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
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
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 )
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:
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
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
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 .
= -$$ t b$ w i t h a ( c ) =
log(a)
%
satisfying
1”0 Cn/(2ic)”’
and s(5)
w i t h Co = -f qrdx,
X
REltARK 9.4
(SPECCRAL AsrlJl’IPCOClCS ARD CALI FLIrWEltXI$).
from [ NE1 ] i s r e l e v a n t here. for
F i n t h e form
A further construction Thus upon p o s i t i n g a n a s y m p t o t i c e x p r e s s i o n
ROBERT CARROLL
122
one p u t s t h i s i n t h e formulas a k F = Q k F and equates t h e c o e f f i c i e n t s o f powers o f >
j
?$ =
= (;)e-jcx
( - ,0) e i g x t x AK-(x,s) t
,,” K(x,s)e-igsds A
xx
$(c.,x)e’cYdc; -i SYd = ( 1 / 2 a ) Jl C ~ ( c , x ) e ’ c Y d c ;K(x,y) == ((1l / Z2 n ))..fft? $ ( c , x ) e -i cyd A A
Let us derive equations e q u a t i o n s f o r KK - and KK- i n tthe h e same manner o p e r a t e on on $ aand i n (9.79) by by ( 11 // 22 ~~ )) //exp(icy)dc e~~x p ( i g y ) d c (( yy y) where 1tK- = (l+K:)-’,=
This i s i n f a c t j u s t t h e r e l a t i o n 1 = ( 1 +
Now 1 + B = (l+K:)-’
=
example so AKNS c o u l d have v a r i o u s f e a t u r e s .
i s n o t t r u e f o r KP f o r
l+K-
We n o t e t h a t GLM equations u
correspond t o 1 +K
= (1 + K t ) ( l +F) w h i l e orthogonal it y corresponds t o (1 +K-)
= (l+K+) -l;f o r KdV t h e s e a r e “ e q u i v a l e n t ” ( c f .
+
(a;
b$),qTo;
I
C131 and §2,3,7
f o r Parse-
Now i n (9.76) i f one w r i t e s f o r example $1) = 0 1 T $2 (l o ) y $ = (JI1 J12) = row v e c t o r ) and expands i n terms
Val formulas e t c . ) . (alA=
A
o f K and K, F and F, i t f o l l o w s t h a t t h e completeness r e l a t i o n i s e q u i v a l e n t
M equations (9.86). ThePe should be an e q u i v a l e n t v e r s i o n i n terms A T T AT o f K-,K-,G, and ^G i f we w r i t e e.g. $ = b$ - a? and $$ 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
1 say l e t
qs
%
1
-
(A/ 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),
...
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 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 = 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
= (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
: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
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
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 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).
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
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
...
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, 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
