Yang-Baxter deformations of WZW model on the Heisenberg Lie group

The Yang-Baxter (YB) deformations of Wess-Zumino-Witten (WZW) model on the Heisenberg Lie group ($H_4$) are examined. We proceed to obtain the nonequivalent solutions of (modified) classical Yang-Baxter equation ((m)CYBE) for the $ h_4$ Lie algebra by using its corresponding automorphism transformation. Then we show that YB deformations of $H_4$ WZW model are splitted into ten nonequivalent backgrounds including metric and $B$-field such that some of the metrics of these backgrounds can be transformed to the metric of $H_{4}$ WZW model while the antisymmetric $B$-fields are changed. The rest of the deformed metrics have a different isometric group structure than the $H_{4}$ WZW model metric. As an interesting result, it is shown that all new integrable backgrounds of the YB deformed $ H_4$ WZW model are conformally invariant up to two-loop order. In this way, we obtain the general form of the dilaton fields satisfying the vanishing beta-function equations of the corresponding $\sigma$-models.


Introduction
The study of integrable two dimensional σ-models and their deformations have always remarkable attentions of people from early times of their presentation [1,2]. Integrable deformations of SU (2) principal chiral model firstly presented in [3][4][5]. Then, YB (or η) deformation of chiral model was introduced by Klimcik [6][7][8] as the generalization of [4,5] while the model proposed in [3] was generalized as λ-deformation in [9]. The relation between these integrable deformations was studied in [10,11]. The YB integrable deformations [6] are based on R-operators satisfying the mCYBE and the generalization to models with R-operators satisfying the CYBE (homogeneous YB deformations) was also studied in [12]. The application of these integrable deformation to string theory specially the AdS 5 ×S 5 string model has presented in [13][14][15] (see also [16]). It has been shown that [17] the homogeneous YB deformed models have a Weyl anomaly unless the R-operators are unimodular (see also [18] up to two-loop). In general, it was shown that the deformed backgrounds solve the generalized supergravity equations [19,20] instead of the supergravities ones [21]. The generalization to YB σ-models with WZW term has also carried out in [22][23][24][25][26]. In most of the works, the models have been constructed on semisimple or compact Lie groups. Only in Ref. [23], the YB models on the Nappi-Witten group was constructed. There, it has been shown that the Nappi-Witten model is the unique conformal theory within the class of the YB deformations preserving the conformal invariance. Here we particularly focus on the YB σ-models with WZW term on the H 4 Lie group obtaining from R-operators satisfying the (m)CYBE. We show that YB deformations of H 4 WZW model are splitted into ten nonequivalent backgrounds including metric and B-field such that some of the metrics of these backgrounds can be transformed to the metric of H 4 WZW model while the antisymmetric B-fields are changed.
The plan of the paper is as follows: In order to present the notations, we review in general the YB deformations of chiral and WZW models in Sec. 2. In Sec. 3, after a review of the construction of WZW model based on the H 4 Lie group [27,28], by using the automorphism group of the h 4 Lie algebra we obtain the solutions of the (m)CYBE, i.e. corresponding nonequivalent classical r-matrices. We prove that in general the equivalent classical r-matrices (r-matrices related by automorphism group) lead to equivalent models. After then, we classify all backgrounds of YB deformed WZW model on H 4 in subSec. 3.3. The use of the convenient coordinate transformations (similar to YB deformed WZW model on the Nappi-Witten group [23]) in order to transform the metrics of some deformed backgrounds to the metric of H 4 WZW model is given at the end of Sec. 3. The one-loop conformal invariance of the deformed models is investigated in subSec. 4.1 in such a way that the corresponding dilaton fields are found. In subSec. 4.2, we immediately check the conformal invariance of the models up to two-loop order and conclude that two-loop beta-function equations are satisfied with the same previous dilaton fields. Some concluding remarks are given in the last section. We tabulate the nonzero components of tensors H µνρ , (H 2 ) µν , R µν and Riemann tensor field related to the backgrounds of YB deformed H 4 WZW model in Appendix A. Finally, in Appendix B, by following our present method we improve the results of [23], in such a way that we classify all nonequivalent classical r-matrices and corresponding YB deformed WZW models based on the Nappi-Witten group; moreover, we show that all deformed backgrounds are conformally invariant up to two-loop order.

A review of YB σ-model and YB deformations of WZW model
Before proceeding to review the YB deformations of WZW model, let us introduce the YB deformation of the principal chiral model on the Lie groups.

