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

Transcription

1 Generalized sine-gordon/massive Thirring models and soliton/particle correspondences José Acosta and Harold Blas Citation: Journal of Mathematical Physics 4, 96 (00); doi: 0.06/ View online: View Table of Contents: Published by the AIP Publishing

2 JOURNAL OF MATHEMATICAL PHYSICS VOLUME 4, NUMBER 4 APRIL 00 Generalized sine-gordonõmassive Thirring models and solitonõparticle correspondences José Acosta and Harold Blas a) Instituto de Física Teórica, Universidade Estadual Paulista, Rua Pamplona 45, São Paulo, S.P., Brazil Received 5 September 00; accepted for publication 8 January 00 We consider a real Lagrangian off-critical submodel describing the soliton sector of the so-called conformal affine sl() () 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 GSG or the massive Thirring GMT models are derivable. The dual description of the model is further emphasized by providing the relationships between bilinears of GMT spinors and relevant expressions of the GSG fields. In this way we exhibit the strong/weak coupling phases and the generalized soliton/particle correspondences of the model. The sl(n) () case is also outlined. 00 American Institute of Physics. DOI: 0.06/ I. INTRODUCTION Integrable theories in two dimensions have been an extraordinary laboratory for the understanding of basic nonperturbative aspects of physical theories and various aspects, relevant in more realistic four-dimensional models, have been tested. In particular the conformal affine Toda models coupled to Dirac matter fields CATM for the sl() () and sl() () cases are discussed in Refs., 4, and 5, respectively. The interest in such models comes from their integrability and duality properties,,4 which can be used as toy models to understand some phenomena; such as a confinement mechanism in quantum chromodynamics QCD,5 and 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 ATM 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, 9 and, in particular, polyacteline molecule systems in connection with fermion number fractionization. 0 Off-critical submodels, such as the sl() ATM, can be obtained at the classical or quantum mechanical level through some convenient reduction processes starting from CATM. 4, In the sl() case, using bosonization techniques, it has been shown that the classical equivalence between the U() 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() ATM theory decouples into a sine-gordon model SG and a free scalar., These facts indicate the existence of a sort of duality in these models involving solitons and particles. 6 The symplectic structure of the sl() ATM model has recently been studied in the context of Faddeev Jackiw FJ and constrained symplectic methods. 4,5 Imposing the equivalence between the U() vector and topological currents as a constraint there have been obtained the SG or the massive Thirring MT model. One of the difficulties with generalizations of complex affine Toda field theories, beyond su() and its associated SG model, has to do with unitarity. Whereas for practical applications a Electronic mail: blas@ift.unesp.br /00/4(4)/96//$ American Institute of Physics

3 J. Math. Phys., Vol. 4, No. 4, April 00 Generalized sine-gordon/massive Thirring models 97 such as low dimensional condensed matter systems see Ref. 6 and references therein and N-body problems in nuclear physics, 7 the properties of interest are usually integrability and nonperturbative results of multifield Lagrangians. Therefore, integrable quantum field theories with several fields bosons and/or fermions 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() case the related sl(n) () 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() () CATM soliton solutions and some of their properties up to general twosoliton. Using the FJ and symplectic methods we show the parent Lagrangian 8 nature of the sl() ATM model from which the generalized sine-gordon GSG or the massive Thirring GMT models are derivable. We thus show that there are at least classically 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 /g and the generalized soliton/particle correspondences in each sl() 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 generalizations. In this way we give a precise field content of both sectors; namely, the correct GMT/GSG duality, first undertaken in Ref. 9. The paper is organized as follows. In Sec. II we define the sl() 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 quantization 4,5 giving 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() () CATM model and its relationship to the two-loop Wess Zumino Novikov Witten WZNW 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 solutions of the model. In general, a few soliton solutions survive the reality constraint, if in addition one also demands positivity. These kind of issues are discussed in Refs. 0. 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 real Toda field for each pair of Dirac fields i and i. This fact allows us to make the identifications i ( i )*, and take real Toda fields. In the case of sl() () CATM theory, this procedure does not spoil the particle soliton correspondence.,4 We consider the sl() () CATM theory see the Appendix with the conformal symmetry gauge fixed by setting 0 and the reality conditions or j j *, j,,, a * a a,,. j j *, j,, *,,, the new a s being real fields,. where an asterisk means complex conjugation. The condition. must be supplied with x x. Moreover, for consistency of the equations of motion A5 A under the reality conditions. and., from Eqs. A6 to A8, A0, A, and A, we get the relationships

