The KP hierarchy in Miwa coordinates1

26 July 1999

Physics Letters A 258 Ž1999. 272–278 www.elsevier.nlrlocaterphysleta

The KP hierarchy in Miwa coordinates Boris Konopelchenko a , Luis Martınez ´ Alonso

1

b

a

b

Dipartimento di Fisica, UniÕersita´ de Lecce, 73100 Lecce, Italy Departamento de Fısica Teorica, UniÕersidad Complutense, E28040 Madrid, Spain ´ ´ Received 7 May 1999; accepted 4 June 1999 Communicated by A.P. Fordy

Abstract A systematic reformulation of the KP hierarchy by using continuous Miwa variables is presented. Basic quantities and relations are defined and determinantal expressions for Fay’s identities are obtained. It is shown that in terms of these variables the KP hierarchy gives rise to a Darboux system describing an infinite-dimensional conjugate net. q 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: KP hierarchy; tau-functions; Miwa variables

1. Introduction The Kadomtsev–PetviashviliŽKP. equation, introduced to describe propagation of shallow water waves under special conditions w1x, is nowadays a principal ingredient in studies of many problems in physics and mathematics. The KP hierarchy is a paradigm of the hierarchies of integrable systems w2,3x, it is used to characterize Jacobian varieties in algebraic geometry w4x and is connected to string theory due to the close relation arising between partition functions of quantum models and KP t-functions w5,7x. The standard variables for the KP hierarchy form an infinite-dimensional vector t s Ž t 1 ,t 2 , . . . ., the three first components of which being the two spatial and one time variables of the KP equation viewed as a hydrodynamical model. The KP hierarchy can be

formulated as the compatibility conditions for the following linear system of equations Žsee e.g. w2,3,9,10x.:

E En c s Pn Ž t , Ex . c ,

Partially supported by CICYT proyecto PB95–0401.

E tn

, n G 2.

Ž 1.

Here PnŽ t, Ex . denotes a set of linear differential operators with respect to the variable x ' t 1 and c s c Ž z, t . is the KP wave function, a complex-valued function defined on the unit circle Ž< z < s 1. which admits a factorization c s c 0 x , where

c 0 s exp

ž

Ý nG1

tn zn

/

,

xs1q

Ý an Ž t . z n .

Ž 2.

nG1

Another remarkable parametrization of the KP hierarchy is provided by the so-called Miwa Õariables defined as tn s

1

En s

1 n

`

Ý pi z in ,

n G 1,

is1

0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 3 7 3 - 4

Ž 3.

B. Konopelchenko, L.M. Alonsor Physics Letters A 258 (1999) 272–278

where pi and z i are discrete Žinteger valued. and continuous Žcomplex valued. Miwa variables, respectively. It is worth-mentioning that the KP hierarchy can be derived from a single finite-difference Hirota equation involving the discrete Miwa variables w4x, which in turn is closely connected with the Fay’s trisecant formula. Miwa variables have been extensively used several years ago in the study of matrix models in string theory and two-dimensional quantum gravity w5–7x. In particular, they were relevant to show that partition functions for matrix models of Kontsevich type are connected with KP t-functions. On the other hand, in a different approach to string theory via the KP hierarchy it was found that Miwa variables pi and z i can be identified with the momenta and Koba–Nielsen variables of strings w12x. At last, it was recently shown w13x that Miwa variables are also useful in the analysis of the Bethe ansatz for quantum integrable systems. From the point of view of mathematics, sets of variables of the types t n and z i connected by Ž3. have been used for a long time in the representation theory of the symmetric group SN w14–16x. In this context t n and z i are referred to as power sums and symmetric variables, respectively. The presence of these variables in the theory of the KP hierarchy is a consequence of the well established relationship between this hierarchy and the representation theory of loop groups w16,17x. A main goal of the present paper is to reformulate the KP hierarchy, its basic objects and relations, using symmetric Žcontinuous. Miwa variables. These variables and the spectral parameters appear on an equal footing in the expressions for t-functions and wave functions. Thus, it follows that several fundamental relations for the KP hierarchy as, for instance, Sato’s equations, the relationship between wave functions and t-functions and the action of Darboux–Backlund transformations take a symmet¨ ric and simple form. Furthermore, Miwa variables are very useful to deal with the bilinear identity for t-functions. In particular, we prove that they provide a compact determinantal form for addition formulae ŽFay’s identities.. We also prove that the simplest Fay’s identity, written in terms of Miwa variables, leads to a Darboux system describing an infinite-dimensional conjugate net.

