A SYSTEM G. B.

OF E Q U A T I O N S

OF S.

P.

NOVIKOV

Shabat

UDC 517.43

The problem of classifying two commuting differential o p e r a t o r s of mutually p r i m e o r d e r s can b e c o n s i d e r e d as solved (cf. G e l ' f a n d - D i k i i [1], Veselov [2]). In this situation it is natural to consider not only two commuting o p e r a t o r s but a whole ring of o p e r a t o r s which commute with them; because commuting operators can be multiplied, the set of o r d e r s of o p e r a t o r s in the ring is closed under addition; the condition that the o r d e r s of the two original o p e r a t o r s be mutually p r i m e means that the ring contains o p e r a t o r s of almost all o r d e r s . It is known that the s p e c t r u m of such a ring is an algebraic curve, and the R i e m a n n - R o c h t h e o r e m i m plies that its genus is equal to the n u m b e r of o r d e r s which do not occur. However, such c r i t e r i a are v e r y special (i.e., they f o r m v a r i e t i e s of high codimension in the moduli s p a c e s of curves), whereas by the theory of K r i c h e v e r [4], e v e r y c u r v e is the s p e c t r u m of a commutative ring of differential o p e r a t o r s . In o r d e r to obtain g e n e r a l c u r v e s as s p e c t r a , it is n e c e s s a r y t o consider commutative subrings of d i f f e r ential o p e r a t o r s such that t h e i r s e m i g r o u p of o r d e r s is generated by m o r e than two g e n e r a t o r s . In this paper we c o n s i d e r the s i m p l e s t case of this type when the s e m i g r o u p of o r d e r s has the three g e n e r a t o r s 3, 4, 5. It is possible to compute which differential equations for the coefficients of a t h i r d - o r d e r differential o p e r a t o r guarantee the existence of f o u r t h - and f i f t h - o r d e r o p e r a t o r s which c o m m u t e with it. In [4], equations of this type a r e called Novikov equations. If an o p e r a t o r has the f o r m L = D 1 + pD + q, these equations are: i p,,t _ 2q" + 2pp' -- 4pq = 3C~p' + 6Clq + 3C2p + C4, 2pIv -- 3q" + 6pp" + 3p'2 -- 3pq' + ~ p~ -- 6q~ -. 9Clq' + 9C2q + C5,

(1) i(2)

pIV + 5pp" + i5p'q + .~5 p~__ 15q2= __ 9C7p' + t8C7q + 9Csp + Clo,

(3)

qIV + tOpOq + iOp'q'+ 5pq"-{- 5p2q -- 15qq'= -- 6C7p"-- 3Cvp2+ 9C,q'+ 9Csq + C~.

(4)

Here the Ci, .... Cii are constants arising f r o m integrating the equations for the coefficients of the o p e r a t o r s M = D 4 + . . . , N = D 6 + ..., commuting with L; the condition on the functions p and q amounts to the e x istence of { C i } f o r which s y s t e m of equations (1)-(4) is satisfied. The o p e r a t o r s M and N have the f o r m . M = D ~ + ' ( _ ~ p + C a ) D Z + ( 2 p , + _ ~ q _ + . C 2 ) D + 2 p , , + T2q ' + T 2 p~+ ~ 2 C~p+C3, ~- ~--~P t T q t'-~P ~- CnP + Cs) D -~-

