e q,n

(m ) n Then

= m f o r q 5 n where 8 * M/InM -t M / I q M i s t h e canonical s u r j e c t i o n . q A q,n' A M* = M. As a n example t a k e formal power s e r i e s 0 as t h e c o m p l e t i o n o f p o l y -

nomials 0 i n 5 , w i t h I = SO 0, I t

2

C

I , and f o r p =

pis, po

t p1 5 t p2s

2

1 pks

%

k

,.. . f o r

p o l y n o m i a l s v a n i s h i n g a t 0.

C l e a r l y nz In =

one has components i n O / I n o f t h e form po, po n = 1,2,.

...

T h i s example i n d i c a t e s t h e maps

c l e a r l y . G e n e r a l l y i f A i s a Noetherian l o c a l r i n g w i t h maximal i d e a l qsn A and t h e m-adic t o p o l o g y o f A m t h e n A i s a l o c a l r i n g w i t h maximal i d e a l 8

i s Hausdorff. L e t us remark a l s o t h a t a graded r i n g A = $An ( n 2 0 ) i n v o l v e s AnAm C ,,A,n,

so An i s an A.

module.

The t y p i c a l example i s A = K[x1,...,xn1

homogeneous p o l y n o m i a l s o f degree n.

w i t h An =

A graded A module M = $Mn i n v o l v e s

A M C Mnm so Mn i s an A. module. One can show t h a t A Noetherian i s e q u i nm v a l e n t t o A Noetherian w i t h A f i n i t e l y generated as an A. a l g e b r a . I f a C 0 n ntl A i s an i d e a l o f a Noetherian r i n g A d e f i n e G r A = Gr,A = $(a / a ), n 2 0. A Then G r A i s Noetherian and G r A 2 G r A a s graded r i n g s . G e n e r a l l y t h e dimens i o n o f a r i n g A ( = d ( A ) ) i s t h e supremum o f l e n g t h s o f ascending c h a i n s o f prime i d e a l s i n A (a f i e l d has dimension 0 ) .

I n t u i t i v e l y f o r v a r i e t i e s one

t h i n k s o f c h a i n s l i k e p o i n t , curve, s u r f a c e , e t c . r e p r e s e n t e d b y prime i d e a l s

...

G e n e r a l l y f o r a Noetherian l o c a l 3 p1 3 p2 3 3 pk % dimension k. 2 r i n g w i t h maximal i d e a l m one has d(A) < - a n d i n f a c t d ( A ) 2 dim(m/m ). I n

po

a l g e b r a i c geometry t h e l o c a l r i n g s o f n o n s i n g u l a r p o i n t s g e n e r a l i z e t o regu2 l a r l o c a l r i n g s which s a t i s f y any o f t h e e q u i v a l e n t c o n d i t i o n s (1) dim(m/m

388

ROBERT CARROLL

= d(A) ( 2 )

m

i s generated by d = d(A) elements.

m

Intuitively i f

-

i d e a l i n A = " r e g u l a r " f u n c t i o n s , o f f u n c t i o n s v a n i s h i n g a t x,then c l a s s e s o f f u n c t i o n s w i t h t h e same l i n e a r terms and r e g u l a r

Q

maxim1

m/m2

% f

dimension o f

t h e v a r i e t y = dimension o f t h e v e c t o r space spanned by g r a d i e n t s .

One notes

4

a l s o t h a t A i s r e g u l a r i f and o n l y i f A i s r e g u l a r .

I n t h e geometrical s i -

A

t u a t i o n A/m 1 K and a t n o n s i n g u l a r p o i n t s A i s a formal power s e r i e s r i n g i n d indeterminants. Now t o d e f i n e a c u r v e i n t h e language o f schemes i s somewhat complicated. By c o n t r a s t i n terms o f v a r i e t i e s i t i s easy t o d e f i n e a curve.

Thus, r o u g h l y ,

i n a p r o j e c t i v e c o n t e x t l e t V be an i r r e d u c i b l e a l g e b r a i c s e t i n Pn ( i . e . V = { p E Pn such t h a t F(p) = 0 f o r FoS=Ifi}=homogeneous

polynomials i n K[xl,

1 and V I union o f 2 s m a l l e r a l g e b r a i c s e t s ) . Then i f I i s t h e (hoyXn+l 1 mogeneous) i d e a l i n K [ x l,...,~nl generated by S, V(1) = V ( S ) , and V-irredue e

c i b l e i f and o n l y i f I ( V ) i s prime. t h a t I(V(1)) = radI. g e b r a i c subset o f Pn. C

The p r o j e c t i v e N u l l s t e l l e n s a t z says

Now d e f i n e a s e t U

c

Pn t o be open i f Pn

-

U i s an a l -

T h i s g i v e s t h e Z a r i s k i t o p o l o g y o n P" and subsets V

Pn a r e g i v e n t h e induced t o p o l o g y .

c l o s e d i f and o n l y i f i t i s a l g e b r a i c .

Thus f o r a v a r i e t y V, a subset o f V i s Let X

c V be open.

It i s a l s o c a l -

l e d a v a r i e t y ( i n t h e induced t o p o l o g y ) , and one w r i t e s K ( X ) f o r t h e f i e l d o f r a t i o n a l f u n c t i o n s on X ( d e f i n e d i n some s u i t a b l e manner).

Now one knows

K ( X ) i s a f i n i t e l y generated e x t e n s i o n o f K and one d e f i n e s dimX = transcendence degree K(X) o v e r K ( c f . Remark 12.4).

When dimX = 1 we have an a l -

gebraic curve. Roughly t o g e t schemes one adds g e n e r i c p o i n t s t o v a r i e t i e s b u t t h e r e s u l t i n g c o m p l e x i t y makes t h e d e f i n i t i o n o f a c u r v e c o n s i d e r a b l y more t e c h n i c a l . For example one can d e f i n e a c u r v e as an i n t e g r a l separated scheme X o f f i n i t e t y p e o v e r C w i t h dimension 1.

A d e t a i l e d d i s c u s s i o n o f dimension i s n o t

necessary here ( t h e dimension i s 1 f o r C i n 512 h e u r i s t i c a l l y and i n fact,

--

a s above f o r v a r i e t i e s , dim X = transcendence degree K ( X ) o v e r K where now K(X)

l o c a l r i n g 0 c o f t h e g e n e r i c p o i n t 5 i s t h e f u n c t i o n f i e l d o f X;

- q u o t i e n t f i e l d o f A o v e r any U = SpecA open i n X

ed v i a A ) .

Now we d e f i n e t h e terms.

-

K(X)

i . e . dimX i s determin-

Thus i n t e g r a l means t h a t f o r e v e r y

open U C X, Ox(U) i s an i n t e g r a l domain (SpecA i s i n t e g r a l i f and o n l y i f

A i s an i n t e g r a l domain).

X i s separated o v e r W means t h e diagonal morphisn

ALGEBRAIC CURVES 6: X

-t

X X

W

parated).

X i s a c l o s e d immersion ( W

N

389

scheme; i f W = SpecZ,X i s c a l l e d se-

Here X Xw Y i s t h e f i b r e d product,

i.e.

t h e f i r s t diagram

commutes and any commuting diagram I 1 f a c t o r s t h r o u g h I i n t h e sense t h a t q1 = plt and q2 = p 2 t f o r some morphism t: Z

+

X Xw Y.

Next one says X i s

quasicompact i f e v e r y open c o v e r i n g has a f i n i t e subcovering and a scheme X ( o v e r K) i s o f f i n i t e t y p e i f X i s quasicompact and f o r U C X open r(U,0,) i s a f i n i t e l y generated K a l g e b r a ( n o t e r ( X , O X ) may n o t be f i n i t e l y generated).

There i s a l s o a n o t i o n o f completeness f o r v a r i e t i e s which g e n e r a l i -

zes t o t h e idea o f p r o p e r f o r schemes.

Thus a v a r i e t y X i s complete i f f o r

a l l v a r i e t i e s Y t h e p r o j e c t i o n p2: X X Y

Y i s c l o s e d ( i . e . maps c l o s e d

Now a scheme X o v e r K = C i s regarded a s a morphism

sets t o closed sets).

+:

-+

X 3 SpecK = i p l ( t h e map i s t r i v i a l here b u t i n v o l v e s a l s o an induced map

o f s t r u c t u r e sheaves

-

note t h a t Ocpl

=

K).

Such a

+ i s proper over W i f i t

is separated, X i s o f f i n i t e type, and f o r any (scheme) morphism Y t h e p r o j e c t i o n p2: X Xw Y X over K - X

+

Y i s c l o s e d (see [ HA1 ;MA2;MU4

o v e r SpecK and X Xw Y + f i b r e

+

I f o r more

-

SpecKNW note

p r o d u c t i n terms o f s e t s ) .

One can develop a d i v i s o r t h e o r y o n schemes v i a C a r t i e r o r Weil d i v i s o r s (which a r e sometimes e q u i v a l e n t ) .

Roughly f o r n o n s i n g u l a r curves

integral

separated schemes X o f f i n i t e t y p e o v e r K, o f dimension 1, w i t h r e g u l a r l o c a l r i n g s , prime d i v i s o r s a r e c l o s e d p o i n t s p . and ( W e i l ) d i v i s o r s a r e sums D =

1 nipi

w i t h degD =

1

1 ni.

P r i n c i p a l d i v i s o r s ( f ) a r e determined by r a -

t i o n a l f u n c t i o n s f ( w i t h s u i t a b l e l o c a l d e f i n i t i o n ) and f o r curves as i n d i c a t e d C a r t i e r d i v i s o r s a r e e s s e n t i a l l y l o c a l l y p r i n c i p a l Weil d i v i s o r s . Note t h a t i n t h e a f f i n e s i t u a t i o n w i t h C a c u r v e and A a r i n g w i t h prime i d e a l s N = maximal i d e a l s , d i v i s o r s nipi Q f r a c t i o n a l i d e a l s pyl .p;N and t h u s

c1

..

t h e idea o f d i v i s o r g e n e r a l i z e s t h a t o f i d e a l ( c f . [ D Q l on a RS t h e ni v i s o r s i n [ DQl

Q

o r d e r o f zeros o r p o l e s .

I).

Recall here a l s o

There i s a n i c e d i s c u s s i o n o f d i -

I where t h e s e t o f d i v i s o r c l a s s e s o v e r s u i t a b l e schemes x i s

3 90

ROBERT CARROLL

shown t o correspond t o

H 1 (X,0*).

L e t (X,Ox)

be an i n t e g r a l prescheme and

R ( U ) = f i e l d o f f r a c t i o n s o f r(U,OX) (U C X open).

The R ( U ) g i v e r i s e t o a

quasicoherent 0x module RX = sheaf o f r a t i o n a l f u n c t i o n s . o r determined b y a c o l l e c t i o n fUE r ( U , R i ) t ( D ) v i a r(U,L(D))

( R; = R X

i f and o n l y i f

YO)).