4 98 J. Math. Phys., Vol. 4, No. 4, April 00 J. Acosta and H. Blas Lj R Rj L e i j 0, j,, L R e i L R e i 0.. Then, the above-given reality conditions and constraints allow us to define a suitable physical Lagrangian. Equations A, A5 A, supplied with. or. and., follow from the Lagrangian k L j 4 j j i j j m j je i j 5 j,.4 where j ( j ) 0,,,, m m m, k is an overall coupling constant and the j are real fields. Equation.4 defines the sl() affine Toda theory coupled to matter fields ATM. Notice that the space of solutions of sl() () CATM model satisfying conditions.. must be solutions of the sl() ATM theory.4. Indeed, it is easy to verify that the three species of one-soliton solutions S -soliton(s -antisoliton) : 5 (, ) S/S, 0, 0, 0, (, ) S/S, 0, 0, 0, and (, ) S/S,, 0, 0 satisfy the equations of motion, i.e., each positive root of sl() reproduces the sl() ATM case.,4 Moreover, these solutions satisfy the above-given reality conditions and constriants.. with. and. for S and S, respectively, and the equivalence between the U() vector and topological currents A9. Then, the soliton/particle correspondences survive the above-given reduction processes performed to define the sl() ATM theory. The class of two-soliton solutions of sl() () CATM 5 behave as follows: i they are given by six species associated with the pair ( i, j ), i j; i, j,,; where the s are the positive roots of sl() Lie algebra. Each species ( i, i ) solves the sl() CATM submodel; ii satisfy the U() vector and topological currents equivalence A9. III. THE GENERALIZED MASSIVE THIRRING MODEL GMT Let us consider the following Lagrangian: k L j 4 j j i j j m j je i j 5 j j m j j j q j j,. where the ATM Lagrangian.4 is supplied with the constraints, (m l l l (m /) j l ), (l,), with the help of the Lagrange multipliers ( )/, q q, q 0. Their total sum bears an intriguing resemblance to the U() vector and topological currents equivalence A9 ; however, the m j 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. The constraints in. will break the left right local symmetries A5 A8 of sl() ATM.4. In order to apply the FJ method we should write. in the first-order form in time derivative, so let us define the conjugated momenta,, 0, 0, j j R R i Rj, j j L L i. Lj. We are assuming that Dirac fields are anticommuting Grasmannian variables and their momenta variables defined through left derivatives. Then, as usual, the Hamiltonian is defined by sum over repeated indices is assumed H c Rj R j Lj L j L..

5 J. Math. Phys., Vol. 4, No. 4, April 00 Generalized sine-gordon/massive Thirring models 99 Explicitly the Hamiltonian density becomes H c j 4 4 J J 4 4 j,x j R R,x j L L,x j j im j e j Rj L j e j Lj R j 0 J 0,x 0 J 0,x,.4 where, J J 4( ), J J 4( ), and J m j l m j, J m j m j, j l l l, l,,..5 Let us observe that each U() Noether current of the sl() ATM theory defined in.4 is conserved separately, i.e., j l 0, l,,. Next, the same Legendre transform. is used to write the first-order Lagrangian L Rj R j Lj L j H c..6 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 J J, J J.7 and the remaining equations lead to two constraints, J 0,x 0, J 0,x 0..8 The Lagrange multipliers and must be replaced back in.6 and the constraints.8 solved. First, let us replace the and multipliers into H c, then one gets H c j J J J J 4 j,x i Rj R,x j j i Lj L,x im j e i j Rj L j e i j Lj R j..9 The new Lagrangian becomes L Rj R j Lj L j H c..0 We implement the constraints.8 by replacing in.0 the fields, in terms of the space integral of the current components J 0, J 0. Then we get the Lagrangian L t x J 0 t x J 0 j Rj R j j Lj L i Rj R,x j i Lj L,x im j e i x 0 J j j Rj L e i x J j 0 Lj R j J J J.J J J,. where J 0 J 0 J 0. Observe that the terms containig the a s in Eq.. 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: R j exp i x J j 0 R j, L j exp i x J j 0 L j, j,,.

