CLASSICAL INTEGRABILITY OF TWO DIMENSIONAL NON LINEAR SIGMA MODELS \footnote{ Preprint number: LTH 543}
N. Mohammedi \footnote{Permanent address: Laboratoire de Math\'ematiques et Physique Th\'eorique
The conditions under which a general two-dimensional non-linear sigma model is classically integrable are given. These requirements are found by demanding that the equations of motion of the theory are expressible as a zero curvature relation. Some new integrable two-dimensional sigma models are then presented.
hep-th0206179.tex

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\title{CLASSICAL INTEGRABILITY OF TWO DIMENSIONAL NON LINEAR SIGMA MODELS \footnote{
Preprint number: LTH 543}}



\author{N. Mohammedi \footnote{Permanent address: 
Laboratoire de Math\'ematiques et Physique Th\'eorique, 
Universit\'e Fran\c{c}ois Rabelais,
Facult\'e des Sciences et Techniques,  
Parc de Grandmont, F-37200 Tours, France.}$^{,}$\footnote{E-mail: 
nouri@celfi.phys.univ-tours.fr}}



\address{\textit{
Division of Theoretical Physics, 
Department of Mathematical Sciences,\\ 
Chadwick Building, 
The University of Liverpool, \\
Liverpool, 
L69 3BX, 
England, U.K.\\ }}

\maketitle






\vspace{5mm}

\begin{abstract}
The conditions under which a general two-dimensional non-linear sigma model is
classically integrable are given. These requirements are found by demanding that
the equations of motion of the theory are expressible as a zero curvature
relation. Some new integrable two-dimensional sigma models are then presented.
 
    
\end{abstract}


\smallskip
\smallskip

\centerline{June  2002}


%\tableofcontents

  
%\newpage 


















\section{Introduction}



As is well-known, a two-dimensional theory is classically integrable if its equations
of motion can be represented as a zero curvature relation. That is, a  
Lax pair $\left({\cal{A}}_0,{\cal{A}}_1\right)$ can be found such that the commutator
\be
\left[\dd_0+{\cal A}_0\left(\lambda\right)\,\,,\,\,
\dd_1+{\cal A}_1\left(\lambda\right)\right] =0
\ee
yields the equations of motion of the model under consideration for all values 
of the spectral parameter $\lambda$. Here, the two-dimensional coordinates are
$\left(\tau,\sigma\right)$ with $\dd_0={\dd\over \dd\tau}$ and $\dd_1={\dd\over \dd\sigma}$.
In the rest of the paper, however, we will use the complex coordinates
$\left(z=\tau +i\sigma\,,\,\bar z=\tau -i\sigma\right)$ together with
$\dd= {\dd\over\dd z}$ and $\bar\dd={\dd\over\dd\bar z}$.
\par
The classical example of this setting is the principal chiral non-linear sigma 
model in two dimensions \cite{pohlmeyer,polyakov}. 
The action of this theory is given by
\be
S=\int {\rm d} z{\rm d}\bar{z}\,\, \eta_{kl}\,e^k_i\left(X\right) e^l_j\left(X\right)
\dd X^i \bar\dd X^j
\,\,\,\,,
\label{chiral}
\ee
where $\eta_{kl}$ is an invertible invariant bilinear form of the Lie algebra ${\cal{G}}$
defined by $\left[T_i\,\,,\,\,T_j\right]=f_{ij}^kT_k$.
That is,  $\eta_{ij}f^j_{kl} +\eta_{kj}f^j_{il}=0$.
The vielbeins $e^k_i$ satisfy the 
Cartan-Maurer relation 
\be
\dd_i e^k_j -\dd_j e^k_i + f^k_{mn} e^m_i e^n_j=0\,\,\,.  
\ee
The equations of motion of the principal chiral sigma model can be cast in the form
\be
\dd \left(e^k_i\bar\dd X^i\right) + \bar\dd \left(e^k_i\dd X^i\right)=0
\,\,\,.
\ee
This expression is reached after using the properties of $\eta_{ij}$ and $e^k_i$.
\par
The Lax pair for this model is constructed in the following manner: Consider the commutator
\bea
&\left[\dd +\left({1\over 2} +\lambda \pm\sqrt{{1\over 4}+\lambda^2}\right)
\left(e^k_i\dd X^i\right)T_k\,\,,\,\,
\bar\dd +\left({1\over 2} -\lambda \pm\sqrt{{1\over 4}+\lambda^2}\right) 
\left(e^l_j\bar \dd X^j\right)T_l\right]&
\nonumber\\
&=\left\{-\lambda
\left[\dd \left(e^k_i\bar\dd X^i\right) + \bar\dd \left(e^k_i\dd X^i\right)\right]
+\left({1\over 2} \pm\sqrt{{1\over 4}+\lambda^2}\right)
\left[\dd\left(e^k_i\bar\dd X^i\right)
-\bar\dd\left(e^k_i\dd X^i\right) +
f_{mn}^k\left(e^m_i\dd X^i\right)\left(e^n_j\bar\dd X^j\right)\right]
\right\}T_k
\,\,\,.&
\label{chirallax}
\eea
If this commutator vanishes for all values of $\lambda$ then the term proportional to
$-\lambda$ yields the equations of motion of the principal chiral sigma model.
The second term is identically zero due to the Cartan-Maurer relations between the vielbeins
$e^k_i$.
\par
In summary, the equations of motion are completely specified in terms of the currents
\be
I^j=e^j_i\dd X^i\,\,\,\,,\,\,\,\,\bar I^j_i=e^j_i \bar\dd X^i \,\,\,.
\ee
These currents satisfy the Bianchi identities $\dd \bar I^i -\bar \dd I^i +f^i_{jk}I^j\bar I^k=0$.
A convenient way of interpreting the Bianchi identities stems from the expressions
\be
\dd X^i=\left(e^{-1}\right)^i_j I^j\,\,\,\,,\,\,\,\,
\bar\dd X^i=\left(e^{-1}\right)^i_j \bar I^j \,\,\,\,,
\ee
The Bianchi identities are nothing else than the integrability conditions of these 
last relations. Namely,
\be
\dd \bar \dd X^i - \bar\dd \dd X^i=0\,\,\,.
\ee
This remark will be of use in what follows. Finally, a curvature which can be written as 
a linear combination of the equations of motion and the Bianchi identities is found.