D e f i n e a sheaf

= module o v e r AU = r(U,OX) generated by fU ( i . e .

The d i v i s o r 0 i s p r i n c i p a l i f and o n l y i f

= AUfu).

-

L e t D be a d i v i s -

L(D) 2 L(D'). plr

D :D '

The sheaves LfD) a r e l o c a l l y f r e e o f r a n k 1

and one r e f e r s t o them o c c a s i o n a l l y i n e.g. t o t h e l i n e bundles LD

L(D) 1 0 x and

r(U,L(D))

112.

There a r e obvious r e l a t i o n s

D used f r e q u e n t l y i n c o n n e c t i o n w i t h RS.

I n general t h e d i v i s o r c l a s s group i s CL(X) = d i v i s o r s modulo p r i n c i p a l d i visors.

An i n v e r t i b l e s h e a f o n X i s a l o c a l l y f r e e 0x module o f r a n k 1 and

PicX i s t h e group o f isomorphism c l a s s e s o f i n v e r t i b l e sheaves on X (under

a).

If0;2 i s t h e s h e a f whose s e c t i o n s o v e r an open U a r e t h e u n i t s i n r(U, 2 H 1 (X,0*) ( t h e remarks above c l a r i f y t h i s ) . Also t h e r e i s a n

O x ) t h e n PicX

isomorphism o f C a r t i e r d i v i s o r s modulo l i n e a r equivalence t o PicX ( f o r t h e s i t u a t i o n of i n t e r e s t here).

Thus much o f t h e Riemann s u r f a c e machinery

w i l l have a scheme v e r s i o n and i n p a r t i c u l a r CLoX 2 J(X) where J ( X ) r e q u i r e s a scheme t h e o r e t i c d e f i n i t i o n here (which we o m i t

RfmARK %.lL

-

c f . [HAl;MU6]).

We have t r i e d t o r e c o r d ( i n t h e t e x t o r appendices) t h e d e f i n i -

t i o n s and ideas needed t o e x p l i c a t e c e r t a i n techniques and r e s u l t s i n s o l i t o n mathematics.

Some i d e a o f what meaning i s a t t a c h e d t o these d e f i n i t i o n s

and ideas i s a l s o presented v i a p r o o f s o r examples b u t n e c e s s a r i l y , g i v e n l i m i t e d space, etc., ends e x i s t ) .

many background m a t t e r s remain fragmentary (and l o o s e

We w i l l t r y h e r e t o g i v e some p e r s p e c t i v e and c l a r i f i c a t i o n

f o r v a r i o u s i t e m s r e l a t i v e t o curves and a l g e b r a i c geometry.

I n terms o f

a p p l y i n g a l g e b r a i c geometry t o K r i z e v e r data and Grassmannians t h e b e s t source i s p r o b a b l y s t i l l [ S E l ],and [MU21 p r o v i d e s some comprehensive background ( c f . §12,19 f o r e x p l i c i t m a t e r i a l from these papers).

However b o t h

o f these papers a r e somewhat d i f f i c u l t f o r a beginner and we have o f t e n approached o r covered some o f t h e i r c o n t e n t o b l i q u e l y o r v i a o t h e r p o i n t s o f view (see e.g.

§11,12,18,19,21,22).

I n p a r t i c u l a r i t i s probably d i f f i c u l t

f o r a beginner t o absorb t h e deluge o f i n f o r m a t i o n about algebra, sheaves, schemes, e t c . ( I assume t h e r e a d e r t o be somewhat more f a m i l i a r w i t h Riemann s u r f a c e s and d i f f e r e n t i a l geometry).

Hence we w i l l make a few a d d i t i o n a l

SHEAVES

391

comments on a l g e b r a i c geometry i n a r a t h e r more l i e s u r e l y manner.

For

sheaves [ GN1;GUlY3,4;SER1 , 3 l seem t o i n v o l v e t h e c l e a r e s t p r e s e n t a t i o n and i n p a r t i c u l a r one can perhaps t h i n k o f [ SERl-31 as p a r t o f " c l a s s i c a l " a l g e b r a i c geometry, b u i l t upon Weil and Z a r i s k i (and many o t h e r s ) , b u t s t i l l i n a prescheme e r a .

One knows e.g.

from [SER31 t h a t i n a s u i t a b l e p r o j e c -

t i v e c o n t e x t c o h e r e n t a l g e b r a i c sheaves correspond b i u n i q u e l y t o c o h e r e n t a n a l y t i c sheaves.

Over a p r o j e c t i v e v a r i e t y X t h e homomorphism 8 : 0x -+ Hx

i s b i j e c t i v e where 0x i s b u i l t from r a t i o n a l f u n c t i o n s ( r e g u l a r f u n c t i o n s ) T h i s may e x p l a i n t h e l i b e r t i e s t a k e n w i t h

and Hx from holomorphic f u n c t i o n s .

0x i n v a r i o u s a r e a s o f mathematics and p h y s i c s (and r e f l e c t e d i n t h i s book). Now f o l l o w i n g [ S E R l 1 i n Kr = X b u i l d 0, i n a NBH o f x, P and Q polynomials

E

v i a r a t i o n a l f u n c t i o n s P/Q,

Q(y) 4 0

Kzxl,...,~rl w i t h 0 t h e corresponding

( c o h e r e n t ) sheaf ( K = C f o r o u r purposes

-

c f . [ SERl 1 f o r d e t a i l s ) .

For Y

l o c a l l y c l o s e d = open I-I c l o s e d ( Z a r i s k i t o p o l o g y ) where F c l o s e d means F = s e t o f zeros o f a f a m i l y o f polynomials P* E K [ x ~ , . . . , x ~ I ,

0

Y

i n a n o b v i o u s way.

Y c X, one forms

For such Y = U n F and I ( F ) = i d e a l o f polynomials

v a n i s h i n g on F one has A = K[x l,...,xr]/I(F)

~

0 An a~l g e b r a~i c v a r~i e t y

( o v e r K) i s d e f i n e d t o be a s e t X w i t h a t o p o l o g y and a s t r u c t u r e sheaf 0, c F(X) = sheaf o f germs o f f u n c t i o n s on X p l u s two axioms ( 1 ) t h e r e e x i s t s

a l o c a l l y f i n i t e covering

1=

(Vi)

o f X such t h a t Vi 2 l o c a l l y c l o s e d s e t

i n an a f f i n e space w i t h sheaf 0u.(2) t h e diagonal A

Ui

C

X x X i s closed.

1

Given an i r r e d u c i b l e a l g e b r a i c v a r i e t y X w i t h U

C

X open w r i t e AU = r(U,OX)

so AU i s an i n t e g r a l domain and K defined v i a q u o t i e n t f i e l d s KU o f AU i s a I n p a r t i c u l a r f o r l o c a l l y c l o s e d Y = U n F a s above K 2 K c o n s t a n t sheaf.

(A)

= q u o t i e n t f i e l d o f A.

each x.

The s e c t i o n s o f K g i v e a f i e l d K ( X ) 2 Kx f o r

The transcendence degree o f K(X) o v e r K i s dimX f o r X i r r e d u c i b l e .

Now l e t V C X = Kr be a c l o s e d s u b v a r i e t y and I x ( V ) = i d e a l o f 0x coming v i a elements f such t h a t f l V = 0 n e a r x (Zx(V) = 0x f o r x i d e a l o f K[xl,.

..,xr]

c o h e r e n t s h e a f Z(V)

v a n i s h i n g o n V, C

V and i f I ( V ) =

Zx(V) 2 I(V)). T h i s g i v e s r i s e t o a

0 o f 0 modules and 0v = 0/Z(V) i s a c o h e r e n t s h e a f .

Extending arguments, any coherent a l g e b r a i c sheaf ( o f modules) on V can be Recall

considered a s a coherent sheaf on X ( s i m i l a r l y i n p r o j e c t i v e space).

here a sheaf o v e r an a l g e b r a i c v a r i e t y V i s c a l l e d a l g e b r a i c i f i t i s a

.

392

ROBERT CARROLL

s h e a f o f 0v modules.

...,x r l )

Now f o r V an ( a f f i n e ) a l g e b r a i c v a r i e t y (C X = K[xl,

and f a r e g u l a r f u n c t i o n o n V ( i . e .

locally f

2,

P/Q), l e t V f = { x

I f V i s i r r e d u c i b l e A = K[xl, ...,x r l / I ( V ) 2 r ( v , 0") a s above and if Q i s a r e g u l a r f u n c t i o n o n X, P a r e g u l a r f u n c t i o n on

E V such t h a t f ( x ) # 01.

X

9'

t h e n f o r l a r g e enough n t h e r a t i o n a l f u n c t i o n QnP i s r e g u l a r on X.

i s used i n p r o v i n g v a r i o u s facts, and F C 0; coherent. variety V the 0

This = 0 for q > 0

i n p a r t i c u l a r t h a t Hq(V,F)

Also f o r F a c o h e r e n t a l g e b r a i c sheaf o n any a f f i n e

module Fx i s generated by elements o f r(V,F).

XYV

Now l o o k a t t h e p r o j e c t i v e s i t u a t i o n ( s t i l l f o l l o w i n g [ SERl

I).

L e t Y = Kr+l - (01 w i t h y E Xy so Pr(K) = X = Y/ I r e l a t i o n = I T ( Y ) . The ith coordinate r+1 f u n c t i o n i s ti (ti(!-i ...,u ) = pi) and Vi C K 2, ti = 0 w i t h Ui = n(Vi). 0) r The Ui c o v e r X and t . / t . determines a f u n c t i o n ( a g a i n t . / t . ) o n Ui which i s J rl J 1 -1 Ui + K . For U open i n X one w r i t e s AU = r ( n (U),Oy) w i t h a bijection A: t h e homogeneous elements o f degree 0. For V ZI U t h e r e i s a homomorphism v o o v 4": A V + A: ( r e s t r i c t i o n ) and (AUy$,,) determine a sheaf Ox. I n o r d e r t h a t

qi:

f, d e f i n e d near x, belong t o 0 i t i s necessary and s u f f i c i e n t t h a t l o c a l x, x l y f = P/Q w i t h P,Q homogeneous p o l y n o m i a l s o f t h e same degree and Q ( y ) # 0

near x.

With t h i s s t r u c t u r e X = Pr(K)

0 f o r 0,.

i s an a l g e b r a i c v a r i e t y and one w r i t e s

An a l g e b r a i c v a r i e t y i s c a l l e d p r o j e c t i v e i f i t i s isomorphic t o

a c l o s e d s u b v a r i e t y o f a p r o j e c t i v e space.

