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

Ali<EMAIL_ADDRESS>Tayebe<EMAIL_ADDRESS>Adel Rezaei-Aghdam333Corresponding author<EMAIL_ADDRESS>
Department of Physics, Faculty of Basic Sciences,
Azarbaijan Shahid Madani University, 53714-161, Tabriz, Iran

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
split 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.
---

###### Contents

  1. 1  Introduction
  2. 2  A review of YB $\sigma$-model and YB deformations of WZW model
    1. 2.1 YB $\sigma$-model
    2. 2.2  YB deformation of WZW model
  3. 3  YB deformations of WZW model based on the $H_{4}$ Lie group and their classification
    1. 3.1  WZW model based on the $H_{4}$ Lie group
    2. 3.2  Classical r-matrices for $h_{4}$ Lie algebra
    3. 3.3 Backgrounds for YB deformations of the $H_{4}$ WZW model
      1. 3.3.1  About of the deformed backgrounds
  4. 4 Conformal invariance of the backgrounds up to two-loop
    1. 4.1 Conditions for one-loop solution
    2. 4.2 Conditions for two-loop solution
  5. 5 Summary and concluding remarks
  6. A Some computational results related to YB deformations of the $H_{4}$ WZW model
  7. B More on YB deformations of the Nappi-Witten WZW model
    1. B.1 Nonequivalent r-matrices
    2. B.2 Backgrounds for YB deformations of the Nappi-Witten WZW model
    3. B.3 Conformal invariance of the backgrounds up to one- and two-loop orders

##  1  Introduction

The study of integrable two dimensional $\sigma$-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 $\eta$) 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 $\lambda$-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${}_{5}\times$S5 string model has presented in [13, 14, 15] (see also [16,
17, 18]). For homogeneous YB deformed models it has been shown that [19] there
is no Weyl anomaly if the $R$-operators are unimodular (see also [20] up to
two-loop, and [21]). In [22], the relationship between unimodularity condition
on $R$-matrices with the divergence-free of the noncommutative parameter
$\Theta$ of the dual noncommutative gauge theory has been discussed; moreover,
it has been shown that [23] the equations of motion of the generalized
supergravity reproduce the CYBE in such a way that $\Theta$ is the most
general $r$-matrix solution built from antisymmetric products of Killing
vectors. The $r$-matrices may be divided into Abelian and non-Abelian, and it
has been proved that Abelian $r$-matrices correspond to T-duality shift
T-duality transformations [24], thus ensuring that the corresponding YB
deformation is a supergravity solution. In the case of non-Abelian
$r$-matrices, the unimodularity condition on the $r$-matrix [19] distinguishes
valid supergravity backgrounds [25] from the generalized supergravity
solutions [26, 27]. The Weyl invariance of bosonic string theories on
generalized supergravity backgrounds was shown at one-loop order by
constructing a local counterterm [28, 29].

The generalization to YB $\sigma$-models with WZW term has also carried out in
[30, 31, 32, 33, 34]. In most of the works, the models have been constructed
on semisimple or compact Lie groups. In Ref. [31], 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. Lately, YB deformation of the Nappi-
Witten background based on the mCYBE has been used in order to find a one-
parameter family of supergravity solutions which contains the Nappi-Witten
background and the flat Minkowski space [35]. Here we particularly focus on
the YB $\sigma$-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 split 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 [36, 37], 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 [31]) 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 $H_{\mu\nu\rho},(H^{2})_{\mu\nu},R_{\mu\nu}$ 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 classify all nonequivalent classical r-matrices and corresponding YB
deformed WZW models based on the Nappi-Witten group [31]; moreover, we show
that all deformed backgrounds are conformally invariant up to two-loop order.

##  2  A review of YB $\sigma$-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.

###  2.1 YB $\sigma$-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 ${\mathscr{G}}$), giving [6]

$\displaystyle S=-\frac{1}{2}\int_{{}_{\Sigma}}d^{2}\sigma
Tr\Big{[}(g^{-1}\partial_{-}g)\frac{1}{1-\eta R}(g^{-1}\partial_{+}g)\Big{]},$
(2.1)

where $\partial_{\pm}=\partial_{\tau}\pm\partial_{\sigma}$ are the derivatives
with respect to the standard lightcone variables
$\sigma^{\pm}=(\tau\pm\sigma)/2$ on the worldsheet $\Sigma$, and
$g^{-1}\partial_{\pm}g$ are components of the left-invariant Maurer-Cartan
one-forms which are defined by means of an element $g:\Sigma\rightarrow G$ in
the following formula

$\displaystyle
g^{-1}\partial_{{}_{\pm}}g\equiv{L_{{}_{\pm}}}={L_{{}_{\pm}}^{i}}~{}T_{{{}_{i}}},$
(2.2)

in which $T_{{{}_{i}}},i=1,...,dim\hskip 1.13809ptG$ are the bases of Lie
superalgebra ${\mathscr{G}}$. In Eq. (2.1), $\eta$ is a real parameter by
which deformation is measured. If one puts $\eta=0$, the action reduces to the
principal chiral model [1, 2]. In addition, the linear operator444One can
associate the $R$-operator to a classical r-matrix
[12].$R:{\mathscr{G}}\rightarrow{\mathscr{G}}$ is the solution of equation
[12]

$\displaystyle[R(X),R(Y)]-R\big{(}[R(X),Y]+[X,R(Y)]\big{)}=\omega[X,Y],$ (2.3)

for all $X,Y\in{\mathscr{G}}$. Here $\omega$ is constant parameter which can
be normalized by rescaling $R$. When $\omega=0$, the equation (2.3) is called
the CYBE. This equation can be generalized to the mCYBE with $\omega=\pm 1$.
The skew-symmetric condition of operator $R$ is written as

$\displaystyle Tr(R(X)Y)+Tr(XR(Y))=0.$ (2.4)

The integrability of the model (2.1) is an important property of the model
such that the corresponding Lax pair is given by555Note that the Lax pair in
(2.5) is the one for $\omega=0$. One can find a general form of the Lax pair
for an arbitrary $\omega$ rather than (2.5), giving [12] $\displaystyle{{\cal
L}}_{\pm}(\lambda)=\frac{1}{1\pm\lambda}\Big{(}1\mp\frac{\lambda\eta(\eta\omega\pm
R)}{1\pm\eta R}\Big{)}{L_{{}_{\pm}}}.$ [12]

$\displaystyle{{\cal
L}}_{\pm}(\lambda)=\frac{1}{1\pm\lambda}\big{(}1-\frac{\lambda\eta R}{1\pm\eta
R}\big{)}{L_{{}_{\pm}}},$ (2.5)

where $\lambda$ is a spectral parameter.

###  2.2  YB deformation of WZW model

In this subsection we shall consider the YB deformation of the WZW model [30].
The corresponding action consists of standard principal chiral model and WZW
term based on a Lie group $G$, giving [30, 31]

$\displaystyle
S^{{}^{YB}}_{{}_{WZW}}(g)=\frac{1}{2}\int_{{}_{\Sigma}}d^{2}\sigma~{}\Omega_{ij}L_{-}^{i}J_{+}^{j}+\frac{\kappa}{2}\
\int_{{}_{B_{3}}}d^{3}\sigma~{}\Omega_{kl}~{}f_{ij}^{~{}l}~{}L_{\xi}^{i}L_{+}^{j}L_{-}^{k},$
(2.6)

in which $\kappa$ is a constant parameter, $B$ is a three-manifold bounded by
worldsheet $\Sigma$, and $\Omega_{kl}$ defined by $\Omega_{kl}=<T_{k},T_{l}>$
is a non-degenerate ad-invariant symmetric bilinear form on Lie algebra
${\mathscr{G}}$ with structure constants $f_{ij}^{~{}~{}k}$ which satisfies
the following relation [38]

$\displaystyle f_{ij}^{\;\;l}\;\Omega_{lk}+f_{ik}^{\;\;l}\;\Omega_{lj}\;=\;0.$
(2.7)

Here the deformed currents $J_{\pm}$ are defined in the following way

$\displaystyle
J_{\pm}=(1+\omega\eta^{2})\frac{1\pm\tilde{A}R}{1-\eta^{2}R^{2}}L_{\pm},$
(2.8)

where $\eta$ and $\tilde{A}$ measure a deformation of WZW model. One can see
that when $\eta=\tilde{A}=0$ and $k=0$($k=1$) we recover the action of the
principal chiral model(undeformed WZW model) [30], and for
$\tilde{A}=\pm\eta,k=0$ one recovers the YB deformation of chiral model [1,
2]. In general, the constant parameter $\omega$ classifies integrable
deformations so that one may consider $\omega=0,\pm 1$ [1, 2]. In [30], it was
shown that in general the model (2.6) is integrable. This model was then
considered for the Nappi-witten group [31]. In the next section, we will
consider the model (2.6) for the $H_{4}$ Lie group.

