Generalized sine-Gordon/massive Thirring models and soliton/particle correspondences

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correspondences

José Acosta and Harold Blas

Citation: Journal of Mathematical Physics 43, 1916 (2002); doi: 10.1063/1.1454186 View online: http://dx.doi.org/10.1063/1.1454186

View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/43/4?ver=pdfcov Published by the AIP Publishing

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Generalized sine-Gordon

Õ

massive Thirring models

and soliton

Õ

particle correspondences

Jose´ Acosta and Harold Blasa)

Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900-Sa˜o Paulo, S.P., Brazil

共Received 25 September 2001; accepted for publication 8 January 2002兲

We consider a real Lagrangian off-critical submodel describing the soliton sector of the so-called conformal affine sl(3)(1) Toda model coupled to matter fields. The theory is treated as a constrained system in the context of Faddeev–Jackiw and the symplectic schemes. We exhibit the parent Lagrangian nature of the model from which generalizations of the sine-Gordon 共GSG兲 or the massive Thirring 共GMT兲 models are derivable. The dual description of the model is further emphasized by providing the relationships between bilinears of GMT spinors and relevant expres-sions of the GSG fields. In this way we exhibit the strong/weak coupling phases and the共generalized兲 soliton/particle correspondences of the model. The sl(n)(1) case is also outlined. © 2002 American Institute of Physics.

关DOI: 10.1063/1.1454186兴

I. INTRODUCTION

Integrable theories in two dimensions have been an extraordinary laboratory for the under-standing of basic nonperturbative aspects of physical theories and various aspects, relevant in more realistic four-dimensional models, have been tested.1In particular the conformal affine Toda models coupled to共Dirac兲 matter fields 共CATM兲2for the sl(2)(1)and sl(3)(1)cases are discussed in Refs. 3, 4, and 5, respectively. The interest in such models comes from their integrability and duality properties,2,4which can be used as toy models to understand some phenomena; such as a confinement mechanism in quantum chromodynamics共QCD兲3,5and the electric–magnetic duality in four-dimensional gauge theories, conjectured in Ref. 6 and developed in Ref. 7. The affine Toda model coupled to matter field 共ATM兲 type systems may also describe some low dimensional condensed matter phenomena, such as self-trapping of electrons into solitons, see, e.g., Ref. 8, tunneling in the integer quantum Hall effect,9and, in particular, polyacteline molecule systems in connection with fermion number fractionization.10

Off-critical submodels, such as the sl(2) ATM, can be obtained at the classical or quantum mechanical level through some convenient reduction processes starting from CATM.4,11 In the

sl(2) case, using bosonization techniques, it has been shown that the classical equivalence

be-tween the U(1) vector and topological currents holds true at the quantum level, and then leads to a bag model like mechanism for the confinement of the spinor fields inside the solitons; in addition, it has been shown that the sl(2) ATM theory decouples into a sine-Gordon model共SG兲 and a free scalar.3,12 These facts indicate the existence of a sort of duality in these models involving solitons and particles.6The symplectic structure of the sl(2) ATM model has recently been studied11in the context of Faddeev–Jackiw共FJ兲13and共constrained兲 symplectic methods.14,15 Imposing the equivalence between the U(1) vector and topological currents as a constraint there have been obtained the SG or the massive Thirring共MT兲 model.

One of the difficulties with generalizations of complex affine Toda field theories, beyond

su(2) and its associated SG model, has to do with unitarity. Whereas for practical applications

a兲_{Electronic mail: blas@ift.unesp.br}

1916

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such as low dimensional condensed matter systems 共see Ref. 16 and references therein兲 and

N-body problems in nuclear physics,17 the properties of interest are usually integrability and nonperturbative results of multifield Lagrangians. Therefore, integrable quantum field theories with several fields 共bosons and/or fermions兲 are of some importance.

In this paper we construct many field generalizations of SG/MT models based on soliton/ particle duality and unitarity. Beyond the well-known sl(2) case the related sl(n)(1)CATM model does not possess a local Lagrangian, therefore we resort to an off-critical submodel Lagrangian with well behaved classical solutions making use of the results of Ref. 5. In Ref. 5 the authors studied the sl(3)(1) CATM soliton solutions and some of their properties up to general two-soliton. Using the FJ and symplectic methods we show the parent Lagrangian18nature of the sl(3) ATM model from which the generalized sine-Gordon 共GSG兲 or the massive Thirring 共GMT兲 models are derivable. We thus show that there are共at least classically兲 two equivalent descriptions of the model, by means of either the Dirac or the Toda type fields. The duality exchange of the coupling regimes g→1/g and the generalized soliton/particle correspondences in each sl(2) ATM submodel will also be clear, which we uncover by providing explicit relationships between the GSG and GMT fields. We also outline the steps toward the sl(n) affine Lie algebra generaliza-tions. In this way we give a precise field content of both sectors; namely, the correct GMT/GSG duality, first undertaken in Ref. 19.

The paper is organized as follows. In Sec. II we define the sl(3) ATM model. Section III deals with the model in the FJ framework, the outcome is the GMT model. In Sec. IV, we attack the same problem from the point of view of symplectic quantization14,15giving the Poisson brackets of the GMT and GSG models. Section V deals with the soliton/particle and strong/weak coupling correspondences. Section VI outlines the relevant steps toward the generalization to sl(n) ATM. In the appendix we present the construction of sl(3)(1) CATM model and its relationship to the

共two-loop兲 Wess–Zumino–Novikov–Witten 共WZNW兲 model.

II. DESCRIPTION OF THE MODEL

In affine Toda type theories the question of whether all mathematical solutions are physically acceptable deserves a careful analysis, especially if any consistent quantization of the models is discussed. The requirement of real energy density leads to certain reality conditions on the solu-tions of the model. In general, a few soliton solusolu-tions survive the reality constraint, if in addition one also demands positivity. These kind of issues are discussed in Refs. 20. Here we follow the prescription to restrict the model to a subspace of classical solutions which satisfy the physical principles of reality of energy density and soliton/particle correspondence.