273

2. KP flows and t-functions The KP hierarchy of integrable systems can be introduced by means of the Sato’s flows w8,9,4x

ES E tn

s y Ž SExn Sy1 . y S,

Ž 4.

for the pseudo-differential operator S:s 1 q

Ý a n Ž t . Eyn x .

Ž 5.

nG1

The solutions of Ž4., Ž5. can be related to KP wave functions by setting the same coefficients a nŽ t . in the expansions Ž2. of x and Ž5. of S. On the other hand, it is well known w9,16x that a KP wave function leads to a flow in an infinite-dimensional Grassmannian. To describe this property we consider the Hilbert space H:s L2 Ž S 1 ,C. of square-integrable complex-valued functions on the unit circle S 1 and the big cell Gr 0 of the Grassmannian Gr of H. For our purposes Gr 0 can be defined as the set of all closed subspaces W of H which admit a dense linear subspace generated by an admissible basis; i.e. a subset  wn s wnŽ z . n G 04 ; W of finite order elements: wn Ž z . s zyn q O Ž zyn q1 . . Each KP wave function c determines an element W of Gr defined as the closure in H of the set of all linear combinations of the form N

Ý cn Ž t n . c Ž z , t n . .

Ž 6.

nG1

Here c n are arbitrary functions of t and t n are arbitrary points of the domain of c in C` . From Ž1. and by using Taylor expansion around any fixed t it follows that: W s span  Exnc Ž z , t . , n G 0, any fixed t 4 ,

Ž 7.

Thus each KP wave function determines a flow in Gr given by W Ž t . :s c 0 Ž z , t .

y1

W

s span  c 0 Ž z , t .

y1

Exnc Ž z , t . , n G 0, 4 .

B. Konopelchenko, L.M. Alonsor Physics Letters A 258 (1999) 272–278

274

For all values of t in the domain of c in C` we have that

c0 Ž z , t .

y1

Exnc Ž z , t . s zyn q O Ž zynq1 . ,

so that W Ž t . g Gr 0 . There is a natural imbedding of Gr in the projective space P Ž H . of the infinite wedge space H :s n `H. It assigns to each W g Gr the ray < W : in H generated by the vectors

c n s e n n P, n s 0," 1," 2, . . . ,

where e n :s zyn , the operator E Ee n denotes the supersymmetric derivative with respect to e n and the operator e n n P is the exterior multiplication operator by e n . They satisfy the canonical anticommutation relations

 bn ,bm 4 s  c n ,c m 4 s 0,  bn ,c m 4 s dn m . Moreover, if we define the vacuum state as
c n
Ž n G 0. . Ž 8.

These fermionic modes determine the following fields on the unit circle: b Ž z . :s Ý bn zyn ,

c Ž z . :s Ý c n z n ,

n

< z < s 1.

n

It can be shown w9x that given a KP wave function c s c 0 x we may write

x Ž z,t . s

t Ž ty w z x. t Ž t.

,

ž

w z x :s z ,

z2 z3 , ,... . 2 3 Ž 9.

/

Here the function t is defined up to a multiplicative constant by the correlation function

t Ž t . :s ²vac < W Ž t . : ,

/

bm c mqn .

msy`

Ž t n y tXn . t Ž t y w z x .

/

nG1

= t Ž tX q w z x .

dz

s 0. Ž 11 . z2 Natural symmetries of this identity are the so called Darboux–Backlund transformations w18x ¨ D Ž p, t . t Ž t . :s c 0 Ž p, t . t Ž t y w p x . , Ž 12 . and their dual analogues y1

D ) Ž p, t . t Ž t . :s c 0 Ž p, t . t Ž t q w p x . . Ž 13 . As a consequence of Ž11. every KP wave function c has an associated adjoint function c ) s Ž c 0 .y1x ) with t Ž tq w z x. x ) Ž z,t . s , Ž 14 . t Ž t. which verifies the bilinear identity dz c Ž z , t . c ) Ž z , t X . 2 s 0. Ž 15 . 1 z S From Ž11. it also follows that:

H

it follows that: bn
ž

Ý

The following bilinear identity constitutes the main property of t-functions: yn

E ,

Ý tn nG1

1

where  wn4n G 0 is any admissible basis of W. A set of fermionic operators on H can be defined by

E en

`

H Ž t . :s

HS exp ž Ý z

w 0 n w 1 n PPP n wn n PPP ,

bn s

where W Ž t . is the KP trajectory of the element W g Gr 0 generated by c . Moreover, it turns out w9x that < W Ž t . : s exp Ž H Ž t . . < W : ,

Ž 10 .