##  3  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 $r^{\prime}$ equivalent if one can be obtained from the
other by means of a change of basis which is an automorphism $A$ of Lie
algebra $\mathscr{G}$. We then calculate all linear $R$-operators
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.

###  3.1  WZW model based on the $H_{4}$ Lie group

In this subsection we shall consider the WZW model on the $H_{4}$ Lie group
[36, 37]. 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

$\displaystyle[T_{1},T_{2}]~{}=~{}T_{2},~{}~{}~{}~{}~{}[T_{1},T_{3}]~{}=~{}-T_{3},~{}~{}~{}~{}~{}[T_{3},T_{2}]~{}=~{}T_{4}.$
(3.1)

The action of ungauged and undeformed WZW model on a Lie group $G$ is given by

$\displaystyle S_{{}_{WZW}}(g)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{{}_{\Sigma}}d\sigma^{+}d\sigma^{-}\;{\Omega}_{ij}L^{i}_{+}\;L^{j}_{-}+\frac{1}{12}\int_{{}_{B}}d^{3}\sigma~{}\varepsilon^{\gamma\alpha\beta}~{}{\Omega}_{ik}\;f_{jl}^{~{}~{}k}~{}L^{i}_{{}_{\gamma}}L^{j}_{{}_{\alpha}}L^{l}_{{}_{\beta}}.$
(3.2)

Accordingly, one needs a non-degenerate bilinear form $\Omega_{ij}$ on Lie
algebra ${\mathscr{G}}$ of $G$. Using (3.1) and also formula (2.7), one can
get the non-degenerate bilinear form on the $h_{4}$, giving [37]

$\displaystyle\Omega_{ij}=\left(\begin{tabular}[]{cccc}$\rho$&0&0&-$\lambda$\\\
0&0&$\lambda$&0\\\ 0&$\lambda$&0&0\\\ -$\lambda$&0&0&0\\\
\end{tabular}\right),$ (3.7)

where $\rho$ and $\lambda$ are some real constants. To construct the WZW
action (3.2) on the $H_{4}$, we parameterize an element of the $H_{4}$ as

$\displaystyle
g=e^{vT_{{}_{4}}}~{}e^{uT_{{}_{3}}}~{}e^{xT_{{}_{1}}}~{}e^{yT_{{}_{2}}},$
(3.8)

where $x^{\mu}=(x,y,u,v)$ stand for the coordinates of the $H_{4}$ group
manifold. Using (3.1) and (3.8) the corresponding left-invariant one-forms
components are obtained to be [37]

$\displaystyle
L^{1}_{\pm}=\partial_{\pm}x,~{}~{}~{}~{}~{}L^{2}_{\pm}=y\partial_{\pm}x+\partial_{\pm}y,~{}~{}~{}~{}~{}L^{3}_{\pm}=e^{x}~{}\partial_{\pm}u,~{}~{}~{}~{}~{}L^{4}_{\pm}=ye^{x}~{}\partial_{\pm}u+\partial_{\pm}v.$
(3.9)

Finally, the WZW action on the $H_{4}$ is found to be of the form [37]

$\displaystyle S_{{}_{WZW}}(g)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
d\sigma^{+}d\sigma^{-}\Big{[}\rho~{}\partial_{+}x\partial_{-}x-\partial_{+}x\partial_{-}v-\partial_{+}v\partial_{-}x+e^{x}\big{(}\partial_{+}y\partial_{-}u+\partial_{+}u\partial_{-}y$
(3.10)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+y\partial_{+}u\partial_{-}x-y\partial_{+}x\partial_{-}u\big{)}\Big{]}.$

Here we have set $\lambda=1$. Identifying the action (3.10) with the
$\sigma$-model of the form666$h_{\alpha\beta}$ and $\epsilon^{\alpha\beta}$
are the induced metric and antisymmetric tensor on the worldsheet,
respectively, such that $h=\det{h_{\alpha\beta}}$ and the indices
$\alpha,\beta$ run over $(\tau,\sigma)$. The dimensionful coupling constant
$\alpha^{\prime}$ turns out to be the inverse string tension.

$\displaystyle S=\frac{1}{4\pi\alpha^{\prime}}\int_{{}_{\Sigma}}\\!d\tau
d\sigma\sqrt{-h}\big{(}h^{\alpha\beta}G_{{}_{\mu\nu}}+\epsilon^{\alpha\beta}B_{{}_{\mu\nu}}\big{)}\partial_{\alpha}x{{}^{{}^{\mu}}}\partial_{\beta}x^{{}^{\nu}},$
(3.11)

we can read off the spacetime metric $G_{{}_{\mu\nu}}$ and the antisymmetric
$B$-field. They are then given by the following relations

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\rho~{}dx^{2}-2~{}dx~{}dv+2e^{x}~{}dy~{}du,$ (3.12)
$\displaystyle B$ $\displaystyle=$ $\displaystyle-ye^{x}~{}dx\wedge du.$
(3.13)

The metric (3.12) 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 (3.12). This
metric admits a seven-dimensional Lie algebra of Killing vectors, which can be
generated by

$\displaystyle K_{1}$ $\displaystyle=$ $\displaystyle-\frac{\partial}{\partial
x}+u\frac{\partial}{\partial u}-\rho\frac{\partial}{\partial
v},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}K_{2}=e^{-x}\frac{\partial}{\partial
y}-u\frac{\partial}{\partial v},$ $\displaystyle K_{3}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
y},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}K_{4}=y\frac{\partial}{\partial
y}-u\frac{\partial}{\partial u},$ $\displaystyle K_{5}$ $\displaystyle=$
$\displaystyle e^{-x}\frac{\partial}{\partial u}-y\frac{\partial}{\partial
v},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}K_{6}=\frac{\partial}{\partial
u},$ $\displaystyle K_{7}$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial v}.$ (3.14)

One can easily check that the Lie algebra spanned by these vectors is

$\displaystyle{[K_{1},K_{2}]}$ $\displaystyle=$ $\displaystyle
K_{2},~{}~{}~{}~{}~{}~{}~{}[K_{1},K_{6}]=-K_{6},~{}~{}~{}~{}~{}~{}~{}[K_{2},K_{4}]=K_{2},~{}~{}~{}~{}~{}~{}~{}~{}[K_{2},K_{6}]=-K_{7},$
$\displaystyle{[K_{3},K_{4}]}$ $\displaystyle=$ $\displaystyle
K_{3},~{}~{}~{}~{}~{}~{}~{}[K_{3},K_{5}]=K_{7},~{}~{}~{}~{}~{}~{}~{}~{}~{}[K_{4},K_{5}]=K_{5},~{}~{}~{}~{}~{}~{}~{}~{}~{}[K_{4},K_{6}]=K_{6},$
(3.15)

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 (3.1).

###  3.2  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
$\mathscr{G}$ with the basis $\\{T_{i}\\}$ [39]

$\displaystyle r=\frac{1}{2}r^{ij}(T_{i}\otimes T_{j}-T_{j}\otimes T_{i}),$
(3.16)

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 [31]

$\displaystyle R(T_{k})=<r,(1\otimes T_{k})>=r^{ij}\Omega_{jk}~{}T_{i}.$
(3.17)

Based on this, the action of $R$ on any element $X=x^{k}T_{k}\in\mathscr{G}$
is written as

$\displaystyle R(X)=x^{k}R(T_{k})=r^{ij}\Omega_{jk}x^{k}~{}T_{i}.$ (3.18)

Considering

$\displaystyle R(T_{k})={R_{k}}^{i}~{}T_{i},$ (3.19)

and then comparing (3.17) and (3.19), one gets

$\displaystyle{R_{k}}^{i}=r^{ij}\Omega_{jk}.$ (3.20)

Now, making use of formulas (2.7) and (3.18) and after some algebraic
calculations, one can write Eq. (2.3) in the following form [31]

$\displaystyle{f_{lm}}^{k}r^{li}r^{mj}+{f_{lm}}^{i}r^{lj}r^{mk}+{f_{lm}}^{j}r^{lk}r^{mi}-\omega{f_{lm}}^{k}\Omega^{li}\Omega^{mj}=0.$
(3.21)

This equation can be used in order to calculate the r-matrices for a given Lie
algebra $\mathscr{G}$. But, for obtaining the nonequivalent r-matrices one
must use the automorphism group of Lie algebra $\mathscr{G}$. The action of
the automorphism $A$ on ${\mathscr{G}}$ is given by the following
transformation

$\displaystyle T^{\prime}_{i}={A}(T_{i})={A}_{i}^{~{}j}~{}T_{j},$ (3.22)

where $T^{\prime}_{i}$ are the changed basis by the automorphism $A$. Since
the automorphism preserves the structure constants, the basis $T^{\prime}_{i}$
must obey the same commutation relations as $T_{i}$, i.e.,

$\displaystyle[T^{\prime}_{i},T^{\prime}_{j}]={f_{{}_{ij}}}^{k}~{}T^{\prime}_{k}.$
(3.23)

Inserting the transformation (3.22) into (3.23) we find that the elements of
automorphism group $A$ satisfy the following relation