6 90 J. Math. Phys., Vol. 4, No. 4, April 00 J. Acosta and H. Blas is used to diagonalize the canonical one-form. Then, the kinetic terms will give additional current current interactions, J ( j j ) J ( j j ). We are, thus, after defining k /g, and rescaling the fields j / k j, left with the Lagrangian L, j i j j j m j j k,l ā kl j k j l, k l. where ā kl ga kl, with a m 8 m, a m m, a ii mi a i mi m 4 m i m, i,. 6 m i, This defines the generalized massive Thirring model GMT. The canonical pairs are ( i Rj, j R ) and ( i, j Lj L ). 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. 4 The geometric structure is defined by the closed pre symplectic two-form where f (0) f (0) ij (0) d (0)i d (0)j, 4. f (0) ij (0) (0)i a (0) j (0) (0)j a (0) i (0) 4. with a (0) ( (0) ) a j (0) ( (0) )d (0)j being the canonical one-form defined from the original first-order Lagrangian L (0) dt a (0) (0) V (0) (0) dt. 4. The superscript 0 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. 4,5 The corresponding Lagrangian gets modified and consequently the superscript also changes. The algorithm terminates once the symplectic 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. Let L, Eq..0, be the zeroth-iterated Lagrangian L (0). Then the first iterated Lagrangian will be L () Rj R j Lj L j V (), 4.4 where the once-iterated symplectic potential is defined by V () H c 0, 4.5

7 J. Math. Phys., Vol. 4, No. 4, April 00 Generalized sine-gordon/massive Thirring models 9 and the stability conditions of the symplectic constraints, and, under time evolution have been implemented by making 0 and 0. Consider the once-iterated set of symplectic variables in the following order (),,,, R, L, R, L, R, L,,, R, L, R, L, R, L, 4.6 and the components of the canonical one-form a (),,,, R, L, R, L, R, L,0,0,0,0,0,0,0, These result in the singular symplectic two-form 8 8 matrix f () AB x,y a a a a x y, 4.8 where the 9 9 martices are 0 0 x 0 im R im L x 0 0 im R a im R im im L im x x im L im R R R 0 im L im im R R , a im L 0 0 m R m m L 0 0 R m L L m R m m L R m L,

8 9 J. Math. Phys., Vol. 4, No. 4, April 00 J. Acosta and H. Blas im m R a L im L m L , 0 m R m L m R m R m L m L a This matrix has the zero modes u m, m,0,0,u R,u L, R, L, m v ()T x u m m R, m u m m L, u m,iu R,iu L,i R,i L,i m u m m R, im u m m L, 4.9 where u and are arbitrary functions. The zero-mode condition gives m, dxv ()T x () dyv () 0. x 4.0 Thus, the gradient of the symplectic potential happens to be orthogonal to the zero-mode v (). 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 () v A () x, A,,...,8. 4. 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. 5 In our case we have two symmetry generators associated with the parameters u and v, so there must be two gauge conditions. Let us choose 0, These conditions gauge away the fields and, 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.

9 J. Math. Phys., Vol. 4, No. 4, April 00 Generalized sine-gordon/massive Thirring models 9 Implementing the consistency conditions by means of Lagrange multipliers and 4 we get the twice-iterated Lagrangian L () R R j L L j 4 4 V (), 4. where V () V () 4 0. Assuming now that the new set of symplectic variables is given in the following order, (),,, 4,,, R, L, R, L, R, L,,, R, L, R, L, R, L, 4.4 and the nonvanishing components of the canonical one-form a (),,, 4,,, R, L, R, L, R, L,0,0,0,0,0,0,0,0, 4.5 one obtains the singular twice-iterated symplectic 0 0 matrix f () AB x,y a a a a x y, 4.6 where the 0 0 matrices are x 0 im R im L x 0 0 im R im L x a im R x im L im R im L ,

10 94 J. Math. Phys., Vol. 4, No. 4, April 00 J. Acosta and H. Blas im im a R im L 0 0 m R m L R im L m R 0 0 m m L m R m L R m L im im a R im R L im L m R m, L m R m L m R m R m L m L a , The zero-modes are