YB σ-model
In order to make the paper somewhat self-contained, let us first start with the YB deformation of the principal chiral model on a Lie group G (with Lie algebra G ), giving [6] where ∂ ± = ∂ τ ± ∂ σ are the derivatives with respect to the standard lightcone variables σ ± = (τ ± σ)/2 on the worldsheet Σ, and g −1 ∂ ± g are components of the left-invariant Maurer-Cartan one-forms which are defined by means of an element g : Σ → G in the following formula in which T i , i = 1, ..., dim G are the bases of Lie superalgebra G . In Eq. (2.1), η is a real parameter by which deformation is measured. If one puts η = 0, the action reduces to the principal chiral model [1,2]. In addition, the linear operator 4 R : G → G is the solution of equation [12] [R(X), for all X, Y ∈ G . Here ω is constant parameter which can be normalized by rescaling R. When ω = 0, the equation (2.3) is called the CYBE [23]. This equation can be generalized to the mCYBE with ω = ±1. The skew-symmetric condition of operator R is written as The integrability of the model (2.1) is an important property of the model such that the corresponding Lax pair is given by [7,22] where λ is a spectral parameter.

YB deformation of WZW model
In this subsection we shall consider the YB deformation of the WZW model [22]. The corresponding action consists of standard principal chiral model and WZW term based on a Lie group G, giving [22,23] in which κ is a constant parameter, B is a three-manifold bounded by worldsheet Σ, and Ω kl defined by Ω kl =< T k , T l > is a non-degenerate ad-invariant symmetric bilinear form on Lie algebra G with structure constants f k ij which satisfies the following relation [29] f l ij Ω lk + f l ik Ω lj = 0. (2.7) Here the deformed currents J ± are defined in the following way where η andÃ measure a deformation of WZW model. One can see that when η =Ã = 0 and k = 0(k = 1) we recover the action of the principal chiral model(undeformed WZW model) [22], and forÃ = ±η, k = 0 one recovers the YB deformation of chiral model [1,2]. In general, the constant parameter ω classifies integrable deformations so that one may consider ω = 0, ±1 [1,2]. In [22], it was shown that in general the model (2.6) is integrable. This model was then considered for the Nappi-witten group [23]. In the next section, we will consider the model (2.6) for the H 4 Lie group.

YB deformations of WZW model based on the H 4 Lie group and their classification
In this section, we shall solve the mCYBE to obtain the classical r-matrices of the h 4 Lie algebra. Since our goal is the classification of all nonequivalent r-matrices, we prove a Proposition. This Proposition states that two r-matrices r and r ′ equivalent if one can be obtained from the other by means of a change of basis which is an automorphism A of Lie algebra G . We then calculate all linear Roperators corresponding to nonequivalent r-matrices in order to construct the YB deformations of the H 4 WZW model. Finally, by performing convenient coordinate transformations on some of the deformed backgrounds we show the invariance of the H 4 WZW model metric under arbitrary YB deformations, up to antisymmetric B-fields. This means that the effect coming from the deformations is reflected only as the coefficient of B-field.

WZW model based on the H 4 Lie group
In this subsection we shall consider the WZW model on the H 4 Lie group [27,28]. Before proceeding to construct model, let us first introduce the h 4 Lie algebra of H 4 . The Lie algebra h 4 is defined by the set of generators (T 1 , T 2 , T 3 , T 4 ) with the following nonzero Lie brackets The action of ungauged and undeformed WZW model on a Lie group G is given by Accordingly, one needs a non-degenerate bilinear form Ω ij on Lie algebra G of G. Using (3.1) and also formula (2.7), one can get the non-degenerate bilinear form on the h 4 , giving [28] where ρ and λ are some real constants. To construct the WZW action (3.2) on the H 4 , we parameterize an element of the H 4 as Finally, the WZW action on the H 4 is found to be of the form [28] S W ZW (g) = Here we have set λ = 1. Identifying the action (3.6) with the σ-model of the form 5 we can read off the spacetime metric G µν and the antisymmetric B-field. They are then given by the following relations The metric (3.8) has an isometry group, where the generators of the corresponding Lie algebra can be expressed in terms of the Killing vectors K i of the target space geometry. Therefore it is crucial for our further considerations to obtain the Lie algebra of Killing vectors of (3.8). This metric admits a seven-dimensional Lie algebra of Killing vectors, which can be generated by (3.10) One can easily check that the Lie algebra spanned by these vectors is with the center K 7 . The generator K 4 can be interpreted as dilation in y, u. As it is seen, the h 4 Lie algebra, e.g. generated by (K 1 , K 2 , K 6 , K 7 ), is a subalgebra of (3.11).