$\displaystyle{A}_{i}^{~{}m}~{}{f_{{}_{mn}}}^{k}~{}{A}_{j}^{~{}n}={f_{{}_{ij}}}^{l}~{}A_{l}^{\;\;k}.$
(3.24)

In order to calculate the elements ${A}_{i}^{~{}j}$ of Lie algebra
$\mathscr{G}$ it would be helpful to further write the matrix form of (3.24),
giving777Here “t” denotes transposition. [40]

$\displaystyle A~{}{\cal Y}^{k}A^{t}={\cal Y}^{l}A_{l}^{\;\;k},$ (3.25)

where $(\mathcal{Y}^{k})_{ij}=-{f_{ij}}^{k}$ are the adjoint representations
of $\mathscr{G}$. It is also useful to obtain matrix form of Eq. (3.21) by
using the adjoint representations $(\mathcal{Y}^{k})_{ij}=-{f_{ij}}^{k}$ and
$({\cal X}_{i})_{j}^{~{}k}=-{f_{ij}}^{k}$. It is then read

$\displaystyle
r\mathcal{Y}^{k}r+r(\mathcal{X}_{l}r^{lk})-(r^{kl}\mathcal{X}_{l}^{t})r=-\omega(\Omega^{-1}\mathcal{Y}^{k}\Omega^{-1}).$
(3.26)

In order to determine the nonequivalent r-matrices for a given Lie algebra
$\mathscr{G}$ we give Proposition 3.1.

Proposition 3.1. Let $r$ and $r^{\prime}$ be two r-matrices as solutions of
the (m)CYBE (3.21). If there exists an automorphism $A$ of $\mathscr{G}$ such
that

$\displaystyle r=A^{t}~{}r^{\prime}~{}A,$ (3.27)

then the matrices $r$ and $r^{\prime}$ of Lie algebra $\mathscr{G}$ are
equivalent.
Proof. Let $\\{T_{i}\\}$ and $\\{T^{\prime}_{i}\\}$ be the bases of
$\mathscr{G}$ such that $T^{\prime}_{i}={A}_{i}^{~{}j}~{}T_{j}$ in which
${A}_{i}^{~{}j}$ is an element of automorphism group $Aut(\mathscr{G})$. Since
an automorphism $A$ of $\mathscr{G}$ preserves the structure constants, one
may use (2.7) to conclude that $\Omega_{ij}=<T^{\prime}_{i},T^{\prime}_{j}>$.
Then, using (3.22) it is simply shown that

$\displaystyle{A}_{i}^{~{}k}~{}\Omega_{kj}=\Omega_{il}~{}(A^{-1})_{j}^{~{}~{}l}.$
(3.28)

On the one hand, according to (3.19) for the changed basis we find
$R(T^{\prime}_{i})={R^{\prime}}_{i}^{~{}j}~{}T^{\prime}_{j}={R^{\prime}}_{i}^{~{}j}~{}{A}_{j}^{~{}k}~{}T_{k}$.
In addition, one can write
$R(T^{\prime}_{i})={A}_{i}^{~{}l}~{}{R}_{l}^{~{}k}~{}T_{k}$. Putting these
relations together, one obtain that

$\displaystyle{R^{\prime}}_{i}^{~{}j}={A}_{i}^{~{}k}~{}{R}_{k}^{~{}l}~{}(A^{-1})_{l}^{~{}j},$
(3.29)

on the other hand, one may use (3.20) and (3.28) to write (3.29) as

$\displaystyle{r^{\prime}}^{jn}~{}\Omega_{ni}$ $\displaystyle=$
$\displaystyle{A}_{i}^{~{}k}~{}\Omega_{kp}~{}{r}^{lp}~{}(A^{-1})_{l}^{~{}j}$
(3.30) $\displaystyle=$
$\displaystyle\Omega_{in}~{}(A^{-1})_{p}^{~{}n}~{}{r}^{lp}~{}(A^{-1})_{l}^{~{}j}.$

Multiplying both sides of the above equation in $\Omega^{im}$, we finalize
that

$\displaystyle{r^{\prime}}^{jm}=(A^{-1})_{l}^{~{}j}~{}{r}^{lp}~{}(A^{-1})_{p}^{~{}m},$
(3.31)

and this is nothing but (3.27). One can show that the $r^{\prime}$ satisfies
the (m)CYBE (3.26) if the $r$ be a solution of (3.26). We note that (3.27) is
an equivalence relation.

In the following, we shall solve the (m)CYBE (3.21) (or equivalently (3.26))
for $h_{4}$ Lie algebra to obtain the corresponding r-matrices. In this
respect, we consider two r-matrices $r$ and $r^{\prime}$ equivalent if one can
be obtained from the other by means of a change of basis which is an
automorphism $A$ of Lie algebra $\mathscr{G}$. Indeed, the solutions that
relate to each other through Eq. (3.27) are equivalent. In fact, one can use
(3.27) 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 (3.25) 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 $(T_{1},\cdots,T_{4})$ as  [41, 42]

$\displaystyle
Aut(h_{4})=\left\\{A_{1}=\left(\begin{tabular}[]{cccc}1&c&d&e\\\ 0&a&0&ad\\\
0&0&b&bc\\\ 0&0&0&ab\\\
\end{tabular}\right),~{}A_{2}=\left(\begin{tabular}[]{cccc}-1&c&d&e\\\
0&0&a&-ac\\\ 0&b&0&-bd\\\ 0&0&0&-ab\\\ \end{tabular}\right);~{}ab\neq
0\right\\}$ (3.40)

for some real constants $a,b,c,d,e$.

In order to solve the (m)CYBE (3.21) for $h_{4}$ Lie algebra, let us assume
that $r^{ij}$ has the following general form:

$\displaystyle
r^{ij}=\left(\begin{tabular}[]{cccc}0&$m_{1}$&$m_{2}$&$m_{3}$\\\
-$m_{1}$&0&$m_{4}$&$m_{5}$\\\ -$m_{2}$&-$m_{4}$&0&$m_{6}$\\\
-$m_{3}$&-$m_{5}$&-$m_{6}$&0\\\ \end{tabular}\right),$ (3.45)

for some real constants $m_{1},\cdots,m_{6}$. By substituting (3.45) into
(3.21) and then by using (3.1) together with (3.7), the general solution of
(3.21) is split into three classes such that the solutions are, in terms of
the constants $\lambda,\omega$ and $m_{1},\cdots,m_{6}$, given by

$\displaystyle r_{{}_{1}}=\left(\begin{tabular}[]{cccc}0&0&0&$m_{3}$\\\
0&0&$\pm\sqrt{-\frac{\omega}{\lambda^{2}}}$&$m_{5}$\\\
0&$\mp\sqrt{-\frac{\omega}{\lambda^{2}}}$&0&$m_{6}$\\\
-$m_{3}$&-$m_{5}$&-$m_{6}$&0\\\
\end{tabular}\right),~{}~{}r_{{}_{2}}=\left(\begin{tabular}[]{cccc}0&$m_{1}$&0&-$\Delta_{16}$\\\
-$m_{1}$&0&$\Delta_{16}$&$m_{5}$\\\ 0&-$\Delta_{16}$&0&$m_{6}$\\\
$\Delta_{16}$&-$m_{5}$&-$m_{6}$&0\\\ \end{tabular}\right),$ $\displaystyle
r_{{}_{3}}=\left(\begin{tabular}[]{cccc}0&0&$m_{2}$&$\Delta_{25}$\\\
0&0&$\Delta_{25}$&$m_{5}$\\\ -$m_{2}$&-$\Delta_{25}$&0&$m_{6}$\\\
-$\Delta_{25}$&-$m_{5}$&-$m_{6}$&0\\\
\end{tabular}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(3.51)

where $\Delta_{16}=\sqrt{m_{1}m_{6}-\frac{\omega}{\lambda^{2}}}$ and
$\Delta_{25}=\sqrt{m_{2}m_{5}-\frac{\omega}{\lambda^{2}}}$ for all $\omega$ in
$\mathbb{R}$. Now by using the automorphisms group elements $A\in
Aut({h_{4}})$ of (3.40) and by employing formula (3.27) of Proposition 3.1,
one concludes that r-matrices given by (3.51) are split into ten nonequivalent
classes such that the results888In Ref. [43], all coboundary Lie bialgebras of
the $h_{4}$ Lie algebra have been obtained and classified into three
multiparametric families. Accordingly, their corresponding r-matrices have
been also found as multiparametric. Here we have exactly found the r-matrices
of $h_{4}$. However, our results are in agreement with those of Ref. [43]. are
summarized in Theorem 3.1.