X

X

HS c Ž z ,u, t . c Ž u , z , t . dz s 0, 1

Ž 16 .

where X c Ž z , zX , t . s c 0 Ž z , t . x Ž z , zX , t . cy1 0 Ž z ,t. is the Cauchy–Baker function introduced in w19x which is determined by 1 t Ž t y w z x q w zX x . X x Ž z , z , t . :s . Ž 17 . z y zX t Ž t.

3. Miwa variables We introduce Miwa variables x s Ž x 1 , x 2 , x 3 , . . . . and y s Ž y 1 , y 2 , y 3 , . . . . < x i <, < yi < - 1 by `

ts

Ý Ž w x i x y w yi x . , is1

Ž 18 .

B. Konopelchenko, L.M. Alonsor Physics Letters A 258 (1999) 272–278

or equivalently tn s

1 n

Ž4. and Ž19. imply that Sato’s flows in Miwa variables can be written as

`

ž

Ý Ž x in y yin . is1

/

,

n G 1.

ES

Our definition, slightly more general than those used in w11x or w14–16x, is also adopted in w6,19,20x. Transformations of the type Ž18. are useful in the theory of symmetric polynomials w14,15x. To show this relationship we just mention that the Jacobian matrix of the transformation Ž18. is given by

E xi E tn

s y Ž y1 .

n

fny1 Ž x 1 , . . . , xu i , . . . ; y 1 , y 2 . . . .

Ł Ž 1 y x irx m . Ł Ž 1 y x irym .

E tn

s Ž y1 .

n

fny1 Ž x 1 , x 2 , . . . ,; y 1 , . . . , yu i , . . . .

Ł Ž 1 y yirym . Ł Ž 1 y yirx m . 1 1

Ý i1-i 2- . . . -i n

,

nG1

ž

1y

zn

/

s

z i1 z i 2

...

1 zin

.

E xi

s x iny1 ,

E tn E yi

1 y yi E x

n Ý Ž yl. fn Ž z1 , z 2 , . . . . ,

nG0

s yyiny1 .

Sy1

Sy1

/

/

S, y

S.

Ž 20 .

y

The vacuum wave function takes the form

iG1

1 y yirz 1 y x irz

.

The undressed KP wave function and its adjoint function are given by

x Ž z ; x ; y. s

t Ž x ; z, y.

,

t Ž x, y.

t Ž z, x ; y. t Ž x, y.

.

Ž 21 .

Analogously, the Cauchy–Baker function reads

and taking into account that

E tn

Ex

x ) Ž z ; x ; y. s

These expressions can be derived by using the following standard identity of the theory of symmetric polynomials:

Ł

E yi

ž

s S

1 y x i Ex

,

where fn are the symmetric functions

l

ES

Ex

m

m/i

fn Ž z 1 , z 2 . . . . s

E xi

ž

sy S

c0 Ž z ; x ; y. s Ł

n

m/i

E yi

275

Ž 19 .

Given a function a Ž t . depending on the standard KP variables we will denote by a Ž x; y . the corresponding function depending on the Miwa variables. In what follows the following obvious properties will be useful: 1. The function a Ž x; y . is invariant under permutations among the components of x or y. 2. a Ž0, x; y . s a Ž x;0, y . s a Ž x; y . 3. For any given finite or infinite-dimensional vector a s Ž a1 ,a 2 , . . . .

a Ž a, x ; a, y . s a Ž x ; y . . Let us rewrite the basic objects of the KP theory in terms of Miwa variables. First of all we notice that

X

x Ž z, z ; x ; y. s

1

t Ž zX , x ; z , y .

z y zX

t Ž x ; y.

.

Ž 22 .