THEOREM. The set of solutions of s y s t e m (1)-(4) is p a r a m e t r i z e d by a rational algebraic v a r i e t y . At f i r s t glance it is unclear whether the s y s t e m in general has a solution differing f r o m the trivial one p = const, q = const, C4 = C 5 = C10 = Cll = 0. The existence of nontrivial solutions is a s s u m e d by the g e n e r a l a l g e b r o g e o m e t r i c theory c o n s t r u c t e d by K r i c h e v e r [4]; in [5] D r i n f e l ' d gave a purely algebraic exposition of this theory; p a p e r s [6-8] can also be r e c o m m e n d e d . According to this theory, the nontrivial solutions of (1)-(4) a r e p a r a m e t r i z e d by t r i p l e s (X, ~o, 6), where X is a c o m p a c t Riemann s u r f a c e of genus 2, ~ a n o n - W e i e r s t r a s s point of X, 6 an effective nonspecial divisor of X of d e g r e e 2 (such triples f o r m a s i x - d i m e n s i o n a l algebraic variety). A solution is called nontrivial if the set of o r d e r s of the o p e r a t o r s commuting with L is p r e c i s e l y N \ {t; 2} One can s t a r t with " s e m i t r i v i a l " solutions c o r r e s p o n d i n g to the set of o r d e r s N\{1} . F o r L = D 3 + pD + 1/2p' the o p e r a t o r K = D 2 + 2/3p c o m m u t e s with L if the K o r t e w e g - d e V r i e s equation p~ + 4pp' = 0 holds; it r e Institute f o r Scientific Data on the Social Sciences, A c a c e m y of Sciences of the USSR. T r a n s l a t e d f r o m Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 14, No. 2, pp. 89-90, A p r i l - J u n e , 1980. Original article s u b mitted D e c e m b e r 21, 1979. 158

0 0 1 6 - 2 6 6 3 / 8 0 / 1 4 0 2 - 0 1 5 8 $07.50 © 1980 Plenum Publishing C o r p o r a t i o n

m a i n s to put M = K 2 , N = K L . F o r this r e a s o n we m a k e the substitution q = 1/2p' + r . The c r u c i a l step in the p r o o f c o n s i s t s in going t o the new s y s t e m in which Eq. (5) is twice Eq. (1), E q . (6) is t w i c e Eq. (2) m i n u s t h r e e t i m e the d e r i v a t i v e of E q . (1), Eq. (7) equals Eq. (3), and Eq. (8) is twice Eq. (4) m i n u s the d e r i v a t i v e of Eq. (3). It is a m a z i n g h e r e that two of the new equations b e c o m e l i n e a r in r, and two do net involve d i f f e r e n t i a l s . We note a n o t h e r m y s t e r i o u s f a c t which is not u s e d below, v i z . , joint r e d u c t i o n of the o r d e r with r e s p e c t to r in E q s . (5) and (8) leads to an e q u a t i o n of the f o r m Ar + B r ' = E, w h e r e A, B, E a r e d i f f e r e n t i a l p o l y n o m i a l s in p; f o r r e a s o n s unknown to the a u t h o r , it thus t u r n s out that A = - B ' . If we i n t r o d u c e Pi = p(i), r i = r(i), then in the s p a c e of v e c t o r s X = (p, Pl, P2, P3; r , r t , r~, r~), E q s . (5) and the d i f f e r e n c e b e t w e e n E q s . (6) and (7) give the a l g e b r a i c r e l a t i o n s F = G = 0, while E q s . (8) and (6) give the single d i f f e r e n t i a l equation X~ = H(X), w h e r e F = 2r~ @ (@ T 6C~)r @ 6clp~ - - 3cep @ c~, 3 p~ @ p~ @ 9C~p~ @ 9Cspa -+- 3r ~ + t8 (C~ - - Ca) r @ Clo - - 2C~, , H =

i5 p~~ -- -~5 p + 9Csp -~ p~, p~_,pa, --5pp~ - - -~

15r~ + t8CTr 4- Ca0; ra, r~, ra, --5pr~ - -

- - ~ PVx--

-- 2

The solutions of the equations X' = H ( X ) f o r m a p a r a m e t r i z a t i o n of the i n t e g r a l lines of the v e c t o r field 0

0

30_4_(__5ppz ~54p:__TF '

+ r~J~- ~- r2 0~ + r,~ 0 :-(--Spry" 5

9Csp@t5r~_]_18Cvr+C,,]