In CATM models associated with the principal gradation of an affine Lie algebra we have a one-soliton solution共real Toda field兲 for each pair of Dirac fields␺i _and_␺_˜i_.2_{This fact allows us} to make the identifications ␺˜i⬃(␺i)*, and take real Toda fields. In the case of sl(2)(1) CATM theory, this procedure does not spoil the particle–soliton correspondence.3,4

We consider the sl(3)(1) CATM theory 共see the Appendix兲 with the conformal symmetry gauge fixed21by setting␩⫽0 and the reality conditions

␺ ˜j_⫽⫺共_␺j_兲_*_, _{共 j⫽1,2,3兲,} _␸ a *_⫽_␸_a _{共a⫽1,2兲,} _共2.1兲 or ␺ ˜j_⫽共_␺j_兲_*_, _j_⫽1,2, _␺_˜3_⫽⫺共_␺3_兲_*_,

␸1,2→␸1,2⫺␲ 共the new ␸a’s being real fields兲, 共2.2兲

where an asterisk means complex conjugation. The condition 共2.2兲 must be supplied with x␮

→⫺x␮_{. Moreover, for consistency of the equations of motion} _{共A15兲–共A23兲 under the reality}

conditions 共2.1兲 and 共2.2兲, from Eqs. 共A16兲 to 共A18兲, 共A20兲, 共A21兲, and 共A23兲, we get the relationships

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␺ ˜ L j_␺ R 3_⫺_␺_˜ R j_␺ L 3

e⫺3i␸j⫽0, j⫽1,2, ␺L1␺R2_e⫺3i␸1⫺␺L2␺R1_e⫺3i␸2⫽0. 共2.3兲

Then, the above-given reality conditions and constraints allow us to define a suitable physical Lagrangian. Equations共A13兲, 共A15兲–共A23兲, supplied with 共2.1兲关or 共2.2兲兴 and 共2.3兲, follow from the Lagrangian 1 kL⫽

兺

j⫽1 3

冋

1 24⳵␮␾j⳵ ␮_␾ j⫹i␺¯j␥␮⳵␮␺j⫺m_␺j␺¯jei␾j␥5␺j

册

, 共2.4兲 where␺¯j⬅(␺j)†␥₀, ␾₁⬅2␸₁⫺␸₂, ␾₂⬅2␸₂⫺␸₁, ␾₃⬅␾₁⫹␾₂, m_␺3⫽m_␺1⫹m_␺2, k is an over-all coupling constant and the␸j are real fields.

Equation共2.4兲 defines the sl(3) affine Toda theory coupled to matter fields 共ATM兲. Notice that the space of solutions of sl(3)(1) CATM model satisfying conditions 共2.1兲–共2.3兲 must be solutions of the sl(3) ATM theory 共2.4兲. Indeed, it is easy to verify that the three species of one-soliton solutions 关S⬅1-soliton(S¯⬅1-antisoliton)兴:5 兵(␸1,␺1)S/S¯,␸2⫽0,␺2⫽0,␺3⫽0其, 兵(␸2,␺2)S/S¯,␸1⫽0,␺1⫽0,␺3⫽0其, and 兵(␸1⫹␸2,␺3)S/S¯,␸1⫽␸2,␺1⫽0,␺2⫽0其 satisfy the equations of motion, i.e., each positive root of sl(3) reproduces the sl(2) ATM case.3,4Moreover, these solutions satisfy the above-given reality conditions and constriants 共2.1兲–共2.3兲关with 共2.1兲 and共2.2兲 for S and S¯, respectively兴, and the equivalence between the U(1) vector and topological currents共A29兲. Then, the soliton/particle correspondences survive the above-given reduction pro-cesses performed to define the sl(3) ATM theory.

The class of two-soliton solutions of sl(3)(1)_CATM5_{behave as follows:}_{共i兲 they are given by} six species associated with the pair (␣i,␣_j), i⭐ j; i, j⫽1,2,3; where the␣’s are the positive roots of sl(3) Lie algebra. Each species (␣i,␣_i) solves the sl(2) CATM submodel;22 共ii兲 satisfy the

U(1) vector and topological currents equivalence共A29兲.

III. THE GENERALIZED MASSIVE THIRRING MODEL„GMT… Let us consider the following Lagrangian:

1 kL⫽j

兺

⫽1 3

冋

1 24⳵␮␾j⳵ ␮_␾ j⫹i␺¯j␥␮⳵␮␺j⫺m␺ j_␺_¯j_ei␾_j␥₅_␺j_⫹␭ ␮j共mj␺¯j␥␮␺j⫺⑀␮␯⳵␯共qj␾j兲兲

册

, 共3.1兲

where the ATM Lagrangian 共2.4兲 is supplied with the constraints, (ml␺¯l␥␮␺l⫹ (m3/2)␺¯3␥␮␺3

⫺⑀␮␯_⳵

␯␾l), (l⫽1,2), with the help of the Lagrange multipliers ␭_␮ j _共␭ ␮ 3_{⬅ (␭} ␮ 1_⫹␭ ␮ 2 )/2, q1⬅q2

⬅1, q3⬅0兲. Their total sum bears an intriguing resemblance to the U(1) vector and topological currents equivalence 共A29兲; however, the mj’s here are some arbitrary parameters. The same procedure has been used, for example, to incorporate the left-moving condition in the study of chiral bosons in two dimensions.23The constraints in共3.1兲 will break the left–right local symme-tries共A25兲–共A28兲 of sl(3) ATM 共2.4兲. In order to apply the FJ method we should write 共3.1兲 in the first-order form in time derivative, so let us define the conjugated momenta

␲1⬅␲␾₁⫽ 1 12共2␾˙1⫹␾˙2兲⫹␭1 1 , ␲2⬅␲␾₂⫽ 1 12共2␾˙2⫹␾˙1兲⫹␭1 2 , 共3.2兲 ␲␭_␮1⫽0, ␲_␭ ␮ 2⫽0, ␲Rj⬅␲_␺ R j _⫽⫺i ␺ ˜ R j , ␲Lj⬅␲_␺ L j _⫽⫺i ␺ ˜ L j .