Theorem 3.1. Any r-matrix of the $h_{4}$ Lie algebra as a solution the (m)CYBE
(3.21) belongs just to one of the following ten nonequivalent classes

$\displaystyle r_{{}_{I}}=\left(\begin{tabular}[]{cccc}0&0&0&0\\\ 0&0&0&1\\\
0&0&0&0\\\ 0&-1&0&0\\\
\end{tabular}\right),~{}r_{{}_{II}}=\left(\begin{tabular}[]{cccc}0&0&0&0\\\
0&0&0&1\\\ 0&0&0&1\\\ 0&-1&-1&0\\\
\end{tabular}\right),~{}r_{{}_{III}}=\left(\begin{tabular}[]{cccc}0&1&0&0\\\
-1&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{tabular}\right),~{}r_{{}_{IV}}=\left(\begin{tabular}[]{cccc}0&0&0&1\\\
0&0&0&0\\\ 0&0&0&0\\\ -1&0&0&0\\\ \end{tabular}\right),$ $\displaystyle
r_{{}_{V}}=\left(\begin{tabular}[]{cccc}0&0&0&-$1$\\\ 0&0&$1$&1\\\
0&-$1$&0&0\\\ $1$&-1&0&0\\\
\end{tabular}\right),r_{{}_{VI}}=\left(\begin{tabular}[]{cccc}0&1&0&-1\\\
-1&0&1&0\\\ 0&-1&0&0\\\ 1&0&0&0\\\
\end{tabular}\right),r_{{}_{VII}}=\left(\begin{tabular}[]{cccc}0&1&0&0\\\
-1&0&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\\\
\end{tabular}\right),r_{{}_{VIII}}=\left(\begin{tabular}[]{cccc}0&0&0&0\\\
0&0&1&0\\\ 0&-1&0&0\\\ 0&0&0&0\\\ \end{tabular}\right),~{}~{}$ $\displaystyle
r_{{}_{IX}}=\left(\begin{tabular}[]{cccc}0&0&0&$q^{2}$\\\ 0&0&1&0\\\
0&-1&0&0\\\ -$q^{2}$&0&0&0\\\
\end{tabular}\right),~{}~{}r_{{}_{X}}=\left(\begin{tabular}[]{cccc}0&0&0&1\\\
0&0&1&0\\\ 0&-1&0&0\\\ -1&0&0&0\\\
\end{tabular}\right),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

where $q^{2}\neq 0,1$.

It should be noted that:

  * •

Both the solutions $r_{{}_{I}}$ and $r_{{}_{II}}$ can be obtained from the
matrix $r_{{}_{3}}$ by putting $\omega=0,m_{2}=m_{6}=0,m_{5}=1$ and
$\omega=0,m_{2}=0,m_{5}=m_{6}=1$, respectively; moreover, one can obtain
$r_{{}_{X}}$ from $r_{{}_{3}}$ by putting
$\omega=-1,\lambda=1,m_{2}=m_{5}=m_{6}=0$. Using (3.27) 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.

  * •

The $r_{{}_{III}}$, $r_{{}_{VI}}$ and $r_{{}_{VII}}$ are just obtained from
the matrix $r_{{}_{2}}$ by putting $\omega=0,m_{5}=m_{6}=0,m_{1}=1$ and
$\omega=-1,\lambda=1,m_{5}=m_{6}=0,m_{1}=1$, and
$\omega=\lambda=1,m_{5}=0,m_{1}=m_{6}=1$, respectively; moreover, the solution
$r_{{}_{V}}$ can be obtained from $r_{{}_{2}}$ by putting
$\omega=-1,\lambda=1,m_{1}=m_{6}=0,m_{5}=1$. We have also checked that all
four of the solutions $r_{{}_{III}}$, $r_{{}_{V}}$, $r_{{}_{VI}}$ and
$r_{{}_{VII}}$ are, under both automorphisms $A_{1}$ and $A_{2}$,
nonequivalent.

  * •

All three of the solutions $r_{{}_{IV}}$, $r_{{}_{VIII}}$ and $r_{{}_{IX}}$
are obtained from the matrix $r_{{}_{1}}$ by putting
$\omega=0,m_{5}=m_{6}=0,m_{3}=1$ and
$\omega=-1,\lambda=1,m_{3}=m_{5}=m_{6}=0$, and
$\omega=-1,\lambda=1,m_{5}=m_{6}=0,m_{3}=q^{2}$, respectively. One can show
that these solutions 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 $\omega=0$ while solutions of the mCYBE
are the $r_{{}_{V}},r_{{}_{VI}},r_{{}_{VII}},r_{{}_{VIII}},r_{{}_{IX}}$ and
$r_{{}_{X}}$ with $\omega=\pm 1$. Now one may use formulas (3.7), (3.19) and
(3.20) 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 $J_{\pm}$ 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 (3.22),
invariant. First of all, the invariance of the left invariant one-forms
$L_{\alpha}$ under (3.22) requires that

$\displaystyle L^{\prime
i}_{{}_{\alpha}}=L^{j}_{{}_{\alpha}}~{}(A^{-1})_{j}^{~{}i}.$ (3.55)

Then, using relations (3.24) and (3.28) one can deduce that the second term
(WZW term) of action (2.6) is invariant with respect to the transformation
(3.22). To investigate the invariance of the first term of (2.6), we need to
know how the currents $J_{\pm}$ change under (3.22). To this end, one may
write down (2.8) in the following form

$\displaystyle
J^{i}_{\pm}-\eta^{2}J^{k}_{\pm}~{}R_{k}^{~{}l}~{}R_{l}^{~{}i}=(1+\omega\eta^{2})\big{[}L^{i}_{{}_{\pm}}\pm\tilde{A}L^{k}_{{}_{\pm}}~{}R_{k}^{~{}i}\big{]}.$
(3.56)

Using (3.29) and (3.55) we find that relation (3.56) does remain invariant
with respect to the transformation (3.22) if the following relation is held

$\displaystyle J^{\prime i}_{\pm}=J^{j}_{\pm}~{}(A^{-1})_{j}^{~{}i}.$ (3.57)

Finally, one verifies the invariance of the first term of (2.6) under (3.22)
by applying formulas (3.28), (3.55) together with (3.57).

###  3.3 Backgrounds for YB deformations of the $H_{4}$ WZW model

As was mentioned earlier, by using (3.7), (3.19) and (3.20) we can obtain all
linear $R$-operators corresponding to the nonequivalent r-matrices of the
$h_{4}$ Lie algebra. Having $R$-operators, we can find the deformed currents
$J_{\pm}$ from Eq. (2.8). In this way, one uses (2.6) to obtain YB
deformations of the $H_{4}$ WZW model. For the sake of clarity the results
obtained in this subsection are summarized in Table 1; we display the deformed
backgrounds including metric and $B$-field, together with the related
comments. It should be noted that the symbol of each background, e.g.
$H^{(\kappa,\eta,\tilde{A})}_{4}.III$, 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 $(\kappa,\eta,\tilde{A})$ indicate the deformation ones of each
background.

####  3.3.1  About of the deformed backgrounds

The backgrounds $H^{(\kappa)}_{4}.I,H^{(\kappa,\eta)}_{4}.II$ and
$H^{(\kappa,\tilde{A})}_{4}.{\small X}$. As it is seen from Table 1, the
metrics of $H^{(\kappa)}_{4}.I$ and $H^{(\kappa,\tilde{A})}_{4}.{\small X}$
have not, under the deformation, been changed, i.e. in these cases, the
$H_{4}$ WZW model metric remains, under the deformation, invariant while
corresponding $B$-fields have been changed. In the case of the background
$H^{(\kappa,\eta)}_{4}.II$, by shifting
$\rho\rightarrow\rho^{\prime}=\rho-2\eta^{2}$ one can easily show that this
background is the same as $H^{(\kappa)}_{4}.I$. But, considering the same
values of $\rho$ in both backgrounds we are faced with a deformed metric of
the $H^{(\kappa,\eta)}_{4}.II$.

The backgrounds
$H^{(\kappa,\eta,\tilde{A})}_{4}.IV,H^{(\kappa,\eta,\tilde{A})}_{4}.V,H^{(\kappa,\eta)}_{4}.VIII$
and $H^{(\kappa,\eta,\tilde{A})}_{{}_{4,q}}.IX$. It is also interesting to
note the fact that under some coordinate transformations one concludes that
all deformed metrics of backgrounds
$H^{(\kappa,\eta,\tilde{A})}_{4}.IV,H^{(\kappa,\eta,\tilde{A})}_{4}.V,H^{(\kappa,\eta)}_{4}.VIII$
and $H^{(\kappa,\eta,\tilde{A})}_{{}_{4,q}}.IX$ 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 (3.12), i.e. (3.1).
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

$\displaystyle
x^{{}^{\prime}}=\frac{1}{1-\eta^{2}}x,~{}~{}~{}~{}~{}y^{{}^{\prime}}=y~{}e^{\frac{-\eta^{2}}{1-\eta^{2}}x},~{}~{}~{}~{}u^{{}^{\prime}}=u,~{}~{}~{}~{}v^{{}^{\prime}}=v,$
(3.58)

and also by applying $\rho^{\prime}=\rho(1-\eta^{2})$, we see that the metric
of the background $H^{(\kappa,\eta,\tilde{A})}_{4}.IV$ 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

$\displaystyle
x^{{}^{\prime}}=x,~{}~{}~{}~{}~{}y^{{}^{\prime}}=y-\frac{2\eta^{2}}{1-\eta^{2}}x,~{}~{}~{}~{}u^{{}^{\prime}}=u,~{}~{}~{}~{}v^{{}^{\prime}}=v,$
(3.59)

and without any shift in $\rho$, one can easily show that the metric of
$H^{(\kappa,\eta,\tilde{A})}_{4}.V$ is nothing but the same (3.12). The
background $H^{(\kappa,\eta)}_{4}.VIII$ can be also simplified by performing a
coordinate transformation

$\displaystyle
x^{{}^{\prime}}=({1-\eta^{2}})x,~{}~{}~{}~{}~{}y^{{}^{\prime}}=y~{}e^{\eta^{2}x},~{}~{}~{}~{}u^{{}^{\prime}}=u,~{}~{}~{}~{}v^{{}^{\prime}}=v.$
(3.60)

After performing the transformation (3.60) and using
$\rho^{\prime}=\rho/(1-\eta^{2})$, the resulting metric takes the same form as
in (3.12).