0 .~_

-~ p~,r-- Sp~r -L 9Csr-- 3~-C,p~ -- 3C,p~ -- 2CI~) 0

C o n s e q u e n t l y , the solutions of s y s t e m (5)-(8) a r e in b i j e e t i v e c o r r e s p o n d e n c e with points of the a l g e b r a i c s e t Y = F - t {0 } ~ G -~ {0 } at which the field v is n o n s i n g u l a r and a r e such that the i n t e g r a l c u r v e s p a s s i n g t h r o u g h t h e m r e m a i n in Y, i.e., v • F = v • G = 0. We have v .F = 4plr + 3c2Pl 4- 6Clp~ + 4prl + 6C~rl + 2r3, v.G = P~P2 @ 3p2pa - - ~/ePJP2 + 9Csp2_~- PP3 + 9Clp3 ~- 3rrl "+- 18CTra - - t8C2r.

The s e t of s o l u t i o n s of our s y s t e m in b i r a t i o n a l l y i s o m o r p h i c to the affine v a r i e t y defined by the equations F =0,G =0, v'F =0,v-G = 0 . We w r i t e t h e m down t o g e t h e r : (9)

2r~ @ (4p ~ 6Ci)r @ 6Cap1 ~ 3C2p @ C4 = O, 3 p~ @ p~ @ 9Clp ~ @ 9Csp 1 @ 3r 2 @ t8 (C7 - - C~) r + Clo - - 2C5 = pp~ - - ..~ 4Plr Jr 3C2pl -~ 6Clp2 -~ 4po ~ 6Clrl -]- 2r3 =

0,

0,

3

piP2 + 3p~pl - - T P~P~ ~- 9Csp2 + Pp8 + 9Clp3 + 3rr~ + 18C7r1 - - t8C2r 1 = O,

(10)

(11) (12)

H e r e the { C i } a r e p a r a m e t e r s . This v a r i e t y is r a t i o n a l , s i n c e in t e r m s of the c o o r d i n a t e s p, Pl; r, r t we can e x p r e s s r a t i o n a l l y the c o o r d i n a t e s P2, r3, and P3 in E q s . (10), (11), and (12), r e s p e c t i v e l y . The t h e o r e m is proved. C O R O L L A R Y . T a k e the m o d u l i s p a c e of c u r v e s of genus 2 and c o n s i d e r a bundle o v e r it with o v e r X equal t o the p r o d u c t of X by the J a c o b i a n of X. This v a r i e W is u n i r a t i o n a l . The a u t h o r is g r a t e f u l to A. B. Shabat f o r explaining how to w o r k with o v e r d e t e r m i n e d s y s t e m s , Yu. A. Manin and A. V. Z e l e v s i n k i i f o r c r i t i c a l r e m a r k s , and O. A. B r y a n d i n s k a y a , R. L. T u r e t s k a y a , and A. K. T s a t u r y a n y f o r help in laying out this a r t i c l e . LITERATURE 1. 2. 3. 4. 5. 6.

CITED

I . M . G e l ' f a n d and L. A. Dikii, F u n k t s . Anal. P r i l o z h e n . , 10, No. 4, 13-29 (1976). A . P . V e s e l o v , F u n l ~ s . Anal. P r i l o z h e n . , 13, No. 1, 1-7 (1979). J . L . B u r c h n a l l and T. W. Chatmdy, P r o c . R. Acad. Sci. London, S e r . A l l S , 557-583 (1928). I . M . K r i c h e v e r , F u n k t s . Anal. P r i l o z h e n . , 1__1,No. 1, 15-31 (1977). V . G . D r i n f e l ' d , F u n k t s . Anal. P r i l o z h e n . , 1~.1, No. 1, 11-14 (1979). Yu. I. Manin, M o d e r n P r o b l e m s in M a t h e m a t i c s [in R u s s i a n ] , Vol. 11, VINY£I, M o s c o w .