F i s a c o h e r e n t a l g e b r a i c sheaf o v e r X t h e r e e x i s t s n ( f )such t h a t f o r n 2. n ( F ) and x E X t h e Ox module F(n), i s generated b y e l -

Now f o r X = Pr(K)

if

ements o f r(X,F(n)).

R e c a l l here F ( n ) i s c o n s t r u c t e d v i a 8 . .(n) = ( t . / t . ) n : 1J J 1 Fj(ui n U . ) -+ Fi(Ui n U.) (si = 0 . . s . ) . For F = 0 we g e t 0 ( n ) 2 0 ' ( n ) where J J IJ J-1 O l ( n ) i s determined v i a A; C A" =T(n (U),0 ) c o n s i s t i n g o f homogeneous f u n c t i o n s o f degree n ( f ( h y ) = Xnf(y), y

E

nyl(U)).

Note a s e c t i o n o f 0 ( n )

o v e r an open U c X i s a system si o f s e c t i o n s o f 0 o v e r Ui n U w i t h si = (t!/t?)s. o v e r U n Ui n U * t h e c o r r e s p o n d i n g gi = t!s. jy 1 J 1 J t i o n s o f degree n. On t h e o t h e r hand elements o f 0;(n) geneous polynomials, degP

-

1

2,

homogeneous func-

%

P / Q w i t h P,Q homo-

degQ = n, and Q(y) # 0 near x; a l s o F ( n ) 2 F f10

To g e t t h e r e s u l t above a b o u t F(n),

b e i n g generated b y (X,F(n)) one 0(n). f i r s t notes t h a t f o r a f f i n e v a r i e t i e s V and c o h e r e n t a l g e b r a i c F, w i t h Q r e gular on

v,

V

Q

= (x;

Q ( x ) # 01, and s a s e c t i o n o f F o v e r V

9'

one can show

SCHEMES t h a t there e x i s t s s '

E

393

r(V,F) such t h a t s ' = Qns o v e r V

9'

Then one a p p l i e s

a s e c t i o n si r e s t r i c t e d t o Ui n U t h i s t o V = Uiy Q = ti/t etc. (cf. jy j' [ SERl I ) . As a c o r o l l a r y e v e r y c o h e r e n t a l g e b r a i c sheaf F o v e r X = Pr(K) i s isomorphic t o a q u o t i e n t o f 0 ( n I P f o r s u i t a b l e n,p.

One remarks i n passing

-

degPi

t h a t s e c t i o n s o f say F ( n ) l o c a l l y i n v o l v e terms fiPi/Qiy

deqQi = n,

so f . i t s e l f c o u l d have p o l e s o f o r d e r n and t h i s i s t h e background o f con1

s t r u c t i o n s i n Remark 12.6 f o r example. We n e x t r e v i s i t schemes and a l g e b r a i c c u r v e s f o l l o w i n g [ OE1 1; w i t h some r e p e t i t i o n t h i s s h o u l d connect v a r i o u s ideas more s y s t e m a t i c a l l y .

There i s a n

u n d e r l y i n g f u n c t o r i a l framework f o r a l l t h i s which Grothendieck emphasizes and which i s most i m p o r t a n t , b u t we choose t o downplay t h i s here s i n c e matThus we r e c a l l l o c a l l y r i n g e d spaces ( X ,

t e r s a r e a l r e a d y a b s t r a c t enough.

O x ) w i t h 0, a l o c a l r i n g w i t h maximal i d e a l field.

A homomorphism (f,$): (X,Ox)

and a sheaf homomorphism $: 0

(m,) ( n o t e

1'(V,f*(0~))

-.y

= r(f

+.

+.

f,(Ox)

mx and

K(x) = Ox/mx t h e r e s i d u e

(Y,Oy) i s a continuous map f : X with $

(m

x f(x) One forms

(V),Ox)).

)

C

mx o r m f ( x )

Y

-f

= $-'

a s usual

and we r e c a l l t h a t a base o f open s e t s i s formed from D ( f ) = f p E SpecA; f

4

p) (where f

E

A).

F C SpecA i s c l o s e d i f t h e r e e x i s t s M

= {p E SpecA; f ( p ) = 0, f o r a l l f

'F Af

E

MI

= I p E SpecA; M C

pl.

C

A such t h a t F

F u r t h e r OSpecA

A ( l i m f o r f $ p ) and r ( S p e ~ A . 0 ~= ~A.~ ~An ~ )a f f i n e P scheme i s t h e n a r i n g e d space isomorphic t o (SpecA,dSpeCA) and (by abuse o f

at p is

n o t a t i o n ) one i d e n t i f i e s i t w i t h t h e spectrum o f i t s r i n g o f g l o b a l sections.

A l o c a l l y r i n g e d space (X,Ox) a f f i n e scheme.

r(u,sx);

i s a scheme i f i t i s l o c a l l y isomorphic t o an

Given a scheme (X,Ox)

and a c l o s e d Y

C

X s e t r(U,J) = I f

E

f ( y ) = 0 f o r f E Y n U l ; here Y i s t h e c l o s e d s e t d e f i n e d b y t h e

i d e a l J C 0x v i a Y = { x E X; 0x #

Jx1

= s u ~ p ( 0 ~ / J=)

i x E X; (0,/J),

# 01.

(One r e f e r s here t o i d e a s o f quasicoherence on p. 384 and n o t e s t h a t f o r X w

= SpecA, J

(Y,(A/I)-IY)

* Ifor

I a n i d e a l i n A w i t h Y = {p; ( A / I )

= Spec(A/I)).

P

4 03 = {p; I C

I n general one w r i t e s (Yred.Oy(red)

p l and

) f o r the

c l o s e d subscheme o b t a i n e d v i a J and t h u s i n p a r t i c u l a r Oy(red) does n o t have 2 n i l p o t e n t elements ( n o t e ( x ) and ( x ) i n A = K I x I determine t h e same Y b u t n o t t h e same subschemes

- red

Now one r e c a l l s ( c f . [ D E l

-

reduced).

I for details) that

y

E

{yl i n

SpecA i f and o n l y i f

3 94

ROBERT CARROLL

-

-

X C Y i n A and i f A i s an i n t e g r a l domain I01 = SpecA. p o i n t s o f SpecA

maximal i d e a l s o f A.

n i l p o t e n t s t h e n X i s i r r e d u c i b l e i f and o n l y Ared c a l l E i s i r r e d u c i b l e means E

#

F

U

Thus t h e c l o s e d

I f % = SpecA and Ared

F ' , F,F'

= A/ideal o f

i s an i n t e g r a l domain

re-

closed, u n l e s s F o r F ' = E )

Every i r r e d u c i b l e c l o s e d p a r t o f a scheme X i s t h e adherence o f a unique generic p o i n t .

For K a (commutative) f i e l d (K

C f o r o u r purposes) one r e -

f e r s t o K schemes X and X i s a l g e b r a i c i f X = f i n i t e u n i o n o f Xi where t h e Ai

a r e K algebras o f f i n i t e type.

a r e Noetherian ( c f . p. 381) and a p o i n t x

= SpecA

Such a l g e b r a i c schemes o v e r K

E X i s closed

i f and o n l y i f t h e

Note here x i s

r e s i d u e f i e l d K(x) i s an e x t e n s i o n o f K o f f i n i t e degree.

c l o s e d i n X i f and o n l y i f i t i s c l o s e d .i n a. l l t h e SpecAi so x must be a maximal i d e a l i n Ai.

Moreover K(x) = A ~ / x A : 1 Frac(Ai/x)

t i o n s and one sees e a s i l y t h a t x type

f

[ Frac(A/x):Kl <

m.

Q

= f i e l d o f frac-

maximal i d e a l o f t h e K a l g e b r a A o f f i n i t e

One assumes K i s a l g e b r a i c a l l y c l o s e d ( f o r us K

= C ) and l e t X(K) = p o i n t s o f X which a r e r a t i o n a l o v e r K ( i . e .

f i e l d K ( x ) = K).

the residue

Then X ( K ) i s t h e s e t o f c l o s e d p o i n t s o f X and such p o i n t s

s u f f i c e t o s t u d y t h e t o p o l o g y o f X ( t h e y a r e " v e r y dense"

-

c f . [ DE1

I). One

can d e f i n e dimension f o r an a l g e b r a i c scheme X o v e r K v i a dimX = suptn; dimX <

m

0 5 i 2 n, X. =

Since X i s Noetherian Xi $ Xi+,}. and t h i s agrees w i t h t h e K r u l l dimension o f A f o r X = SpecA ( i t

t h e r e e x i s t s Xi,

1

a l s o agrees w i t h t h e cohomological dimension - c f . t G N 1 I). F i n a l l y ( c f . [ DE1 1) one can d e f i n e an a l g e b r a i c c u r v e o v e r K t o be an a l g e b r a i c scheme o f

dimension 1 o v e r K.

T h i s i s somewhat more a c c e s s i b l e t h a n o u r p r e v i o u s de-

f i n i t i o n (we do n o t make comparisons h e r e ) and one n o t e s t h a t c l a s s i c a l a l g e b r a i c v a r i e t i e s a r e always a l g e b r a i c schemes. L e t us a l s o make a few f u r t h e r remarks about d i v i s o r s f o l l o w i n g 1 DE1 1. First g i v e n a Noetherian scheme (X,0,) f o r any a f f i n e open U tions o f A o f Ox,, on X.

-

c f . [ RW1

C

t h e r e e x i s t s a unique sheaf K X such t h a t

X, T(U,KX)

= Fr(r(U,OX))

I). The s t a l k a t x

(FrA = t o t a l r i n g o f f r a c -

E X o f KX i s t h e r i n g o f f r a c t i o n s

and t h e g l o b a l s e c t i o n s KX o f KX form a r i n g o f r a t i o n a l f u n c t i o n s For X a reduced i r r e d u c i b l e a l g e b r a i c scheme o v e r K,

s i o n f i e l d o f K w i t h transcendence degree = dimX.

KX i s an exten-

One w r i t e s K!

( r e s p . 0;)

f o r t h e subsheaves o f i n v e r t i b l e s e c t i o n s o f KX (resp. Ox); t h u s e.g.

c X, r(U,K!)