We are assuming that Dirac fields are anticommuting Grasmannian variables and their mo-menta variables defined through left derivatives. Then, as usual, the Hamiltonian is defined by

共sum over repeated indices is assumed兲

Hc⫽␲1␾˙1⫹␲2␾˙2⫹␺˙R j ␲R j_⫹_␺_˙ L j ␲Lj_⫺L. _共3.3兲

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Explicitly the Hamiltonian density becomes Hc⫽2共␲j兲2⫹4共␭1 1_兲2_⫹4共␭ 1 2_兲2_⫺␭ 1 1_J 1⫺␭1 2_J 2⫺4共␭1 1_␭ 1 2_兲⫹ 1 24共␾j,x兲2⫺␲R j ␺R,x j _⫹ ␲L j ␺L,x j ⫹im_␺j_共e⫺␾_j_␺_˜ R j_␺Lj_⫺e␾_j_␺_˜ L j_␺Rj_兲⫹␭ 0 1_关J 1 0_⫺_␾ 1,x兴⫹␭0 2_关J 2 0_⫺_␾ 2,x兴, 共3.4兲 where␲3⬅␲1⫺␲2, J1⬅J1 1_⫹4(2 ␲1⫺␲2), J2⬅J2 1_⫹4(2 ␲2⫺␲1), and J₁␮⫽ m1j_l␮⫹m 3 2 j3 ␮_, _J 2 ␮_⫽m2_j 2 ␮_⫹m3 2 j3 ␮_, _j l ␮_⬅_␺¯l_␥␮_␺l_, _l_⫽1,2,3. _共3.5兲

Let us observe that each U(1) Noether current of the sl(3) ATM theory defined in共2.4兲 is conserved separately, i.e.,⳵_␮j_l␮⫽0, l⫽1,2,3.

Next, the same Legendre transform共3.3兲 is used to write the first-order Lagrangian

L⫽␲1␾˙1⫹␲2␾˙2⫹␺˙R j_␲Rj_⫹_␺_˙

L j_␲Lj_⫺H

c. 共3.6兲

Our starting point for the FJ analysis will be this first-order Lagrangian. Then the Euler– Lagrange equations for the Lagrange multipliers allow one to solve two of them

␭1

1_⫽2J1⫹J2 12 , ␭1

2_⫽2J2⫹J1

12 共3.7兲

and the remaining equations lead to two constraints,

⍀1⬅J1 0_⫺

␾1,x⫽0, ⍀2⬅J2 0_⫺

␾2,x⫽0. 共3.8兲

The Lagrange multipliers␭₁1 and␭₁2 must be replaced back in共3.6兲 and the constraints 共3.8兲 solved. First, let us replace the␭₁1 and␭₁2 multipliers intoHc, then one gets

Hc

⬘

⫽2共␲j兲2⫺ 1 12兵共J1兲2⫹共J2兲2⫹共J1J2兲其⫹ 1 24共␾j,x兲 2_⫹i_␺_˜ R j_␺R,xj _⫺i_␺_˜ L j_␺L,xj ⫹im_␺j 共e⫺i␾j␺˜ R j_␺ L j_⫺ei␾_j_␺_˜ L j_␺Rj_兲. _共3.9兲

The new Lagrangian becomes

L

⬘

⫽␲1␾˙1⫹␲2␾˙2⫹␺˙R j ␲Rj_⫹_␺_˙ L j ␲L j_⫺H c

⬘

. 共3.10兲

We implement the constraints共3.8兲 by replacing in 共3.10兲 the fields␾1, ␾2 in terms of the space integral of the current components J₁0, J₂0. Then we get the Lagrangian

L

⬙

⫽␲1⳵t

冕

x

J₁0⫹␲2⳵t

冕

x

J₂0⫹␺˙_Rj␲Rj⫹␺˙_Lj␲Lj⫺i˜␺_Rj␺R,xj ⫹i␺˜_Lj␺L,xj ⫺im_␺j共e⫺i兰xJj

0 ␺ ˜ R j_␺ L j ⫺ei兰x_J j 0 ␺ ˜ L j_␺Rj_兲⫺1 12共共J1兲2⫹共J2兲2⫹J1.J2兲⫹␲1J1 1_⫹_␲ 2J2 1_, _共3.11兲

where J₃0⬅J₁0⫹J₂0. Observe that the terms containig the␲a’s in Eq.共3.11兲 cancel to each other if one uses the current conservation laws. Notice the appearances of various types of current–current interactions. The following Darboux transformation:

␺Rj_→exp

冉

_⫺i 2

冕

x J_j0

冊

␺Rj , ␺Lj→exp

冉

i 2

冕

x J0_j

冊

␺Lj, j⫽1,2,3 共3.12兲

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is used to diagonalize the canonical one-form. Then, the kinetic terms will give additional current– current interactions, ⫺1

2关J1•( j1⫹ j3)⫹J2•( j2⫹ j3)兴. We are, thus, after defining k⬅1/g, and rescaling the fields␺j→1/

冑

This defines the generalized massive Thirring model 共GMT兲. The canonical pairs are (⫺i␺˜_Rj ,␺Rj) and (⫺i␺˜_Lj,␺Lj).

IV. THE SYMPLECTIC FORMALISM AND THE ATM MODEL A. The„constrained…symplectic formalism

We give a brief overview of the basic notations of symplectic approach.24 The geometric structure is defined by the closed共pre兲symplectic two-form

f(0)⫽12fi j (0)_共 ␰(0)_兲d_␰(0)i_∧_d_␰(0) j_, _共4.1兲 where f_{i j}(0)共␰(0)兲⫽ ⳵ ⳵␰(0)iaj (0)_共_␰(0)_兲⫺ ⳵ ⳵␰(0) jai (0)_共_␰(0)_兲 _共4.2兲

with a(0)(␰(0))⫽a(0)_j (␰(0))d␰(0) j being the canonical one-form defined from the original first-order Lagrangian

L(0)dt⫽a(0)共␰(0)兲⫺V(0)共␰(0)兲dt. 共4.3兲

The superscript共0兲 refers to the original Lagrangian, and is indicative of the iterative nature of the computations. The constraints are imposed through Lagrange multipliers which are velocities, and in such case one has to extend the configuration space.14,15The corresponding Lagrangian gets modified and consequently the superscript also changes. The algorithm terminates once the sym-plectic matrix turns out to be nonsingular.

B. The generalized massive Thirring model„GMT…

Next, we will consider our model in the framework of the symplectic formalism. LetL

⬘

, Eq.

