Two-dimensional integrable systems and self-dual Yang-Mills equations

F. Guil and M. Maiiasa) Departamento de Fisica Teo’rica, Universidad Complutense, 2804OMadrid, Spain

(Received 8 June 1993; accepted for publication 11 January 1994)

The relation between two-dimensional integrable systems and four-dimensional self-dual Yang-Mills equations is considered. Within the twistor description and the zero-curvature representation a method is given to associate self-dual Yang- Mills connections with integrable systems of the Korteweg-de Vries and nonlinear Schrodinger type or principal chiral models. Examples of self-dual connections are constructed that as points in the moduli do not have two independent conformal symmetries.

1. INTRODUCTION

As a system of partial differential equations, the self-dual Yang-Mills (SDYM) equations are invariant under the action of the group of conformal transformations acting on the four space-time coordinates. It is well known that invariant solutions by the action of a subgroup with two conformal generators satisfy a differential equation in two variables since each one-dimensional subgroup reduces by one the number of independent variables. These solutions are called self- similarity solutions of the original equations. It has been observed that this procedure allows one to describe the corresponding invariant solutions in terms of a two-dimensional integrable system. This is the case for the principal chiral modelim3 [for the classical Euclidean 0(1,3) nonlinear c model this was already noticed in Ref. 41 as well as for the Korteweg-de Vries (KdV) and nonlinear Schrodinger (NLS) equations. 5,6 A review of these results can be found in Ref. 7. It is a nontrivial aspect of the reduction problem to describe the resulting two-dimensional equations. For the principal chiral model this description is based in the formulation of SDYM equations in terms of the Yang matrix. An alternative approach consists in describing the SDYM equations as a compatibility condition for two linear differential equations, this allows one to identify the reduced solutions in the KdV and NLS cases.

It seems, however, that any reasonable relation between two integrable equations shouldbe based in their definitions as integrable systems. The SDYM equations describe a connection for a bundle over the Grassmannian of two-dimensional subspaces of the twistor space. Integrability for a SDYM connection means that its curvature vanishes on certain two-planes in the tangent space of the Grassmannian. As proved by Ward, 8*9 this allows one to characterize SDYM connections in terms of the splitting problem for a transition function in a holomorphic bundle over the Riemann sphere, i.e., the trivialization of the bundle. For two-dimensional integrable systems the situation is quite analogous. An important feature of equations such as the KdV, NLS, or the principal chiral model consists of the possibility of constructing solutions through a factorization problem in the circle. This is equivalent to the splitting problem for the SDYM case if we choose the transition function having the form required by the factorization problem of a two-dimensional integrable system.

The main result of this article is the construction of a map taking arbitrary solutions of a two-dimensional integrable system into solutions of the SDYM equations. This map comes from the zero-curvature formulation of a two-dimensional integrable system and represents an extension of the correspondence derived from the factorization problem. A generic property of the relation

a)Permanent address: The Mathematical Institute, Oxford University, 24-29 St. Giles’ Oxford OX1 3LB, United Kingdom.

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0 1994 American Institute of Physics

F. Guil and M. Maiias: 2-d integrable systems and self-dual Yang-Mills equation 2903

for a principal chiral model, the KdV, and NLS equations is the appearance of arbitrary functions. In particular, this generalizes the self-similarity solutions under a group generated by two trans- lations. Section II is devoted to the analysis of these properties. The analysis of the correspon- dence for a Yang matrix is given in Sec. III where we study its relation with a chiral field. As a special case one gets Ward’s construction 8,1~2 which we mentioned before. Finally in Sec. IV we present examples of SDYM connections derived from two-dimensional integrable systems that cannot be obtained by symmetry reduction by two conformal generators.

Note that we do not present a characterization of the image of this map as was done in Ref. 5, there the SDYM connections associated to KdV or NLS equations had two particular translational symmetries. The construction in that article is an example of ours and among the SDYM connec- tions associated to the integrable equations we are interested in we find examples that cannot have two independent conformal symmetries. This shows that our generalization cannot be character- ized as in, for example, Ref. 5.

