On some integrable deformations of the Wess-Zumino-Witten model

Lie algebra valued equations translating the integrability of a general two-dimensional Wess-Zumino-Witten model are given. We found simple solutions to these equations and identified three types of new integrable non-linear sigma models. One of them is a modified Yang-Baxter sigma model supplemented with a Wess-Zumino-Witten term.


Introduction
The search for integrable two dimensional non-linear sigma model has known various developments. The early attempts dealt mostly with deformations of the principal chiral sigma model and examples based on the Lie algebra SU(2) were found [1,2]. Later other integrable Wess-Zumino-Witten models, involving the Lie algebra SU(2), were constructed [3][4][5]. The revival of the subject came after the work of Klimčík on the so-called Yang-Baxter deformation of the principal chiral model [7]. More recently Sfetsos presented a method for constructing integrable deformation of the Wess-Zumino-Witten model [8]. Various issues were treated later in the literature  and a nice account of these can be found in [33] and references within. Our interest in integrable non-linear sigma models is motivated by their relation to string theories [34]. The hope is to find more solvable string theories and their spectrum in non-trivial backgrounds along the lines in [35][36][37].
In [38,39] we have given the conditions for the most general non-linear sigma model to be integrable. These were specified in terms of the geometry and the structure of the target space manifold. A general two-dimensional non-linear sigma model is given by the action 1 S = dzdz [G ij (ϕ) + B ij (ϕ)] ∂ϕ i∂ ϕ j . (1.1) The invertible metric G ij and the anti-symmetric tensor B ij are the backgrounds of the bosonic string theory. The equations of motion of this theory arē where Γ k ij and H k ij = 1 2 G kl (∂ l B ij + ∂ j B li + ∂ i B jl ) are, respectively, the Christoffel symbols and the torsion.
The equations of motion can be cast, for all values of the parameter µ, in the form of a zero curvature relation if the space manifold is equipped with two sets of matrices K i (ϕ) and L i (ϕ) satisfying (1.4) The last two equations determine the structure of the space manifold of the non-linear sigma model. On the other hand, the first two relations indicate that the non-linear sigma model is symmetric under a global isometry transformation [40,41] with J = (K i − L i ) ∂ϕ i and 1 The two-dimensional coordinates are (τ, σ) with ∂ 0 = ∂ ∂τ and ∂ 1 = ∂ ∂σ . In the rest of the paper, however, we will use the complex coordinates (z = τ + iσ ,z = τ − iσ) together with ∂ = ∂ ∂z and∂ = ∂ ∂z . Our conventions are such that the alternating tensor is ǫ zz = +1. J = (K i + L i )∂ϕ i being the conserved currents. The zero curvature relation is then the same as the two equations ∂J +∂J = 0 and ∂J −∂J + J ,J = 0.
Athough the conditions (1.4) specify the geometry of the manifold [39], their general solutions are not yet known. In this note, we continue this program and consider simpler non-linear sigma models. Namely, the most general integrable deformation of the Wess-Zumino-Witten (WZW) model. The conditions (1.4) are now more tractable. They are in the form of a Lie algebra valued relation which generalises the Yang-Baxter equation used in [7] and the integrable deformations of the principal chiral model [42]. We are able to find a solutions to this integrability condition. This leads to an integrable two-dimensional non-linear sigma model in the form of a two-parameter family of integrable deformations of the Wess-Zumino-Witten model. Our result might be a generalisation of the two-parameter integrable deformations of the WZW model found in [43]. Indeed, the two constructions coincide for a special case and we conjecture that our work contains more integrable models.
The paper is organised as follow: In the next section we give in details the steps leading to the equivalent relation to (1.4) for the case of the general Wess-Zumino-Witten model with a summary of the results at the end. For completeness, we show in section 3 how the Yang-Baxter integrable sigma model is obtained as a particular case of our construction. In section 4, we construct the solution to the integrability conditions and give, in section 5, the corresponding integrable non-linear sigma models.