共3.10兲, be the zeroth-iterated Lagrangian L(0)_{. Then the first iterated Lagrangian will be}

L(1)_⫽_␲ 1␾˙1⫹␲2␾˙2⫹␺˙R j ␲Rj_⫹_␺_˙ L j ␲Lj_⫹_␩ ˙1⍀1⫹␩˙2⍀2⫺V(1), 共4.4兲

where the once-iterated symplectic potential is defined by

V(1)_⫽H c

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and the stability conditions of the symplectic constraints,⍀1 and⍀2, under time evolution have been implemented by making␭₀1→␩˙1 _and_␭

0

2_→_␩_˙2_{. Consider the once-iterated set of symplectic} variables in the following order

␰(1)_⫽共_␩1_,_␩2_,_␾ 1,␾2,␺R 1 ,␺L1,␺_R2,␺L2,␺_R3,␺L3,␲1,␲2,␲R 1 ,␲L1,␲R2,␲L2,␲R3,␲_L3兲, 共4.6兲

and the components of the canonical one-form

a(1)⫽共⍀1,⍀2,␲1,␲2,⫺␲R 1

,⫺␲L1,⫺␲R2,⫺␲L2,⫺␲R3,⫺␲L3,0,0,0,0,0,0,0,0兲. 共4.7兲

These result in the singular symplectic two-form 18⫻18 matrix

f_AB(1)共x,y兲⫽

冉

a₁₁ a₁₂

a₂₁ a₂₂

冊

␦共x⫺y兲, 共4.8兲

where the 9⫻9 martices are

˜␺L 3

冊

, 共4.9兲

where u and␷are arbitrary functions. The zero-mode condition gives

冕

dxv(1)T共x兲 ␦

␦␰(1)_共x兲

冕

dyV

(1)_⬅0. _共4.10兲

Thus, the gradient of the symplectic potential happens to be orthogonal to the zero-mode v(1). Since the equations of motion are automatically validated no symplectic constraints appear. This happens due to the presence of the symmetries of the action

␦␰A (1)_⫽v

A

(1)_{共x兲, A⫽1,2, . . . ,18.} _共4.11兲

So, in order to deform the symplectic matrix into an invertible one, we have to add some gauge fixing terms to the symplectic potential. One can choose any consistent set of gauge fixing conditions.15In our case we have two symmetry generators associated with the parameters u and

v, so there must be two gauge conditions. Let us choose

⍀3⬅␾1⫽0, ⍀4⬅␾2⫽0. 共4.12兲

These conditions gauge away the fields␾1 and␾2, so only the remaining field variables will describe the dynamics of the system. Other gauge conditions, which eventually gauge away the spinor fields␺i, will be considered in Sec. IV C.

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Implementing the consistency conditions by means of Lagrange multipliers␩3and␩4 we get the twice-iterated Lagrangian

L(2)_⫽_␲ 1␾˙1⫹␲2␾˙2⫹␺˙R␲R j_⫹_␺_˙_L␲Lj_⫹_␩ ˙1⍀1⫹␩˙2⍀2⫹␩˙3⍀3⫹␩˙4⍀4⫺V(2), 共4.13兲 where V(2)_⫽V(1)_兩 ⍀3⫽⍀4⫽0.

Assuming now that the new set of symplectic variables is given in the following order,

␰(2)_⫽共_␩1_,_␩2_,_␩3_,_␩4_,_␾ 1,␾2,␺R1,␺L 1_,_␺R2_,_␺ L 2_,_␺R3_,_␺ L 3_,_␲ 1,␲2,␲R1,␲L1,␲R2,␲L2,␲R3,␲L3兲, 共4.14兲

and the nonvanishing components of the canonical one-form

a(2)_⫽共⍀ 1,⍀2,⍀3,⍀4,␲1,␲2,⫺␲R 1_,_⫺_␲ L 1_,_⫺_␲R2_,_⫺_␲ L 2_,_⫺_␲R3_,_⫺_␲L3_{,0,0,0,0,0,0,0,0}_兲, 共4.15兲

one obtains the singular twice-iterated symplectic 20⫻20 matrix

冉

a11 a12

a21 a22

冊

␦

共x⫺y兲, 共4.16兲

where the 10⫻10 matrices are

a11⫽

冢

0 0 0 0 ⳵x 0 im1_␺_˜ R 1 _im1_␺_˜ L 1 ₀ ₀ 0 0 0 0 0 ⳵x 0 0 im2_␺_˜ R 2 _im2_␺_˜ L 2 0 0 0 0 ⫺1 0 0 0 0 0 0 0 0 0 0 ⫺1 0 0 0 0 ⳵x 0 1 0 0 0 0 0 0 0 0 ⳵x 0 1 0 0 0 0 0 0 im1␺˜_R1 0 0 0 0 0 0 0 0 0 im1␺˜_L1 0 0 0 0 0 0 0 0 0 0 im2␺˜_R2 0 0 0 0 0 0 0 0 0 im2␺˜_L2 0 0 0 0 0 0 0 0

冣

,

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a12⫽

¨

im3 2 ˜␺R 3 im 3 2 ˜␺L 3 ₀ ₀ _m1_␺ R 1 m1␺L1 0 0 m 3 2 ␺R 3 m 3 2 ␺L 3 im3 2 ˜␺R 3 im 3 2 ˜␺L 3 ₀ ₀ ₀ ₀ _m2_␺R2 _m2_␺L2 m 3 2 ␺R 3 m 3 2 ␺L 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⫺1 0 0 0 0 0 0 0 0 0 0 ⫺1 0 0 0 0 0 0 0 0 0 0 ⫺1 0 0 0 0 0 0 0 0 0 0 ⫺1 0 0 0 0 0 0 0 0 0 0 ⫺1 0 0 0 0 0 0 0 0 0 0 ⫺1 0 0

©

, a21⫽

¨

im3 2 ˜␺R 3 im 3 2 ˜␺R 3 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ im3 2 ˜␺L 3 im 3 2 ˜␺L 3 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 m1␺_R1 0 0 0 0 0 ⫺1 0 0 0 m1␺L1 0 0 0 0 0 0 ⫺1 0 0 0 m2␺R2 0 0 0 0 0 0 ⫺1 0 0 m2_␺ L 2 ₀ ₀ ₀ ₀ ₀ ₀ ₀ _⫺1 m3 2 ␺R 3 m 3 2 ␺R 3 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀ m3 2 ␺L 3 m 3 2 ␺L 3 ₀ ₀ ₀ ₀ ₀ ₀ ₀ ₀

©

, a22⫽

冉

0 0 0 ¯ 0 ⫺1 0 0 0 0 ¯ 0 0 ⫺1 0 0 0 ¯ 0 0 0 ... 0 0 0 ¯ 0 0 0 ⫺1 0 0 ¯ 0 0 0 0 ⫺1 0 ¯ 0 0 0

冊

.