= { i n v e r t i b l e elements o f r ( U , K X ) l .

for U

A ( C a r t i e r ) d i v i s o r on X

CURVES

395

i s defined as a global section of the sheaf Ki/0; so the group of divisors i s Div(X) =r(X,K;70;). One writes Div(fg) = Div(f) t Div(g) f o r f E r(x, K;) and Div(f) = 0 i f a n d only i f f E r(X,0;). There i s a sheaf O X ( D ) c KX associated t o D E D i v ( X ) defined a s follows. D induces a germ Dx E K; / ,x a n d there e x i s t s a n open s e t U 3 x and f E T(U,Ki) such t h a t f induces D on U . f i s called a local equation for D on U and suitable collections of such f i on a covering { U i ) of X determine D. One defines O X ( D ) via O X ( D ) x = f - 1 O x , x = { $ E K X Y x ; $fx E O x , x l . O X ( D ) i s locally isomorphic to 0x ( v i a

Oi,x

f": O x l U 0,(D)lU) a n d invertible (a sheaf of 0x modules i s invertible i f i t i s l o c a l l y free of rank 1 - i . e . l o c a l l y isomorphic t o Q x ) . In fact i f 1; C KX i s invertible then there e x i s t s a unique D such t h a t 1; = O x ( D ) . One 0 A ~divisor . D i s positive i f notes t h a t 1;-l = Hom ( C , O x ) and 1;-l 8 1; ~ @X the local equations f f o r D a r e (regular) sections of O x ; equivalently Q X C 0 X (D) C K X o r O X ( - D ) i s a sheaf of ideals o f O x . Hence for any D E Div(X), -+

r(x,sx(D)) = t f E r ( X , K X ) ; D t Div(f) 0 1 . Given a system I f x } of local equations, suppD i s the closed subspace I D 1 C X composed of points where f,

#

0 i Y x ( i . e . where 1 i s n o t a local equation of D ) . For a ringed space X one defines PicX as the s e t of equivalence classes of invertible dX modules 1 with g r o u p law via tensor product a n d PicX 14. H (X,0;) i f say X i s paracompact. For a Noetherian scheme X a group morphism Div(X) PicX (r(XYK;/0;) 1 1 + H (X,0;)) i s defined via a : r(X,K;/0;) H (X,0;) (coming from 0 -+ 0; -+ -f

-f

K; K;/0; -+ 0 ) . Here a i s generally not surjective b u t will be so i f X i s irreducible for example. Generally D ~kD' ( l i n e a r equivalence) i f there exi s t s f E r(X,K;) such t h a t D = D' + Div(f) o r equivalently O X ( D ) 2 0 , ( D ' ) o r f.OX(D) = 0,(D') for some f E r(X,K;). -f

We mention a few more items, s t i l l from [DEl I.

Given a n irreducible alge-

braic scheme with generic p o i n t 5 one has the following equivalences: ( 1 ) C i s a curve ( 2 ) every point d i f f e r e n t from 5 i s closed ( 3 ) for every closed point 0x has Krull dimension 1 ( 4 ) i f SpecA i s a n o n v o i d open s e t in C then dimA = 1 ( 5 ) every non void open subset of C has dimension 1 (and contains 6 ) . Next one shows t h a t the closed s e t s of an irreducible curve C a r e C a n d f i n i t e subsets of C - { E l a n d i f C i s a reduced irreducible scheme over K algebraically closed with d = di1nHO(C,0~)< m (over K ) then d = 1 . For C a reduced irreducible curve over K one h a s KC = 0 = Frac(0,) for a l l x

E

C

3 96

ROBERT CARROLL

(KC i s a constant sheaf here).

KC i s a n extension f i e l d of f i n i t e type over

K with transcendence degree 1 .

A point

x

E

C i s regular i f 0x i s a discrete

valuation ring a n d C i s regular i f a l l i t s points ar e regular (we will n o t discuss valuations here - c f . [DEl;HAl I ) . For a regular irreducible curve C the group Div(C) = r(C,K*/0;) 2 @I Z (@I over x E C - 151) = free abelian group generated by the closed points of C . In the projective situation X = P r ( K ) i s a reduced irreducible scheme a n d KX = f i e l d of rational functions constructed e a r l i e r a r e invertible mo, I , K ( x l , . . . y ~ r ) . The sheaves sX(n) dules a n d any invertible module 1; on X i s isomorphic t o 0,(n) f or some n 1 = n. (uniquely i f r >/ 1 ) . Thus PicX = H (X,0;) 2 Z.1f C ( D ) Z aX(n)),deg(D) Much more material is in [ DE1 1, including Riemann-Roch, duality, residues, e t c., which i s c l e a rl y presented a n d accessible,and we recommend i t for background reading o n curves. Let us f i n a ll y follow [ DC1;O0ly2;RB1;SE1 1 t o make a few comments on the o r i gin of torsion fre e sheaves in discussing curves and Jacobians in Remarks 19.4-19.6.for example. Thus l e t X be a projective integral curve over a n algebraically closed K a n d P = space of isomorphism classes o f l i n e bundles of degree 0 on X . Recall here for smooth X , L - L ( D ) with deg(0) = 0 for Pico(X) % JacX so x ( L ) = dimHo(X,L) - dimH1 ( X , L ) = 1-g = ~ ( x . 0 ~ )T.h u s for smooth X , P ,I, JacX {invertible sheaves I; on X such t h a t x ( L ) = x(Ox)1. When X has si ngula ri ti e s t h i s general ized Jacobian i s more complicated a n d we refer t o [ DCl;RB1;001,2] for discussion. One shows there i s a natural compactification 3 P, which i s a projective scheme, a n d P 2 Crank 1 torsion f r ee sheaves F on X such t h a t x(F) = x(SX)1 ( t h i s idea apparently goes back to Mumford a n d Mayer a n d the construction plus much more i s in [DCl I ) . The d e t a i l s a r e quite complicated however and we will refrain from going int o t h i s . One notes t h a t the sheaves a rising from Grassmannians as in [SEl I ( c f . Remark 19.4) a r e automatically torsion free.

397

REFERENC ES

There a r e a l o t o f r e f e r e n c e s and many e n t r i e s from c e r t a i n j o u r n a l s and p u b l i s h e r s ; hence t h e f o l l o w i n g a b b r e v i a t i o n s a r e v e r y expedient: AAM = Acta Applicandae Math.; AMS = American Math. Society; ASPM = Advanced s t u d i e s i n pure mathematics (Academic Press); CPAM = Communications i n Pure and A p p l i e d k t h . ; CMP = Communications i n Math. Physics; DAN = Doklady Akad. Nauk USSR; FAA = F u n c t i o n a l A n a l y s i s and A p p l i c a t i o n s (Russian); I V = Invent i o n e s Math.; JMP = Journal o f Math. Physics; JPSP = Journal o f t h e P h y s i c a l S o c i e t y o f Japan; LMP = L e t t e r s i n Math. Physics; LNM = L e c t u r e n o t e s i n mathematics ( S p r i n g e r ) ; LNP = L e c t u r e n o t e s i n p h y s i c s ( S p r i n g e r ) ; NH = N o r t h H o l l a n d p u b l i s h e r s ; NPB = Nuclear Physics B; PLA = Physics L e t t e r s A; PLB = Physics L e t t e r s B; PRM = Pitman r e s e a r c h notes i n math. (Longman publ i s h e r s ) ; PTP = Progress i n T h e o r e t i c a l Physics; R I M S = P u b l i c a t i o n s RIMS, Kyoto Univ.; SAM = S t u d i e s i n A p p l i e d Math.; S I A M = S o c i e t y f o r I n d u s t r i a l and A p p l i e d Math.; SP = S p r i n g e r Verlag; SPM = Symposia i n pure math. (AMS); TMF = T h e o r e t i c a l and Math. Physics (Russian); UMN = Uspekhi Mat. k u k ; WS = World S c i e n t i f i c p u b l i s h e r s ; E E l s e v i e r pub1 i s h e r s . A AA

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ABRAHAM, R. and MARSDEN, J. ( 1 ) Foundations o f mechanics, AddisonWesley, 1978 ALVAREZ, 0. ( 1 ) NPB 216 (1983), 125-184 - ( 2 ) and WINDEY, P., Mathem a t i c a l aspects o f s t r i n g theory, WS, 1987, pp. 76-94 ABLOWITZ, M. (1) and KAUP, 0, NEWELL, A., and SEGUR, H., SAM 53 (1974), 249-315 - ( 2 ) and FOKAS, A., LNP 189, 1983, pp. 3-24 - ( 3 ) and NACHMAN, A., Physica 180 (1986), 223-241 - ( 4 ) and BAR YAACOV, D., and FOKAS, A., SAM 69 (1983), 135-143 - ( 5 ) and SEGUR, H., S o l i t o n s and t h e i n v e r s e s c a t t e r i n g transform, SIAM, 1981 - ( 6 ) and SEGUR, H., Phys. Rev. L e t t . , 38 (1977), 1103-1106 ( 7 ) and RAMANI, A. and SEGUR, H., JMP 21 (1980), 715-721 and 1006-1015 - ( 8 ) and SEGUR, H., and RAMANI, A., L e t t . NC, 23 (1978), 333-338 - ( 9 ) SAM, 58 (1978), 17-94 - (10) and KRUSKAL, M., and SEGUR, H., JMP 20 (19 79), 999-1003 - (11) and SEGUR, H., SAM, 57 (1977), 13-44 - (12) and CLARKSON, P., S o l i t o n s , n o n l i n e a r e v o l u t i o n equations, and i n v e r s e s c a t t e r i n q , Cambridge Univ. Press, 1991 ADAMS, M. ( 1 ) and HARNAD, J . , and PREVIATO, E., CMP, 117 (1988), 451500 - ( 2 ) and RATIU, T., and SCHMID, R., I n f i n i t e dimensional groups w i t h a p p l i c a t i o n s , SP, 1985, pp. 1-70 - ( 3 ) and HARNAD, J., and HURTUBISE, J. , I n t e g r a b l e and s u p e r i n t e g r a b l e systems, WS, 1990, pp. 232-256 - ( 4 ) and BERGVELT, M., Heisenberg algebras, Grassmannians, and i s o s p e c t r a l curves, t o appear - ( 5 ) and HARNAD, J., and HURTUBISE, J., CRM, Montrebl, 1648, 1989 - ( 6 ) and HURTUBISE and HARNAD, J., CMP, 34 (1990), 555-585 - ( 7 ) and HARNAD, J . and HURTUBISE, J., Dual moment maps i n t o l o o p algebras, t o appear - ( 8 ) and HARNAD, J. and HURTUBISE, J., CRM Workshop o n Hamiltoni a n systems, Univ. Montreal , 1990, pp. 19-32