The general construction
We consider the two-dimensional non-linear sigma model as defined by the action where M is a three-dimensional ball having x µ , with µ = 1, 2, 3, as coordinates and ∂M is the boundary of this ball with coordinates z andz. The bi-linear form < , > G is the Killing-Cartan form on the Lie algebra G and the field g(z ,z) is an element of the Lie group corresponding to G. The Lie algebra is of dimension n. The Wess-Zumino-Witten term comes with a parameter λ. The Lie algebra G is defined by the commutation relations [T a , T b ] = f c ab T c . For a semi-simple Lie algebra the Killing-Cartan form is η ab = f d ac f c bd and we have < T a , T b > G = η ab = Tr (T a T b ). However, for a non semi-simple Lie algebra the bi-linear form is such that < T a , T b > G = η ab with η ab an invertible matrix satisfying η ab f b cd + η cb f b ad = 0. The two quantities M and N are linear operator acting on the generators of the Lie algebra G. They are required to satisfy the relation for any two elements X and Y in the Lie algebra G. In other words, M is symmetric while N is anti-symmetric with respect to < , > G .
Putting indices, the action of M and N on the generators {T a } of the Lie algebra G is where η ab is the bi-linear form corresponding to the Lie algebra G as stated above.
It is useful to introduce the two quantities In terms of A andĀ, the equations of motion of the model take the form Multiplying this equation by g on the left and g −1 on the right, we get the conservation equation where we have defined the two currents J andJ as (2.7) Here the two linear operators P −1 and Q −1 , acting on A andĀ only, are defined as where h is a constant group element. It is, of course, assumed that the two linear operators P and Q are invertible. Hence, the inversion of (2.7) gives (2.10) However, the two currents A andĀ satisfy the Cartan-Maurer identity In terms of the currents J andJ, after a use of (2.10) and (2.11), one finds the identity (2.12) We have added and substracted the term proportional to the constant ε. At this stage ε is just a bookkeeping device but will later join the constant λ to form one of the deformation parameters λε.
In order to have an identity that is suitable for the concept of integrability, we demand that the linear operators P and Q are such that the last four terms in (2.12) vanish. That is, (2.13) Since the quantities g −1 Jg and g −1J g take values in the Lie algebra G, this last equation is equivalent to requiring that for any two Lie algebra elements X and Y . Notice that the constant ε can be absorbed by a rescaling of the two operators P and Q (which amounts to a rescaling of the two currents J andJ in (2.7)). When this last relation holds, the currents obey the identity If in addition, the operator (Q + P ) is invertible then the two currents J andJ obey the two relations Therefore, in addition of being on-shell conserved, the currents J andJ have zero curvature. These last two equations are the consistency conditions of the linear differential system Here Ψ (z,z, µ) is a matrix valued field. The requirement that this linear differential system is consistent, for all values of the spectral parameter µ, leads to the equations of motion of the non-linear sigma model (2.16). This is preciseley the statement of the classical integrability of a two-dimensional non-linear sigma model [44].
Finally, in terms of the linear operators P and Q, the relation (2.2) involving the bi-linear form < , > G becomes upon using (2.8) (2.18) By writting X = P Z and Y = QW , where X, Y , Z, and W are in the Lie algebra G, this last relation becomes

Summary :
Given two linear operators P and Q (we assume that P , Q and P + Q are invertible) on a Lie algebra G and satisfying, for any two elements X and Y in G, the two relations then the two-dimensional non-linear sigma model defined by the action is classically integrable. We have used (2.8) to write M + N = Q −1 + λI. The equations of motion stemming from this action are written in (2.16) in terms of two the currents J andJ and are equivalent to the consistency conditions of the linear system (2.17).