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v(2)T共x兲⫽

冉

u, ␷, ␻, ␹, 0, 0, m1u␺1_R, m1u␺L1, m2␷␺R2, m2␷␺L2, m 3 2 共u⫹␷兲␺R 3_, m3 2 共u⫹␷兲␺L 3_{, u}

_⬘

_⫹_␻_, _␷

_⬘

_⫹_␹_{, im}1_␺_˜ R 1_{u, im}1_␺_˜ L 1_{u, im}2_␺_˜ R 2_␷_,im2_␺_˜ L 2_␷_, im3 2 ˜␺R 3_共u⫹_␷_{兲, i}m 3 2 ˜␺L 3_共u⫹_␷_兲

冊

. 共4.17兲

The zero-mode condition gives no constraints, implying the symmetries of the action ␦␰A

(2)_⫽v

A

(2)_{共x兲, A⫽1,2, . . . ,20.} _共4.18兲

Now, let us choose the gauge conditions

⍀5⬅␲1J1 1_⫹1

2J1•共 j1⫹ j3兲⫽0, ⍀6⬅␲2J2 1_⫹ 1

2J2•共 j2⫹ j3兲⫽0, 共4.19兲

and impose the consistency conditions with the Lagrange multipliers ␩5,␩6, then

L(3)_⫽_␲ 1␾˙1⫹␲2␾˙2⫹␺˙R␲R j_⫹_␺_˙_L␲Lj_⫹_␩_˙1_⍀ 1⫹␩˙2⍀2⫹␩˙3⍀3⫹␩˙4⍀4⫹␩˙5⍀5⫹␩˙6⍀6⫺V(3), 共4.20兲 where V(3)_⫽V(2)_兩 ⍀5⫽⍀6⫽0, 共4.21兲 or explicitly V(3)_⫽ 1 12共共J1兲2⫹共J2兲2⫹J1•J2兲⫹ 1 2关J1•共 j1⫹ j3兲⫹J2•共 j2⫹ j3兲兴⫹i␺˜R j_␺R,xj _⫺i_␺_˜ L j_␺L,xj _⫹im ␺ j_␺_¯j_␺j_. 共4.22兲

The symplectic two-form for this Lagrangian is a nonsingular matrix, then our algorithm has come to an end. Collecting the canonical part and the symplectic potentialV(3) one has

One can choose other gauge fixings, instead of共4.12兲, to construct the twice-iterated Lagrang-ian. Let us make the choice

⍀3⬅J1

0_{⫽0, ⍀}

4⬅J2

0_⫽0, _共4.24兲

which satisfies the nongauge invariance condition as can be verified by computing the brackets 兵⍀a,Jb

0

其⫽0; a,b⫽1,2. The twice-iterated Lagrangian is obtained by bringing back these

con-straints into the canonical part ofL(1), then

L(2)_⫽_␲

1␾˙1⫹␲2␾˙2⫹␺˙R␲R

j_⫹_␺_˙_L␲Lj_⫹_␩

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where the twice-iterated symplectic potential becomes

V(2)_⫽V(1)_兩

⍀3⫽⍀4⫽0. 共4.26兲

Considering the set of symplectic variables in the following order: ␰A(2) ⫽共␩1_,_␩2_,_␩3_,_␩4_,_␾ 1,␾2,␺R 1 ,␺L 1 ,␺R2,␺L 2 ,␺R3,␺L 3 ,␲1,␲2,␲R 1 ,␲L 1 ,␲R2,␲L2,␲R3,␲L3兲共4.27兲

and the components of the canonical one-form

a_A(2)⫽共⍀1,⍀2,⍀3,⍀4,␲1,␲2,⫺␲R1,⫺␲L 1_,_⫺_␲R2_,_⫺_␲ L 2_,_⫺_␲R3_,_⫺_␲ L 3_{,0,0,0,0,0,0,0,0}_兲, 共4.28兲

, im1␺˜_R1共u⫹␻兲, im1␺˜_L1共u⫹␻兲, im2˜␺_R2共␷⫹␹兲, im2␺˜_L2共␷⫹␹兲, im 3 2 ˜␺R 3_共u⫹_␷_⫹_␻_⫹_␹_兲, im3 2 ˜␺L 3_共u⫹_␷_⫹_␻_⫹_␹_兲

冊

, 共4.30兲

where u, ␷,␻, and␹are arbitrary functions. The zero-mode condition becomes

冕

dx v(2)T共x兲 ␦

␦␰(2)

冕

dy

⬘

V(2)⫽

冕

dx Ja 1

⳵xfa⬅0, fa⬅共␻,␹兲, a⫽1,2. Since the functions fa are arbitrary we end up with the following constraints:

⍀5⬅J1

1_{⫽0, ⍀}

6⬅J2

1_⫽0. _共4.31兲

Notice that by solving the constraints,⍀3⫽⍀4⫽⍀5⫽⍀6⫽0, Eqs. 共4.24兲 and 共4.31兲, we may obtain

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␺ ˜ R j_⫽_␺ R j , ␺˜_Lj⫽␺Lj. 共4.32兲

So, at this stage, we have Majorana spinors, the scalars␾1 and␾2, and the auxiliary fields. Next, introduce a third set of Lagrange multipliers intoL(2), then

L(3)_⫽_␲ 1␾˙1⫹␲2␾˙2⫹␺˙R␲Rj⫹␺˙L␲Lj⫹␩˙1⍀1⫹␩˙2⍀2⫹␩˙3⍀3⫹␩˙4⍀4⫹␩˙5⍀5⫹␩˙6⍀6⫺V(3), 共4.33兲 where V(3)_⫽V(2)_兩 ⍀5⫽⍀6⫽0 共4.34兲 or V(3)_⫽ 1 24␾j,x 2 _⫹i_␺ R j_␺R,xj _⫺i_␺Lj_␺L,xj _⫹im ␺ j_␺Rj_␺ L

, 共4.37兲

where a₁⫹⬅u⫹␻⫹y, a₂⫹⬅␷⫹␹⫹z, a₃⫹⬅u⫹␻⫹y⫹␷⫹␹⫹z, a₁⫺⬅u⫹␻⫺y, a₂⫺⬅␷⫹␹⫺z,

a₃⫹⬅u⫹␻⫹y⫹␷⫺␹⫺z, and u,␷,␻,␹, y , and z are arbitrary functions. The relevant zero-mode condition gives no constraints. Then the action has the following symmetries:

␦␰A(3)_⫽v

A

(3)_{共x兲, A⫽1,2, . . ., 22.} _共4.38兲

These symmetries allow us to fix the bilinears i␺Rj␺_Lj to be constants. By taking ␺_Rj

⫽⫺iCj␪¯_j _and_␺

L j_⫽_␪

j ( j⫽1,2,3) with Cj being real numbers, we find that i␺R j_␺Lj

indeed be-comes a constant. Note that ␪j and¯␪j are Grassmannian variables, while¯␪j␪j is an ordinary commuting number.