Table 1. YB deformed backgrounds of the $H_{4}$ WZW model∗ Background symbol
Backgrounds including metric and $B$-field Comments $H^{(\kappa)}_{4}.I$
$ds^{2}=\rho dx^{2}-2dxdv+2e^{x}dydu,$ $B=\kappa ye^{x}du\wedge dx$
$\omega=0,~{}\lambda=1$ $H^{(\kappa,\eta)}_{4}.II$
$ds^{2}=(\rho-2\eta^{2})dx^{2}-2dxdv+2e^{x}dydu,$ $B=\kappa ye^{x}du\wedge dx$
$\omega=0,~{}\lambda=1$ $H^{(\kappa,\eta,\tilde{A})}_{4}.{\small III}$
$ds^{2}=\rho dx^{2}-2dxdv+2e^{x}dydu-\rho\eta^{2}e^{2x}du^{2},$ $B=\kappa
ye^{x}du\wedge dx+\tilde{A}e^{x}dv\wedge du$ $\omega=0,~{}\lambda=1$
$H^{(\kappa,\eta,\tilde{A})}_{4}.IV$ $ds^{2}=\frac{1}{1-\eta^{2}}\big{[}\rho
dx^{2}-2dxdv-2\eta^{2}ye^{x}dxdu\big{]}+2e^{x}dydu,$
$B=(\kappa-\frac{\tilde{A}}{1-\eta^{2}})ye^{x}du\wedge dx$
$\omega=0,~{}\lambda=1$ $H^{(\kappa,\eta,\tilde{A})}_{4}.V$ $ds^{2}=\rho
dx^{2}-2dxdv+2e^{x}dydu-\frac{4\eta^{2}}{1-\eta^{2}}e^{x}dxdu,$
$B=(\kappa+\tilde{A})ye^{x}du\wedge dx$ $\omega=-1,~{}\lambda=1$
$H^{(\kappa,\eta,\tilde{A})}_{4}.VI$ $ds^{2}=\rho
dx^{2}-2dxdv+2e^{x}dydu-\frac{2\rho\eta^{2}}{1-\eta^{2}}e^{x}dxdu-\frac{\rho\eta^{2}}{1-\eta^{2}}e^{2x}du^{2},$
$B=(\kappa+\tilde{A})ye^{x}du\wedge dx+\tilde{A}e^{x}dv\wedge du$
$\omega=-1,~{}\lambda=1$ $H^{(\kappa,\eta,\tilde{A})}_{4}.VII$
$ds^{2}=\frac{\rho}{1+\eta^{2}}dx^{2}-2dxdv+2e^{x}dydu-\frac{\rho\eta^{2}}{1+\eta^{2}}e^{2x}du^{2},$
$B=\kappa ye^{x}du\wedge dx+\tilde{A}e^{x}dv\wedge du$ $\omega=1,~{}\lambda=1$
$H^{(\kappa,\eta)}_{4}.VIII$ $ds^{2}=(1-\eta^{2})(\rho
dx^{2}-2dxdv)+2e^{x}dydu+2\eta^{2}ye^{x}dxdu,$ $B=\kappa ye^{x}du\wedge dx$
$\omega=-1,~{}\lambda=1$ $H^{(\kappa,\eta,\tilde{A})}_{{}_{4,q}}.IX$
$ds^{2}=\frac{1-\eta^{2}}{1-\eta^{2}q^{4}}(\rho
dx^{2}-2dxdv)+2e^{x}dydu+\frac{2\eta^{2}(1-q^{4})}{1-\eta^{2}q^{4}}ye^{x}dxdu,$
$B=\Big{[}\kappa-\frac{\tilde{A}q^{2}(1-\eta^{2})}{1-\eta^{2}q^{4}}\Big{]}ye^{x}du\wedge
dx$ $\omega=-1,~{}\lambda=1$ $H^{(\kappa,\tilde{A})}_{4}.{\small X}$
$ds^{2}=\rho dx^{2}-2dxdv+2e^{x}dydu,$ $B=(\kappa-\tilde{A})ye^{x}du\wedge dx$
$\omega=-1,~{}\lambda=1$

∗Here we have ignored the total derivative terms that appeared in the
$B$-fields part.

Finally, we find that the metric of background
$H^{(\kappa,\eta,\tilde{A})}_{{}_{4,q}}.IX$ can be equal to (3.12) if one
applies the transformation

$\displaystyle
x^{{}^{\prime}}=\frac{1-\eta^{2}}{1-\eta^{2}q^{4}}x,~{}~{}~{}~{}~{}y^{{}^{\prime}}=y~{}e^{\frac{\eta^{2}(1-q^{4})}{1-\eta^{2}q^{4}}x},~{}~{}~{}~{}u^{{}^{\prime}}=u,~{}~{}~{}~{}v^{{}^{\prime}}=v,$
(3.61)

and also $\rho^{\prime}=\rho(1-\eta^{2}q^{4})/(1-\eta^{2})$. 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
$H^{(\kappa,\eta,\tilde{A})}_{4}.III,H^{(\kappa,\eta,\tilde{A})}_{4}.VI$ and
$H^{(\kappa,\eta,\tilde{A})}_{4}.VII$. In order to clarify the structure of
the metrics of
$H^{(\kappa,\eta,\tilde{A})}_{4}.III,H^{(\kappa,\eta,\tilde{A})}_{4}.VI$ and
$H^{(\kappa,\eta,\tilde{A})}_{4}.VII$ 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 (3.1).
Accordingly, these backgrounds cannot be turned into the $H_{4}$ WZW model.

##  4 Conformal invariance of the backgrounds up to two-loop

In the $\sigma$-model context, the conformal invariance conditions of the
$\sigma$-model are provided by the vanishing of the beta-function equations
[25]. 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 $G_{\mu\nu}$, $B_{\mu\nu}$ and dilaton field $\Phi$ of the
string effective action [25]. The dilaton field is only one more massless
degree of freedom of the bosonic string theory. This gives a contribution to
the action (3.11) in the form of $\frac{1}{8\pi}\int d\tau d\sigma
R^{{}^{(2)}}\Phi(x^{\mu})$ 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 fields999Notice 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 [19, 20] 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 [19], because of the existence of a WZW term..

###  4.1 Conditions for one-loop solution

The conditions for conformal invariance to hold in the $\sigma$-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 [25]

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
R_{{\mu\nu}}-(H^{2})_{\mu\nu}+{\nabla}_{\mu}{\nabla}_{\nu}\Phi,$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle-{\nabla}^{\lambda}H_{{\lambda\mu\nu}}+H_{{\mu\nu}}^{~{}\;\lambda}~{}{\nabla}_{\lambda}\Phi,$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
2\Lambda+{\nabla}^{2}\Phi^{\prime}-({\nabla}\Phi^{\prime})^{2}+\frac{2}{3}H^{{2}},$
(4.1)

where $R_{{\mu\nu}}$ is the Ricci tensor of the metric $G_{\mu\nu}$,
$H_{{}_{\mu\nu\rho}}$ defined by

$\displaystyle
H_{{\mu\nu\rho}}=\frac{1}{2}\big{(}\partial_{\mu}B_{\nu\rho}+\partial_{\nu}B_{\rho\mu}+\partial_{\rho}B_{\mu\nu}\big{)},$
(4.2)

is the torsion of the antisymmetric $B$-field, and $\Lambda$ is a cosmological
constant which vanishes for critical strings. We have also used the
conventional notations
$(H^{2})_{\mu\nu}=H_{{\mu\rho\sigma}}H^{{\rho\sigma}}_{~{}~{}\nu}$ and
$H^{2}=H_{{\mu\nu\rho}}H^{{\mu\nu\rho}}$. 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.

