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Proceedings of the International Congress of Mathematicians August 16-24, 1983, Warszawa L. A. TAKHTAJAN Integrable Models in Classical and Quantum Field Theory Eecent papers of Faddeev, Sklyanin and the author ([25], [27], [28], [33]) contain a quantum version of the inverse scattering method (see also the reviews and lectures [7]-[9], [15], [23]). This is a new method of exact solution of the models in 1 + 1 dimensional quantum field theory and in classical statistical mechanics on a two-dimensional lattice. The profound papers of Baxter [l]-[3] have also played an important role in the formation of the method. In the present talk I will try to explain the underlying ideas and basic constructions of this new domain of modern mathematical physics and also to point out its connections with other parts of mathematics and theoretical physics. Our exposition (as it can be seen from the references) will be mostly based on the results obtained in the Leningrad Branch of the Steklov Mathematical Institute. I. Classical theory 1. Formulation of the inverse scattering method. This method applies to equations which can be represented as the zero curvature condition [Po> Pi] = 0 for the connection Vß = djdxß— TJ^Çx, A), p = 0 , 1 ; x0 = t, xx = x defined in a trivial fibre bundle with the base JB2 (the space-time) and the fibre Cm (the auxiliary space).^Matrix elements of the matrices TJ^x, t, A) depend on the classical fields y?(x, t), a = 1, . . . , M, involved in the equation considered, and on the variable A e C1 called the spectral parameter. The zero curvature condition must hold for all A. U331] 1332 Section 13: L. A. Takhtajan A fundamental role in the method is played by the equation of parallel translation along the #-axis dF dx = TJ(x9 X)F. (1) Here we put TJ = Ux and reduce the dependence on t. In the case of periodic initial data ipa(x+2L) = ipa(x) there naturally emerge the transition matrix T(x9 y9 A) = exp j U(x'9 X)dx* which is a solution of equation (1) y with the initial condition T(x9 y9 A) \Xesy = I and the monodromy matrix TL(X) for the interval (~L9L)9 TL(K) = T(L9 -L9X). Here I denotes the unit matrix in C"1 and the integral is understood as a multiplicative one. The time dependence of these objects is determined by the equation exp J Updibp = I (2) Y which holds for any closed contour y cz R2, due to the zero curvature condition. In particular, the functionals trî^UA),"k= 1 , . . . , m— 1, where tr stands for the trace in Cm9 do not depend on t and play the role of generating functions for the integrals of motion. In the rapidly decreasing case, where lim yf*(x) = ^ ± , there appear œ-j-dboo the reduced monodromy matrix T(X), relating the left and the right Jost solutions, and the characteristics of the discrete spectrum. In these terms the dynamics becomes completely transparent and the matrix TJ(x, A) is uniquely determined by them. This procedure is based on the formalism of the Eiemann problem, i.e., the problem of analytic factorization of matrix-valued functions. Here we have a connection with the theory of functions. These are the basic/items of the classical inverse scattering method (see [9], [12], [38]; our exposition follows [9], [12]). It is applied to such well-known equations as the Korteweg-de Vries equation (KdV), the nonlinear Schrödinger equation (N3), the Sine-Gordon equation (SG), the Heisenberg magnet equation (HM), and others. 2. The Hamiltonian approach. The most elegant formulation of the method is its Hamiltonian formulation originating from paper [37]. Its present form is based on the concept of a classical r-matrix introduced in [24]. A classical r-matrix is defined as a matrix r(X) in Cm® C™ which enables Integrarle Models in Classical and Quantum Field Theory 1333 one (if possible) to write down all the Poisson brackets of matrix elements of U(x9 A) in the following compact form {U(x,X)®TJ(y,[i)} =[r(A-/i), U(x, A)® L+L® U(x9 (i)lô(x-y). * (3) Here {A®B} denotes the matrix in Cm®Cm of the same structure as * A®B with the products of matrix elements replaced by their Poisson brackets. As an illustrating example