Sine–Gordon Model in the Homogeneous Higher Grading

A construction of equations and solutions for the sine–Gordon model in the homogeneous grading as an example of higher grading affine Toda models are considered.


Introduction and preliminaries
The problem of construction of exactly solvable models and their solutions is a very important problem in the theory of integrable models in general and in application to real dynamical systems in Physics. One of the ways to proceed is to us the the Lie-algebraic method to construct non-linear exactly solvable models in classical regions. This method is very well known and elaborated [10]. Applying the zero-curvature conditions on elements of connection containing Lie algebra generators in appropriate grading subspaces, we obtain systems of equations of motion associated to a specific Lie algebra.
The main motivation to use the theory of Lie algebras in exactly solvable models is its effectiveness. We are able not only to re-construct the equations of motion but also to find exact solutions starting from internal algebraic symmetries based on deep algebraic symmetries of systems under consideration.
In [4] the higher grading generalization to the conformal affine Toda models was considered. Elements of the higher (then number one) grading subspaces are taking into account while connection elements are constructed. The main example was the principal grading case.
In this paper we conside an alternative, new case, which is corresponds to the homogeneous grading of the Lie algebra. We derive the systems of equations generalizing the case of the sine-Gordon equation and provide quantum group solutions.

Affine Kac-Moody Lie algebras
In this subsection we recall some facts about affine Kac-Moody algebras [7], [4]. Consider an untwisted affine Kac-Moody algebra G endowed with an integral grading G = n∈Z Z G n , and denote G ± = n>0 G ±n . By an affine Lie algebra we mean a loop algebra corresponding to a finite dimensional simple Lie algebra G of rank r, extended by the center C and the derivation D. According to [7], integral gradings of G are labelled by a set of co-prime integers s = (s 0 , s 1 , . . . s r ), and the grading operators are given by Here H s ≡ r a=1 s a l v a · H 0 , N s ≡ r i=0 s i m ψ i , ψ = r a=1 m ψ a α a , m ψ 0 = 1. H 0 is an element of Cartan subalgebra of G; α a , a = 1, 2, . . . r, are its simple roots; ψ is its maximal root; m ψ a the integers in expansion ψ = r a=1 m ψ a α a ; and l v a are the fundamental co-weights satisfying the relation α a · l v b = δ ab . The principal grading operator Q ppal is given by and α (m) are positive roots of height m, and C is the center. The element B is parameterized as B = e ϕ·H 0 e ν C e ηQ ppal = e ϕ·H 0 eν C e ηQ ppal , wherẽ H 0 was defined in [4] as H 0 , and l a being the fundamental weights of G. Let us denote by H n , E n ± , D, C the Chevalley basis generators of sl 2 . The commutation relations are where C is the center. The grading operator for the principal grading (s = (1, 1)) is 1.2. Quantized universal enveloping algebra U q( sl 2 ) In the spirit of [2], [5], the quantised enveloping algebra U q (sl 2 ) is an associative algebra generated by X + , It possesses a Hopf algebra structure with the deformed adjoint action (ad X ± ) q a = X ± aq H/2 − q ∓1 q H/2 aX ± , (ad H ) q a = Ha − aH, for all a ∈ U q (sl 2 ). Let us recall the second Drinfeld realization of the quantized universal enveloping algebra U q ( sl 2 ), (i.e., U q ( sl 2 ) without grading operator) [2], [6], which is a natural quantum analogue of the algebra sl 2 in the loop realizations. U q ( sl 2 ) is an associative algebra generated by {x ± k , k ∈ Z Z; a n , n ∈ {Z Z−0}; γ ± 1 2 , K}, where γ ± 1 2 belong to the center of the algebra, satisfying the commutation relations [K, a k ] = 0, [a k , a l ] = δ k,−l The generators φ k and ψ −k , k ∈ Z Z + are related to a k and a −k by means of the expressions It is easy to define the grading operators corresponding to the principal and homogeneous grading of U q ( sl 2 ) by analogy with the grading of U q (G) where G is a simple Lie algebra. The principal grading can be realized with the help of the operator and λ is an affinization parameter. The power of λ is denoted by the subscript of U q ( sl 2 ) generators. Then the grading subspaces are q G 0 = {K, γ}, The grading operator for the homogeneous grading is D h x = 2λ d dλ x, so that the grading subspaces are n , x − n , a n , n ∈ {Z Z − 0}}. The level one irreducible integrable highest weight representation of U q ( sl 2 ) can be constructed in the following way [6]. Let P = Z Z α 2 , Q = Z Zα be the weight/root lattice of Its submodules are isomorphic to irreducible highest weight modules V (Λ n ) with the highest vectors |Λ n = |1 ⊗ e αn 2 , n = 0, 1.