The two form f_AB(3)(x,y ), Eq. 共4.36兲, in the subspace (␾1,␾2,␲␾₁,␲␾₂) defines a canonical symplectic structure modulo canonical transformations. The coordinates␾aand␲_␾

a (a⫽1,2) are not unique. Consider a canonical transformation from (␾a,␲␾_a) to (␾ˆa,␲ˆ␾ˆ_a) such that ␾a

⫽⳵F/⳵␲_␾

a and␾

ˆ_a_⫽_⳵_F/_⳵␲_ˆ_␾ˆ

a. Then, in particular if␾a⫽␾

ˆ_a _{one can, in principle, solve for the}

function F such that a manifestly covariant kinetic term appears in the new Lagrangian. Then choosing k⫽1/g as the overall coupling constant, we are left with

L

⬙

⫽

兺

j_⫽1 3

冋

1 24g⳵␮␾j⳵ ␮_␾ j⫹ Mj g cos␾j

册

⫹␮1⳵x␾1⫹␮2⳵x␾2, 共4.39兲

where M_j⫽m_␺jC_j. This defines the generalized sine-Gordon model共GSG兲. In addition we get the terms multiplied by chemical potentials␮₁and␮₂(␩˙1, 2→⫺␮_{1, 2}). These are just the topological charge densities, and are related to the conservation of the number of kinks minus antikinks

Q_topola ⫽1/␲兰_⫺⬁⫹⬁dx⳵x␾a.

In the above-mentioned gauge fixing procedures the possibility of Gribov-type ambiguities deserves a careful analysis. See Ref. 11 for a discussion in the sl(2) ATM case. However, in Sec. V, we provide indirect evidence of the absence of such ambiguities, at least for the soliton sector of the model.

V. THE SOLITONÕPARTICLE CORRESPONDENCES

The sl(2) ATM theory contains the sine-Gordon共SG兲 and the massive Thirring 共MT兲 models describing the soliton/particle correspondence of its spectrum.3,11,12 The ATM one-共anti兲soliton solution satisfies the remarkable SG and MT classical correspondence in which, apart from the Noether and topological currents equivalence, MT spinor bilinears are related to the exponential of the SG field.25The last relationship was exploited in Ref. 4 to decouple the sl(2) ATM equations of motion into the SG and MT ones. Here we provide a generalization of that correspondence to the sl(3) ATM case. In fact, consider the relationships

␺R1_␺_˜ L 1 i ⫽⫺ 1 4⌬ 关共m␺ 1_p 1⫺m␺ 3_p 4⫺m␺ 2_p 5兲ei(␸2⫺2␸1)⫹m␺ 2_p 5e3i(␸2⫺␸1)⫹m␺ 3_p 4e⫺3i␸1⫺m␺ 1_p 1兴, 共5.1兲

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␺R2_␺_˜ L 2_⫽⫺ 1

4⌬ 关共m␺ 2_p

2⫺m_␺1p5⫺m_␺3p6兲ei(␸1⫺2␸2)⫹m_␺1p5e3i(␸1⫺␸2)⫺m_␺3p6e⫺3i␸2⫺m_␺2p2兴,

共5.2兲 ␺ ˜ R 3_␺ L 3 i ⫽⫺ 1 4⌬ 关共m␺ 3 p3⫺m_␺ 1 p4⫹m_␺ 2 p6兲ei(␸1⫹␸2)⫹m_␺ 1 p4e3i␸1⫺m_␺ 2 p6e3i␸2⫺m_␺ 3 p3兴, 共5.3兲

where ⌬⬅a¯11¯a22¯a33⫹2a¯12¯a23¯a13⫺a¯11(a¯23)2⫺(a¯12)2¯a33⫺(a¯13)2a¯22; p1⬅(a¯23)2⫺a¯22¯a33; p2

⬅(a¯13)2⫺a¯11¯a33; p3⬅(a¯12)2⫺a¯11¯a22; p4⬅a¯12¯a23⫺a¯22¯a13; p5⬅a¯13¯a23⫺a¯12¯a33; p6⬅a¯11¯a23

⫺a¯12¯a13 and the a¯i j’s being the current–current coupling constants of the GMT model 共3.13兲. Relationships 共5.1兲–共5.3兲 supplied with the conditions 共2.1兲–共2.3兲 and conveniently substituted into Eqs.共A13兲 and 共A15兲–共A23兲 decouple the sl(3)(1)CATM equations into the GSG共4.39兲 and

GMT共3.13兲 equations of motion, respectively.

Moreover, one can show that the GSG共4.39兲 Mj parameters and the GMT共3.13兲 couplings

a

¯i j are related by

2⌬M1

g共m_␺1兲2⫽a¯22

冉

⫺

m_␺3

m_␺1¯a13⫹a¯33

冊

⫹a¯23

冉

⫺a¯23⫹ m_␺3

m_␺1¯a12

冊

, 共5.4兲

2⌬M2

g共m_␺2兲2⫽a¯11

冉

⫺ m_␺3

m_␺2¯a23⫹a¯33

冊

⫹a¯13

冉

⫺a¯13⫹ m_␺3

m_␺2¯a12

冊

, 共5.5兲

2⌬M3

g共m_␺3兲2⫽⫺ m_␺1m_␺2

共m␺3兲2共a¯12¯a33⫺a¯13¯a23兲⫺a¯11¯a22⫹共a¯12兲

2_. _共5.6兲

Various limiting cases of the relationships共5.1兲–共5.3兲 and 共5.4兲–共5.6兲 are possible. First, let us consider

a ¯jk→

再

⬁ j⫽k⫽l 共for a given l兲

finite other cases 共5.7兲

then one has

␺Rl_␺_˜ L l i ⫽ m_␺l 4 共e ⫺i␾l⫺1兲, ␺Rj␺˜ L j_{⫽0, j⫽l} _共5.8兲

for a¯ll⫽␦lg (␦1,2⫽1,␦3⫽⫺1). The three species of one-soliton solutions of the sl(3) ATM theory 共2.4兲, found in Ref. 5 and described in Sec. II, satisfy the relationship 共5.8兲.4 Moreover, from Eqs.共5.4兲 to 共5.6兲 taking the same limits as in 共5.7兲 one has

