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.
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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