159

7.

D. M u m f o r d , " A n a l g e b r o g e o m e t r i c c o n s t r u c t i o n of c o m m u t i n g o p e r a t o r s and of s o l u t i o n s of the T o d a l a t t i c e e q u a t i o n , K o r t e w e g - d e V r i e s e q u a t i o n , and r e l a t e d e q u a t i o n s , " P r e p r i n t . J . L . V e r d i e r , S e m . B o u r b a k i , No. 512 {1977-78).

8.

VECTOR ON

FIELDS

AND

DIFFERENTIAL

EQUATIONS

SUPE RMANIFOLDS V.

N.

Shander

UDC 517.9

In the l a t e s t p a p e r s in p h y s i c s devoted to s u p e r g r a v i t a t i o n , the g e n e r a l i z e d Y a n g - M i l l s e q u a t i o n s , e t c . , a n i m p o r t a n t r o l e is p l a y e d by odd v e c t o r f i e l d s of the f o r m 8 / ~ + ~ ~ / S u , c o n s i d e r e d as " s q u a r e r o o t s " of the s h i f t g e n e r a t o r 8 / 8 u (cf. [1]). In t h i s note a t h e o r e m on r e c t i f i a b l e v e c t o r f i e l d s is p r o v e d , which shows that the f i e l d 8 / 8 ~ + ~ 8 / 8 u h a s a s i m p l e i n v a r i a n t c h a r a c t e r i z a t i o n ; on the b a s i s of it d i f f e r e n t i a l e q u a t i o n s a r e d e f i n e d on s u p e r m a n i f o l d s f o r w h i c h an e x i s t e n c e and u n i q u e n e s s t h e o r e m for t h e s o l u t i o n s is p r o v e d . All the p r e l i m i n a r y i n f o r m a t i o n is c o n t a i n e d in [2, 3]. 1. Let ,Z~ be a s u p e r d o m a i n of d i m e n s i o n (p, q). In c o o r d i n a t e s x = (u, ~) on ,,~l , e a c h v e c t o r f i e l d D, o b v i o u s l y c a n be d e s c r i b e d in the f o r m D = ZD{u i) 8 / 8 u i + Z D ( ~ j ) 8 / 8 ~ j . We c a l l the f i e l d D w e a k l y n o n d e g e n e r a t e at the point ,,~ ~ J ! , if n o t all the c o e f f i c i e n t s of D v a n i s h at the p o i n t m , and n o n d e g e n e r a t e if D 1o~: C~ (?l) ~ C~ (?l) iS a n e p i m o r p h i s m f o r s o m e n e i g h b o r h o o d o2l of the point m . T H E O R E M 1. Let the f i e l d D be n o n d e g e n e r a t e at the p o i n t ~ ~ .,~/ . T h e n t h e r e e x i s t s a c o o r d i n a t e s y s t e m x = ( u , ~) in a n e i g h b o r h o o d of the p o i n t m , i n w h i c h D = 2 / ~ u 1 , w h e r e D i s e v e n a n d D = 8 / 8 ~ 1 + ~ 1 8 / 8 u 1 if D is odd. P r o p o s i t i o n 1. Let the field D be w e a k l y n o n d e g e n e r a t e at the p o i n t rn ~ J . T h e n if D is e v e n , t h e n D is n o n d e g e n e r a t e , and if D is odd, t h e n in s o m e n e i g h b o r h o o d ol~ of the p o i n t m t h e r e e x i s t s a c o o r d i n a t e s y s t e m in D I~ = a/a~ ÷ ~L , w h e r e L is an e v e n f i e l d on ~. 2. P r o o f . We d e n o t e by J the i d e a l i n c ~¢ (, I~) g e n e r a t e d by a l l odd f u n c t i o n s . I n d u c t i o n on k shows t h a t a n e v e n w e a k l y n o n d e g e n e r a t e f i e l d c a n be r e d u c e d to the f o r m 8 / S u l ( m o d j k ) . T h e o r e m 1 and P r o p o s i t i o n 1 f o l l o w s f r o m the f a c t t h a t Jq+~ = 0, and f r o m the f a c t that D 2 = ( l / 2 ) [ D , D] is an e v e n n o n d e g e n e r a t e v e c t o r f i e l d . 3. It is k n o w n (cf. [4]) that by a n 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 it is c o n v e n i e n t to u n d e r s t a n d a v e c t o r f i e l d on a m a n i f o l d M, d e p e n d i n g on t i m e , a o n e - d i m e n s i o n a l m a n i f o l d T; by a s o l u t i o n of t h i s e q u a t i o n is m e a n t a T - f a m i l y of d i f f e o m o r p h i s m s of the m a n i f o l d M ( t r a n s l a t i o n s a l o n g i n t e g r a l c u r v e s ) . A n a l o g o u s l y , let ( ~ c ~0, ~, 1)m) , w h e r e Z is a s u b s u p e r d o m a i n in the s u p e r s p a c e a r e c o o r d i n a t e s and a v e c t o r f i e l d on J , and one has one of the two q u a d r u p l e s : a) (I~,0 c ~ , ° , t,