consider the BM model with the equation of motion ds dœ* dt 8(xt ,*)e R3 > A 8; <8,8> == 1. w We have ([30]) 1 #0 = dx Oil) ZiA. 1 1 8, 2U* 2'IA~ Vi = * •8, ~2Î 2A 8 ••= <s, *>, (5) where —- oa, a = 1, 2, 3, are the generators of the fundamental repre2% sentation of the Lie algebra su(2): aa are called the Pauli matrices. The Poisson brackets and the Hamiltçnian of the model are {BM, Sb(y)} = eahc8c(y)à(x-y), _ 1 r / dB dB\2 J \ dx ' dx j ' (6) v ' —& where sab0 are the structure constants of su(2). The r-matrix is r(À) =P/2A, (8) 2 2 where P is the permutation operator in C ®C . The r-matrix formalism is based on the following result, though simple, yet important. THEOEEM 1. Suppose that relation (3) holds. Then for the transition matrix of equation (1) we have . {T(x, y, A) ®T(x9 y, p)} = [rtf-/*), X{a9 y9 A) ®T(x9 y9 /*)], > where —L^y*^x<L. (9) 1334 Section 13: L. A. Takhtajan As a corollary we obtain that {tr2^( A), t r l 7 ^ ^ ) } = 0 ; ft, I =1, ...,m —1, which is the involution property of the families of integrals of motion including the Hamiltonian of the model. Using Theorem 1 one can establish complete integrability in the rapidly decreasing case by constructing a canonical transformation to variables of action-angle type ([9], [12]). One of the characteristic features of the Hamiltonian formulation is t h a t the r-matrix replaces the zero curvature condition ([25], [9], [12], [35]). THEOREM 2. Suppose that for the matrix U(x, A) in equation (1) the condition (3) holds. Then for the generic Hamiltonian equation ^ ^ L = { t r ^ M , W"(x)}, a=l9...,M, (10) where \i is a parameter, the zero curvature condition holds with TJx(x, A) = U(x, A) and U0(x,X,p) =t?x[(TL(L,x,iJL)®I)r(p-X)(TL(x^ ~L9i*)®I)). (11) Here trx stands for the trace in the first factor in C™® Gm. 3 . Symplectic structure associated with an ^-matrix. Formula (3) can also be interpreted as the way of defining the symplectic structure on the phase space parametrized by the functions y>a(x), a = 1 , . . . , M. The skewsymmetry of the Poisson brackets is provided by the condition r(X) = —Pr(—X)P and the relation {rX2(X-p), r13(A)] + [r 1 2 (A-^), r 2 3 ( ^ ] + [r13(A), r23(p)] = 0 (12) guarantees the Jacobi identity. Here P is the permutation matrix in Cm®Cm and r12 denotes the matrix in Cm®Cm®Cni which acts trivially in the third factor and coincides with r in the product of the first two (analogously for r13 and r 23 ). The relation (12) is called the classical YangBaxter equation (or the classical triangle equation) and is quite popular nowadays'([5], [22]). Thus, in [5] the solutions of (12) associated with simple Lie algebras were constructed. As a by-product of the* study of this equation a new object, the Lie-Hamilton group, has appeared in [6]. 4. Geometrical interpretation of the r-matrix Poisson brackets. There exists a very elegant interpretation of the r-matrix in the language of the representation theory ([9], [11]): the Poisson brackets defined by an Integrable Models in Classical and Quantum Field Theory 1335 r-matrix of the form (8) are just the Lie-Poisson brackets for an infinitedimensional Lie algebra. More precisely, let g be the finite-dimensional semisimple Lie algebra. The Lie-Poisson bracket on the phase space g* is {/,<?}(£) = y°a>c^-§f„ dL ôh ài fefif, (13) where öabß are the structure constants of g. The brackets (13) can be naturally lifted to define the Poisson brackets for the functionals on </* —the dual space of the current algebra g. The latter is just the Lie algebra of the Laurent series in the variable A with coefficients in g. These brackets are also defined on the dual space of the subalgebra gf+ consisting of the Laurent series in the negative powers of A; the same holds for the complementary subalgebra gL. Moreover, let Kab be the matrix of the Killing form in the basis Xa in g, let Kab be its inverse and n = ^KabXa®Xb. (14) a,b Then we have the following theorem ([9], [11]). THEOREM 3. The Poisson brächet for the generic element TJ(l) eg\ is {U(A)®U(V)} where r(X) = [r(A—i»)f 17(A)®I + I® U(t*)]9 (15) =nß. Introducing the ^-dependence in g (roughly speaking, considering g = f] ®g)> we obtain