The Yang-Baxter sigma model
The so-called Yang-Baxter non-linear sigma model is obtained as a special case of our construction. Indeed, let us first assume that the two linear operators are of the form where R is a linear operator acting on the generators of the Lie algebra G and κ and ζ 2 = −κ (κ + ε) > 0 are two constants. The parameters κ and ε are such ζ 2 is strictly positive. We also put the Wess-Zumino-Witten term in the action to zero. That is, When ζ 2 = −κ (κ + ε), the two relations in (2.20) and (2.21) become then respectively The last relation is known as the modified Yang-Baxter equation while the firt equation says that the linear operator R is anti-symmetric with respect to the bi-linear form. A solution to these relations is given in [6,7] and is briefly recalled in the next section. The corresponding action is obtained upon replacing Q −1 in (2.22) and is given by This is precisely the action found in [7].

Constructing a solution
Our main concern now is to find solutions to (2.20) and (2.21). We start by recalling the commutation relations of a Lie algebra in the Cartan-Weyl basis Here Σ is the set of roots 2 . The generators are normalised such that the Killing form (the bi-linear form) is Since we will use the linear operator R, defined in (3.3), we start by giving its action on the generators of the Lie algebra in the Cartan-Weyl basis as found in [6,7]. This is where Σ + is the set of positive roots and i 2 = −1 (not to be confused with the index i used above). The action of the linear operator R on the generators of the Lie algebra in the basis Here H is the Cartan subalgebra of the Lie algebra G.
It is instructive to illustrate the action of the linear operator R on the generators of the Lie algebra SU (3). The generalisation to other Lie algebras can be figured out in a similar manner. The SU(3) Cartan-Weyl basis is constituted as Using (4.4), one finds that the operator R acts on the SU(3) generators {T a } as (4.6) It is then clear that the matrix R 2 is diagonal with entries equal to either −1 or 0 (zero corresponds to the action of R 2 on the elements of the Cartan subalgebra). The operator R 2 will be needed later.
Lut us now return to the linear operators P and Q. We assume that they act on the generators of the Lie algebra in the Cartan-Weyl basis as where no summation over the repeated index i is implied. The constants σ i and ξ i are real while p and q are complex. In the basis (H i , E α , E −α ), the matrices associated to the operators P and Q are diagonal. Using the commutation relations (4.1), the Killing form (4.2) and the action of the linear operators as in (4.7), the relations (2.20) and (2.21) 10) p = q * − 2λpq * . (4.11) The last two equations give simply σ i in terms of ξ i and p in terms of q (4.13) Upon reporting (4.12) and (4.13) in (4.9) and (4.8) one finds (4.14) Here i 2 = −1 and τ j is defined as We have therefore determined q in terms of ξ j . The two equations in (4.14) are always compatible. Next, the parameter p is calculated from (4.13). Now two paramaters ξ i and ξ j , say, must lead to the same value of q according to (4.14). This means that we must have also Therefore, the parameters ξ i are such that This means that they fall into two sets {ξ 1 , . . . , ξ r−l } and {ξ r−l−1 , . . . , ξ r }, 0 ≤ l ≤ r, and the members of a set are identical. A set could be empty (if l = 0). The corresponding expressions for the parameters σ i are found from (4.12). These two choices for the constants ξ i suggests the splitting of the Cartan subalgebra of G as where H r−l contains the first r −l elements of H and H l the remaining l elements (0 ≤ l ≤ r).