Let us observe that the present construction does not exhaust all possible relations between SD equations and integrable systems. This is the case for the Nahm’s equations that can be associated to SDYM equations in an alternative way to that followed in this article.”

II. INTEGRABLE SYSTEMS AND SELF-DUAL YANG-MILLS EQUATIONS

The compatibility condition for the linear system of two first-order differential equations on the vector ti

*x= U(h)& $4= V(A)@, implies for the two matrix functions U(x,t,X), V(x,t,A), depending on a complex variable X, the nonlinear equation

u,-v,+[u, V]=O. (2.1)

Under an appropriate choice of the functions V(X) and V(A) one can represent through Eq. (2.1) a wide class of nonlinear partial differerential equations in the variables x and t known as inte- grable systems. I1 Concrete exa m p les for these functions that we shall consider below are as follows. ’

Let U and V be given by the expressions

u(x):=xu~+uo, V(A):=A2U1+AUO+V0.

Then Eq. (2.1) implies for the coefficients Ui and Vu the relations

uo,t-vo,x+[uo~ Vol=O, ~l,,-uo,++[u,~ Vol=O, u1,,=0.

In particular if we take

we obtain the Korteweg-de Vries (KdV) equationI for the scalar function u(x,t)

4u,= u,,,+ 6uu,.

Letting now U and V be given by

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2904 F. Guil and M. Mat?as: 2-d integrable systems and self-dual Yang-Mills equation

uh)=( :;* <,), w[ ;r+i;; -;;-j;;,*) 2 : 2

the system above results in the nonlinear Schriidinger (NLS) equation for the complex scalar field P(W>

iPt= i Pxxf IP12P*

The equations of a principal chiral field for functions u(x,y), u(x,y) with values in a Lie algebra g

u,+&, rJ]=O, u,-$4, u]=O (2.2)

can equally be represented with the aid of Eq. (2.1) if we choose in this case

U(X)= -fi , VW=& 3

with the substitution t + y.t3 The preceding formulas can be conveniently represented as the zero curvature condition for a

connection

w:c + A’(M,g), x - w(X),

where X is a spectral parameter and o(X) is a g-valued one-form over the two-dimensional time manifold M having as local coordinates the variables x,t (or x and y). In terms of local coordi- nates x,y (or x,t) of the manifold M we can write the connection as

o(X)= U(X)dx+ V(X)dy

for which the vanishing of the curvature

i-l(X):=do(X)-&J(h), o(h)]

is equivalent to the relation (2.1). Self-dual Yang-Mills (SDYM) equations constitute another important example of nonlinear

partial differential equations which are also integrable. Now, we shall be dealing with a four- dimensional system instead of the two-dimensional cases previously analyzed. If z = (zAA ), A =O,l, A ’ =0’, 1’ represents a 2X2 complex matrix describing the local coordinates of a four- dimensional complex manifold and @ =QAA f dzAA ’ is a g-valued connection described by the set of functions @AA r (z), then the curvature two-form F = dQ, - ;[a, @] can be written as

F=F,,,,,t dzAA’r\dzBB’.

A self-duality condition’ for the Yang-Mills fields FAA ‘BB , follows from the decomposition FAArBBt=~AB~ArBr+ ~~~~~~~~ where E is the Levi-Civita tensor. Thus, the connection @ is self-dual (SD) if

(PA’B’- -0.

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Given a SD connection @ any gauge transformation @ + Qg = dg . g -‘+ Ad g @ gives again a SD connection. A conformal transformation of coordinates z. H z = (A. z + B) . (C . z + 0) - ‘, with (“,g) ESL(~,C), generates a local diffeomorphism 4 such that ~$*a is also a SD connection. We say that two SD connections are equivalent if there exists a gauge and/or a conformal trans- formation taking one into the other. When we introduce certain topological requirements this space of equivalence classes of SD connections is called the moduli space.

The structure of this set of nonlinear partial differential equations and its relation with the two-dimensional integrable systems presented before are best understood through the geometry of a complex vector bundle over the Riemann sphere8*9 as we shall explain briefly.