from (15) the relation (3). iceR1 This approach leads to various integrable models if one considers suitable orbits in g\ (or in f 1). Thus the HMmodel corresponds to g = su(2) and the simplest orbit consisting of the points U(x9 A) = 8(x)ß9 82(x) = const I. Here we have a relationship with the representation theory and the method of orbits. If g has nontrivial automorphism <r, then the phase space can be reduced by considering quasiperiodic elements 00 l7(a>,A)= JT AnU{x,K + nco)A-n, tt=*-oo 32 — Proceedings..., t. II \ (16) 1336 Section 13: L. A. Takhtajan where A is the representation of a. If the matrix II commutes with A ® A9 then for V(x9 A) the relation (3) holds, with the r-matrix ZJ v A + wco ' Thus, in particular, the r-matrix of the SG model is obtained from the r-matrix of the HM model. For the Lie algebras of An type a second averaging is possible. Thus one can obtain r-matrices expressed in terms of elliptic functions. 5. The lattice case. Classical models on a lattice play an intermediate role in quantizing classical continuous models ("quantization of the auxiliary space"). The matrix U(x9 A) is replaced by the transition matrix Ln(X) from the wth lattice site to the (w+l)th. I n the continuous limit Ln(X) = I+AU(x9 A) + 0(A2)9 where A is the lattice spacing. The monodromy matrix is given by the ordered product N TNW = [JLn(X) = LN(A)... LX(X), (18) where JV is the number of sites of the lattice. Since Ln(X) is the transition matrix for one lattice site, formula (3) is uniquely transferred to the lattice case as follows: {Ln(X)®Lm(p)} = [ r t f - j i ) , Ln(X)®Lm(p)1ònm (19) and in the continuous limit (19) goes back into (3). All continuous models have their lattice variants with the same rmatrices. The averaging procedure also works for the lattice case. I n contrast with the continuous case, in the lattice case the matrix Ln(X) is represented as an ordered product ([11]) and not as in (16). Here we have a connection with the theory of analytic matrix-valued functions. As a result of the study of equation (19), in [26] there were introduced quadratic Poisson brackets algebras. These brackets are nontrivial deformations of the Lie-algebraic Poisson brackets. Here we have an interesting example of a deformation of algebraic structures. Integrarle Models in Classical and Quantum Field Theory 1337 II. Quantum theory We begin with lattice theories, which necessarily a.rise in quantizing compact models, i.e. models with Poisson brackets algebras associated with compact Lie algebras (e.g. the HM model). Moreover, introduction of the lattice plays the role of ultraviolet regularization of continuous models. 1. The fundamental relation with the quantum JS-matrix. Instead of the classical fields yf(x) we consider the field operators W^, which act irreducibly on the Hilbert space l)n, the space of quantum states at the nth lattice site. The complete Hilbert space of the model on a lattice with N M sites is $N = f] ®1)n. In quantization it is natural to replace Ln(X) by the matrix-operator Ln(X) which is a matrix in Cm with matrix elements belonging to the ring generated by ¥£ and depending on the spectral parameter A. The problem of the right generalization of our main relation (19) to the quantum case is far from being trivial. The study of concrete models ([25], [28], [33]) suggests the following generalization of (19), explicitly introduced in [28], [33]: B(A-t*)(Ln(?i) ®Ln(fi)) = (LJp) ®Ln(A))B(A-p). (20) Here the tensor product refers only to the auxiliary space Cm and B(A) is a matrix in Cm®Cm called the quantum B-matrix. If B(A) =P(I — — ihr (A)) + 0 (%2) as Ti->0, where % is the Planck constant, then, in the quasiclassical limit relation, (20) goes back into (19). The uniqueness of such a deformation of the Poisson brackets as that given by (19) is an open question. As in the classical case, relation (20) leads to a generalization of standard mathematical objects. In [26] it was used to introduce some nontrivial deformations of the universal enveloping algebras of Lie algebras, the so-called SMyanin quadratic algebras. The quantum monodromy matrix TN(À) is introduced by formula (18), where matrices Ln(X) are replaced by LU(A). A remarkable property of the quantum U-matrix is that it gives a compact form of all commutation relations of the matrix elements of TN(A). The following simple result holds. 