Table 2. The dilaton fields making the $H_{4}$ deformed backgrounds conformal
up to one-loop order∗
Background symbol Dilaton fields Comments $H^{(\kappa)}_{4}.I$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{4}(1-\kappa^{2})x^{2}+c_{1}x+c_{2}$
$H^{(\kappa,\eta)}_{4}.II$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{4}(1-\kappa^{2})x^{2}+c_{1}x+c_{2}$
$H^{(\kappa,\eta,\tilde{A})}_{4}.III$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{4}\big{(}1-\kappa^{2}\big{)}x^{2}+c_{1}x+c_{2}$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{A}}=0$
$H^{(\kappa,\eta,\tilde{A})}_{4}.IV$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{4}\Big{[}\frac{1}{(\eta^{2}-1)^{2}}-\big{(}\kappa+\frac{\tilde{A}}{\eta^{2}-1}\big{)}^{2}\Big{]}x^{2}+c_{1}x+c_{2}$
$H^{(\kappa,\eta,\tilde{A})}_{4}.V$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{4}\Big{[}1-(\kappa+\tilde{A})^{2}\Big{]}x^{2}+c_{1}x+c_{2}$
$H^{(\kappa,\eta,\tilde{A})}_{4}.VI$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{4}\big{(}1-\kappa^{2}\big{)}x^{2}+c_{1}x+c_{2}$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{A}}=0$
$H^{(\kappa,\eta,\tilde{A})}_{4}.VII$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{4}\big{(}1-\kappa^{2}\big{)}x^{2}+c_{1}x+c_{2}$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\tilde{A}}=0$
$H^{(\kappa,\eta)}_{4}.VIII$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{4}\big{[}(1-\eta^{2})^{2}-\kappa^{2}\big{]}x^{2}+c_{1}x+c_{2}$
$H^{(\kappa,\eta,\tilde{A})}_{{}_{4,q}}.IX$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{4}\Big{[}(\frac{1-\eta^{2}}{1-\eta^{2}q^{4}})^{2}-\big{(}\kappa-\frac{\tilde{A}q^{2}(1-\eta^{2})}{1-\eta^{2}q^{4}}\big{)}^{2}\Big{]}x^{2}+c_{1}x+c_{2}$
$H^{(\kappa,\tilde{A})}_{4}.X$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{4}\big{[}1-(\tilde{A}-\kappa)^{2}\big{]}x^{2}+c_{1}x+c_{2}$

∗ Here $c_{1}$ and $c_{2}$ are some arbitrary constants.

###  4.2 Conditions for two-loop solution

In order for the fields $(G,B,\Phi)$ to provide a consistent string background
at low-energy up to two-loop order, they must satisfy the following equations
[44, 45]

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
R_{{\mu\nu}}-(H^{2})_{\mu\nu}+{\nabla}_{\mu}{\nabla}_{\nu}\Phi+\frac{1}{2}\alpha^{\prime}\Big{[}R_{{\mu\rho\sigma\lambda}}R_{\nu}^{{~{}\rho\sigma\lambda}}+2R_{{\mu\rho\sigma\nu}}(H^{2})^{\rho\sigma}$
$\displaystyle+2R_{{\rho\sigma\lambda(\mu}}H_{\nu)}^{~{}\lambda\delta}H^{\rho\sigma}_{~{}~{}_{\delta}}+\frac{1}{3}({\nabla}_{\mu}H_{\rho\sigma\lambda})({\nabla}_{\nu}H^{\rho\sigma\lambda})-({\nabla}_{\lambda}H_{\rho\sigma\mu})({\nabla}^{\lambda}H^{\rho\sigma}_{~{}~{}\nu})$
$\displaystyle+2H_{{\mu\rho\sigma}}H_{{\nu\lambda\delta}}H^{{\eta\delta\sigma}}H_{\eta}^{~{}~{}\lambda\rho}+2H_{{\mu\sigma\lambda}}H_{{\nu\rho}}^{~{}~{}\lambda}(H^{2})^{\rho\sigma}\Big{]}+{\cal
O}(\alpha^{\prime 2}),$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle{\nabla}^{\lambda}H_{{\lambda\mu\nu}}-({\nabla}^{\lambda}\Phi^{\prime})H_{{\mu\nu\lambda}}+\alpha^{\prime}\Big{[}{\nabla}^{\lambda}H^{\rho\sigma}_{~{}~{}[\mu}R_{{}_{\nu]}\lambda\rho\sigma}-({\nabla}_{\lambda}H_{\rho\mu\nu})(H^{2})^{\lambda\rho}$
$\displaystyle-2({\nabla}^{\lambda}H^{\rho\sigma}_{~{}~{}[\mu})H_{{}_{\nu]}\rho\delta}H_{\lambda\sigma}^{~{}\;\delta}\Big{]}+{\cal
O}(\alpha^{\prime 2}),$ $\displaystyle 0$ $\displaystyle=$ $\displaystyle
2\Lambda+{\nabla}^{2}\Phi^{\prime}-({\nabla}\Phi^{\prime})^{2}+\frac{2}{3}H^{{2}}-\alpha^{\prime}\Big{[}\frac{1}{4}R_{{\mu\rho\sigma\lambda}}R^{{\mu\rho\sigma\lambda}}$
(4.3) $\displaystyle~{}~{}~{}-$
$\displaystyle\frac{1}{3}({\nabla}_{\lambda}H_{\mu\nu\rho})({\nabla}^{\lambda}H^{\mu\nu\rho})-\frac{1}{2}H^{\mu\nu}_{~{}~{}\lambda}H^{\rho\sigma\lambda}R_{\mu\nu\rho\sigma}-R_{\mu\nu}(H^{2})^{\mu\nu}+\frac{3}{2}(H^{2})_{\mu\nu}(H^{2})^{\mu\nu}$
$\displaystyle~{}~{}~{}+$
$\displaystyle\frac{5}{6}H_{\mu\nu\rho}H^{\mu}_{~{}~{}\sigma\lambda}H^{\nu\sigma}_{~{}~{}\delta}H^{\rho\lambda\delta}\Big{]}+{\cal
O}(\alpha^{\prime 2}),$

where ${R}_{{\mu\nu\rho\sigma}}$ is the Riemann tensor field of the metric
$G_{{\mu\nu}}$,
$(H^{2})^{\mu\nu}=H^{\mu\rho\sigma}H_{\rho\sigma}^{~{}~{}\nu}$, and in second
equation of (4.2) $\Phi^{\prime}=\Phi+\alpha^{\prime}qH^{2}$ for some
coefficient $q$ [44]. 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 $\sigma$-model with a
non-vanishing field strength $H$ and the dilaton field in the absence of a
cosmological constant $\Lambda$. The field equations (4.2) are satisfied for
all backgrounds of Table 1 with the same dilaton fields given in Table 2.

##  5 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 $\Lambda$. 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 is also straightforward to determine the dilaton in the YB deformed
Nappi-Witten model [31] 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 $\mathscr{G}$ of $G$.
Fortunately, the classical r-matrices related to some of the Lie superalgebras
are available [46, 47, 48, 49] (see also [50]). 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.

### Acknowledgements

The authors would like to thank the anonymous referee for invaluable comments
and criticisms. This work has been supported by the research vice chancellor
of Azarbaijan Shahid Madani University under research fund No. 97/231.

##  Appendix A Some computational results related to YB deformations of the
$H_{4}$ WZW model

In this appendix, we tabulate the nonzero components of tensors
$H_{\mu\nu\rho},(H^{2})_{\mu\nu},R_{{}_{\mu\nu}}$ and Riemann tensor field
related to the backgrounds of YB deformed $H_{4}$ WZW model representing in
Table 1. We note that for all backgrounds one quickly finds that $R=H^{2}=0$;
moreover, the only nonzero component of $R_{{}_{\mu\nu}}$ is $R_{{}_{xx}}$
which is indicated for all backgrounds in Table 3.