\section{Generalisation}


We would like now to generalise the previous discussion to
any two-dimensional non-linear sigma model. Our aim is to determine 
the conditions under which a theory is integrable. 
We start from the action
\be
S=\int {\rm d} z{\rm d}\bar{z}\,\, Q_{ij}\left(X\right)\dd X^i \bar\dd X^j
\,\,\,\,.
\ee
The tensor $Q_{ij}$ has symmetric and anti-symmetric parts.
The equations of motion resulting from the variation of this action are given by
\be
{\cal{E}}_l\equiv\dd _lQ_{ij}\dd X^i \bar\dd X^j - \dd\left(Q_{lj}\bar\dd X^j\right)
- \bar\dd\left(Q_{il}\dd X^i\right)=0\,\,\,.
\ee
By contacting the equations of motion with the matrix $K^{ml}_n\left(X\right)$, one gets
\be
K^{ml}_n{\cal{E}}_l=
\left(K^{ml}_n\dd_lQ_{ij} + Q_{lj}\dd_iK^{ml}_n +Q_{il}\dd_j K^{ml}_n\right)
\dd X^i\bar\dd X^j - \dd\left(K^{ml}_nQ_{lj}\bar\dd X^j\right)
- \bar\dd\left(K^{ml}_nQ_{il}\dd X^i\right)=0\,\,\,.
\ee
%We notice that if $K^{l}_a$ is a Killing vector of the tensor $Q_{ij}$ then the
%term proportional to $\dd X^i\bar\dd X^j$ vanishes. This takes place when the 
%action has the global symmetry $X^i\longrightarrow X^i +\epsilon^a K^{i}_a$, where
%$\epsilon^a$ is a constant. 
In general, $K^{ml}_n$ could also
depend on some parameters which will be interpreted as the spectral parameters.
\par
Our next step is the introduction of the two currents defined by
\be
\dd X^i =\alpha^i_j\left(X\right) A^j\,\,\,\,\,\,,\,\,\,\,\,\,
\bar\dd X^i =\beta^i_j\left(X\right) \bar A^j\,\,\,\,,
\ee
where $\alpha^i_j$ and $\beta^i_j$ are two invertible matrices.
In terms of these currents, the equations of motion take the form
\be
K^{ml}_n{\cal{E}}_l=
\left(K^{ml}_n\dd_lQ_{ij} + Q_{lj}\dd_iK^{ml}_n +Q_{il}\dd_j K^{ml}_n\right)
\alpha^i_p\beta^j_q A^p\bar A^q
- \dd\left(K^{ml}_nQ_{lj} \beta^j_p \bar A^p\right)
- \bar\dd\left(K^{ml}_nQ_{il} \alpha^i_q A^q\right)=0\,\,\,.
\ee
There are various ways of expressing the equations of motion. However, this last
form is the most convenient for our purpose.
\par
The above definition of the two currents $A^i$ and $\bar A^i$ leads to some Bianchi identities.
These are found from the integrability condition 
\be
{\cal{B}}^i\equiv
\dd\bar\dd X^i-\bar\dd\dd X^i=
\dd\left(\beta^i_q \bar A^q\right)-\bar\dd\left(\alpha^i_p A^p\right)=0
\,\,\,\,.
\ee
By contracting this last equation with the matrix $L^m_{ni}\left(X\right)$, one gets
\be
L^m_{ni}{\cal{B}}^i =
 \dd\left(L^m_{ni}\beta^i_q\bar A^q\right) - \bar\dd\left(L^m_{ni}\alpha^i_pA^p\right)
-\left(\dd_jL^m_{ni} -\dd_iL^m_{nj}\right)\alpha^j_p\beta^i_qA^p\bar A^q=0
\,\,\,.
\ee
Like $K^{ml}_n$, the tensor $L^m_{ni}$ could in principle depend on some parameters too.
We have therefore expressed the equations of motion in terms of two currents
subject to some Bianchi identities. We now turn our attention to the construction of the
Lax pair.
\par
Let us consider the two differential operator as defined by