Classical r-matrices for h 4 Lie algebra
According to the formulas (2.6) and (2.8), to obtain the YB deformations of the H 4 WZW model one needs the linear operators R associated to classical r-matrices of the h 4 Lie algebra. Before proceeding to this, let us consider the general form of classical r-matrix of a given Lie algebra G with the basis where r ij is an antisymmetric matrix. One may associate a linear operator R to a r-matrix that satisfies the mCYBE (2.3). This operator can be defined in the following way [23] Based on this, the action of R on any element X = x k T k ∈ G is written as Considering and then comparing (3.13) and (3.15), one gets Now, making use of formulas (2.7) and (3.14) and after some algebraic calculations, one can write Eq. (2.3) in the following form [23] f lm k r li r mj + f lm i r lj r mk + f lm j r lk r mi − ωf lm k Ω li Ω mj = 0. This equation can be used in order to calculate the r-matrices for a given Lie algebra G . But, for obtaining the nonequivalent r-matrices one must use the automorphism group of Lie algebra G . The action of the automorphism A on G is given by the following transformation where T ′ i are the changed basis by the automorphism A. Since the automorphism preserves the structure constants, the basis T ′ i must obey the same commutation relations as T i , i.e., Inserting the transformation (3.18) into (3.19) we find that the elements of automorphism group A satisfy the following relation In order to calculate the elements A j i of Lie algebra G it would be helpful to further write the matrix form of (3.20), giving 6 where (Y k ) ij = −f ij k are the adjoint representations of G . It is also useful to obtain matrix form of Eq. (3.17) by using the adjoint representations In order to determine the nonequivalent r-matrices for a given Lie algebra G we give Proposition 3.1.
Proposition 3.1. Let r and r ′ be two r-matrices as solutions of the (m)CYBE (3.17). If there exists an automorphism A of G such that then the matrices r and r ′ of Lie algebra G are equivalent.
On the one hand, according to (3.15) for the changed basis we find R( Putting these relations together, one obtain that Multiplying both sides of the above equation in Ω im , we finalize that and this is nothing but (3.23). One can show that the r ′ satisfies the (m)CYBE (3.22) if the r be a solution of (3.22). We note that (3.23) is an equivalence relation.
In the following, we shall solve the (m)CYBE (3.17) (or equivalently (3.22)) for h 4 Lie algebra to obtain the corresponding r-matrices. In this respect, we consider two r-matrices r and r ′ equivalent if one can be obtained from the other by means of a change of basis which is an automorphism A of Lie algebra G . Indeed, the solutions that relate to each other through Eq. (3.23) are equivalent. In fact, one can use (3.23) to obtain all nonequivalent r-matrices. Before proceeding further, let us calculate the automorphism group of the particular Lie algebra h 4 . Using the structure constants given by (3.1) and then applying (3.21) the automorphism A can be easily obtained. The result is given by the following statement.
Proposition 3.2. The automorphism groups of the h 4 Lie algebra are expressed as matrices in basis (T 1 , · · · , T 4 ) as [32,33] for some real constants a, b, c, d, e.
In order to solve the (m)CYBE (3.17) for h 4 Lie algebra, let us assume that r ij has the following general form: for some real constants m 1 , · · · , m 6 . By substituting (3.29) into (3.17) and then by using (3.1) together with (3.3), the general solution of (3.17) is splitted into three classes such that the solutions are, in terms of the constants λ, ω and m 1 , · · · , m 6 , given by 30) where ∆ 16 = m 1 m 6 − ω λ 2 and ∆ 25 = m 2 m 5 − ω λ 2 for all ω in R. Now by using the automorphisms group elements A ∈ Aut(h 4 ) of (3.28) and by employing formula (3.23) of Proposition 3.1, one concludes that r-matrices given by (3.30) are splitted into ten nonequivalent classes such that the results 7 are summarized in Theorem 3.1.
It should be noted that: • Both the solutions r I and r II can be obtained from the matrix r 3 by putting ω = 0, m 2 = m 6 = 0, m 5 = 1 and ω = 0, m 2 = 0, m 5 = m 6 = 1, respectively; moreover, one can obtain r X from r 3 by putting ω = −1, λ = 1, m 2 = m 5 = m 6 = 0. Using (3.23) we have checked that all three of the solutions r I , r II and r X are, under both automorphisms A 1 and A 2 , nonequivalent.
According to above explanations the r-matrices r I , r II , r III and r IV of the h 4 Lie algebra are all solutions of CYBE with ω = 0 while solutions of the mCYBE are the r V , r V I , r V II , r V III , r IX and r X with ω = ±1. Now one may use formulas (3.3), (3.15) and (3.16) to obtain all linear R-operators corresponding to the nonequivalent r-matrices. R-operators are one of the basic tools for calculating the deformed currents J ± and then constructing the YB deformed WZW models. In the next subsection, we will classify all YB deformations of the H 4 WZW model.
Before closing this subsection, it is useful to comment on the fact that the YB deformed WZW model (2.6) is, under the automorphism transformation (3.18), invariant. First of all, the invariance of the left invariant one-forms L α under (3.18) requires that Then, using relations (3.20) and (3.24) one can deduce that the second term (WZW term) of action (2.6) is invariant with respect to the transformation (3.18). To investigate the invariance of the first term of (2.6), we need to know how the currents J ± change under (3.18). To this end, one may write down (2.8) in the following form Finally, one verifies the invariance of the first term of (2.6) under (3.18) by applying formulas (3.24), (3.31) together with (3.33).