1338 Section 13: L. A. Takhtajan THEOREM 4 (a quantum version of Theorem 1). It follcnosfrom the relatim .(20) that B(X-p)(TN{X)®TNM) = (TN(V)®TN(A))B(A-V). (21) In particular, one has ltTTN(A),tvTN(t*)]=0, (22) where tr denotes the trace in the auxiliary space Cm. Thus the operators trT^A) form a commutative family, which is the family of quantum^ integrals of motion. From the associative property of the tensor product one obtains a sufficient condition for admissible JE-matrices (I®i2(A-^))(i2(A)®I)^ (23) This relation is called the Yang-Baxter equation (or the triangle equation and also the factorization equation). It occurs in statistical mechanics [1] as the commuting condition for transfer-matrices and in the scattering theory [39] as the factorization condition for ^-matrices (see also [33], [22]). Lately this equation has become rather popular; its solutions and methods for constructing them can be found in [4], [21], [22]. This equation also has connections with algebraic geometry [19]. We shall now consider the case of m = 2 more thoroughly. The simplest solution of equation (23) has the form w) ~ i^+aL (24, A + rç and corresponds to the models HM and NS. Here rj is a parameter (the coupling constant). Other solutions can be obtained by a quantum analogue of the averaging procedure. Namely, we put 01(A) =B(A)P and , 00 MW = fi ((r^IWA + ^Ktv»®!), (25) n=—oo i(A) = Y\ K®I)@{X + nco2)(crïn®I) (26) Integrable Models in Classical and Quantum Field Theory 1339 for Im(o)2/co1) > 0 . Then the matrices B(X) ^0t(X)P and B(X) = m(X)P satisfy the Yang-Baxter equation and correspond to the S G model and the Xr^-Heisenberg model, respectively (see below). 2. The local vacuum and the algebraic Bethe Ansatz. In addition to the matrix-operator Ln(X) and the quantum JK-matrix, a local vacuum is another important object of the quantum inverse scattering method. This is a vector con e ì)n characterized by the property a(X) * coni 0 0(A) t Ln(X)con (27) where a(X) and (5(A) are some functions of A. The reference state QN N = fj ® con has an analogous property with respect to the monodromy matrix. Using the existence of the E-matrix and local vacuum it is possible to give a general procedure for diagonalizing the operators trTN(X) ([28], [33]), which is the algebraic background of the method. Namely, the following statement holds. THEOREM 5 (the algebraic Bethe Ansatz). Let TNW = YAM BN{X)' 0N(X) BN(X). (28) and suppose that there exist a local vacuum and a reference state. Moreover9 suppose that in the basis in C2 ® C29 associated with the basis in C2 where (27) holds9 the B-matrix has the form B(X) = -1 0 0 0 0 6(A) o(A) 0 0 c(X) &(A) 0 .0 0 0 (29) 1 where 6(A)/c(A) is an odd function of A. Then the vectors XFN(XX9..., Xj) = BN(XX)... BN(Xl)QN (Bethe vectors) are the eigenvectors for the operators tvTN(X) = AN(X) + JDN(X) with the eigenvalues + ^ùié^ ^n 1 c{X}-X) 1 1 o(X — L) (30) 1340 Section 13: L. A. Takhtajau if the parameters XX9..., Xx satisfy the system of equations <5"(A,) - l l c(A,-A,)' 3- 1 '-' 1 - (31> k& This theorem can also be generalized to the case m > 2. As we have pointed out before, txTN(X) form a family of commuting quantum integrals of motion. I t contains also the Hamiltonian of the model. The simplest expression for the Hamiltonian occurs in the case of fundamental models, where the quantum space is isomorphic to the auxiliary space. I n this case there exists a point X = X0 for which the operator trTN(X) is proportional to the cyclic shift operator i n § ^ (the quasimomentum operator). The Hamiltonians with interaction of Jc + 1 nearest neighbors on the lattice are expressed in terms of the operators jffÂ== "^IOgtrTiv(A)lAs=;l«, *=l,2,--.,tf-l. I n addition, ô(XQ) = 0, and so the spectrum of Hk is additive. For nonfundamental models the construction of local Hamiltonians requires additional tools ([8], [20], [36]). 3 . Characteristic examples. 