The integrable non-linear sigma model
The linear operators P and Q acting on the basis {T a } of the Lie algebra G are deduced from (4.7) and the solution (4.19). It might be helpful to work out their action on the Lie algebra SU(3) first. For instance, QE α (1) = q E α (1) = (τ ∓ iω) E α (1) implies that Q T 1 = τ T 1 ± ωT 2 and Q T 2 = τ T 2 ∓ ωT 1 , and so on. If we partition the SU(3) Cartan subalgebra as H = H r−l ∪ H l = T 3 ∪ T 8 then we have The matrix corresponding to the operator P can be determined in a similar manner. For the sake of condensing the expressions, we introduce the notation (5. 2) The operators P and Q are given by The linear operator R is still that in (4.4), I is the identity operator and the action of the linear operators Z r−l and Z l on the basis {T a } is The operators Z r−l and Z l act only on the elements of the Cartan subalgebra H = H r−l ∪ H l with 0 ≤ l ≤ r. The next step in our construction is the computation of the inverses of the two operators P and Q. These are block diagonal matrices having either 2 × 2 or 1 × 1 matrices along the diagonal and are easily inverted. Indeed, we have Explicitly, these expressions give The two constants α and β are defined as By eliminating the parameter ξ between α and β, we find that In terms of the parameters α and β, the operators P and Q are as given in (5.3) where We notice that the parameters (τ ′ , ω ′ , γ ′ , ρ ′ ) are obtained from (τ , ω , γ , ρ) by the change λ −→ −λ.
There is another way of writing the operators P −1 and Q −1 . Let Z r be the operator that acts as That is, Z r acts on all the generator in the Cartan subalgebra H. It satisfies the relation Furthermore, it can be seen that Using this last relation, we can write the operators P −1 and Q −1 in the form These are precisely the two situations which are not allowed as can be seen from the solution (4.19). Using the expression of Q −1 in (5.13), our action (2.22) takes then the form (5.14) The parameters α and β are related by (5.8) and λε is another free parameter ( 1 ε is an overall factor). This is the main result of this paper. The above two dimensional non-linear sigma model is integrable. The two current J = g [P −1 (g −1 ∂g)] g −1 andJ = g Q −1 g −1∂ g g −1 , with P −1 and Q −1 as given in (5.13), are conserved and have a vanishing curvature on-shell.
At this stage a remark is due : In the case when l = 0, that is when the set H l = H 0 is an empty set (consequently Z 0 T a = 0 for all T a in the Lie algebra G), the action S 0 (g) is precisely that constructed in ref. [43]. Their parameters, in this case, are related to ours as With this identification, their relation A = η 1 − k 2 1+η 2 is exactly that written in (5.8). In order to explore the novelty of our construction, we find it convenient to rewrite our final action as In reaching this simplified version we have made use of (5.11) and (5.12) and the action of the linear operators Z l and Z r−l is as defined in (5.4). The Cartan subalgebra is split as H = H r−l ∪ H l with 0 ≤ l ≤ r and H 0 is the empty set. As mentioned above, the case l = 0 is already treated in ref. [43] and their integrable non-linear sigma model is given by the action The linear operator Z r = I + R 2 , given in (5.10), acts on all the generators in the Cartan subalgebra H.
In the next section we will point out, by considering specific examples, that the action S l (g), for Lie algebras with rank r ≥ 2, contains deformations of the Wess-Zumino-Witten model that are not accounted for by the action S 0 (g) (the non-linear sigma model of ref. [43]). Hence, this article is a generalisation of the work of ref. [43].

The deformed SU (2) WZW model and beyond
It is instructive to illustrate our construction by first considering the Lie algebras SU (2). For this purpose, let us call the deformation operator. We will also consider α and β as our free parameters instead of α and λε. In terms of α and β, (5.8) gives The deformed WZW action (5.16) is then written as Notice that the coefficient of the WZW term is symmetric under the exchange α ↔ β.
In the case of the SU (2) Lie algebra, with generators {T 1 , T 2 , T 3 } and H = {T 3 }, there are two deformation operators and their action is given by These differ by their action on the generator T 3 . However, by the parameter redefinition α ↔ β, the deformation operators D 0 and D 1 are mapped to each other 3 and, therefore, lead to the same integrable non-linear sigma model. Despite the fact that we have established that the deformations operator D 0 and D 1 are the same (up to a parameter redefinition), we will for completeness give the action for the deformed SU (2) WZW model. The SU (2) group element g is parametrised as For the bi-linear form we take < , > = Tr. The non-linear sigma model corresponding to the deformation operator D 0 is given, up to a total derivative, by the action The deformation operator D 1 yields the same action with the replacement α −→ β. Next, we consider the Lie algebra SU(3). Its Cartan subalgebra is H = {T 3 , T 8 }. The three deformation operators are where A = ∓ √ −αβ as in the dictionary (5.15). We see that D 2 and D 0 are related by the parameter redefinition α ↔ β. However, D 1 and D 0 cannot be related by any parameter redefinition. It seems, therefore, that there are two independent deformations of the SU(3) WZW model, namely S 0 (g) and S 1 (g). This remains though to be verified by an explicit calculation.
In general, one may decompose the Maurer-Cartan one-form along the Cartan-Weyl basis (4.1) as Here ϕ a (z,z) are the n local fields and the index γ runs over the positive roots Σ + . The Cartan subalgebra is partitioned as H = H r−l ∪ H l with 0 ≤ l ≤ r. The indices i (r−l) = 0, . . . , l and i (l) = l + 1, . . . , r are such that H i (r−l) ∈ H r−l and H i (l) ∈ H l . The vielbiens are functions of ϕ a (z,z) and e i (0) a = 0. Using the bi-linear form < , > as given in (4.2) and the action of the operator R in (4.3) together with the action of the operators Z r−l and Z l as defined in (5.4), we find that We see that the non-linear sigma models defined by S l (g), l = 0 . . . r, share the same antisymmetric tensor field (coming from the last two terms) but differ by their target space metric (coming from the first term). It is clear that the two models S 0 (g) and S r (g) are related by the parameter redefinition α ↔ β. Apart from this, we are inclined to conjecture that there are r different integrable models given by S l (g), l = 0 . . . r − 1.