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AH A1

AJ AK

AL AM AN

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A0

AR

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AS

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AT

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REFERENCES

BA BB BC BD BE

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400

ROBERT CARROLL and LAODOMAOA, C., PEMPINELLI, F., and TU, G., JMP, 24 (1983), 2085-2041; Physica 90 (1983), 425-432 ( 4 ) and GERDZIKOV, V . and PEMPINELLI, F., PTP, 75 (1986), 1111-1141 ( 5 ) and LEON, J., MARTINA, L., and PEMPINELLI, F., PLA 123 (1987), 340-344 - ( 6 ) and TU, G., NC 75B (1983), 145-159 - ( 7 ) and LEON, J. and PEMPINELLI,F., JMP 31 (1990), 2612-2618 - ( 8 ) and LEON, J., MARTINA, L., and PEMPINELLI, F., PLA 132 (1988), 432-439 ( 9 ) and LEON, J . , MARTINA, L., and PEMPINELLI, F., NEEDS 89, SP 1990, pp. 26-28 - ( 1 0 ) and LEON, J. and PEMPINELLI, F., PLA 141 (1989), 96-100 and 101-107 (11) I n v e r s e problems, Academic Press, 1987, pp. 359-368 ( 1 2 ) and and LEON, J., PEMPINELLI, F., and POGREBKOV, A., M o n t p e l l i e r pre(13) and LEON, J. and PEMPINELLI, F., p r e p r i n t PM/89-26, 1989 p r i n t s , M o n t p e l l i e r , 1990 - ( 1 4 ) and LEON, J . , MARTINA, L., and PEMPINELLI, F., p r e p r i n t s , Lecce, 1990 BLAU, M. (1) LMP 1 3 (1987), 83-92 BUSLAEV, V. (1) and FADOEEV, L., DAN 32 (19601, 13-16 - ( 2 ) and FADDEEV, L. and TAKHTAJAN, L., Physica 18D (1986), 255-266 BRYANT, P. (1) LMP, 1 9 (1990), 97-108 BOTT, R. 1 ) and TU, L., D i f f e r e n t i a l forms i n a l g e b r a i c topology, SPY 1982 1 ) and FINKEL, A., Jour. D i f f . Eqs., 55 (1984), 257-275 BUYS, M. ( 1 ) and PERELOMOV, A., CMP, 74 (1980), 129-140 BEREZIN, F BANKS, T. ( 1 ) and DOUGLAS, M., SEIBERG, N., and SHENKER, S., PLB 238 (1 990 , 279-286 - ( 2 ) TASI 1987, WS 1988, pp. 572-627 BARBASHOV, B. (1) and NESTERENKO, V., I n t r o d u c t i o n t o r e l a t i v i s t i c s t r i n g theory, WS, 1990 BAGGER, J. ( 1 ) TASI 1987, WS, 1988, pp. 688-724 BAXTER, R. (1 ) I n t e g r a b l e systems i n s t a t i s t i c a l mechanics, WS, 1985, pp. 5-64 BABELON, 0. (1) Knots, topology, and quantum f i e l d theory, WS 1990, pp. 447-468 BEAUVILLE, A. ( 1 ) Acta Math., 164 (1990), 211-235 BERNARD, D. ( 1 ) NPB 309 (1988), 145-174 BISMUT, J. ( 1 ) and FREED, D., CMP, 106 (1986), 159-176 BOGOLIUBOV, N. ( 1 ) and IZERGIN, A. and KOREPIN, V., LNP 242, 1985, pp. ( 2 ) and IZERGIN, A. and KOREPIN, V., NPB 275 (1986), 220-316 687-705 BOGOYAVLENSKIJ, 0. (1) I z v e s . Akad. Nauk SSSR, 33 (19891, 39-65 BORDEMANN, M. ( 1 ) CMP, 135 (1990), 201-216 BREZIN, E. ( 1 ) and KAZAKOV, V., PLB 237 (1990), 43-46 BULLOUGH, R. ( 1 ) and PILLING, D . , TIMONEN, J., CHUNG, Y., and CHEN, Y. N o n l i n e a r physics, SPY 1990, pp. 98-110 - ( 2 ) and OLAFSSON, S., D i f f e r e n t i a l geometric methods i n t h e o r e t i c a l physics, WS, 1989, pp. 295-309 BURTSEV, S. ( 1 ) and ZAKHAROV, V., and MIKHAILOV, V., TMF, 70 (1987), 323- 342 ( 2 ) and DELIC, G., CARROLL, R. ( 1 ) Mathematical physics, NH, 1988 Math. Methods Appl. Sci., 11 (1989), 43-63 ( 3 ) CF Royal SOC. CA Canada, 9 (1987), 237-242 ( 4 ) LNM 1394, 1989, pp. 11-17 (5) A p p l i c a b l e Anal., 26 (1987), 61-85 - ( 6 ) Proc. L.K. Hua Sympos. B e i j i n g , Aug. 1988, t o appear ( 7 ) Conf. D i f f . Eqs., Ohio Univ.

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cs CY

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ROBERT CARROLL PREVIATO, E. ( 1 ) Duke Math. Jour., 52 (1985), 329-377 - ( 2 ) Contem. Math., AMS, 64 (19871, 153-180 ( 3 ) and WILSON, G., SPM 49 (1989) ( 4 ) Comm. Math. Helv., 62 (1987), 292-310 ( 5 ) CRM 553-569 workshop on Hamil t o n i a n systems, Univ. Montrgal , 1990, pp. 57-67 PESKIN, M. Proc. TASI 1986, WS, 1987, pp. 277-407 (1) Generalized c o h e r e n t s t a t e s and a p p l i c a t i o n s , SP, PERELOMOV, A. ( 2 ) I n t e g r a b l e systems o f c l a s s i c a l mechanics and L i e a l g e 1986 bras, Birkhauser, 1990 PALMER, J. (1) AAM, 1 8 (1990), 199-223 ( 2 ) and TRACY, C., Contem. ( 3 ) and TRACY, C., Advan. Math. Math. 108, AMS, 1989, pp. 119-131 4 (1983), 46-102; 2 (1981), 329-388 ( 4 ) CMP, 102 (1985), 175-206 ( 5 ) CMP 120 (1989), 547-574 ( 6 ) CMP, 104 (1986), 353-385 POST, G. ( 1 ) LMP 11 (1985), 253-257 ( 2 ) KP r e d u c t i o n s Proef( 3 ) LNP 313, 1988, pp. 57-64 s c h r i f t , 1988 PRESSLEY, A. ( 1 ) and SEGAL, G., Loop groups, Oxford Univ. Press, 1986 ( 2 ) I n f i n i t e dimensional groups w i t h a p p l i c a t i o n s , SP, 1985, pp. 285-306 POSCHEL, J. (1 ) and TRUBOWITZ, E., I n v e r s e s p e c t r a l theory, Academic Press, 1987 PETERSON, D. ( 1 ) LNM 933, 1982, pp. 166-175 PU, F. (1) and SATTINGER, D., I M A p r e p r i n t 510, 1989 (1) Gauge f i e l d s and s t r i n g s , Harwood, 1987 ( 2 ) Mod. POLYAKOV, A. Phys. L e t t . A, 11 (1987), 893-898 ( 3 ) I n t e r . Jour. Mod. Phys., 5 (1990), 833-842 POLYA, G. (1) and SZEGO, G., Aufgaben und Lehrsatze aus d e r Analysis, 11, SPY 1974 ( 1 ) FAA, 1 9 (1985), 37-41 ( 2 ) Topology, 24 (1985), 89-95 QUILLEN, D. RAUCH, H. ( 1 ) and LEBOWITZ, A., E l l i p t i c f u n c t i o n s , t h e t a f u n c t i o n s , and Riemann surfaces, W i l l i a m s and W i l l e n , 1973 RODBERG, L. (1) and THALER, R., I n t r o d u c t i o n t o t h e quantum t h e o r y o f s c a t t e r i n g , Academic Press, 1967 REYMAN, A. ( 1 ) Jour. Sov. Math., 1 9 (1982), 1507-1545 ( 2 ) and SEMENOV TIAN SHANSKY, Jour. Sov. Math., 31 (1985), 3399-34-10 (3) ( 4 ) and SEMENOV TIAN SHANJour. Sov. Math., 30 (1985), 2319-2325 SKY, DAN 21 (1980), 630-634 - ( 5 ) and SEMENOV TIAN SHANSKY, I V , 54 ( 6 ) and SEMENOV T I A N SHANSKY, (1979), 81-100; 63 (1981), 423-432 PLA 130 (1988), 456-460 - ( 7 ) and SEMENOV TIAN SHANSKY, M., Jour. ( 8 ) Jour. Sov. Math., 131 (1983), Sov. Math., 133 (1984), 212-227 118-127 ( 9 ) and SEMENOV TIAN SHANSKY, M., and FRENKEL, I.,DAN, 247 (1979), 811-814 (10) and SEMENOV TIAN SHANSKY, M., Z a p i s k i NauE. Sem. LOMI, 164 (1987), 176-188 (1 ) Holomorphic f u n c t i o n s and i n t e g r a l r e p r e s e n t a t i o n s i n RANGE, R. several complex v a r i a b l e s , SPY 1986 RASETTI, M. (1) I n t e r . Jour. Theor. Phys., 14 (1975), 1-21 ( 2 ) and d'ARIAN0, G. and MONTOSI, A,, NC 110 (1989), 19-29 RAMANI, A. (1) and GRAMMATICOS, B., and BOUNTIS, T., Phys. Reports 180 1989, pp. 159-245 RAINA, A. ( 1 ) EXPOS. Math. 8 (1990), 227-245 RAMOND, P. ( 1 ) TASI 1987, WS, 1988, pp. 501-571 RODIN, Yu. ( 1 ) The Riemann boundary problem on Riemann s u r f a c e s , Reidel , 1988