Table 3. The nonzero components of tensors
$R_{{}_{\mu\nu}},R_{{}_{\mu\nu\rho\sigma}},H_{{}_{\mu\nu\rho}}$ and
$(H^{2})_{{}_{\mu\nu}}$ related to the backgrounds represented in Table 1
Background symbol $R_{{}_{xx}}$ $R_{{}_{\mu\nu\rho\sigma}}$
$H_{{}_{\mu\nu\rho}}$ $(H^{2})_{{}_{\mu\nu}}$ $H^{(\kappa}_{4}.I$ \-
$\frac{1}{2}$ $R_{{}_{xyxu}}=\frac{-e^{x}}{4}$ $H_{{}_{xyu}}=\frac{\kappa
e^{x}}{2}$ $(H^{2})_{{}_{xx}}=\frac{-\kappa^{2}}{2}$
$H^{(\kappa,\eta)}_{4}.II$ -$\frac{1}{2}$ $R_{{}_{xyxu}}=\frac{-e^{x}}{4}$
$H_{{}_{xyu}}=\frac{\kappa e^{x}}{2}$
$(H^{2})_{{}_{xx}}=-\frac{\kappa^{2}}{2}$
$H^{(\kappa,\eta,\tilde{A})}_{4}.III$ -$\frac{1}{2}$
$R_{{}_{xyxu}}=\frac{-e^{x}}{4},$ $H_{{}_{xyu}}=\frac{\kappa e^{x}}{2}$
$(H^{2})_{{}_{xx}}=\frac{-\kappa^{2}}{2}$
$R_{{}_{xuxu}}=\frac{5\rho\eta^{2}e^{2x}}{4}$
$H_{{}_{xuv}}=\frac{-\tilde{A}e^{x}}{2}$
$(H^{2})_{{}_{xu}}=\frac{-\kappa\tilde{A}}{2}e^{x}$
$(H^{2})_{{}_{uu}}=\frac{-{\tilde{A}}^{2}}{2}e^{2x}$
$H^{(\kappa,\eta,\tilde{A})}_{4}.IV$ -$\frac{1}{2(1-\eta^{2})^{2}}$
$R_{{}_{xyxu}}=\frac{-e^{x}}{4(1-\eta^{2})^{2}}$
$H_{{}_{xyu}}=\frac{1}{2}(\kappa-\frac{\tilde{A}}{1-\eta^{2}})e^{x}$
$(H^{2})_{{}_{xx}}=\frac{-1}{2}\big{(}\kappa-\frac{\tilde{A}}{1-\eta^{2}}\big{)}^{2}$
$H^{(\kappa,\eta,\tilde{A})}_{4}.V$ -$\frac{1}{2}$
$R_{{}_{xyxu}}=-\frac{e^{x}}{4}$
$H_{{}_{xyu}}=\frac{1}{2}(\kappa+\tilde{A})e^{x}$
$(H^{2})_{{}_{xx}}=\frac{-(\kappa+\tilde{A})^{2}}{2}$
$H^{(\kappa,\eta,\tilde{A})}_{4}.VI$ -$\frac{1}{2}$
$R_{{}_{xyxu}}=\frac{-e^{x}}{4},$
$H_{{}_{xyu}}=\frac{1}{2}(\kappa+\tilde{A})e^{x}$
$(H^{2})_{{}_{xx}}=\frac{-(\kappa+\tilde{A})^{2}}{2}$
$R_{{}_{xuxu}}=\frac{5\rho\eta^{2}e^{2x}}{4(1-\eta^{2})}$
$H_{{}_{xuv}}=\frac{-\tilde{A}e^{x}}{2}$
$(H^{2})_{{}_{xu}}=\frac{-(\kappa+\tilde{A})\tilde{A}e^{x}}{2}$
$(H^{2})_{{}_{uu}}=\frac{-{\tilde{A}}^{2}}{2}e^{2x}$
$H^{(\kappa,\eta,\tilde{A})}_{4}.VII$ -$\frac{1}{2}$
$R_{{}_{xyxu}}=\frac{-e^{x}}{4},$ $H_{{}_{xyu}}=\frac{\kappa e^{x}}{2}$
$(H^{2})_{{}_{xx}}=\frac{-\kappa^{2}}{2}$
$R_{{}_{xuxu}}=\frac{5\rho\eta^{2}e^{2x}}{4(1+\eta^{2})}$
$H_{{}_{xuv}}=\frac{-\tilde{A}e^{x}}{2}$
$(H^{2})_{{}_{xu}}=\frac{-\kappa\tilde{A}e^{x}}{2}$
$(H^{2})_{{}_{uu}}=\frac{-{\tilde{A}}^{2}}{2}e^{2x}$
$H^{(\kappa,\eta)}_{4}.VIII$ -$\frac{(1-\eta^{2})^{2}}{2}$
$R_{{}_{xyxu}}=-\frac{e^{x}(1-\eta^{2})^{2}}{4}$ $H_{{}_{xyu}}=\frac{\kappa
e^{x}}{2}$ $(H^{2})_{{}_{xx}}=-\frac{\kappa^{2}}{2}$
$H^{(\kappa,\eta,\tilde{A})}_{{}_{4,q}}.IX$
-$\frac{1}{2}\big{(}\frac{1-\eta^{2}}{1-\eta^{2}q^{4}}\big{)}^{2}$
$R_{{}_{xyxu}}=-\frac{1}{4}\big{(}\frac{1-\eta^{2}}{1-\eta^{2}q^{4}}\big{)}^{2}e^{x}$
$H_{{}_{xyu}}=\frac{e^{x}}{2}\big{(}\kappa-\frac{\tilde{A}q^{2}(1-\eta^{2})}{1-\eta^{2}q^{4}}\big{)}$
$(H^{2})_{{}_{xx}}=-\frac{1}{2}{\big{(}\kappa-\frac{\tilde{A}q^{2}(1-\eta^{2})}{1-\eta^{2}q^{4}}\big{)}^{2}}$
$H^{(\kappa,\tilde{A})}_{4}.X$ \- $\frac{1}{2}$
$R_{{}_{xyxu}}=\frac{-e^{x}}{4}$
$H_{{}_{xyu}}=\frac{1}{2}(\kappa-\tilde{A})e^{x}$
$(H^{2})_{{}_{xx}}=\frac{-1}{2}(\kappa-\tilde{A})^{2}$

##  Appendix B More on YB deformations of the Nappi-Witten WZW model

###  B.1 Nonequivalent r-matrices

In this appendix, using automorphism group of the Nappi-Witten Lie algebra
[42, 41] we find all nonequivalent r-matrices as solutions of the (m)CYBE
(3.26). We then find all YB deformations of WZW model on the Nappi-Witten Lie
group. Before proceeding to get nonequivalent r-matrices, let us introduce the
Nappi-Witten Lie algebra. It is spanned by the set of generators
$T_{{}_{i}}=(P_{{}_{1}},P_{{}_{2}},J,T)$ which fulfill the following nonzero
commutation rules [38]:

$\displaystyle[J,P_{{}_{1}}]=P_{{}_{2}},~{}~{}~{}~{}~{}~{}[J,P_{{}_{2}}]=-P_{{}_{1}},~{}~{}~{}~{}~{}~{}[P_{{}_{1}},P_{{}_{2}}]=T.$
(B.1)

This algebra is a central extension of the 2D Poincaré algebra to which it
reduces if one sets $T=0$. Using (B.1) together with (2.7), one obtains the
non-degenerate ad-invariant bilinear form $\Omega_{ij}$ on the Nappi-Witten
Lie algebra, giving

$\displaystyle\Omega_{ij}=\begin{pmatrix}1&~{}0~{}&~{}0~{}&0\;\\\ 0&1&0&0\\\
0&0&b&1\\\ 0&0&1&0\end{pmatrix},$ (B.2)

where $b$ is a real constant. In order to calculate the left-invariant one-
forms ${L}_{\alpha}$ we parameterize the Nappi-Witten group with coordinates
$x^{\mu}=(a_{{}_{1}},a_{{}_{2}},u,v)$ so that its elements can be written as

$\displaystyle
g=\exp\bigl{(}a_{{}_{1}}P_{{}_{1}}+a_{{}_{2}}P_{{}_{2}}\bigr{)}\exp\bigl{(}u\,J+v\,T\bigr{)}.$
(B.3)

We then obtain

$\displaystyle L_{\alpha}=g^{-1}~{}\partial_{\alpha}g$ $\displaystyle=$
$\displaystyle\big{(}\cos u\,\partial_{\alpha}a_{1}+\sin
u\,\partial_{\alpha}a_{2}\big{)}\,P_{{}_{1}}+\big{(}\cos
u\,\partial_{\alpha}a_{2}-\sin u\,\partial_{\alpha}a_{1}\big{)}\,P_{{}_{2}}$
(B.4)
$\displaystyle+\partial_{\alpha}u\,J+\big{(}\partial_{\alpha}v+\frac{1}{2}\,a_{2}\,\partial_{\alpha}a_{1}-\frac{1}{2}\,a_{1}\,\partial_{\alpha}a_{2}\big{)}\,T.$

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

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
2dudv+b\,du^{2}+d{a_{{}_{1}}}^{2}+d{a_{{}_{2}}}^{2}-a_{{}_{1}}\,da_{{}_{2}}du+a_{{}_{2}}\,da_{{}_{1}}du,$
$\displaystyle B$ $\displaystyle=$ $\displaystyle u~{}da_{{}_{1}}\wedge
da_{{}_{2}}.$ (B.5)

According to (2.6), (2.8) and (3.17) to construct the YB deformation of WZW
model on the Nappi-Witten group we need to find the corresponding
nonequivalent r-matrices. Using relations (B.1) and (B.2) one may obtain the
general solution of the (m)CYBE (3.26) as follows [31]

$\displaystyle
r^{ij}=\left(\begin{tabular}[]{cccc}0&$\pm\sqrt{\omega}$&0&$m_{3}$\\\
$\mp\sqrt{\omega}$&0&0&$m_{5}$\\\ 0&0&0&$m_{6}$\\\
-$m_{3}$&-$m_{5}$&-$m_{6}$&0\\\ \end{tabular}\right),$ (B.10)

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 (3.25) the automorphism groups of the
Nappi-Witten algebra are expressed as matrices in the following form [41, 42]