Conclusions and outlook
We have presented in this work an integrable two-dimensional non-linear sigma model. It is a two-parameter deformation of the Wess-Zumino-Witten model. We have found a simple solution to the main integrability equations (2.20) and (2.21) of this article. It remains to see if these relations admit other solutions. The renormalisability of the sigma model studied here and its possible connection to string theories is another interesting subject to be explored. There is a strong link between integrability and gauging as shown in [8,51]. This property is not very neat here. Indeed, the general WZW model (2.22) is related to another theory as follows: The non-linear sigma model as defined by the action S (g , h) = λ ∂M dzdz < g −1 ∂g , g −1∂ g > G is invariant under the constant left multiplication h −→ l h. This can be gauged by introducing a two components gauge field B µ , with µ = z ,z, transforming as B µ −→ lB µ l −1 − ∂ µ ll −1 .
The gauging is carried out by replacing h −1 ∂ µ h with h −1 (∂ µ + B µ ) h. The choice of the gauge h = 1 leads then, after the use of (2.18), to the action S (g , B µ ) = λ ∂M dzdz < g −1 ∂g , g −1∂ g > G + λ 6 M d 3 x ǫ µνρ < g −1 ∂ µ g , g −1 ∂ ν g , g −1 ∂ ρ g > G (7. 2) The equations of motion of the non-dynamical fields B andB are B = (P −1 − 2λI) (g −1 ∂g) andB = Q −1 g −1∂ g . Substituting these into (7.2) we recover our general WZW action (2.22). Now, the equations of motion corresponding to the original action (7.1) are where a = h −1 ∂h andā = h −1∂ h and A andĀ are as defined in (2.4). The operator (P −1 − 2λI) is obtained from the expression of P −1 by simply changing λ to −λ as can be seen from (2.8). These equations of motion, assuming that P and Q obey (2.20) and (2.21), do not seem to derive from some zero curvature conditions. Yet, the gauge fixed action (7.2) leads to integrable non-linear sigma model. This issue deserves to be investigated. As a matter of fact, this remark is true for most of the integrable sigma models found in the literature.
Note added: After the completion of this work we became aware of the existence of ref. [46] where (2.21) was also established. Their construction makes the formulations in [43,47] more compact and is inspired by the works of Klimčík [48][49][50]. Their assumption on the anti-symmetric operator R is that it solves the homogeneous or inhomogeneous classical Yang-Baxter equation. In the case of the usual Drinfel'd-Jimbo solution, the R matrix satisfies the important relation R 3 = −R. They showed, in this particular case, that their integrable non-linear sigma model is precisely that found in [43] (see their section 3.2). Since our R matrix obeys also R 3 = −R, we conjecture that our models with l = 1 , . . . , r − 1 are not covered by the construction of ref. [46].