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REFERENCES RQ RR RS RT RY

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41 5

RUIJSENAARS, S. (1 ) I n t e g r a b l e and s u p e r i n t e g r a b l e systems, WS, 1990, D D . 165-206 RYDER;‘ L. (1) Quantum f i e l d theory, Cambridge U n i v . Press, 1985 (1) A I P Conf. Proc. 88, 1982, pp. 339-346 ROOS, A. RATIU, T. (1) LNM 775, 1980, pp. 219-257 - ( 2 ) Trans. AMS, 264 ( 1 9 8 l ) , 321 -329 RICHTMYER, R. (1 ) P r i n c i p l e s o f advanced mathematical physics, Vol. 1 2, SPY 1985 ROWEN, L. (1) Ring theory, Academic Press, 1991 (1 ) and SZ. NAGY, B., F u n c t i o n a l a n a l y s i s , Ungar, 1955 RIESZ, F, SATO, M. ( 1 ) and JIMBO, M. and MIWA, T., Proc. Japan Acad., 53 (1977), 6-10, 147-152, 153-157, 183-218, 219-224; 54 (1978), 1-5, 36-41, 221-225, 263-268, 309-31 3; 55 (1979), 6-9, 73-77, 267-272, 317-322 ( 2 ) and SATO, Y., Num. Appl. ( w i t h Y. M O R I ) , 115-120, and 157-162 Anal., 5 (1982), 259-271 ( 3 ) and JIMBO, M. and MIWA, T., RIMS, 14 (1977), 223-267; 15 (1979), 201-278, 577-629, and 871-972; 1 6 (1980), 531-584; 1 7 (1981), 137-151 ( 4 ) SPM 49, 1989, pp. 51-66 ( 5 ) and JIMBO, M. and MIWA, T., LNP 126, 1980, pp. 429-491 ROGERS, C. (1) and SHADWICK, W., Ba”ck1und t r a n s f o r m a t i o n s and t h e i r a p p l i c a t i o n s , Academic Press, 1982 SABATIER, P. ( 1 ) Conf. I n v e r s e S c a t t e r i n g , SIAM, 1983, pp. 75-99 (2) (3) Inverse I n v e r s e problems, Academic Press, 1987, pp. 1-18 Probs., 6 (1990), L29-32 and L47-53 SEGUR, H. ( 1 ) AIP Conf. 88, 1982, pp. 211-228 STRANG, G. (1 ) I n t r o d u c t i o n t o a p p l i e d mathematics, W e l l s l e y , 1986 SCHMID, R. (1 ) I n f i n i t e dimensional Hamil t o n i a n systems, Bib1 i o p o l is, Naples, 1987 ( 1 ) and WILSON, G., Publ. I H E S 61, 1984, pp. 5-65 (2) D i f SEGAL, G. f e r e n t i a l geometric methods i n t h e o r e t i c a l physics, K1 uwer, 1988, pp. 165-171 ( 1 ) Helv. Phys. Acta, 24 (1965), 573-605 SCHARF, G. ( 2 ) Proc. AMS, 98 (1986), ( 1 ) CPAM, 40 (1987), 455-522 SCHILLING, R. ( 3 ) A s y s t e m a t i c approach t o t h e s o l i t o n equations 671-675 ( 4 ) JMP, 30 (1989), 1487-1501 t o appear SHIOTA, T. (1 I V Y 83 (1986), 333-382 - ( 2 ) Sugaku, 3 (1990), 183-211 ( 2 ) CMP, 128 (1990), 1-37 SMIT, D. CMP, 114 (1988), 645-685 ( 2 ) JMP, 29 (1988), ( 1 ) Thesis, Univ. Georgia, 1987 SZMIGIELSKI, J 336-346 SCHLICHENMAIER, M. (1 ) An i n t r o d u c t i o n t o Riemann surfaces, a1 g e b r a i c ( 2 ) LMP 1 9 (1990), 151curves, and m o d u l i spaces, LNP 322, 1989 165 and 327-336 SEMENOV TIAN SHANSKY, M. ( 1 ) RIMS, 21 (1985), 1237-1260 ( 2 ) LNM 970, 1982, pp. 173-185 ( 3 ) FAA, 1 7 (1983), 259-272 ( 4 ) Z a p i s k i Nauz. Sem. LOMI, 123 (1983), 77-91 ( 5 ) Z a p i s k i NauE. Sem. LOMI, 150 (1986), 119-142 SKLYANIN, E. (1 ) I n t e g r a b l e and s u p e r i n t e g r a b l e systems, WS, 1990, pp. 8- 33 SANTINI, P. ( 1 ) and ABLOWITZ, M. and FOKAS, A., JMP, 25 (1984), 26142619 ( 2 ) I n v e r s e Probs., 5 (1989), 67-86 ( 3 ) I n v e r s e Probs., 6 (1990), 99-114 and 665-679 ( 4 ) and FOKAS, A., CMP, 116 (1988), .375-419 and 449-474 - ( 5 ) and FOKAS, A., NEEDS 89, SPY 1990, pp. 2-13 ( 6 ) I n v e r s e Probs., 6 (1990), 665-679

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SH SG SH

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SL SM

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41 6

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( 1 ) SIAM Jour. Math. Anal., 20 (1989), 966-986 ( 2 ) CPAM, ZHOU, X. 42 (1989), 895-938 - ( 3 ) CMP, 128 (1990), 551-564 ZAGRODZINSKI, J . ( 1 ) Jour. Phys. A, 15 (1982), 3109-3118 ZHANG, T. (1) and VENAKIDAS, S., P e r i o d i c l i m i t s o f i n v e r s e s c a t t e r i n g , t o appear ( 1 ) Rev. Math. Phys., 1 (1990), 197-234 ZAMOLOOCIKOV, A. ZHUANG, D. ( 1 ) and L I , Y., N o n l i n e a r physics, SP, 1990, pp. 92-96 ADDITIONAL ITEMS

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ALBER, S. (1) JMP, 32 (1991), 916-922 GROSS, D. ( 2 ) and KLEBANOV, I . , NPB 352 (1991), 671-688 NEUBERGER, H. (1) NPB 352 (1991), 689-722 BAUERLE, G. ( 1 ) and deKERF, E., L i e algebras, P a r t 1, NH, 1990 ( 3 ) and M I Y A J I M A , (2) P r e p r i n t RIMS-733, Dec. 1990 NAKAYASHIKI, A. T., and TAKASAKI, K., P r e p r i n t RIMS-689, March 1990 MIKHALEV, V . ( 1 ) Physica 40D (1989), 421-432 ( 2 ) CMP, 134 (1990), 633-646 BONORA, L. ( 6 ) and BABELON, O., PLB 253 (1991), 365-372 - ( 7 ) and MARTELLINI, M., and ZHANG, Y., PLB 253 (1991), 373-379 - ( 8 ) and BABELON, O., and TDPPAN, F., Exchan e a l g e b r a and t h e D r i n f e l d (97 and XIONG, C., SISSA/ISASSokolov theorem, p r e p r i n t 1991 187/90/EP GODEMENT, R. ( 1 ) Topologie a l g g b r i q u e e t t h e b r i e des faisceaux, Herrman, Paris, 1958 OORT, F. ( 1 ) and STEENBRUCH, J., A l g e b r a i c geom., Anvers, 1979, S i j t ( 2 ) Math. Annalen. 147 (1962), hoff-Noordhoff, 1980, pp. 157-204 277-286 DELALE, J . ( 1 ) Courbes a l g g b r i q u e s , Ecole Polytech., Paris, 1968 D'SOUZA, J. (1) Proc. I n d i a n Acad. Sci., 88A (1979), 419-457 EHLERS, F. ( 1 ) and KNORRER, H., Comm. Math. Helv., 57 (1982), 1-10 GIRAUD, J. (1) I n t r o d u c t i o n A l a g6ometrie a l g i b r i q u e , Louvain, 1974 FROHLICH, J. ( 1 ) New problems, methods, and techniques i n QFT and s t a t i s t i c a l mechanics, WS, 1990, pp. 93-122 ( 1 ) Ann. Sci. Ecole Norm. Sup., 1 3 (19801, 211-223 REGO, C. VERDIER, J. ( 2 ) Se'm. ENS, 1977-78 and 1980-81 (1) and VERDIER, J., Grothendieck F e s t s c h r i f t , Vol. 3, TREIBICH, A. Birkhauser, 1990, pp. 437-480

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421

I ND EX

We have t r i e d t o provide a r a t h e r extensive i n d e x s i n c e one can argue t h a t this may be t h e most important p a r t of a technical book such as this. For some frequently used items such a s Riemann surface, Jacobi v a r i e t y , AbelJacobi map, d i f f e r e n t i a l s of t h e f i r s t kind, e t c . we have l i s t e d only the f i r s t few occurances o r places of d e f i n i t i o n and e.g. f o r KdV equations we have simply omitted reference s i n c e the item will occurs so frequently. Tau functions have been 1 i s t e d frequently t o show how t h i s theme dominates. Abel-Jacobi map, 37,47,51 Abel ian t o r i , 51 Abel ian v a r i e t y , 375,380 Ac tion-angl e variabl es, 63,338,362 Adjoint wave function, 76,153 Ad1 er-Kostant-Symes 1 emma, 120,366 Affine scheme, 174,379,382 Affine subvariety, 44 AKNS conservation laws, 150 AKNS systems , 109,114,129 Algebraic curve, 374,394 Algebraic s e t , 388 Almost graded, 295 Almost periodic, 28 A1 t e r n a t i n g element, 344 A1 t e r n a t i n g map, 346 A n n i h i 1 a t i o n opera t o r s , 102,142 ,21 1 , 215.273.306.323 Anomaly, 195,311 An t i d e r i va t i o n , 351 A n t i p a r t i c l e , 319 Atlas, 341 Asymptotics, 17,45,73,121,241,245, 254,337 Baker Akhiezer (BA) function, 34,40, 55,158,283,286,295 Baker-Campbell -Hausdorff, 325 Bands, 28 Bare operator, 89 Bergman kernel, 54 Bethe Ansatz, 237,332 Betti number, 370 Bianchi i d e n t i t y , 66

B i g c e l l , 276,286 Bihamil tonian s t r u c t u r e , 62 Birkhoff decomposition, 73,144,145, 166,170 BKP hierarchy, 98 Bogol i u bov t r a n s format i o n , 31 4,319 Borel subgroup, 267 Borel Weil theorem, 265 Bose gas, 236,333 Boson fermion correspondence, 103, 276,307 Bosonic r e p r e s e n t a t i o n , 102 Bound s t a t e , 4 Boundary conditions, 329 Boundary values, 216,285 Bousinesq equation, 169 Bruhat decomposition, 276 Bruhat-Kac- Peterson decomposition, 144 Bu r c hna 1 -Chaundy theory, 284,291 C number 335 Caloqero system, 22 Canonical c l a s s , 51 Canonical d i v i s o r , 372 Canonical fac t o r i za t i o n , 90,96,112, 331 Canonical 1 ine bundle, 311,314,325, 37 6 Canonical v a r i a b l e s , 65 Cartan matrix, 139 Cartan subalgebra, 116,143 Cartan subgroup, 145,266 C a r t i e r d i v i s o r , 289,394

4 22

ROBERT CARROLL

Cauchy Riemann equations, 256 Cauchy Riemann operator, 316,326,330, 331 Cauchy sequence, 387 Cay1 e y Hamil t o n theorem, 285 C e l l s , 350 C e n t r a l charge, 260,292 C e n t r a l element, 141,147 C e n t r a l extension, 101,139,144,302, 304 ,31 7 C e n t r a l i z e r , 114 Chain, 350,370 Character, 186,225 C h a r a c t e r i s t i c , 50 Charge, 100,258,269,281 Chart, 341 Chemical p o t e n t i a l , 242 Chern c l a s s , 266,320,376,379 C h e v a l l e y generators, 140 C h i r a l determinants, 292 C h i r a l f i e l d s , 292 C h i r a l i t y , 261 C l a s s i c a l mechanics, 58 C1 i f f o r d a1 gebra, 104,210,213,303 C1 i f f o r d group, 205,214,220,229,237, 31 0 C1 i f f o r d o p e r a t o r s , 242 Closed form, 347 Closed map, 389 C o a d j o i n t a c t i o n , 11 3,120,266,363 C o a d j o i n t o r b i t s , 366 Coboundary, 378 Cochain, 378 Cocycl e , 55,148,311 ,31 8,324,376 Coherent sheaf, 289,380,392 Coherent s t a t e s , 266 Cohomology, 174,350,372,379 Commutative a1 gebras, 167,287,290 Complementary s e r i e s , 35 Complete, 389 Completely i n t e g r a b l e , 73 Compl e t i o n , 174,289,387 Completion a l o n g p, 175 Completeness r e l a t i o n s , 6,7,132,135, 95,156,157 Complex m a n i f o l d , 369 Conformal blocks, 263,265,267 Conformal f i e l d theory, 255,267 Conformal gauge, 297 Connection, 266,330 Connection matrices, 220,241