11 J. Math. Phys., Vol. 4, No. 4, April 00 Generalized sine-gordon/massive Thirring models 95 v x ()T u,,,, 0, 0, m u R, m u L, m R, m L, m m u R, u L, u,, im u, im u, im,im, R L R L im u, i m R u. 4.7 L The zero-mode condition gives no constraints, implying the symmetries of the action Now, let us choose the gauge conditions A () v A () x, A,,..., J J j j 0, 6 J J j j 0, 4.9 and impose the consistency conditions with the Lagrange multipliers 5, 6, then L () R R j L L j V (), 4.0 where V () V () 5 6 0, 4. or explicitly V () J J J J j j J j j J j j i Rj R,x i Lj L,x im j j j. 4. 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 potential V () one has L, j i j j j m j j k,l ā kl j k. j l m l l j 0 l, l k l 4. where ( )/. We have made the same choice, k /g, and the field rescalings j / k j as in the last section. This is the same GMT Lagrangian as.. As a bonus, we get the chemical potentials l m l l (,, ) times the charge densities. These terms are related to the charges Q l F / dxj 0 l (t,x), and their presence is a consequence of the symplectic method. C. The generalized sine-gordon model GSG One can choose other gauge fixings, instead of 4., to construct the twice-iterated Lagrangian. Let us make the choice J 0 0, 4 J 0 0, 4.4 which satisfies the nongauge invariance condition as can be verified by computing the brackets a,j b 0 0; a,b,. The twice-iterated Lagrangian is obtained by bringing back these constraints into the canonical part of L (), then L () R R j L L j 4 4 V (), 4.5

12 96 J. Math. Phys., Vol. 4, No. 4, April 00 J. Acosta and H. Blas where the twice-iterated symplectic potential becomes V () V () Considering the set of symplectic variables in the following order: A (),,, 4,,, R, L, R, L, R, L,,, R, L, R, L, R, L 4.7 and the components of the canonical one-form a A (),,, 4,,, R, L, R, L, R, L,0,0,0,0,0,0,0,0, 4.8 the degenerated 0 0 symplectic matrix is found to be where f () AB x,y a a a a x y, x 0 im R im L x 0 0 im R im R a im R 0 im R im a im L im L im R im L x x im L 0 im L im R 0 im R im 0 im L 0 im L R im L 0 0 m R m L R im L m R im R im L 0 0 m R m L m m L m R m L R m L m R m L im im R L m R m m L R m L, ,

13 J. Math. Phys., Vol. 4, No. 4, April 00 Generalized sine-gordon/massive Thirring models 97 v ()T x im im a R im R im R im R im L im L im L L m R 0 m R m L 0 m, L m R 0 m R m L 0 m L m R m R m R m R m L m L m L m L a Its zero modes are u,,,, 0, 0, m u R, m u L, m R, m L, m m u R, u L, u,, im u, R im im u, im, im, L R L u, R im u, 4.0 L where u,,, and are arbitrary functions. The zero-mode condition becomes dx v ()T x () dy V () dx J a x f a 0, f a,, a,. Since the functions f a are arbitrary we end up with the following constraints: 5 J 0, 6 J Notice that by solving the constraints, , Eqs. 4.4 and 4., wemay obtain

14 98 J. Math. Phys., Vol. 4, No. 4, April 00 J. Acosta and H. Blas Rj R j, Lj L j. 4. So, at this stage, we have Majorana spinors, the scalars and, and the auxiliary fields. Next, introduce a third set of Lagrange multipliers into L (), then L () R R j L L j V (), 4. where V () V () or V () 4 j,x i j j R R,x i j j L L,x im j j R j L e i j e i j. 4.5 The new set of symplectic variables is assumed to be ordered as A (),,, 4, 5, 6,,, R, L, R, L, R, L,,, R, L, R, L, R, L. The components of the canonical one-form are a A (),,, 4, 5, 6,,, R, L, R, L, R, L,0,0,0,0,0,0,0,0. After some algebraic manipulations we get the third-iterated symplectic two-form f () AB x,y a a a a x y, 4.6 where x 0 im R im L x 0 0 im R im R im L im R im R im L 0 a im R 0 im R 0 im R im R, x x im L 0 im L 0 im L im R 0 im R 0 im R

15 J. Math. Phys., Vol. 4, No. 4, April 00 Generalized sine-gordon/massive Thirring models 99 0 im im im L a 0 im R L 0 0 m R m L im R L m R im im L im im R L 0 0 m R m L im R L m R 0 im R im L 0 0 m R im L m L im R im L m R m m L m R m R m m m L m R m R m L L L L 0 0 m R m L m L m m R L im a 0 im L 0 im L 0 im L im im R im R im R R im R im R im L im L im L im L im L L m R 0 m R 0 m R , m L 0 m L 0 m L m R 0 m R 0 m R m L 0 m L 0 m L m m R m R m R R m R m R m m L m L m L m L m L L a , It can be checked that this matrix has the zero modes