Let us denote by T a four-dimensional complex linear space, the twistor space,14 and consider its decomposition as a direct sum of a pair of two-dimensional subspaces S and S’, T=S@S’. We associate to this pair of subspaces a coordinate chart of the Grassmannian manifold of two- dimensional subspaces of T as follows. If V is a subspace of T that does not intersect S’, we assign to V the linear map z:S’ + S having the graph V. Then, one is allowed to identify the coordinates Z AA’ on which the connection @=@,&,A’ dzAA’ depends with the coordinate functions of these points in the Grassmannian. If a point in T has coordinates (XA,XA’) adapted to the representation T=S@S’, the linear subspace V determined by the linear transformation z is characterized by the relations xA = zAA’ XA’.

Let XA’ denote the coordinates of a point in the dual of S’ and define the vector fields on the Grassmannian

aA :=aAA’A A’

given in terms of the local coordinates z by the differential operators JAAl : = d/dzAA’ that generate the tangent space. The SD equations for the connection Q, are then equivalent to the requirement that F vanishes over the two-plane c{aA}A=O,l

F(dA ,dg)=O. (2.3)

One has

from which our assertion follows. The connection Cp satisfies the equation

[aA-QA, aB-@B]=aBQ?A-aA@B+[@A) aB]=o

that we get from Eq. (2.3) with QA : = @( aA> =Q)AAAhAA’ and the SD equations are

F oo,lo,=Foo,,,r+F,,,,,,=F,,,,,,=O. (2.4)

All relations involving the coordinates XA’ are homogeneous in these variables, they are therefore well defined on the projective XA’ plane. With the standard covering by the two charts C, , d=-

c, :={X”/AO’,A”’ # O}, c- :={AO’/A”,X” # 0)

the induced equations are

[a;-@;, a;-Q;]=o, a; :‘kaAot+~Alt, @‘A :=A@AOt+@Alr, AEB=-.

Then, we have trivializations I,& :C+ -+ G defined by

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2906 F. Guil and M. Matias: 2-d integrable systems and self-dual Yang-Mills equation

mf+=@A+$+, a,-$-=(D;$-, that on C, IX- solve the splitting problem for the transition function $, $+ = @. #- .8*9,15 This was just the situation for the two-dimensional integrable systems considered before and tells us about the possibility of describing SD connections by means of integrable two-dimensional nonlinear partial differential equations. This is precisely the case if we let o(A) be defined on the tangent vectors ,&A’, equivalently the coordinates x,t (or x,y) are functions of z and A belongs to C, or C- . For suitable functions x(z),t(z) we define the SD connection @ associated to o by the relations

wAhdaA)- (2.5)

The zero-curvature condition for w, dw- 1/2[w, o]=O, on the vectors aA ,ds , implies the SD equations for @

O=(dw-&% wl)(aA,aB)=[aA-w(aA), dB-daB)]=[8A-@(aA)9 aB-%%?)l=F@A ,aB).

The one-form w induces a self-dual connection if its contraction with 8, depends on A as @(aA) does. Explicit expressions for o are obtained in each case by imposing the condition (2.5) upon the A-dependent one-form o. Now, we are in a position to formulate the relation between self-dual connections and integrable systems of KdV and NLS type.

Theorem 2.1: Let w=U(A)dx+V(A)dt be the zero-curvature one-form

w(A)=(AU1+Uo)dx+(A2Ul+XUo+Vo)dt.

Then, there exists a self-dual connection Cp asociated to w(A) in C, if

t(z) =m(zoO’,zq,

for arbitrary functions m and n. The coeficients of the self-dual connection are given by

aAO’= uoaAo’x+ voaAO’t, ~Al’=ulaAo’x~

Proof: On C, we have ai :=aAa’fAdAtt and

Then, Eq. (2.5) implies that the coefficients of X3 and A2 must vanish and this gives the desired result. cl

The second class of SD connections arising from two-dimensional integrable systems we shall consider is related to the equations of the principal chiral field. The proof of the following theorem reproduces the preceding one.

Theorem 2.2: Let the one-form

w(A): = -& dx+

have zero curvature for [Al<1 in C- and x,y be functions of the form

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then the associated connection a( aA> = w( aA) with components given by

is self-dual.