Conformal group, 256 Conformal t r a n s f o r m a t i o n . 256 Co n f o rma 1 we ig ht s , 265 Conservation laws, 1 7 Conservation o f energy, 361 Conserved charges, 297 Conserved q u a n t i t i e s , 121,150 Continuous spectrum, 118,119 Continuum l i m i t , 252 Cotangent space, 58,342 C o n t r a c t i o n o p e r a t o r , 103,214 C o n t r a v a r i a n t form, 103 Coroots, 140 C o r r e l a t i o n f u n c t i o n , 229,235,244, 254 ,257 ,292 Cosmolngical constant, 251 Cotangent bundle, 360,343,376 Covariant, 257 C r e a t i o n o p e r a t o r , 102,142,211,215, 273,276,306 C r i t i c a l temperature, 228 Crum- Da rboux t r a n s form, 42 C u r r e n t o p e r a t o r s , 274,276,309 Curvature, 330 Curve, 388 Cycle, 350 D module, 195 Darboux C h r i s t o f f e l formula, 250 D e f o r m t i o n equations, 234,241 Deformation o f D modules, 195,199 Deformation parameters, 218,219,241 Degree, 100,269 D e l t a f u n c t i o n p o t e n t i a l , 333 D e n s i t y m a t r i x , 237 DeRham complex, 350 D e r i v a t i o n , 351 D e r i v e d algebra, 139 Descendent f i e l d s , 260 Descendent s t a t e s , 262 Determinant, 396 Determinant bundle, 190,191,195,316, 326,328 Determinant methods, 20,80,119,124, 157,161 D i a l a t i o n , 257 D i f f e r e n t i a l s o f f i r s t k i n d (DFK), 38,52,310 D i f f e r e n t i a l s o f second and t h i r d kinds, 52,280,310,313,373

INDEX Dimension, 387,395 Dirac e l e c t r o n s , 99 Dirac e qua t i o n , 230,232,315 Dirac o p e r a t o r , 261 Dirac s e a , 314 Di re c t i ona l d e r i v a t i v e , 342 Dirichlet kernel , 77 Divisor, 41,48,51,312,371,394 Divisor c l a s s group, 390 Dominant i n t e g r a l weight, 141 Double e i ge n v alu es , 35 Doubly p e r i o d i c , 371 Dressing kernels, 85,133,161,184 Dressing methods, 89,112 Dual determinant bundle, 195,318, 322,332 Dual Fock space, 269 E ffe c t i ve , 169,177 E l e c t ron, 100 E l l i p s o i d a l co o rd in ates, 43 Emptiness formation, 243 Energy, 100,269 Energy momentum ten s o r, 260,276,297 Enveloping a l g e b r a , 141 Equal time commutator, 258 Equi variance, 31 8,322 Euler c h a r a c t e r i s t i c , 271,289,370 Euler e qua t i o n s, 1 2 Exact form, 348 Exact sequence, 378 Exponents o f monodromy, 220 Exponential map, 355 E xt e ri or product, 343 Fay's t r i s e c a n t i d e n t i t y , 205,282 Fermi momentum, 242 Fermi p a r t i c l e , 185 Fermi s e a , 250,304 Fermions, 102 Fermion o p e r a t o r s , 273 Feynma n pro pa ga t o r , 205,300 Fi bre product, 272 Fibred product, 389 F i l t r a t i o n , 202,268,271 Finite band p o t e n t i a l s , 31 Finite gap o p e r a t o r , 40 F i n i t e type, 388 Finite zone p o t e n t i a l , 50

423

First q u a n t i z a t i o n , 236,299 F l a t , 321 Flow, 351 Fluxes, 121 Fock r e p r e s e n t a t i o n , 223 Fock space, 102,213 Formal c a l c u l u s o f v a r i a t i o n s , 251 Formal geometry, 163 Frame, 269 Frame m a t r i c e s , 200 Frechet d e r i v a t i v e , 60,348 Fredholm determinant, 81,91 ,244 Fredholm e q u a t i o n , 238 Fredholm minor, 238 Fredhol m opera t o r , 152,286,288,310, 31 6 Free fermion f i e l d s , 209 Free fermion o p e r a t o r s , 206,237,261, 223,300,314,319 Frequency module, 28 F u b i n i Veneziano c o o r d i n a t e s , 301 Fundamental group, 206,370 Fundamental weights, 143 Gaps, 28 Gateaux d i f f e r e n t i a b i l i t y , 60 Gauge f i x i n g , 272 Gauge o p e r a t o r , 71,72,167,180 Gauge transform, 73,109,331 Gelfand D i k i i p o t e n t i a l s , 252 Gel fand Levitan MarEenko e q u a t i o n s , 5,81,111,136,241 General p o s i t i o n , 51 Generating f u n c t i o n s , 104,194 Generating f u n c t i o n a l , 243,300 Generic p o i n t , 383 Genus, 370 Geodesic flow, 24 Geometric qua n t i za t i o n , 265 ,2 66 Geometry o f KdV, 33 Geometry o f s c a t t e r i n g , 47 Genus, 341 Ghosts, 261,263,311 Gradient, 29 Gram Schmidt method, 317 Grassmannian, 137,158,182,187,224, 225,283,285,286,292,310,315,318, 327 Green's f u n c t i o n , 6,28,264,300 Green's o p e r a t o r , 46

4 24

ROBERT CARROLL

Ground s t a t e , 238 Half o r d e r d i f f e r e n t i a l , 53,54 Hamil t oni a n group a c t i o n , 364 Hamiltonian phase flow, 361 Hamil t oni a n v ecto r f i e l d , 360,364 Hamil t o n ' s e q u atio n s , 359 Hamil t o n ' s p r i n c i p l e , 358 Hardy space, 6,137,157 Heisenberg a l g e b r a , 31 9 Hierarchy, 120,224 Highest kei g h t - r e p r e s e n t a t i o n , 99, 141.260 Highest weight s t a t e s , 262 Hi1 b e r t problem, 207 Hi1 b e r t Schmidt o p e r a t o r , 152,286, 31 8 Hill I s e qua t i o n , 21,27 Hill I s o p e r a t o r , 46 Hill ' s s u r f a c e , 35 Hirota b i l i n e a r i d e n t i t y , 74,87,152, 153,161,188,206,226,277,307,309 Hirota e qua t ian s, 79,86,107,122,146, 188,195,225,268,276,282 Hodge * o p e r a t o r , 357 Hodge t he ory, 350 Holomorphic d i f f e r e n t i a l , 52 Holomorphic f i e l d , 262 b l o n o m i c quantum f i e l d s , 205 Holonomy, 331 Homogeneous Heisenberg subal gebra, 142 Homogeneous ideal , 384 Homogeneous l i n e b u n d l e , 267 Homogeneous polynomial , 374 Homogeneous r e a l i z a t i o n , 142,151 Homogeneous zero o r d e r p a r t , 174 Homology, 370 Homology marking, 271 HomotoDic. 370 Hyperei 1 i p t i c curve, 26,35,283, 287,374 Imaginary r o o t , 141,143 Impenetrable bosons, 237,239,246 Index r a i s i n g and lowering, 357 Infra re d divergence, 257 I n s e r t i o n o f o p e r a t o r s , 260,293, 309,315

I n t e g r a b l e , 330 Integra b i l i t y , 63,362 I n t e g r a b l e module, 141 I n t e g r a l , 388 I n t e g r a l curve, 354 I n t e g r a l manifold, 366 I n t e r p o l a t i n g polynomial , 46 I n t e r s e c t i o n product, 370 I n s t a b i l i t y , 35 Inverse 1 i m i t , 387 Inverse s c a t t e r i n g , 1,111 I n v e r t i b l e sheaf , 390,395 I r r e d u c i b l e , 383 I r r e g u l a r s i n g u l a r i t y , 218,235,240, 255 I s i n g model, 227 Isomonodromy problem, 249 I s o s p e c t r a l c l a s s , 36 I s o s p e c t r a l deformation, 166,168,180 I s o s p e c t r a l manifold, 48,83,164 Jacobi i d e n t i t y , 362 Jacobi i n v e r s i o n Droblem. 52 Jacobi v a r i e t y , 4i ,167,1i8,180,374, 380.396 J o s t s o l u t i o n , 1,47,159

Kac Moody a1 gebra, 99,138 Kac Moody groups, 1 4 3 Kac Peterson cocycle, 195 KP 70,107,169 KP o r b i t s , 176 K4hler manifold, 266 Ghler m e t r i c , 51 Ghler structure, 318 Kepler problem, 22 K i l l i n g form, 139 K i l l i n g v e c t o r f i e l d , 364 K i r i l l o v b r a c k e t , 365 K i r i l l o v structure, 366 Koba Nielsen v a r i a b l e s , 205,305,306 KdV Neumann connection, 43 Kostant-Kirillov form, 267 KriEever data , 165,178,270,283,287, 298,292,316 Krizever map, 272,278,290 Kric'ever Novi kov a1 gebra, 294

I NO EX Lagrange equations, 66,358 Laplace B e l t r a m i operator, 358 L a t t i c e , 371 Lax equation, 13,72,199 Lax K i r i l l o v bracket, 120 Lax p a i r , 13,169 Lax o p e r a t o r , 20,169 L e f t d e r i v a t i v e , 212 L e f t mover, 261,353 Legendre Fenchel transform, 359 L e i b n i t z r u l e , 167,363,182 Levinson theorem, 5 Lenard r e c u r s i o n , 116 L i e algebra, 351 L i e b r a c k e t , 351 L i e group, 351 L i e Poisson s t r u c t u r e , 363 L i g h t cone coordinates, 297 L i m i t i n g curve, 47 L i m i t p o t e n t i a l s , 49 L i n e bundles, 40,53,180,266,278,289, 291,313,316,320,376,379,390 L i o u v i l l e A r n o l d theorem, 362 L i o u v i l l e form, 360,364 L i o u v i l l e theorem, 361 Local r i n g , 382 Local r i n g e d space, 382 L o c a l l y f r e e , 377 L o c a l i z e d r i n g , 173 Loop algebra, 150 Loop group, 152,161 L o r e n t z group, 353 L o r e n t z m e t r i c , 357 L o r e n t z t r a n s f o r m a t i o n , 256 Majorana f i e l d , 229 M a n i f o l d , 341 Marcenko equation, 8,119,134 Mass, 269 M a t r i x models, 249 Maximal i d e a l , 271 Maximal o r b i t , 167,177 Maya diagram, 182,185,268,272 Mayer V i e t o r i s sequence, 175,316,380 Mean value, 30,243 Meromorphic d i f f e r e n t i a l , 372,52 M i n i m i z a t i o n , 10 Miura transform, 4,18,19 Mobius t r a n s f o r m a t i o n , 256,260,263 Modes, 261