Backgrounds for YB deformations of the H 4 WZW model
As was mentioned earlier, by using ( .III, indicates the YB deformed background derived by r III ; roman numbers I, II etc. distinguish between several possible deformed backgrounds of the H 4 WZW model, and the parameters (κ, η,Ã) indicate the deformation ones of each background.

About of the deformed backgrounds
The backgrounds H .X. As it is seen from .II, by shifting ρ → ρ ′ = ρ − 2η 2 one can easily show that this background is the same as H .V III and H (κ,η,Ã) 4,q .IX can be turned into the same metric of the H 4 WZW model, while corresponding B-fields are changed. One may show that the Lie algebra of Killing vectors corresponding to metrics of these backgrounds is isomorphic to those of (3.8), i.e. (3.11). Accordingly, it would be interesting to try to reveal the relation between the above backgrounds and H 4 WZW model.
By performing the following coordinate transformation (3.34) and also by applying ρ ′ = ρ(1 − η 2 ), we see that the metric of the background H (κ,η,Ã) 4 .IV turns into the same metric of the H 4 WZW model, while B-field have been changed as mentioned above. In like manner, by using the linear transformation .V III can be also simplified by performing a coordinate transformation After performing the transformation (3.36) and using ρ ′ = ρ/(1 − η 2 ), the resulting metric takes the same form as in (3.8). Finally, we find that the metric of background H (κ,η,Ã) 4,q .IX can be equal to (3.8) if one applies the transformation (3.37) and also ρ ′ = ρ(1 − η 2 q 4 )/(1 − η 2 ). Thus, we showed that, in some cases of the deformed backgrounds, the H 4 WZW model metric is, under arbitrary YB deformations, invariant up to antisymmetric B-fields.
The backgrounds H .V II one may find isometry group of the metrics, where the generators of the corresponding Lie algebra can be expressed in terms of the Killing vectors. One immediately find that the metrics of these backgrounds admit a six-dimensional Lie algebra of Killing vectors, which it cannot evidently be isomorphic to those of (3.11). Accordingly, these backgrounds cannot be turned into the H 4 WZW model.

Conformal invariance of the backgrounds up to two-loop
In the σ-model context, the conformal invariance conditions of the σ-model are provided by the vanishing of the beta-function equations [21]. The study of the conformal invariance has led to the covering of string theory, since one-and two-loop domains in string theory correspond to formulating on worldsheets of nontrivial topology. It is well known that the conditions for conformal invariance can be interpreted as effective field equations for G µν , B µν and dilaton field Φ of the string effective action [21]. The dilaton field is only one more massless degree of freedom of the bosonic string theory. This gives a contribution to the action (3.7) in the form of 1 8π dτ dσR (2) Φ(x µ ) in which R (2) is the curvature scalar on the string worldsheet. This term breaks Weyl invariance on a classical level as do the one-loop corrections to G and B. Below, we shall solve the one-and two-loop beta-function equations for all YB deformed backgrounds of Table 1 to obtain the corresponding dilaton fields 8 . 8 Notice that there is a one-to-one correspondence between the r-matrices r I , r II , r IV as solutions of the CYBE and two-dimensional Abelian subalgebra. These solutions satisfy the unimodularity condition of [17,18] while for the case of r III , two-dimensional subalgebra is non-Abelian; accordingly, the unimodularity condition is not satisfied. Anyway we still have a solution for which the conformal invariance condition is satisfied at one-loop level, as well as two-loop. Also, one can check the two-loop conformal invariance conditions for YB deformed backgrounds constructed from the matrices r V , ..., r X . Here we do not have the condition of [17], because of the existence of a WZW term.