@ AO!=o, @Al!=uaAld+uaAlty

We observe that the coordinates x,y on which a principal chiral field depends determine a SD connection if they are defined on the image under z of the projective lines C(O,O,l,- 1) and C(O,O, 1, l), respectively.

Integrable systems of the type considered in Theorem 2.1 form a large family. Let us mention the modifications of KdV and NLS equations, the Fordy-Kulish NLS equations in homogeneous spaces, or the Burgers equation. Similarly, the principal chiral model contains several integrable systems as those derived from the theory of harmonic maps, not only in Lie groups, but in general in Grassmannians.16 Also it has a number of reductions, such as u models, the Gross-Neveu model, and others.17 Even in the simplest case g=sI(2,C) this model has interesting reductions. Let us mention the sit&Gordon equation, see Ref. 18 for an analysis of the relations of this equation with SDYM and harmonic maps, the massive Thirring model, and the self-induced transparency equations. As examples, we shall write down explicitly the details for the two first mentioned integrable systems: the sinh-Gordon equation and the massive Thirring model. Consider the action of the homographic transformation X -+ (A- l)/(h+ 1) on the zero-curvature one-form w of Theo- rem 2.2, the result is the zero-curvature representation for harmonic maps used in Ref. 16. Now an arbitrary gauge transformation gives a new zero-curvature one-form

o(A)=(AL1+Lo)dx+(Mo+A-‘M,)dy.

In the chart obtained by a left Lorentz transformation of coordinates

1 1 -1 Z’JZ 1 1 ‘2

( i

we have a SD connection

Let {E,H,F} be the standard Cartan-Weyl basis of s((2,C). For the sinh-Gordon equation we have

&=exp( -r)E+z H, Ll=exp(r)F,

Mo=exp( -r)F+? H, Ml = exp( r) E,

where the function r satisfies the sinh-Gordon equation

r,,=2 sinh(2r).

If we now take

Lo=2uE-ilu12H, L,=iH-2u*F,

Mo=-2u*F-ilu12H, M,=iH+2uE

we obtain for the functions u and u the equations of the massive Thirring model

iu,=2ulu~2-2u, iuX= -2uluj2-2u.

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2908 F. Guil and M. Mafias: 2-d integrable systems and self-dual Yang-Mills equation

Conversely, if we let x and t be two functions of z as in Theorem 2.1 for which

A:= a(x,t) a(zoo’,zlo’)

=d00’Xd*0’f-d*O’Xd00’r f 0 (2.6)

and we construct the differential one-form w(X)= U(X)dx+ V(X)dr, where we define Uo, U,, V. according to the formulas

1 uo:=~ (<P,,~d,o~t-~,o~aoo~t),

1 u, : =r (*poo,&J tX-~~o’doo~X), (2.7)

1 V&‘h (~,,~d~o~t-~~*~a,,~r),

then we obtain the inversion of the formulas given in Theorem 2.1 defining the connection Q, in terms of the coefficients Uo, U, , V, of o.

Proposition 2.1: Let o(X) be the one-form given by ~(X)=(klJt+U~)dx +(X2U,+XU0+V,Jdt, where U,,, Ve, Ut are defined in Eq. (2.7) and depend on z through the

junctions x,t of Theorem 2.1. Then the curvature dw -M[w, w]=O if @ is self-dual and satisfies the gauge condition

Proof From the expressions for @AA r in Theorem 2.1 and the relations satisfied by x and t, we get the following expressions for the curvature components:

Foo,lor=A(Uo,r-Vo,x+[Uo, Vol), Fww=AU~,,,

Foo,~l,+Fo~rlo,=A(U~,,-Uo,,-[Vo, u,l). The curvature Q=dw - 1/2[w, 01 becomes

n(A)=dxr\dt{X’u,,,+X(Uo,x-U,,t-[U,, Vol)+vo,,-uo,,-Wo, vol} that vanishes upon the SD equations.

For the principal chiral fields one has Proposition 2.2: Let x and y be as in Theorem 2.2 and dejne

1 1 u:=~(~,,fd,,fy-cp,,~a,,fy), 7x=-r ((Polrd,1,X-Q)1irdo1rX),

where

Suppose that Cp is chosen such that u and v depends on z through the functions x and y defined in Theorem 2.2, then u and v are solutions of the equations of a principal chiral$eld if@ is self-dual and satisjes the gauge condition QA0f =O.