$\displaystyle A_{1}=\left(\begin{tabular}[]{cccc}a&b&0&-ac -bd\\\
-b&a&0&-ad+bc\\\ c&d&1&e\\\ 0&0&0&$a^{2}+b^{2}$\\\
\end{tabular}\right),~{}~{}~{}~{}A_{2}=\left(\begin{tabular}[]{cccc}a&b&0&ac
+bd\\\ b&-a&0&-ad+bc\\\ c&d&-1&e\\\ 0&0&0&-$(a^{2}+b^{2})$\\\
\end{tabular}\right),~{}~{}a^{2}+b^{2}\neq 0,$ (B.19)

where $a,b,c,d$ and $e$ are some arbitrary constants. Ultimately, by employing
formula (3.27) of Proposition 3.1, the r-matrices for the Nappi-Witten algebra
are split into the following six nonequivalent families

$\displaystyle r^{\prime}_{{}_{1}}=\left(\begin{tabular}[]{cccc}0&0&0&1\\\
0&0&0&0\\\ 0&0&0&0\\\ -1&0&0&0\\\
\end{tabular}\right),~{}~{}~{}r^{\prime}_{{}_{2}}=\left(\begin{tabular}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\\\
\end{tabular}\right),~{}~{}~{}r^{\prime}_{{}_{3}}=\left(\begin{tabular}[]{cccc}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&-1\\\ 0&0&1&0\\\ \end{tabular}\right),$ $\displaystyle
r^{\prime}_{{}_{4}}=\left(\begin{tabular}[]{cccc}0&1&0&0\\\ -1&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\
\end{tabular}\right),~{}~{}~{}r^{\prime}_{{}_{5}}=\left(\begin{tabular}[]{cccc}0&1&0&0\\\
-1&0&0&0\\\ 0&0&0&$p^{2}$\\\ 0&0&-$p^{2}$&0\\\
\end{tabular}\right),~{}~{}~{}r^{\prime}_{{}_{6}}=\left(\begin{tabular}[]{cccc}0&-1&0&0\\\
1&0&0&0\\\ 0&0&0&-$p^{2}$\\\ 0&0&$p^{2}$&0\\\ \end{tabular}\right),$ (B.33)

where $p$ is a nonzero constant.

###  B.2 Backgrounds for YB deformations of the Nappi-Witten WZW model

Hence it is straightforward to study YB deformations of the Nappi-Witten WZW
model. Similar to the YB deformations of the $H_{4}$ WZW model in Sec. 3, we
use formulas (3.19), (3.20) and (B.2) to obtain all linear $R$-operators
corresponding to the nonequivalent r-matrices of the Nappi-Witten algebra.
Then, by using (2.6) together with (2.8) one obtains all YB deformed
backgrounds of the Nappi-Witten WZW model. The deformed backgrounds including
metric and $B$-field are summarized in Table 4.

Table 4. YB deformed backgrounds of the Nappi-Witten WZW model∗ Nonequivalent
r-matrices Backgrounds Comments $r^{\prime}_{{}_{1}}$
$ds^{2}=da_{1}^{2}+da_{2}^{2}+(b-\eta^{2})du^{2}+2dudv+a_{2}da_{1}du-
a_{1}da_{2}du,$ $B=\kappa\,u\,da_{1}\wedge da_{2}$ $\omega=0$
$r^{\prime}_{{}_{2}}$
$ds^{2}=da_{1}^{2}+da_{2}^{2}+\frac{1}{1-\eta^{2}}\big{[}bdu^{2}+2du\,dv+a_{2}da_{1}du-
a_{1}da_{2}du\big{]},$ $B=\kappa\,u\,da_{1}\wedge
da_{2}+\frac{\tilde{A}}{2(1-\eta^{2})}\big{[}a_{2}\,du\wedge
da_{1}+a_{1}\,da_{2}\wedge du\big{]}$ $\omega=0$ $r^{\prime}_{{}_{3}}$
$ds^{2}=da_{1}^{2}+da_{2}^{2}+\frac{1}{1-\eta^{2}}\big{[}bdu^{2}+2du\,dv+a_{2}da_{1}du-
a_{1}da_{2}du\big{]},$ $B=\kappa\,u\,da_{1}\wedge
da_{2}-\frac{\tilde{A}}{2(1-\eta^{2})}\big{[}a_{2}\,du\wedge
da_{1}+a_{1}\,da_{2}\wedge du\big{]}$ $\omega=0$ $r^{\prime}_{{}_{4}}$
$ds^{2}=da_{1}^{2}+da_{2}^{2}+(1+\eta^{2})\big{[}bdu^{2}+2du\,dv+a_{2}da_{1}du-
a_{1}da_{2}du\big{]},$ $B=\kappa\,u\,da_{1}\wedge da_{2}$ $\omega=1$
$r^{\prime}_{{}_{5}}$
$ds^{2}=da_{1}^{2}+da_{2}^{2}+\frac{1+\eta^{2}}{1-\eta^{2}p^{4}}\big{[}bdu^{2}+2du\,dv+a_{2}da_{1}du-
a_{1}da_{2}du\big{]},$ $B=\kappa\,u\,da_{1}\wedge
da_{2}+\frac{\tilde{A}p^{2}(1+\eta^{2})}{2(1-\eta^{2}p^{4})}\big{[}a_{2}\,du\wedge
da_{1}+a_{1}\,da_{2}\wedge du\big{]}$ $\omega=1$ $r^{\prime}_{{}_{6}}$
$ds^{2}=da_{1}^{2}+da_{2}^{2}+\frac{1+\eta^{2}}{1-\eta^{2}p^{4}}\big{[}bdu^{2}+2du\,dv+a_{2}da_{1}du-
a_{1}da_{2}du\big{]},$ $B=\kappa\,u\,da_{1}\wedge
da_{2}-\frac{\tilde{A}p^{2}(1+\eta^{2})}{2(1-\eta^{2}p^{4})}\big{[}a_{2}\,du\wedge
da_{1}+a_{1}\,da_{2}\wedge du\big{]}$ $\omega=1$

∗Here we have ignored the total derivative terms that appeared in the
$B$-fields part.

###  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).

Table 5. The dilaton fields making the Nappi-Witten deformed backgrounds
conformal up to the one- and two-loop orders
Nonequivalent r-matrices Dilaton fields $r^{\prime}_{{}_{1}}$
$~{}~{}~{}~{}~{}\frac{1}{4}(\kappa^{2}-1)u^{2}+c_{1}u+c_{2}$
$r^{\prime}_{{}_{2}}$
$~{}~{}~{}~{}~{}\frac{1}{4}\big{[}\kappa^{2}-\frac{2\kappa\tilde{A}}{\eta^{2}-1}-\frac{(1-\tilde{A}^{2})}{(\eta^{2}-1)^{2}}\big{]}u^{2}+c_{1}u+c_{2}$
$r^{\prime}_{{}_{3}}$
$~{}~{}~{}~{}~{}\frac{1}{4}\big{[}\kappa^{2}+\frac{2\kappa\tilde{A}}{\eta^{2}-1}-\frac{(1-\tilde{A}^{2})}{(\eta^{2}-1)^{2}}\big{]}u^{2}+c_{1}u+c_{2}$
$r^{\prime}_{{}_{4}}$
$~{}~{}~{}~{}~{}\frac{1}{4}\big{[}\kappa^{2}-(1+\eta^{2})^{2}\big{]}u^{2}+c_{1}u+c_{2}$
$r^{\prime}_{{}_{5}}$
$~{}~{}~{}~{}~{}\frac{1}{4(1-\eta^{2}p^{4})^{2}}\Big{[}\kappa^{2}(1-\eta^{2}p^{4})^{2}+2\kappa\tilde{A}p^{2}(1+\eta^{2})(1-\eta^{2}p^{4})-(1-\tilde{A}^{2}p^{4})(1+\eta^{2})^{2}\Big{]}u^{2}+c_{1}u+c_{2}$
$r^{\prime}_{{}_{6}}$
$~{}~{}~{}~{}~{}\frac{1}{4(1-\eta^{2}p^{4})^{2}}\Big{[}\kappa^{2}(1-\eta^{2}p^{4})^{2}-2\kappa\tilde{A}p^{2}(1+\eta^{2})(1-\eta^{2}p^{4})-(1-\tilde{A}^{2}p^{4})(1+\eta^{2})^{2}\Big{]}u^{2}+c_{1}u+c_{2}$

From solving Eqs. (4.1) we find the general form of the dilaton fields that
make the YB deformed backgrounds conformal up to the one-loop order. The
results obtained for dilaton fields are represented in Table 5. It would also
be interesting to consider the conformal invariance of the Nappi-Witten
deformed backgrounds up to the two-loop order. To this end, we solve the field
equations (4.2) and show the YB deformed backgrounds that are conformal at
one-loop remain conformal at two-loop with the same dilaton fields given in
Table 5. In this way, the cosmological constant vanishes.

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