Conditions for one-loop solution
The conditions for conformal invariance to hold in the σ-model in the lowest nontrivial approximation are the vanishing of the one-loop beta-function. The equations for the vanishing of the one-loop beta-function are given by [21] where R µν is the Ricci tensor of the metric G µν , H µνρ defined by is the torsion of the antisymmetric B-field, and Λ is a cosmological constant which vanishes for critical strings. We have also used the conventional notations (H 2 ) µν = H µρσ H ρσ ν and H 2 = H µνρ H µνρ . We now solve the field equations (4.1) for all YB deformed backgrounds of Table 1. In this way, we find the dilaton fields that guarantee the conformal invariance of the backgrounds at one-loop level. In all cases, the cosmological constant vanishes. In order to get more clarity, the results obtained for the dilaton fields are summarized in Table 2.

Conditions for two-loop solution
In order for the fields (G, B, Φ) to provide a consistent string background at low-energy up to two-loop order, they must satisfy the following equations [35,36] where R µνρσ is the Riemann tensor field of the metric G µν , (H 2 ) µν = H µρσ H ν ρσ , and in second equation of (4.3) Φ ′ = Φ + α ′ qH 2 for some coefficient q [35]. We note that round brackets denote the symmetric part on the indicated indices whereas square brackets denote the antisymmetric part. Using the above equations we check the conformal invariance conditions of the backgrounds of Table 1 up to two-loop order. In fact, we introduce some new solutions for two-loop beta-function equations of the σ-model with a non-vanishing field strength H and the dilaton field in the absence of a cosmological constant Λ. The field equations (4.3) are satisfied for all backgrounds of Table 1 with the same dilaton fields given in Table 2.

Summary and concluding remarks
Using automorphism group of the h 4 Lie algebra we have classified all corresponding classical r-matrices as the solutions of (m)CYBE. Then, we obtained all YB deformed WZW models based on the H 4 Lie group. We have, in some cases, shown that the metric of the H 4 WZW model is invariant under possible YB deformations while the antisymmetric B-fields are changed. We have also shown that all new integrable backgrounds of YB deformed H 4 WZW model are conformally invariant up to two-loop in the absence of a cosmological constant Λ. In this respect, we have derived the general form of the dilaton fields satisfying the vanishing beta-function equations. In fact, the YB deformed backgrounds that are conformal at one-loop remain conformal at two-loop with the same dilaton fields. Most importantly, it has been shown that the H 4 WZW model is a conformal theory within the class of the YB deformations preserving the conformal invariance up to two-loop order. It would be straightforward to improve the results presented in Ref. [23] to YB deformations of the Nappi-Witten WZW model by following our present analysis and method. As it has been indicated in Appendix B, we have classified all nonequivalent r-matrices of the Nappi-Witten Lie algebra in order to study the corresponding YB deformation of WZW model.
As a future direction, it would be interesting to generalize the YB deformation formulation of WZW model from Lie groups to Lie supergroups. As we know already, in order to construct the YB deformations of WZW model on a Lie group G one needs the r-matrices of Lie algebra G of G. Fortunately, the classical r-matrices related to some of the Lie superalgebras are available [37][38][39][40] (see also [41]). One can use these to construct new backgrounds of YB deformed WZW models. We hope that in future it will be possible to find YB deformed WZW models even for physically interesting backgrounds. The generalization of YB deformation of WZW model to Lie supergroups is currently under investigation. Table 3. The nonzero components of tensors R µν , R µνρσ , H µνρ and (H 2 ) µν related to the backgrounds represented in Table 1 Background symbol We then obtain L α = g −1 ∂ α g = cos u ∂ α a 1 + sin u ∂ α a 2 P 1 + cos u ∂ α a 2 − sin u ∂ α a 1 P 2 Using the above results together with the general form of the WZW model action (3.2), the spacetime metric and antisymmetric B-field are, respectively, found to be for some real constants m 3 , m 5 , m 6 . As was mentioned earlier, to obtain the nonequivalent r-matrices one must use the automorphism group of the Nappi-Witten algebra. Using (B.1) and (3.21) the automorphism groups of the Nappi-Witten algebra are expressed as matrices in the following form [32,33]  where a, b, c, d and e are some arbitrary constants. Ultimately, by employing formula (3.23) of Proposition 3.1, the r-matrices for the Nappi-Witten algebra are splitted into the following six nonequivalent families where p is a nonzero constant.

B.3 Conformal invariance of the backgrounds up to one-and two-loop orders
In order to guarantee the conformal invariance of the YB deformed backgrounds of the Nappi-Witten WZW model of Table 4, at least at the one-loop level, one must show that they satisfy the vanishing beta-function equations (4.1).