16 90 J. Math. Phys., Vol. 4, No. 4, April 00 J. Acosta and H. Blas v x () u,,,, y, z, 0, 0, m a R, m a L, m a R, m a L, m a L, u,, im a R, im a L, im a R, im a L, m a R, i m a R, i m a L, 4.7 where a u y, a z, a u y z, a u y, a z, a 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 () v A () x, A,,...,. 4.8 These symmetries allow us to fix the bilinears i j j j R L to be constants. By taking R ic j j and j L j ( j,,) with C j being real numbers, we find that i j j R L indeed becomes a constant. Note that j and j are Grassmannian variables, while j j is an ordinary commuting number. The two form f () AB (x,y), Eq. 4.6, in the subspace (,,, ) defines a canonical symplectic structure modulo canonical transformations. The coordinates a and a (a,) are not unique. Consider a canonical transformation from ( a, a ) to ( ˆ a, 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 /g as the overall coupling constant, we are left with L j 4g j j M j g cos j x x, 4.9 where M j m j C j. This defines the generalized sine-gordon model GSG. In addition we get the terms multiplied by chemical potentials and (,, ). These are just the topological charge densities, and are related to the conservation of the number of kinks minus antikinks a Q topol / dx x a. In the above-mentioned gauge fixing procedures the possibility of Gribov-type ambiguities deserves a careful analysis. See Ref. for a discussion in the sl() 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() ATM theory contains the sine-gordon SG and the massive Thirring MT models describing the soliton/particle correspondence of its spectrum.,, The ATM one- anti 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. 5 The last relationship was exploited in Ref. 4 to decouple the sl() ATM equations of motion into the SG and MT ones. Here we provide a generalization of that correspondence to the sl() ATM case. In fact, consider the relationships R L i 4 m p m p 4 m p 5 e i( ) m p 5 e i( ) m p 4 e i m p, 5.

17 J. Math. Phys., Vol. 4, No. 4, April 00 Generalized sine-gordon/massive Thirring models 9 R L 4 m p m p 5 m p 6 e i( ) m p 5 e i( ) m p 6 e i m p, 5. R L i 4 m p m p 4 m p 6 e i( ) m p 4 e i m p 6 e i m p, 5. where ā ā ā ā ā ā ā (ā ) (ā ) ā (ā ) ā ; p (ā ) ā ā ; p (ā ) ā ā ; p (ā ) ā ā ; p 4 ā ā ā ā ; p 5 ā ā ā ā ; p 6 ā ā ā ā and the ā ij s being the current current coupling constants of the GMT model.. Relationships supplied with the conditions.. and conveniently substituted into Eqs. A and A5 A decouple the sl() () CATM equations into the GSG 4.9 and GMT. equations of motion, respectively. Moreover, one can show that the GSG 4.9 M j parameters and the GMT. couplings ā ij are related by M g m ā m ā m ā ā ā m ā m, M g m ā m ā m ā ā ā m ā m, m M g m m m ā ā ā ā ā ā ā Various limiting cases of the relationships and are possible. First, let us consider j k l for a given l ā jk 5.7 finite other cases then one has R l Ll i m l 4 e i l, R j Lj 0, j l 5.8 for ā ll l g (,, ). The three species of one-soliton solutions of the sl() ATM theory.4, found in Ref. 5 and described in Sec. II, satisfy the relationship Moreover, from Eqs. 5.4 to 5.6 taking the same limits as in 5.7 one has M l m l, M j 0, j l. 5.9 Therefore, relationships incorporate each sl() ATM submodel particle/soliton weak/strong coupling phases, i.e., the MT/SG correspondence. 4, Then, the currents equivalence A9, relationships 5. 5., and conditions.. 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. 5 It could be interesting to obtain the counterpart of Eqs for the N S solitons, e.g., along the lines of Ref. 5. For N S, Eq. A9 still holds; 5 and Eqs... are satisfied for the species ( i, i ). Second, consider the limit