We would like to emphasize that the spectral parameter z appears in the t-function on the equal footing as the Miwa variables. It is a y-type variable in the expression for the wave function, an x-type variable in the case of the adjoint wave function, and it takes the role of both types in the case of the Cauchy– Baker function. Thus, it is only when we introduce the wave functions we mark one of the Miwa variables of the t-function, call it spectral parameter and break its symmetry with the remaining Miwa variables of the same type. As we will discuss elsewhere, this symmetry between spectral parameters and Miwa coordinates at the t-function level is very useful for dealing with symmetries of the KP hierarchy of both standard and non-isospectral types in the same manner. Furthermore, it can be shown w6,19,20x that the t-function can be expressed as

t Ž x ; y . s D Ž x ; y . det

ž

²vac < c Ž x i . b Ž yj . < W : yj ²vac < W :

/

,

Ž 23 .

B. Konopelchenko, L.M. Alonsor Physics Letters A 258 (1999) 272–278

276

where

4. Additional formulae and Darboux equations

Ł Ž yi y x j . D Ž x ; y . :s

i, j

Ł Ž ym y yn . Ł Ž x n y x m .

n)m

Let us consider the bilinear identity Ž11., which written in terms of Miwa variables takes the form

.

n)m

HS iG1 Ł

Therefore Ž23. allows us to write

t Ž x ; y . s D Ž x ; y . det

ž

1

t Ž x i ; yj . yj y x i

/

.

Ž 1 y yirz . Ž 1 y xXirz . t Ž x ; z, y. Ž 1 y yiXrz . Ž 1 y x irz .

= t Ž z , xX ; yX .

Ž 24 .

dz z2

s 0.

Ž 27 .

If we set x ™ Ž s, x ., xX ™ Ž sX , x .,

Thus, we see that the value of the t-function at a point Ž x; y . g C 2` in Miwa space can be expressed in terms of its values at two-dimensional single points Ž x i , yj . g C 2 . Notice also that the building block of this expression for t is closely related to the Cauchy–Baker function as shows the identity

yX s y, where

sX s Ž sX1 , sX2 , . . . , sXM . ,

s s Ž s1 , s 2 , . . . , s N . , N y M G 2,

and all the components of s and sX are assumed to be different, then the bilinear identity becomes M

t Ž x i ; yj . yj y x i

Ł Ž 1 y sXirz .

s t Ž 0 . x Ž y j , x i ;0 . .

HS

Miwa variables are very suitable for dealing with the symmetry operations of the KP hierarchy too. For instance, let us notice that the Darboux–Backlund ¨ transformations Ž12. take the form D Ž p; x : y . t Ž x ; y . s

Ł

nG1

1 y ynrp 1 y x nrp

t Ž x ; p, y . ,

t Ž s, x ; z , y . t Ž z , sX , x ; y .

Ł Ž 1 y sirz .

dz z2

is1

s 0.

Ž 28 .

Thus by calculating the integral as the sum of residues at si , Ž i s 1, . . . , N . we get a set of Fay’s identities N

Ý Ž 25 .

is1

r Ž si .

Ł Ž si y s j .

tˆ it i s 0,

Ž 29 .

j/i

where

and D ) Ž p; x : y . t Ž x ; y . s

1

is1 N

Ł

nG1

1 y x nrp 1 y ynrp

M

t Ž p, x ; y . .

Ž 26 . By taking advantage of the factorized form of this expression one easily proves the usual transformation properties of the wave functions w18x. For example, we have D Ž p; x : y . c Ž z ; x ; y . s y

z p

D Ž p; x : y . c ) Ž z ; x ; y . s y

c Ž z ; x ; p, y . , p z

D Ž p; x : y . c Ž z , zX ; x ; y . s y

c ) Ž z ; x ; p, y . , z

zX

c Ž z , zX ; x ; p, y . .

r Ž z . :s z NyMy2 Ł Ž z y sXi . , is1

and

tˆ i :s t Ž s1 , . . . , sui , . . . , sN , x ; y . , t i :s t Ž si , sX , x ; y . . We note that these identities can be rewritten as 1

s1

...

s1Ny 2

r Ž s1 . t 1tˆ 1

1 ... ... 1

s2 ... ... sN

... ... ... ...

s2Ny2 ... ... sNNy 2

r Ž s2 . t 2tˆ 2 s 0. ... ... r Ž sN . t N tˆ N

Ž 30 .