a,'at)

~0

, and x and Dx

, w h e r e I 1"0 is an i n t e r v a l c o n t a i n i n g 0;

b) (z~,l c ~,~, (t, ~), a/a~ ÷ ~a/at), w h e r e I~,~ = I~,° × ~0,~ We s h a l l c a l l ~- t i m e and

D x

d i f f e r e n t i a t i o n with r e s p e c t to t i m e .

L e t .Z be a s u p e r m a n i f o l d , ~l be an open s u b s u p e r m a n i f o l d i n J t , and ~- b e t i m e . We c a l l ~L×~~ a c y l i n d e r o v e r ~ , and an open s u b s u p e r m a n i f o l d ~ ~ J~ × ~ , r e p r e s e n t a b l e as a u n i o n of c y l i n d e r s and c o n t a i n i n g ,~ × {0) a p s e u d o c y l i n d e r . (If ,;~ is c o m p a c t , t h e n b e l o w p s e u d o e y l i n d e r c a n b e r e p l a c e d by c y l i n d e r . ) Let ~ , ~ - be the p r o j e c t i o n s of ; ~ × ~ onto .~/~ and ~ , r e s p e c t i v e l y , ~: .# ~ J ~ × ~- b e the i n c l u s i o n d e f i n e d by the p r o j e c t i o n ~*: c ~¢ ( ~ × ~ ) -~ c ~ (J~ × ~)/{1: ! I~×10~ = 0} -~ c~ ( ~ ) . By /)~- we d e n o t e the f i e l d on * $ ~ × ~" , u n i q u e l y d e f i n e d by the c o n d i t i o n s n~- o D~- = D m ° ~m, zT~ o .~* = 0. that D

By a d i f f e r e n t i a l e q u a t i o n on JZ with t i m e ~- we s h a l l m e a n a f i e l d D on a p s e u d o c y l i n d e r o v e r ~ , s u c h o~ = n~- o D ~ - . By a ~ - f a m i l y of d i f f e o m o r p h i s m s of the s u p e r m a n i f o l d ~ we s h a l l m e a n a d i f f e o -

Moscow State U n i v e r s i t y . T r a n s l a t e d f r o m F u n k t s i o n a l ' n y i A n a l i z i E g o P r i l o z h e n i y a , V o l . 14, No. 2, pp. 9 1 - 9 2 , A p r i l - J u n e , 1980. O r i g i n a l a r t i c l e s u b m i t t e d J u n e 15, 1979.

160

0 0 1 6 - 2 6 6 3 / 8 0 / 1 4 0 2 - 0160 $07.50 © 1980 P l e n u m P u b l i s h i n g C o r p o r a t i o n