18 9 J. Math. Phys., Vol. 4, No. 4, April 00 J. Acosta and H. Blas i k j for a chosen j; j, ā ik, 5.0 finite other cases one gets M j 0 and 4 R l Ll i m ā l m l ā e i l m ā l e i l m l ā, l j, 4 R L m i ā ll m l ā l e i m l ā l e i l m ā ll, R j Lj 0, where 4(ā ll ā (ā l ) ). The parameters are related by (m ) ā ll M l m l (m ā l m l ā )M. In the case M l M M and redefining the fields as l g(a B), j gb in the GSG sector, one gets the Lagrangian L BL A B M g cos 4gA cos 7gB, 5. which is a particular case of the Bukhvostov Lipatov model BL. 6 It corresponds to a GMT-type theory with two Dirac spinors. The BL model is not classically integrable, 7 and some discussions have appeared in the literature about its quantum integrability. 8 Alternatively, if one allows the limit ā one gets R L 0, and additional relations for the, spinors and the a scalars. The parameters are related by M m ā M m ā M m m ā. Then we left with two Dirac spinors in the GMT sector and all the terms of the GSG model. The later resembles the -cosine model studied in Ref. 9 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) () furnished with the principal gradation. According to the construction of Ref., these models have soliton solutions for an off-critical submodel, possess a U() vector current proportional to a topological current, apart from the conformal symmetry they exhibit a (U() R ) n (U() L ) n left right local gauge symmetry, and the equations of motion describe the dynamics of the scalar fields a,, (a,...,n ) and the Dirac spinors j, j j,...,n; N (n/) (n ) number of positive roots j of the simple Lie algebra sl(n) with one- anti soliton solution associated with the field j n a a a, a simple roots of sl(n) for each pair of Dirac fields j, j. 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..., i.e., the new s real and the identifications j ( j)* up to signs. To apply the symplectic analysis of sl(n) ATM one must impose (n ) constraints in the Lagrangian, analogous to., due to the above-given local symmetries. The outcome will be a parent Lagrangian of a generalized massive Thirring model GMT with N Dirac fields and a generalized sine-gordon model GSG with (n ) fields. The decoupling of the Toda fields and Dirac fields in the equations of motion of sl(n) () CATM, analogous to A and A5 A, could be performed by an extension of the relationships and...

19 J. Math. Phys., Vol. 4, No. 4, April 00 Generalized sine-gordon/massive Thirring models 9 VII. DISCUSSIONS AND OUTLOOK We have shown, in the context of FJ and symplectic methods, that the sl() ATM.4 theory is a parent Lagrangian 8 from which both the GMT. and the GSG 4.9 models are derivable. From. and 4.9, it is also clear the duality exchange of the couplings: g /g. The various soliton/particle species correspondences are uncovered. The soliton sector satifies the U() vector and topological currents equivalence A9 and decouples the equations of motion into both dual sectors, through the relationships supplied with... Relationships contain each sl() 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. 0 In Ref. 0, 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, the fermion bilinears are identified with bosonic operators. Whereas, in Abelian bosonization there exists an identification between the massive fermion operator charge nonzero sector and a nonperturbative bosonic soliton operator. Recently, it has been shown that symmetric space sine-gordon models bosonize the massive non-abelian free fermions providing the relationships between the fermions and the relevant solitons of the bosonic model. 4 The ATM model allowed us to establish these types of relationships for interacting massive spinors in the spirit of particle/soliton correspondence. We hope that the quantization of the ATM theories and the related WZNW models, and in particular relationships A4, 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 see Ref. 5 and references therein. Moreover, two-dimensional models with four-fermion interactions have played an important role in the understanding of QCD see, e.g., Ref. 6 and references therein. Besides, the GMT model contains explicit mass terms: most integrable models such as the Gross Neveu, SU, and U Thirring models all present spontaneous mass generation, the exception being the massive Thirring model. A GMT submodel with a ii 0, a ij (i j) and equal m j s defines the so-called extended Bukhvostov Lipatov model BL and has recently been studied by means of a bosonization technique. 7 Finally, BL type models were applied to N-body problems in nuclear physics. 7 ACKNOWLEDGMENTS H.B. thanks Professor L. A. Ferreira and Professor A. H. Zimerman for discussions on integrable 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: THE sl CATM MODEL We summarize the construction and some properties of the CATM model relevant to our discussions. 8 More details can also be found in Ref. 5. Consider the zero curvature condition A A A,A 0. The potentials take the form with A BF B, A BB F, A F E F F, F E F F, A