Ml⫽

共m␺l兲2

2 , Mj⫽0, j⫽l. 共5.9兲

Therefore, relationships共5.1兲–共5.3兲 incorporate each sl(2) ATM submodel 共particle/soliton兲 weak/strong coupling phases, i.e., the MT/SG correspondence.4,11

Then, the currents equivalence 共A29兲, relationships 共5.1兲–共5.3兲, and conditions 共2.1兲–共2.3兲 satisfied by the one-soliton sector of CATM theory allowed us to establish the correspondence between the GSG and GMT models, thus extending the MT/SG result.25It could be interesting to obtain the counterpart of Eqs.共5.1兲–共5.3兲 for the NS⭓2 solitons, e.g., along the lines of Ref. 25. For NS⫽2, Eq. 共A29兲 still holds;

5

and Eqs.共2.1兲–共2.3兲 are satisfied for the species (␣i,␣i). Second, consider the limit

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a ¯ik→

再

⬁ i⫽k⫽ j 共for a chosen j; j⫽1,2兲

finite other cases , 共5.10兲

one gets Mj⫽0 and

4⌬¯␺R l_␺_˜ L l i ⫽共m␺ 3 a ¯l3⫺m␺ l a ¯33兲e⫺i␾l⫺m␺ 3 a ¯l3e⫺3i␸l⫹m␺ l a ¯33, l⫽ j, 共5.11兲 4⌬¯␺R 3_␺_˜ L 3 i ⫽共m␺ 3 a ¯ll⫺m_␺ l a ¯l3兲e⫺i␾3⫹m_␺ l a ¯l3e⫺3i␸l⫹m_␺ 3 a ¯ll, ␺Rj_␺_˜ L j_⫽0, _共5.12兲

where ⌬¯⬅4(a¯ll¯a33⫺(a¯l3)2). The parameters are related by (m␺ 3 )2¯allMl⫽m␺ l (m_␺3a¯l3 ⫺m_␺l a

¯33) M3. In the case Ml⫽M3⫽M and redefining the fields as ␾l⫽

冑

12g(A⫹B),␾j

⫽⫺

冑

12gB in the GSG sector, one gets the Lagrangian

LBL⫽ 1 2共⳵␮A兲 2_⫹1 2共⳵␮B兲 2_⫹2M

g cos

冑

24gA cos

冑

72gB, 共5.13兲

which is a particular case of the Bukhvostov–Lipatov model共BL兲.26It corresponds to a GMT-type theory with two Dirac spinors. The BL model is not classically integrable,27and some discussions have appeared in the literature about its quantum integrability.28

Alternatively, if one allows the limit a¯33→⬁ one gets␺R 3_␺_˜

L

3_{⫽0, and additional relations for} the␺1,␺2 spinors and the␸a scalars. The parameters are related by

M1 共m␺1兲2¯a22 ⫽ M2 共m␺2兲2¯a11 ⫽⫺ M3 m_␺1m_␺2a¯12 .

Then we left with two Dirac spinors in the GMT sector and all the terms of the GSG model. The later resembles the 2-cosine model studied in Ref. 29 in some submanifold of its renormalized parameter space.

VI. GENERALIZATION TO HIGHER RANK LIE ALGEBRA

The procedures presented so far can directly be extended to the CATM model for the affine Lie algebra sl(n)(1) furnished with the principal gradation. According to the construction of Ref. 2, these models have soliton solutions for an off-critical submodel, possess a U(1) vector current proportional to a topological current, apart from the conformal symmetry they exhibit a (U(1)R)n⫺1丢(U(1)L)n⫺1left–right local gauge symmetry, and the equations of motion describe the dynamics of the scalar fields ␸a, ␩,˜ (a␯ ⫽1, . . ., n⫺1) and the Dirac spinors ␺␣j_, ␺˜␣j 共j

⫽1, . . ., N; N⬅ (n/2) (n⫺1)⫽number of positive roots␣jof the simple Lie algebra sl(n)兲 with one-共anti兲soliton solution associated with the field ␣_j•␸ជ 共␸ជ⫽兺_an_⫽1⫺1␸a␣a, ␣a⫽simple roots of

sl(n)兲 for each pair of Dirac fields 共␺␣j_, ␺˜␣j兲.2_{Therefore, it is possible to define the off-critical} real Lagrangian sl(n) ATM model for the solitonic sector of the theory. The reality conditions would generalize Eqs.共2.1兲–共2.3兲, i.e., the new␸’s real and the identifications␺˜␣j⬃(␺␣j₎_*共up to

⫾ signs兲. To apply the symplectic analysis of sl(n) ATM one must impose (n⫺1) constraints in

the Lagrangian, analogous to共3.1兲, due to the above-given local symmetries. The outcome will be a parent Lagrangian of a generalized massive Thirring model 共GMT兲 with N Dirac fields and a generalized sine-Gordon model共GSG兲 with (n⫺1) fields. The decoupling of the Toda fields and Dirac fields in the equations of motion of sl(n)(1)CATM, analogous to共A13兲 and 共A15兲–共A23兲, could be performed by an extension of the relationships共5.1兲–共5.3兲 and 共2.1兲–共2.3兲.

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VII. DISCUSSIONS AND OUTLOOK

We have shown, in the context of FJ and symplectic methods, that the sl(3) ATM共2.4兲 theory is a parent Lagrangian18from which both the GMT共3.13兲 and the GSG 共4.39兲 models are deriv-able. From共3.13兲 and 共4.39兲, it is also clear the duality exchange of the couplings: g→1/g. The various soliton/particle species correspondences are uncovered. The soliton sector satifies the

U(1) vector and topological currents equivalence共A29兲 and decouples the equations of motion

into both dual sectors, through the relationships共5.1兲–共5.3兲关supplied with 共2.1兲–共2.3兲兴. Relation-ships 共5.1兲–共5.3兲 contain each sl(2) ATM submodel soliton solution. In connection to these points, recently a parent Lagrangian method was used to give a generalization of the dual theories concept for non-p-form fields.30In Ref. 30, the parent Lagrangian contained both types of fields, from which each dual theory was obtained by eliminating the other fields through the equations of motion.