1. The isotropic HM of spins s, 2s e Z (XXXmodel; see [8], [9], [20], [21], [23], [32], [34]). Here ì)n = C 2 s + 1 and the matrix Ln(X) looks like (31) 1 where — 8", a =1,2,3 are generators of an irreducible representation of sn(2) in \)n and 8% = J8£±*<8£. The ^-matrix has the form (24), where tj — i. The Bethe vectors are the highest weights [34] with respect to the action of su(2) in $)N and the system of multiplets associated with them is complete [16], [32]. In proving this there arise nontrivial combinatorie identities [16]. The Hamiltonian has the form ([20], [21]) Hg** = s %fs«Sn, #»+1», (32) Integrable Models in Classical and Quantum Field Theory 1341 where 8N+1 = Sx and f8 is a polynomial of degree 2s, characterized by 2s the conditions f8(l(l+l)l2-s(s+l)) = £ (Ijh), I = 0 , 1 , . . . , 2s. In the xx quasiclassical limit H$ ) goes into the Hamiltonian of t h e lattice HM model. 2. The lattice ITS model ([13], [15]). Here l)n = ^ ( f i 1 ) and the matrix Ln(X) is obtained from (31) by a left multiplication by CTQ. The operators 8% are now t h e generators of tho irreducible infinite-dimensional representation of su(2) of spin s = —2\KA, where n is the coupling constant and A is the lattice spacing. The operators 8% are expressed in a standard way in terms of t h e usual creation-annihilation operators in J)n. The J2-matrix has the form (24), where rj = — in and the Hamiltonian is given by (32), where <$ w , 8n+1} should be replaced by (cr8n, 8n+1}, a being an involution of su(2). The function fs naturally interpolates the polynomial from (32) to the case of nonintegral 2s ([36]). 3. The lattice SG model ([14], [15], [28]). Here the field operators are unitary operators un and vn satisfying Weyl's commutation relation u v n n = exP(*V)v7i^»j where y = ß2/8, ß being the coupling constant; i)n - Se2(R£\2TzZ) if y ^ 2izp\q and \% = C* otherwise. The matrix LJX) has the form Ln(X)- (33) where m2A2 / , ^ mA \1/2 t x 9(vi A) = —7- ( 0 0 . . — e*v) and m plays t h e role of mass. The JE-matrix has the form (29), where .... imiy sh(A + iy) 7 N ' shA Bh(X + iy) In .contrast to the previous examples, a local vacuum exists only for the product LnJhl(X)Ln(X) and not for individual matrix Ln(X). The Hamil- 1342 Section 13: L. A. Takhtajan tonian has a more complicated form than (32), but in the continuous limit goes into the regularized version of the Hamiltonian of the SG model ^'- /(T*+T(£)' + £-< I —«h m where [cp(x)9 uz(y)] = id(x—y). 4. The anisotropic HM model of spins J (XYZ model; see [2], [3], [31], [33]). This model is naturally related to the eight-vertex model of the classical statistical mechanics on the two-dimensional lattice [1], [33]. Its Hamiltonian is equal to H(XYZ) ^ ^(JÄ^4+1+J,^^+1+Ja^4+1), (35) where the periodic boundary conditions are assumed. The matrices B (X) and Ln(X) for these models have a more complicated form than (29) and (31) and are expressed in terms of elliptic functions (see [1], [31], [33]). The diagonalization procedure for trTN(X) requires more complicated technical tools, but expressions for the eigenvalues of trTN(X) and the system of» equations (30) in Theorem 5 retain their algebraic form ([3], [31], [33]). 4. Thermodynamic limit. Here we shall briefly consider the thermodynamic limit which is the limit as JV->oo for compact models and as L->oo, A->0 for noncompact ones. The behavior of models in this limit is most interesting from the physical point of view. The main problem here is to define the ground state — the eigenvector of the Hamiltonian with the minimal eigenvalue (the lowest energy vector) and to describe the Hilbert space of states near it — the space of low-lying excitations. There are two possibilities. 