20 94 J. Math. Phys., Vol. 4, No. 4, April 00 J. Acosta and H. Blas where E m H 6 (m m )H (m m )H and the F i s and B contain the spinor fields and scalars of the model, respectively, F im R E 0 im R E 0 im E R, A F im R E 0 im E R im E R, A4 F im L E im 0 E L im 0 E L, A5 F im L E im L E im 0 E L, A6 B e i H 0 0 i H e C e Q ppal, A7 where E n i, H n, H n, and C (i,,; n 0, ) are some generators of sl() () ; Q ppal being the principal gradation operator. The commutation relations for an affine Lie algebra in the Chevalley basis are H a m,h b n mc a K ab m n,0, A8 H m n a,e K a E m n, r E m n,e l a H m n a a mc m n,0, A9 A0 E m,e n, E m n ; if is a roo, A D,E n ne n, D,H a n nh a n, A where K a. a / a n b K ba, with n a and l a being the integers in the expansions n a a and / l a a / a, and (, ) the relevant structure constants. Take K K and K K as the Cartan matrix elements of the simple Lie algebra sl(). Denoting by and the simple roots and the highest one by ( ), one has l a (a,), and K K. Take (, ) (, ),, (, ),, (, ) and, (, ). One has Q ppal a s a v a.h D, where v a are the fundamental co-weights of sl(), and the principal gradation vector is s (,,). 9 The zero curvature condition gives the following equations of motion a 4i e m e i a Rl L l e i a Ll R l m e i R L e i L R, a,, A 4 im e R L im e R L im e R L m e, A4 L m e i R, L m e i R, A5 R m e i L i m / m im e R e i L R L e i, A6

21 J. Math. Phys., Vol. 4, No. 4, April 00 Generalized sine-gordon/massive Thirring models R m e i L i m / m im e R e i L R L e i, L m e i R i m / m im e L R e i L R e i, 95 A7 A8 R m e i L, m R e i, A9 L m L e i i m / m R im e L e i R L R e i, A0 m L e i i m / m R im e L e i R L R e i, A m R e i, L m L e i, A R m R e i i m / m L im e e i e i, A R L R L 0, A4 where,,. Apart from the conformal invariance the above-presented equations exhibit the (U() L ) (U() R ) left right local gauge symmetry a a a x a x, a,, A5,, A6 i e i( 5 ) i i (x ) i( 5 ) (x ) i, A7 i e i( 5 )( i i )(x ) i( 5 )( )(x ) i, i,,, A8,,. One can get global symmetries for a a constants. For a model defined by a Lagrangian these would imply the presence of two vector and two chiral conserved currents. However, it was found only half of such currents. 5 This is a consequence of the lack of a Lagrangian description for the sl() () CATM; however see the following. The gauge fixing of the conformal symmetry, by setting the field to a constant, is used to stablish the U() vector, J j m j j j, and topological currents equivalence., Moreover, it has been shown that the soliton solutions are in the orbit of the solution 0. The remarkable equivalence is j m j j j m m, m m m, m i 0. A9 The CATM theory has a local Lagrangian in terms of the B and the two-loop WZNW fields. The relations between their fields can be obtained from F B MM B, F B N NB, A0

22 96 J. Math. Phys., Vol. 4, No. 4, April 00 J. Acosta and H. Blas where provided that the following constraints are imposed: and M exp s 0 s, N exp s 0 s, A MM B m H B, MM 0. A N N B m H B, N N 0. A In A and A and A s and the subscripts denote the principal gradation structure of the relevant group elements. The relationships are im L e i, im e i L, im e i L, im R e i, im R e i, im R e i, im L e e i,, e i, im L e e i,, e i, A4 im e e i L,, e i, im R e e i,, e i, im R e e i,, e i, im R e e i,, e i. i i We observe that the WZNW fields,, i i, (i,,) are nonlocal in terms of the spinors and scalars i, i,,,, and. Then the CATM model Lagrangian must be nonlocal when written in terms of its fields. E. Abdalla, M. C. B. Abdalla, and K. D. Rothe, Non-perturvative Methods in Two-dimensional Quantum Field Theory World Scientific, Singapore, 99. L. A. Ferreira, J-L. Gervais, J. Sánchez Guillen, and M. V. Saveliev, Nucl. Phys. B 470, H. Blas and L. A. Ferreira, Nucl. Phys. B 57, H. Blas, Nucl. Phys. B 596, ; Proceedings of the seventh Hadron Physics 000 Workshop, Caraguatatuba, SP, Brazil, April 000, pp. 0 5 hep-th/ A. G. Bueno, L. A. Ferreira, and A. V. Razumov, hep-th/ C. Montonen and D. I. Olive, Phys. Lett. 7B, ; P. Goddard, J. Nuyts, and D.I. Olive, Nucl. Phys. B 5, C. Vafa and E. Witten, Nucl. Phys. B 4, 994 ; N. Seiberg and E. Witten, ibid. 4, ; 46, ;A. Sen, Phys. Lett. B 9, ; Int. J. Mod. Phys. A 9, S. Brazovskii, J. Phys. IV 0, ; also in cond-mat/000655; A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Wu, Rev. Mod. Phys. 60, D. G. Barci and L. Moriconi, Nucl. Phys. B 48, R. Jackiw and C. Rebbi, Phys. Rev. D, ; J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47, ; J. A. Mignaco and M. A. Rego Monteiro, Phys. Rev. D, H. Blas and B. M. Pimentel, Ann. Phys. Leipzig 8,