425

Modular embedding, 279 Modular group, 381 Modular t r a n s f o r m a t i o n , 281 Moduli, 380 Moduli space, 51,270,280,272,380 Momentum map, 364,365 Monic, 167 Monodromy, 205,206,229,236,253 Monodromy group, 206 Monodromy r e p r e s e n t a t i o n , 206,330 M u l i t l i n e a r , 346 Mu1 t i p 1 i c a t i v e set, 381 Nambu Goto a c t i o n , 296 Nerve, 378 Ne uma nn p r o b l em , 2 3 ,25 ,32 ,4 2 ,44 ,284 Neumann s e r i e s , 216 Newton S a b a t i e r methods, 83,85 N i j e n h u i s tensor, 68 N i l p o t e n t , 142 Noetherian, 381,394 N o n l i n e a r Schrodinger e q u a t i o n (NLS), 109,237 Nonsingular, 387 No ns pec i a 1 d i v i s o r , 40 ,42 ,52,283,287 Normalized e i g e n f u n c t i o n , 35 Novikov c o n j e c t u r e , 166 Normal o r d e r , 210,273,302 N r o p e r a t o r , 21 1 ,225 Number o p e r a t o r , 243 One parameter group, 14,354,361 One p o i n t completion, 174 Opera t o r bosoni za t i o n 292 Operator f i e l d s , 254 Operator p r o d u c t expa s i o n , 219,231 265 Operator v a l u e d s p e c t r a l data, 336 O r b i f o l d, 381 O r b i t o f vacuum, 107,224,242 Order, 371 O r i enta t i on, 350,370 Orthogonal group, 21 0 Orthogonal i t y , 136 Orthogonal t r a n s f o r m a t i o n , 21 6 O s c i l l a t o r algebra, 102 O s c i l l a t o r expansion, 298,322

,

426

ROBERT CARROLL

P a i n l e d e q u a t i o n , 205,220,234,241, 249 Paley Wiener theorem, 6 Pa ra 1 1 el t r a nsl a t io n , 331 Parametri x, 329 P a r t i t i o n s , 186,264 P a r t i t i o n f u n c t i o n , 249 Paul i m a t r i c e s , 109,335 Period m a t r i x , 373 Period parallelogram, 371 P e r i o d i c spectrum, 35 Peter Weyl t h e o r y , 187 Phase, 48 Picard v a r i e t y , 41,167,181,271,272, 371,379,395 Plemel j - L i d s k i j formulas, 98,159 P l k k e r c o o r d i n a t e s , 182,192 P l k k e r map, 278 P1 kker,rel a t i o n s , 187,190,270 Poincare, group, 353 Poincare lemma, 350 Poincare' t r a n s f o r m a t i o n , 256,298 Point s p l i t t i n g , 251 Poisson b r a c k e t , 59,241,298,361, 365 Poisson Jensen formula, 5,17 Po 1a ri za ti on , 37 3 Polyakov a c t i o n , 258,296 Polyhedron, 350 Presheaf, 377 Prime form, 52,53,281,292,311, 31 3,323 Primary f i e l d , 257 Prime i d e a l , 381 P r i n c i p a l d i v i s o r , 51,371,376,389 P r i n c i p a l Heisenberg subalgebra, 143 Pr i nci pal idea 1 doma i n , 381 P r i n c i p a l p o l a r i z a t i o n , 373 P r i n c i p a l s e r i e s , 35 Pro j , 166,173,384,385 P r o j e c t i v e r e p r e s e n t a t i o n , 101 P r o j e c t i v e s t r u c t u r e , 53 Propagator, 261,300 Quantum c o r r e c t i o n , 193 Quantum g r a v i t y , 249 Quantum inverse s c a t t e r i n g , 332 Quasicoherent s h e a f , 175,384 Quasicompact, 389 Quasiperiodic, 28,310

Quasiprimary f i e l d s , 256

R m a t r i c e s , 334,339 Radial o r d e r , 258 Radial q u a n t i z a t i o n , 258 Rank one s h e a f , 180 Real Jacobian, 38 Real p o s i t i o n , 38 Recursion o p e r a t o r , 66 R e f l e c t i o n c o e f f i c i e n t , 3,l 6 Regular l o c a l r i n g , 387 Repulsive i n t e r a c t i o n , 237 Resolvant, 28,251 Resolvant decomposition, 147 Resolvant kernel , 239,246 Resol vant series, 127 R i c c a t i e q u a t i o n , 16,30,65,73 Riemann c l a s s , 293 Riemann c o n s t a n t s , 52,278,312,325 Riemann f u n c t i o n , 6 Riemann Hi1 bert problem, 107,110,205, 286,330 Riemann matrix, 51 Riemann problem, 207,218,242 Riemann Roch theorem, 289,313,371,376 Riemann sphere, 370 Riemann s u r f a c e , 34,369 Riemann vanishing theorem, 52,56,312 Riemannian m e t r i c , 357 Right d e r i v a t i v e , 212 R i g h t mover, 261 Ringed space, 382,393 Roots, 140 Root v e c t o r s , 140 Rotation group, 356 Sato e q u a t i o n , 184,199 Sato grassmannian ( U G M ) , 152,268,309 S c a l i n g limit, 228,251 Schemes, 173,383 Schl e s i n g e r e q u a t i o n s , 208,218,220, 240,254 Schottky problem, 51,166,178 Schwartz space, 1,116,127,157 Schwinger term, 274 Second quantized wave f u n c t i o n , 250 Secondary f i e l d , 257 S e c t i o n , 345 S e c t o r , 241,255

INDEX

Se c t or boundary, 11 7 Semi i n f i n i t e monomial , 100,104 Separated, 388 S e r r e dual i t y , 31 4,377 Sheaf o f s e c t i o n s , 378 Sheaves, 377 S h i f t o p e r a t o r , 182 Shur p a i r s , 290 Shur polynomials, 79, 194,277 Shur f u n c t i o n s , 105 Siege1 h a l f space, 51,54,375,380 Simple r o o t s , 140 Simplex, 370 Simp1 i c i a l homology, 370 Sine Gordon eq u atio n , 245 Sobolev space, 327 S o l i t o n , 15,307 Spec, 173,287,381 Spec,,,, 285 Special i za t i o n , 1 72,174 ,383 Spe c t ra l asymptotics, 121,128 Spe c t ra l da t a, 85,123 Spectral form f o r kernels, 95,132 Special d i v i s o r , 52 Spin bundle, 55,326,331 Spin c o r r e l a t i o n fu n ctio n , 228 Spin o p e r a t o r , 228,318 Spin module, 303 Spin structures, 54,278,292,311 Spi nors, 55,311,314,322 S t a b i l i t y , 35 S t a l k , 377 S t a t e space, 94 Stokes formulas, 372 Stokes m a t rices , 253 Stokes multipliers, 219,220 Stokes theorem, 350 S t r e s s energy ten so r, 258 S t r i n g ampl itude, 281 S t r i n g s , 295,307 S t r i n g e qua tio n s , 251,252 S t r i n g f i e l d s , 322 S t r u c t u r e e q u atio n s, 196,199 S t r u c t u r e s h e a f , 382 Sugawa ra c o n s t r u c t i o n , 274 Symmetric element, 344 Symmetric group, 186 Symplectic form, 66 Symplectic geometry, 358 Symplectic group, 373 Symplectic structure, 360

SzegG kernel

427

, 54,282 ,323

Tangent bundle, 343 Tangent space, 269 Tangent v e c t o r , 346,347 Tau f u n c t i o n , 76,83,91 ,119,121 ,132, 145,160,217,224,236,241 ,242,248, 249,252,267,268,276,277,278,280, 286,292,305,310,318,282,311,321, 324 TeichmJller space, 381 Tensor product, 343 Theta d i v i s o r , 312 Theta c h a r a c t e r i s t i c , 55 Theta f u n c t i o n s , 39,50,159,280,292, 31 2,313,321 ,324,375 Tied spectrum, 35 Time e v o l u t i o n , 13,111 Toda AKNS t h e o r y , 138,151 Toda l a t t i c e , 150 Torelli space, 271,381 T o r e l l i ' s theorem, 51,374 Torsion, 287,289,291,396 Torus, 37,371 Trace formula, 37,49 Transference, 42 Transcendence degree, 171 ,388,394 Tr a n s f e r m a t r i x , 228 Tr a n s l a t i o n o p e r a t o r , 142,146 Transmission c o e f f i c i e n t , 3,110 Transmutation, 112 Tree ampl itude, 306 Triangularity, 8 Twisting s h e a f , 175,291,385,392 Vacuum e x p e c t a t i o n , 103,146,205,209, 21 2,274,300,306 Vacuum space, 1 4 3 Vacuum s t a t e , 185 Vacuum v e c t o r , 142 Valuation, 272 Vandermonde determinant, 189,249 Va r i a t i o nal deri va ti ve , 30 Vector bundle, 348 Verma module, 256,264 Vertex opera t o r , 76,104,122,205,276, 292,304 ,306 Vertex o p e r a t o r equation, 77,83,86, 92,132,160,189

428

ROBERT CARROLL

V i r a s o r o a1 gebra, 256,260,262,275 V i r a s o r o commutator, 311 V i r a s o r o o p e r a t o r s , 298

W r e p r e s e n t a t i o n , 199 Ward i d e n t i t i e s , 255,258,264 Wave f u n c t i o n , 12,71,132,152,231 , 233,237,252,266,281 ,333 Wedge produce, 346 Wedging o p e r a t o r , 103 W e i e r s t r a s s P f u n c t i o n , 371,380 W e i e r s t r a s s p o i n t , 52,287 W e i e r s t r a s s system, 272 Weight, 257 Weil d i v i s o r , 389 Weyl group, 140 Weyl r e s c a l i n g , 297 Wiener algebra, 144 Wiener Hopf equation, 111 Wick r o t a t i o n , 228,314 Wick's theorem, 205,211,212,214,303, 309,315 Young diagram, 182,185,225,268,272 Zakharov Shabat equation, 72,184,199 Z a r i s k i topology, 379,382 ,388 Zero c u r v a t u r e equation, 109,147,247 ADDITIONAL ITEMS A l g e b r a i c scheme, 394,395 A7 gebra i c v a r i e t y , 391 R a t i o n a l p o i n t s , 394 Reduced scheme, 393 Residue f i e l d , 393 Ring o f f r a c t i o n s , 394

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