\be
{\cal{D}}^i_j=\delta^i_j\dd + V^i_j\,\,\,\,\,\,,\,\,\,\,\,\,\,
\bar{\cal{D}}^i_j=\delta^i_j\bar\dd + \bar V^i_j\,\,\,\,
\ee
where $V^i_j$ and $\bar V^i_j$ are two gauge connections. The curvature of these two
operators is given by
\be
F^i_k\equiv {\cal{D}}^i_j\bar{\cal{D}}^j_k-\bar{\cal{D}}^i_j{\cal{D}}^j_k
=\dd \bar V^i_k -\bar\dd V^i_k + V^i_j \bar V^j_k -\bar V^i_j V^j_k\,\,\,\,.
\ee
{}Following the case of the principal chiral sigma model, we require that
this curvature satisfies
\be
F^i_k= K^{il}_k{\cal{E}}_l + L^i_{kl}{\cal{B}}^l\,\,\,\,,
\label{lax}
\ee
\par
By matching the terms involving the two derivatives $\dd$ and 
$\bar\dd$ on both sides of equation (\ref{lax}), one is forced to 
choose the two gauge fields as 
\bea
V^i_k&=&\left(K^{il}_k Q_{jl}+L^i_{kj}\right)\alpha^j_m A^m
=\left(K^{il}_k Q_{jl}+L^i_{kj}\right)\dd X^j
\nonumber\\
\bar V^i_k&=&\left(-K^{il}_k Q_{lj}+L^i_{kj}\right)\beta^j_n\bar A^n
=\left(-K^{il}_k Q_{lj}+L^i_{kj}\right)\bar\dd X^j
\,\,\,.
\label{V}
\eea
Substituting  back $V^i_k$ and $\bar V^k_i$ in (\ref{lax}), leads to the 
following differential equation
\bea
&\left(K^{il}_k\dd_lQ_{mn} + Q_{ln}\dd_mK^{il}_k +Q_{ml}\dd_n K^{il}_k\right)
-\left(\dd_mL_{kn}^i-\dd_nL_{km}^i +L_{jm}^iL_{kn}^j-L_{jn}^iL_{km}^j\right)&
\nonumber\\
&=\left(K^{il}_j K^{jt}_k-K^{it}_j K^{jl}_k \right)Q_{ln}Q_{mt}
+\left(K^{il}_jL^j_{kn}- L^i_{jn}K^{jl}_k\right)Q_{ml}
+\left(K^{il}_jL^j_{km}- L^i_{jm}K^{jl}_k\right)Q_{ln}\,\,\,.&
\label{diff}
\eea
It is remarkable that this last equation is independent of $\alpha^i_j$ and $\beta^i_j$.
Therefore the choice of $A^i$ and $\bar A^i$ has no influence on the construction.
We conclude that the curvature $F^i_j$, corresponding to the 
gauge fields $V^i_j$ and $\bar V^i_j$ in  (\ref{V}),  can be written as a linear combination
of the equations of motion and the Bianchi identities if equation (\ref{diff}) is
fulfilled.
Furthermore,  the quantities $K^{ij}_k$ and $L_{jk}^i$ must depend on 
at least one spectral parameter $\lambda$. 
If the equation $F^i_k=0$ holds for all values of $\lambda$ then 
one obtains ${\cal{E}}_i=0$ and ${\cal{B}}^i=0$. 
Notice that none of the tensors $Q_{ij}$, $K^{ij}_k$ and $L^i_{jk}$ 
is, a priopri, known. Therefore, the class of two-dimensional non-linear sigma model 
which is classically integrable is specified by all the tensors $Q_{ij}$ satisfying (\ref{diff})
for some quantities  $K^{ij}_k$ and $L^i_{jk}$.
\par
So far there is no Lie algebra structure in our construction. This is due
to the fact that our gauge connections $V^i_j$ and $\bar V^i_j$ are not necessarily 
evaluated in a Lie algebra. Let us now choose a gauge connection that takes value in 
the Lie algebra ${\cal{H}}$ generated by $\left[H_i\,,\,H_j\right]=h^k_{ij}H_k$. Namely,
$W=W^iH_i$ and $\bar W=\bar W^i H_i$. In a similar manner, we require that 
\be
\left[\dd +W\,,\, \bar \dd +\bar W\right]=