Higher grading affine Toda system
In this and the next sections we recall [4] The components A ± are the following (we keep notations of [4]) for k = ±: Here B is a mapping M −→ G 0 ( G 0 is a group with the Lie algebra G 0 ) and where E ±l are some fixed elements of G ±l and F ± m ∈ G ±m , (1 ≤ m ≤ l − 1). Substituting 3 into 2 one arrives at the equations of motion Since Q s , C ∈ G 0 then B can be parameterized as B = b e η Qs e ν C where b is a mapping to G 0 , the subgroup of G 0 generated by all elements of G 0 other than Q s and C. Substituting B into the equations of motion (4-5) one has 2.1. The case l = 1 Now consider the case l = 1. Let us parameterize the element B in the homogeneous grading of G, [4]. From the equations (6-9) for an infinite dimensional Lie algebra G in the principal grading we obtain the affine Toda field theory systems of equations Let define for representation vectors |ξ i (see below) and a group-like element g. The formal general solution to the above equation was introduced in [12]: The general solutions to the matter fields F ± i may be written in the following form. For m = 1 in (6-9) one has [4] Here |i; i denotes an element of the Verma module which is result of the action of the lowering generator on the highest state vector. The fact that (11) is indeed a solution to (10) may be checked by using the representation theory of G. A map g : M −→ G appearing in the gradient form of the flat connection A ± = g −1 ∂ ± g, may be factorized (according to the Lie The grading condition provides the holomorphic property of µ ± , i.e., they satisfy the initial value problem with arbitrary functions Φ ±m α (z ± ) determining the general solution to the system. Note that the summations in (13) are performed over the set of positive roots ∆ + m of G = m∈Z Z G m in the subspace G m .

Soliton solution for the sine-Gordon in homogeneous grading
Another way to construct soliton solutions [13] to the sine-Gordon equation is to consider the formal general solution (10) in the homogeneous grading and to use vertex operators [7] which are related to the homogeneous Heisenberg subalgebra of sl 2 . Take the general solution (11) to the affine Toda system (10). In the homogeneous grading the mappings γ ± can be parameterized as γ ± = e dφ d e cφc e φ ± 0 x ± 0 , where d is the grading operator, c is the center of sl 2 and x ± k are generators of the subspaces G k corresponding to the homogeneous grading. The mappings µ ± satisfy (12) where κ ± (z ± ) = a ±1 + φ ± x ± 1 . In order to obtain a soliton solution we put φ ± = 0, φ ± 0 = 0. Then the general solution reduces to e −βφ(z + ,z − ) = J(g a,µ , |Λ 1 ) where g a,µ = e a +1 z + µ(0)e a −1 z − The following group element µ(0) in (14) x 2n /n) = exp log(1 − x 2 ) . When x = 1, X(x) vanishes which results in Φ(ζ)·Φ(ζ) = 0. Therefore the exponential of Φ(ζ) operator terminates after the first order.

Homogeneous higher grading generalization of the affine Toda model
The Lie-algebraic way to construct non-linear exactly solvable models in classical regions is very well known and elaborated [10]. Applying the zero-curvature conditions on elements of connection containing Lie algebra generators in appropriate grading subspaces, we obtain systems of equations of motion associated to a specific Lie algebra. In [4] (of which we keep notations) the higher grading generalization to the conformal affine Toda models was considered. Elements of the higher (then number one) grading subspaces are taking into account while connection elements are constructed. The main example of [4] is the principal grading case. In this paper we consider an alternative, the homogeneous grading case. We derive the systems of equations generalizing the case of the sine-Gordon equation and provide quantum group solutions.
We start with the equations (6-9) of [4] (see subsection 2). Consider the case l = 1. In the principal grading we obtain from (6) the sine-Gordon equation. Recall that In the homogeneous grading of G the grading subspaces are G n = H n , E n ± . We take Consider a particular case when we parameterize the group element b as b = e φH 0 .
Then, substituting (15) and (16) into (6-8) we get the following system of equations ∂ ± φ = e η e −2φ − e 2φ , ∂ ± ν = −e η e 2φ + e −2φ , ∂ ± η = 0, i.e., in the first equation is again the sine-Gordon equation. The solution to the field φ is then the standard classical solution (11), [12] (see subsection 2). Now consider the case l = 2. The equations corresponding to the principal grading can be found in [4]. Here again we take b = e φH 0 though it this is not the most general choice of the group element parameterization since it does not contain dependence on the E 0 ± elements from G, i.e., we have send corresponding fields near those generators to zero. Let us also put Then the system (6-8) gives the following system of equations: The formal general solution to (17-19) age given in [4]: for the φ field and for the F ± In the homogeneous grading case, taking into account the parameterization of b element, we get Here we have made use of the properties of the i-th fundamental representation (corresponding to the homogeneous grading) of the Lie algebra sl 2 . Thus, using (20) we get