B. Konopelchenko, L.M. Alonsor Physics Letters A 258 (1999) 272–278

The case N s 3, M s 1 is the usual Fay’s identity s1 y sX1

Ž s1 y s 2 . Ž s1 y s 3 . = t Ž s2 , s3 , x ; y . t Ž s1 , sX1 , x ; y . Ž s 2 y s1 . Ž s 2 y s 3 . = t Ž s1 , s3 , x ; y . t Ž s2 , sX1 , x ; y .

1 s xi yx j

s3 y sX1

q

s1 y s 3

t s t Ž x ; y . ™ t˜ Ž x ; y . :s t Ž y; x . , is a symmetry of Ž11., it can be proved that the functions

x Ž s2 , s1 ; sX1 , x ; y .

b˜i j Ž x ; y . :s yc Ž yj , yi ; x ; y . ,

sX1 y s2

satisfy the Darboux system

q

s3 y s2

x Ž s 2 , s1 ; s 3 , x ; y .

q Ž sX1 y s3 . x Ž s2 , s3 ; sX1 , x ; y . x Ž s3 , s1 ; s3 , x ; y . s 0. We notice that a similar equation in terms of the usual KP coordinates t n has been derived in w21x. By taking the limit sX1 ™ s3 on this expression it follows that: E s 2 y s1 y x Ž s 2 , s1 ; s 3 , x ; y . E s3 Ž s1 y s 3 . Ž s 2 y s 3 .

ž

/

s x Ž s 2 , s 3 ; s 3 , x ; y . x Ž s 3 , s1 ; s 3 , x ; y . . Notice that this means that for any three different i, j,k we have

E E xi

y

1 y ynrx i

. Ł Ł Ž 1 y x nrx i . nG1 1 y ynrx j

Analogously, by taking into account that

By setting y ™ Ž s2 , y . and dividing by t Ž sX1 , x ; y . t Ž s3 , x ; y . , we get sX1 y s1

Ł Ž 1 y x nrx j .

n/j n/i

Ž s 3 y s1 . Ž s 3 y s 2 . = t Ž s1 , s2 , x ; y . t Ž s3 , sX1 , x ; y . s 0.

ž

For example if we take t ' 0, then c s c 0 and we get the following solution of the Darboux equations:

bi j Ž x ; y .

s2 y sX1

q

277

x j yxk

Ž xk yxi . Ž x j yxi .

/

x Ž x j , xk ; x ; y.

Eb˜jk E yi

s b˜ji b˜i k .

Ž 34 .

The fact that the Darboux system is associated with the one-component KP hierarchy has been already mentioned in w21x within a different approach. We finally indicate how to get from the KP hierarchy in continuous Miwa variables to the discrete KP hierarchy. To this end, it is enough to constraint all the continuous Miwa coordinates Ž x i , yi .,Ž i G 1. to take values on a fixed finite set  a1 , . . . ,a N 4 ,Ž N G 2.. In this way, a set of discrete Miwa variables Ž l 1 , . . . ,ł N . can be introduced by `

N

Ý Ž w x i x y w yi x . s Ý l n w a n x .

ts

is1

sx Ž x j , xi ; x ; y. x Ž xi , xk ; x ; y. . Thus, it follows at once that: bi j Ž x ; y . :s c Ž x i , x j ; x ; y . s lim c Ž x i Ž 1 q e . , x j Ž 1 q e . ; x ; y . , e™0

Ž 33 .

Ž 35 .

ns1

Hence functions a Ž x; y . become functions depending on the discrete variables l i . If we now consider Ž30. with s s Ž a1 , . . . ,a N ., sX s 0, then we get

Ž 31 .

1

a1

...

a1Ny 2

a1Ny2t 1tˆ 1

satisfy the system of Darboux equations for an infinite-dimensional conjugate net w23x Eb jk s b ji bi k . Ž 32 . E xi

1 ... ... 1

a2 ... ... aN

... ... ... ...

a 2Ny2 ... ... a NNy 2

a2Ny2t 2tˆ 2 s 0, ... ... Ny2 a N t N tˆ N

Ž 36 .

278

B. Konopelchenko, L.M. Alonsor Physics Letters A 258 (1999) 272–278

where

tˆ i :s t Ž l 1 q 1, . . . ,l iy1 q 1,l i ,l iq1 q 1, . . . ,l N q 1 . , t i :s t Ž l 1 , . . . ,l iy1 ,l i q 1,l iq1 , . . . ,l N . . These equations constitute the discrete KP hierarchy w22x.

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