23 J. Math. Phys., Vol. 4, No. 4, April 00 Generalized sine-gordon/massive Thirring models 97 E. Witten, Nucl. Phys. B 45, L. Faddeev and R. Jackiw, Phys. Rev. Lett. 60, ; R. Jackiw, Diverse Topics in Theoretical Physics, st ed. World Scientific, Singapore, C. Wotzasek, Ann. Phys. Leipzig 4, ; H. Montani and C. Wotzasek, Mod. Phys. Lett. A 8, ; J. Barcelos-Neto and C. Wotzasek, ibid. 7, ; Int. J. Mod. Phys. A 7, H. Montani, Int. J. Mod. Phys. A 8, H. Saleur and P. Simonetti, Nucl. Phys. B 55, J. Sakamoto and Y. Heike, Prog. Theor. Phys. 00, ; L. V. Belvedere and R. L. P. G. Amaral, Phys. Rev. D 00, S. E. Hjelmeland and U. Lindström, hep-th/ M. B. Halpern, Phys. Rev. D, Z. Zhu and D. G. Caldi, J. Math. Phys. 6, ; G. Takacs and G. Watts, Nucl. Phys. B 547, ; S.P. Khastgir and R. Sasaki, Prog. Theor. Phys. 95, The auxiliary fields and of the CATM theories are associated with the topological character of the soliton masses and to the conformal symmetry, respectively. The classical and quantum reductions CATM ATM can be treated as in Refs. and 4, respectively. sl() ATM -solitons satisfy an analogous equation to A9 ; for real and ( )* constraints.. ;. is trivially satisfied since j j 0 for j i one has, soliton soliton SS, SS bounds, and no SS bounds Ref.. W. Siegel, Nucl. Phys. B 8, ; M. Henneaux, Proceedings of the Summer School in High-Energy Physics and Cosmology ICTP, Trieste, These are given for point mechanics, the extension to field theory is self-evident. 5 S. J. Orfanidis and R. Wang, Phys. Lett. 57B, ; S. J. Orfanidis, Phys. Rev. D 4, A. P. Bukhvostov and L. N. Lipatov, Nucl. Phys. B 80, M. Ameduri and C. J. Efthimiou, J. Nonlinear Math. Phys. 5, H. Saleur, J. Phys. A, ; M. Ameduri, C. J. Efthimiou, and B. Gerganov, Mod. Phys. Lett. A 4, B. Gerganov, Nucl. Phys. B 567, H. Casini, R. Montemayor, and L. F. Urrutia, Phys. Lett. B 507, E. Witten, Commun. Math. Phys. 9, S. Coleman, Phys. Rev. D, S. Mandelstam, Phys. Rev. D, Q.-Han Park and H. J. Shin, Nucl. Phys. B 506, R. K. Kaul and R. Rajaraman, Int. J. Mod. Phys. A 8, J. F. Bennett and J. A. Gracey, Nucl. Phys. B 56, J. Sakamoto and B. R. Poudel, Prog. Theor. Phys. 04, Our notations follow that of Ref., except that the s below have been multiplied by i. 9 V. G. Kac, Infinite Dimensional Lie Algebras, rd ed. Cambridge University Press, Cambridge, 990.