Dirac equations
Let's switch notations similarly to Dirac field components, i.e., Now use the extra conditions (19), substituting them into (17). Then we see that the second summands in the first two formulae in (17) vanish, i.e., the final equations are i.e., equations (21) do not differ much from the corresponding equations with l = 1. Now suppose that η = η 0 = const. Then substitute the last four equations of (18) on f = ψ fields into the first two on κ ± fields. Then we get that can be rewritten as where ω R,L = ψ R,L + ψ R,L , τ R,L = ψ R,L − ψ R,L . The upshot is that using such a parametrization b = e φH 0 we arrive at three systems of sine-Gordon like systems when η is a constant. where (29) When κ + = 0 and κ − = 0 the system of equations is more complicated.

Solitonic solutions from general solutions
In [12] it was shown how to extract solitonic solutions from the formal general solutions of the affine Toda field equations. Let's take γ ± 0 = 1 in (11) to be a constant function. Then the mappings µ ± are µ ± = µ 0 ± e z ± E ± with µ 0 ± being some fixed mappings independent of z ± . Next takeẼ ± in 13 as E ± ≡ E ±l + l−1 N =1 c ± N E ±N where E ± are elements of a Heisenberg subalgebra of G, namely [E + , E − ] = ΩC. One can consider principal of homogeneous Heisenberg subalgebras for that purpose. In this paper we only deel with the principal case while the homogeneous case will be discussed elsewhere. Thus, we arrive at a special solution to (10) for g E,µ = e x ± E ± µ 0 e x ± E ± . In order to compute these solutions explicitly we have to remove E ± -dependence from (30) moving E + to the right and E − to the left . Then we should find such a µ 0 = N i=1 e V i so that V i would be eigenvectors with respect to the adjoint action of E ± , i.e., Then it turns out [12] that resulting expressions provide us with solitonic solutions to the equations under considerations while parameters ω (i) ± characterize solitons.

Quantum group soliton solution for sine-Gordon in homogeneous grading
As in [11] one can show that the affine Toda models are co-invariant with respect to the lightcone quantization. Namely, the equation of motion are preserved in form though a standard normal ordering has to be introduced as well as some infinite constant comming from quantum versions of Lax pair to generate equations using Lie algebra elements in quantum case. At the same time infinite constants do not appear in final formal solutions to the light-cone quantized versions of equations. In order to find quantum solutions, one has to replace [3], [8], [9] group elements as well as state vectors formal general solutions by their quantum group counterparts.
In this subsection we write examples of quantum group solutions to the quantized affine Toda model in the specific case of the higher grading sine-Gordon equation (the cases l = 1, 2, 3). Recall [13], that the homogeneous grading subspaces of U q ( sl 2 ) are n , x − n , a n , n ∈ {Z − 0}}.
7.1. The case l = 1 From the commutation relations for x ± m and a m (see subsection 1.2) it follows that in this realization of the quantum group U q ( sl 2 ), the generators x ± m , a m ∈ G m , x ± 0 ∈ G 0 . The solution and the homogeneous grading quantum vertex operator is .
Using the fact that [5] k ζ −k , φ − , k > 0, we commute exp(−a 1 z + ) with exp(Qφ − ) to the right and exp(Qφ − ) with exp(a −1 z − ) to the left. The commutation of exp(−a 1 z + ) withexp(a −1 z − ) gives exp(−z + z − [2])). Thus it follows that (recall that exp(−a 1 z + ) and exp(a −1 z − ) act on |Λ j and Λ j | as identities). Then we expand exp Qe −q [2k] q − 5k 2 ζ −k , to the left and to the right. Powers of operators φ − act on the second part of tensor product as follows: Thus we have for g Q = e −z + z − [2] exp Qe −q In the limit q → 1 we obtain ordinary soliton solutions.

Conclusions
In this paper we considered the alternative case of the homogeneous grading of the symmetry algebra of the affine Toda systems and, in particular, the sine-Gordon equations. The Liealgebraic method helps us to understand the use pure algebraic nature of exact integrability of the dynamical system under consideration. We show explicitly that even in the new case of homogeneous grading it is possible to derive the equations of motion and provider explicit solutions both in classical and quantum regions. The knowledge of the quantum group structures underlying the homogeneous higher grading case leads to specific form of the quantum group soliton solutions for the sine-Gordon equation. As possible way to develop, we could mention the search for the explicit solutions associated to all higher grading subspaces of the affine Lie algebra.