On the other hand, in non-Abelian bosonization of massless fermions,31the fermion bilinears are identified with bosonic operators. Whereas, in Abelian bosonization32 there exists an identifi-cation between the massive fermion operator 共charge nonzero sector兲 and a nonperturbative bosonic soliton operator.33Recently, it has been shown that symmetric space sine-Gordon models bosonize the massive non-Abelian共free兲 fermions providing the relationships between the fermi-ons and the relevant solitfermi-ons of the bosonic model.34The ATM model allowed us to establish these types of relationships for interacting massive spinors in the spirit of particle/soliton correspon-dence. We hope that the quantization of the ATM theories and the related WZNW models, and in particular relationships共A34兲, would provide the bosonization of the nonzero charge sectors of the GMT fermions in terms of their associated Toda and WZNW fields. In addition, the above-given approach to the GMT/GSG duality may be useful to construct the conserved currents and the algebra of their associated charges in the context of the CATM→ ATM reduction. These currents in the MT/SG case were constructed treating each model as a perturbation on a conformal field theory共see Ref. 35 and references therein兲.

Moreover, two-dimensional models with four-fermion interactions have played an important role in the understanding of QCD 共see, e.g., Ref. 36 and references therein兲. Besides, the GMT model contains explicit mass terms: most integrable models such as the Gross–Neveu, SU共2兲, and

U共1兲 Thirring models all present spontaneous mass generation, the exception being the massive

Thirring model. A GMT submodel with aii⫽0, ai j⫽1 (i⬎ j) and equal m␺ j

’s defines the so-called extended Bukhvostov–Lipatov model共BL兲 and has recently been studied by means of a bosoniza-tion technique.37 Finally, BL type models were applied to N-body problems in nuclear physics.17

ACKNOWLEDGMENTS

H.B. thanks Professor L. A. Ferreira and Professor A. H. Zimerman for discussions on inte-grable models and Professor M. B. Halpern and Professor R. K. Kaul for correspondences and valuable comments on the manuscript. The authors thank Professor B. M. Pimentel for discussions on constrained systems and Professor A. J. Accioly for encouragement. The authors are supported by FAPESP.

APPENDIX: THEsl„3…„1…_{CATM MODEL}

We summarize the construction and some properties of the CATM model relevant to our discussions.38 More details can also be found in Ref. 5. Consider the zero curvature condition ⳵⫹A⫺⫺⳵⫺A⫹⫹关A⫹,A⫺兴⫽0. The potentials take the form

A_⫹⫽⫺BF⫹B⫺1, A_⫺⫽⫺⳵_⫺BB⫺1⫹F⫺, 共A1兲 with

(20)

where E⫾3⬅m•H⫾⫽1 6关(2m␺ 1_⫹m ␺ 2_)H 1 ⫾1_⫹(2m ␺ 2_⫹m ␺ 1_)H 2 ⫾1_{兴 and the F} i

⫾_{’s and B contain the}

spinor fields and scalars of the model, respectively,

_im ␺3␺˜L 3 E_⫺␣ 3 0 , 共A6兲 B⫽ei␸1H1 0 ⫹i␸2H2 0 e␯˜Ce␩Qp pal_, 共A7兲 where E_␣ i n

, H₁n, H₂n, and C (i⫽1,2,3; n⫽0,⫾1) are some generators of sl(3)(1); Qp palbeing the principal gradation operator. The commutation relations for an affine Lie algebra in the Chevalley basis are 关Ha m_,H b n_兴⫽mC 2 ␣a 2Kab␦m_⫹n,0, 共A8兲 关Ha m ,E_⫾␣n 兴⫽⫾K_␣aE_⫾␣m⫹n, 共A9兲关E␣m,E⫺␣n 兴⫽_a

兺

⫽1 r l_a␣H_am⫹n⫹ 2 ␣2mC␦m⫹n,0, 共A10兲关E_␣m

,E_␤n兴⫽␧共␣,␤兲Em_␣⫹␤⫹n; if ␣⫹␤ is a roo, 共A11兲

关D,E␣n兴⫽nE␣n, 关D,Ha n 兴⫽nHa n , 共A12兲 where K_␣a⫽2␣.␣a/␣a 2_⫽n b ␣_K

ba, with na␣ and la␣ being the integers in the expansions ␣⫽na␣␣a and␣/␣2⫽l_a␣␣a/␣_a2, and␧(␣,␤) the relevant structure constants.

Take K11⫽K22⫽2 and K12⫽K21⫽⫺1 as the Cartan matrix elements of the simple Lie alge-bra sl(3). Denoting by␣1and␣2 the simple roots and the highest one by␺(⫽␣1⫹␣2), one has

l_a␺⫽1(a⫽1,2), and K_␺1⫽K_␺2⫽1. Take ␧(␣,␤)⫽⫺␧(⫺␣,⫺␤),␧_1,2⬅␧(␣₁,␣₂)⫽1, ␧_⫺1,3

⬅␧(⫺␣1,␺)⫽1 and ␧⫺2,3⬅␧(⫺␣2,␺)⫽⫺1.

One has Q_{p pal}⬅兺_a2_⫽1s_a␭_av.H⫹3D, where ␭_av are the fundamental co-weights of sl(3), and the principal gradation vector is s⫽(1,1,1).39

The zero curvature condition gives the following equations of motion ⳵2_␸a 4i e␩⫽m␺ 1_关e␩_⫺i␾_a_␺_˜ R l_␺ L l_⫹ei␾_a_␺_˜ L l_␺ R l_兴⫹m ␺ 3_关e⫺i␾₃_␺_˜ R 3_␺L3_⫹e␩⫹i␾₃_␺_˜ L 3_␺ R 3_{兴, a⫽1,2, 共A13兲} ⫺⳵ 2_˜_␯ 4 ⫽im␺ 1_e2␩⫺␾₁_␺_˜ R 1_␺ L 1_⫹im ␺ 2_e2␩⫺␾₂_␺_˜ R 2_␺L2_⫹im ␺ 3_e␩⫺␾₃_␺_˜ R 3_␺L3_⫹m2_e3␩_, _共A14兲 ⫺2⳵⫹␺L 1_⫽m ␺ 1_e␩_⫹i␾₁_␺R1_, _⫺2_⳵ ⫹␺L 2_⫽m ␺ 2_e␩_⫹i␾₂_␺R2_, _共A15兲 2⳵_⫺␺_R1⫽m_␺1e2␩⫺i␾1␺ L 1_⫹2i

冉

m␺ 2 m_␺3 im_␺1

冊

1/2 e␩共⫺␺R3␺˜_L2ei␾2⫺␺˜ R 2_␺ L 3 e⫺i␾3兲, 共A16兲