\left(\dd \bar W^i -\bar\dd W^i +h^i_{jk}W^j\bar W^k\right)H_i
=\left(\mu^{il}{\cal{E}}_l +\rho_l^i{\cal{B}}^l\right)H_i
\label{curvature}
\,\,\,,
\ee
where $\mu^{il}$ and $\rho^i_l$ are the equivalent of $K^{il}_j$ and $L^i_{jl}$,
respectively. The contracted equations of motion, $\mu^{ml}{\cal{E}}_l$,
and the contracted Bianchi identities, $\rho^m_{i}{\cal{B}}^i$,  are
given by
\bea
\mu^{ml}{\cal{E}}_l &=&
\left(\mu^{ml}\dd_lQ_{ij} + Q_{lj}\dd_i\mu^{ml} +Q_{il}\dd_j \mu^{ml}\right)
\alpha^i_p\beta^j_q A^p\bar A^q
- \dd\left(\mu^{ml}Q_{lj} \beta^j_p \bar A^p\right)
- \bar\dd\left(\mu^{ml}Q_{il} \alpha^i_q A^q\right)=0\nonumber\\
\rho^m_{i}{\cal{B}}^i &=&
 \dd\left(\rho^m_{i}\beta^i_q\bar A^q\right) - \bar\dd\left(\rho^m_{i}\alpha^i_pA^p\right)
-\left(\dd_j\rho^m_{i} -\dd_i\rho^m_{j}\right)\alpha^j_p\beta^i_qA^p\bar A^q=0
\,\,\,.
\eea
A direct inspection of equation (\ref{curvature}) leads to the gauge connections
\bea
W^i &=&\left(\mu^{il}  Q_{jl}+\rho^i_j\right)\alpha^j_m A^m
=\left(\mu^{il}  Q_{jl}+\rho^{i}_{j}\right)\dd X^j
\nonumber\\
\bar W^i&=&\left(-\mu^{il}  Q_{lj}+\rho^{i}_{j}\right)\beta^j_n\bar A^n
=\left(-\mu^{il}  Q_{lj}+\rho^{i}_{j}\right)\bar\dd X^j
\,\,\,.
\label{W}
\eea
The differential equation we obtain in this case is given by
\bea
&\left(\mu^{il}\dd_lQ_{mn} + Q_{ln}\dd_m\mu^{il} +Q_{ml}\dd_n \mu^{il}\right)
-\left(\dd_m\rho^{i}_{n}-\dd_n\rho^{i}_{m}+h^i_{jk}\rho^j_m\rho^k_n
\right)&
\nonumber\\
&=h^i_{jk}\mu^{jl}\mu^{kt} Q_{ln}Q_{mt}
+h^i_{jk}\mu^{jl}\left(\rho^{k}_{m}Q_{ln}+\rho^{k}_{n}Q_{ml}\right)\,\,\,. &
\label{liediff}
\eea
Here also, the dependence on the spectral parameter $\lambda$ could only be in the tensors
$\mu^{ij}$ and $\rho^{i}_j$.
\par
There are some interesting geometrical structures appearing in equation (\ref{diff})
(and similarly in equation (\ref{liediff})). These structures are the Lie derivative
of $Q_{ij}$ with respect to $K^{il}_k$ and the curvature of the differential 
operator $\left(\nabla_i\right)^j_k= \delta^j_k \dd_i + L^j_{ki}$. Using the properties 
of these two geometrical objects, one can derive some consistency conditions for the
validity of our differential equations. According to our investigation, 
the form of these conditions is too involved and does not lead to any new insight.
Finally, it is worth mentioning that equation (\ref{liediff}) is a generalisation
of an equation that appears in the context of Poisson-Lie duality in two-dimensional
sigma models \cite{klimcik1,klimcik2}. 
%This equation is given by 
%\bea
%E^{l}_a\dd_lQ_{mn} + Q_{ln}\dd_m E^{l}_a +Q_{ml}\dd_n E^{l}_a
%=h^i_{jk}\mu^{ja}\mu^{kb} K^l_aK^t_bQ_{ln}Q_{mt}
%\eea 
%and has been solved in []. 
This suggests a possible link between classical integrability
an T-duality in sigma models. This relashionship has been noticed 
and exploited in \cite{balog1,evans,balog2} for a very particular class of classically integrable 
sigma models.