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F. Guil and M. Marias: 2-d integrable systems and self-dual Yang-Mills equation 2909

Proof The choice made for u, u, x, and y together with the gauge condition aAof =O imply the relations

F ,,or,o~=O, F~,~=A(u~-v,+[u, VI), Foo,~~r+Fg~,,o,=A(~~+ux)

and the result follows. 0 All the 2D integrable equations cited above have a common feature, their zero-curvature

formulation has a rational dependence in the spectral parameter X. But there exists 2D integrable systems with an elliptic dependence in the spectral parameter such as the Landau-Lifshitz and the Krichever-Novikov equations. It is an open question whether they are related to the standard SDYM equations or there exists an elliptic deformed version of the SDYM equations.

III. THE YANG MATRIX AND CHIRAL FIELDS

The equation of a principal field (2.2) can be equally written as a pair of conditions

uy--rJx+[u, u]=O, uy+u,=o.

The first of them represents the vanishing of the curvature for the connection u dx + u dy and this allows one to introduce the chiral field s related to its currents u and u by the formulas

u=sys-‘, u=sy.s-‘.

Here s represents a function of x and y with values in the Lie group under consideration. Then, upon substitution in the second equation, we get for s the new condition

(s,.s-‘),+(s,.s-‘>,=o.

The situation we have just described has a precise analog for the self-duality equations previously considered. As follows from Theorem 2.2, one can construct a SD connection in terms of a principal chiral field with the gauge condition

<p *ot=O.

This type of SD connection can be conveniently represented by means of a function J(z), the Yang matrix, with values in the corresponding Lie group.4*19 The SD equations Fool lo’ = F,, r 11 r = 0 in terms of the connection (PA,.,! are

[4)0~-@00~, 4o~-@P,o~l=O, [a,,! -@o,r 9 d11r -@*,r]=O.

This pair of zero-curvature conditions allows us to find functions qot and cplr for which

Q, AO’=d*O’qJor’(pO,l, ~A”=d*“(P’+J;,l. We can write the connection 4) as

Q,=aAA, dzAA’=~AO,po,-(p;,’ dzAor+~A,‘(P~‘-~;,’ dz*”

=d~or.cp,l+Ad ~O,(dAt’J’J-’ dzA1’),

where we have set

J:=cpo”q+.

Thus it appears that every SD connection is gauge equivalent to the one given by the formulas

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2910 F. Guil and M. Maiias: 2-d integrable systems and self-dual Yang-Mills equation

Q, AO,=O, CpAl,=dAltJ.J-’ dz*”

provided J satisfies the equation

which is equivalent to the SD condition FOO~ll I + Fol r1 of =O. Moreover, from the relation con- tained in Theorem 2.2 we obtain

which tells us that for a given chiral field s(x,y), we can take J(z) =s(x(z),y(z)) as a Yang matrix if the coordinate functions x,y are those prescribed by Theorem 2.2. In fact one finds the explicit relation

with

A= 4X,Y)

c?(zO”,z*“) .

IV. REDUCTION BY SYMMETRIES

The conformal group represents the symmetry group for the SDYM equations. Invariant solutions under the action of a subgroup with two generators are desc$bed by a system of equa- tions containing two independent variables instead of the variables zAA appearing in the original equations. Thus, it is possible to describe SD connections possessing two translational symmetries in terms of KdV and NLS equations.5 These connections correspond to a real form R272 of the twistor space T and the choice m(z)=zoo and n(z)= -2” for the two functions in the formula of Theorem 2.1. An analogous result for the principal c,hiral field 14,1,‘8 follows from Theorem 2.2 when we define x(z)=zoo’-zol and y(z)=~‘~ +z” in that case.

In this section we present examples of connections derived from our general construction and not having two independent conformal symmetries.