1) The ferromagnetic case : the reference state is the ground state (this occurs for the NS model and the HM model in the case e > 0). Then in 00 the Fock space $F cz g^ adjoining to the vector Q = [J ®coniox XeR1 in a weak sense there exist limits A(X) ='lim a"~N(X)AN(X)9 B(X) N-+OQ Integrable Models in Classical and Quantum Field Theory 1343 = lim BN(X). These operators satisfy the commutation relations N-+OQ [AW,A((i)l=0, A MBM = „, [JS(A),5(^)]=0, \ ,-m BMAWi (36) c(fi~ A — *0) A(X)Q = Q. With the help of these formulae the spectrum and eigenvectors of the operators A (A) are easily obtained ([25], [27]). The commuting family log.A (A) has an additive spectrum and contains the limiting Hamiltonian of the model HF = lim (HN — FQ(N)IN). Here BQ(N) is the ground state JV-+0O energy and IN is the unit operator in §>N, 2) The antiferromagnetic case: to the ground state there corresponds a special distribution of X19 ..., Xx in the Bethe vector (this occurs for the SG model and the HM model in the case e < 0). As J^-^oo, parameters Xx,..., Xt become uniformly distributed on the real axis with the density g (A). This is "the filled Dirac Sea". (This situation occurs in the quantum field theory and in solid state physios and is called "filling of the vacuum".) The function Q(X) satisfies a linear integral equation which follows from the system (30) oo 27ce(A)+ f 0(i-rìe([i)dii=fw, (37) — 00 where % aX c( —A) % aX a(A) Excitations above this ground state are characterized by the density n 1 Q(X, XX, ..., AJ = Q(X) + — • ^?o(X — Xj), where cr(X — [i) is the resolvent kernel of equation (37), and parameters Xx,..., Xn appear as the holes in the Dirac Sea. In these terms the ground state and the excitation creation operator, respectively, have the form ^rcmna = K m exp J5T f logBN(X)e(X)dX Q (38) 1344 Section 13: L. A. Takhtajan and B(X) = lim exp J'logBN(p)a(X-p)äft (39) N-+CO The operator A(X) is defined as in the previous case and A(X) and B(X) satisfy the commutation relations of the type (36) with c(X) replaced by c(X) = exp J logc([j)cs(X—ii)d[i (40) As before, the commuting family logA(A) has an additive spectrum and contains the renormalized Hamiltonian of the model. Using this method, it is also possible to describe the scattering of excitations and to calculate the ^-matrices. The in- and out-states are constructed with the help of operators Z(X) = B(X)A~1(X) (Z(X) ~B(X)Ä~l(X)) satisfying simple commutation relations which follows from (36) (see [34]). I n papers [15], [17], [25], [28], [32]-[34] this general scheme was applied to the detailed investigation of concrete models. I n particular, in [28] it was used to obtain an exact non-pertubative solution of the SG model. With proper modifications this approach can be generalized to the case of models with the auxiliary space Cm, m>2. Finally, let us mention the latest achievements of the quantum inverse scattering method: (a) The exact calculation of norms of Bethe vectors [18], which gives us a hope to obtain explicit expressions for Green's functions; (b) The quantum variant of the equations of inverse problem [29], which provides an expression of Heisenberg field-operators W* in terms of the operators A (A) and -B(A); (c) A new approach towards the integrability of the quantum 0 (3) nonlinear cr-model [10]. References [1] Baxter E. J., Partition function of the eight-vertex lattice model, Ann. Phys. 70, No. 1 (1972), pp. 193-228. [2] Baxter E. J., One-dimensional anisotropic Heisenherg chain, Ann. Phys. 70, Ko. 2 (1972), pp. 323-337. [3] Baxter E. J., Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain, I-III, Ann. Phys. 76, No. 1 (1973), pp. 1-71. [4] Belavin A. A., Discrete groups and integrability of quantum systems, Funk. anal, i pril. 14, No. 4 (1980), pp. 18-26. [5] Belavin A. A. and Drinfeld V. G-., On the solutions of classical Yang-Baxter equation for simple Lie algebras, Funk. anal, i pril. 16, No. 3 (1982), pp. 1-29. Integrable Models in Classical and Quantum Field Theory 1345 [6] Drinfeld V. Gr., Hamiltonian structures on Lie algebras, Lie bialgebras and geometrical meaning of Yang-Baxter equations, DAN SSSB 268, No. 2 (1983), p p . 285-287. [7] Faddeev L. D., Quantum completely integrable models in field theory, Sov. Sci. Bev. CI (1980), pp. 107-155, ser. Oontem. Math. Phys. , [8] Faddeev L. J)., Eecent developments of QST, BIMS, Kopyuprolcu 469 (1982), pp. 53-71. [9] Faddeev L. D., Integrable models inl-\-\ dimensional quantum field theory, Saclay preprint S. P h . 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LENINGRAD BRANCH OF THE V. A. STEKLOV MATHEMATICAL INSTITUTE OF THE ACADEMY OF SOIENOES USSR