The group of transformations preserving up to a conformal factor the symmetric bilinear form

g=gAA,BB, dz**’ dzBB’,dz’O’ dZol’-&” dzoo’

coincides with the conformal group defined in Sec. II. If a conformal transformation has as fundamental vector field X= CA*’ XAA aAAl, then the coefficients XAA (z) are rational functions of ZAA’ and are characterized by the condition

where Zx is the Lie derivative operator along X and A represents the infinitesimal conformal factor. In components, the condition above reads

or in a more explicit form

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F. Guil and M. Manas: 2-d integrable systems and self-dual Yang-Mills equation 2911

d”~X”‘+dOOIX o”‘=dO1,Xo”+dlOIX’o’=A.

Suppose that the connection @ has a conformal symmetry generated by X. Then it follows2o that Q, satisfies an equation of the form

L&(Q))=dW+[W, G’]

for some W: T -+ g that under a gauge transformation generated by g transforms according to w t+ X(g)-g-1 +Ad g W. For the curvature F we have the condition

L&F=[W, F]

and for any Ad-invariant bilinear form B in the Lie algebra g we obtain

L&B(F,F)=B([W, F],F)+B(F,[W, F])=O. (4.1)

This is a necessary condition in order for X to generate a conformal symmetry of the connection a.

A simple computation proves that

B(F,F)=.R2,

where n = dz OO’~dzO”~dz’O’~dz”’ E A4( T) is the standard volume form on T, and therefore

ZxB(F,F)=(XF+Fdivo X)0,

where diva X= CA* r aA* 1X** ’ is the standard divergence of a vector field that for a conformal vector field is diva X=2A. Equation (4.1) for the vector field X can now be written as the equivalent condition

XY+.Tdivo X=0. (4.2)

Let us now consider the particular solution to the principal chiral field model given by

Here A and g are constants and elements in g and G, respectively, with the normalization condi- tion B(A,A)=l. For the functions u=s~~s-’ and u=sY.s-’ we obtain u = u =A, and the asso- ciated connection of Theorem 2.2 becomes

@ AO’=O, @A, ,=A~A~,(x+Y). (4.3)

Proposition 4. I: Define

x(z)= -exp(z OO’-zO1’)-exp(z’O’-z”‘), y(z)=exp(zOO’+zO”)+exp(z’o’+zll’).

Then, the symmetry group of conformal transformations for the SD connection (4.3) is one dimen- sional.

Proof For the function .P in Eq. (4.2) we find the expression

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2912 F. Guil and M. Maiias: 2-d integrable systems and self-dual Yang-Mills equation

9=(exp(zoo’-zo” )+exp(z00’+z01’))(exp(z1o’-z11’)+exp(z10’+z11’)).

Upon substitution in Eq. (4.2) we obtain for X the conditions

xA”=o , A=O,l, Xoo’+X1o’=divn X,

which follows from the rational character of the coefficients XAA’(z) of the vector field X. These conditions and the differential equations satisfied by any conformal field X imply that X is pro- portional to d oor -diet. This proves that this SD connection obtained from the principal chiral field model through the map prescribed by Theorem 2.2 has at most a conformal symmetry. El

Analogous considerations show that the same is true for the KdV and NLS type equations. For if we let the zero-curvature one-form be defined as o+(X)= U,(dx+ Xdt) for a constant element U1 in g such that B(lJ,,V,)=l and take

x(z)= - 1/2(z0”(z00’)2+z11’(z’0’)2)+exp(z00’)+exp(z10’),

we obtain as the conformal symmetry group, for the SD connection associated to this solution of a NLS type equation, the one-dimensional subgroup generated by

I I d001+6)‘o~+Zo’ do”+Z” a,,, .

Therefore this SD connection associated to a NLS type equation as in Theorem 2.1 has at most a conformal symmetry.

Finally, in the KdV case, for the constant vectors Uo, U, Eg satisfying B(U,,U,)=B(U,,U,)=O and B([Uo, V,],[V,, U,])=l, we define the zero-curvature one- form

w+(A)=(xu’+uo)(dx+x dt)

and the functions

x(z)=-zol’ exp(zoo’)-z”’ exp(z10’)+exp(-z00’)+exp(3z10’).

The symmetry group of conformal transformations for the associated SD connection is generated . . m this case by dOl~--dllr.

ACKNOWLEDGMENT

This work was partially supported by CICYT proyecto PB89-0133.

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