\section{Examples}


It is easier to consider the case described by the gauge connections $W^i$ and $\bar W^i$
in (\ref{W}) 
and the differential equation (\ref{liediff}). 
Let us first check that one obtains some known integrable two dimensional non-linear
sigma models. Indeed, the Lax pair construction (\ref{chirallax})
for the principal chiral sigma model is 
found by taking
\bea
\mu^{il}&=&\lambda \eta^{ik}\left(e^{-1}\right)^l_k\nonumber\\
\rho^i_n&=&\left({1\over 2}\pm\sqrt{{1\over 4}+\lambda^2}\right)e^i_n\nonumber\\
h^i_{jk}&=&f^i_{jk}
\,\,\,\,\eea
where $\eta^{ij}$ is the inverse of the bilinear form $\eta_{ij}$.
Similarly, all the Lax pairs corresponding to the
other known integrable sigma models can be constructed in a similar manner.  
\par
By examining equation (\ref{liediff}) one can realise that there are new
integrable sigma models. 
A straight forward solution to this equation is found by taking
\bea
&& \mu^{ij}=\omega^{ik}v^j_k\nonumber\\
&& v^l_k\dd_lQ_{mn} + Q_{ln}\dd_m v^l_k +Q_{ml}\dd_n v^l_k=0\nonumber\\
&& h^i_{jk}=0\nonumber\\
&& \rho^i_j=\dd_j \Lambda^i\,\,\,\,,
\eea
where $\Lambda^i\left(X\right)$ is an arbitrary function and $\omega^{ik}$ is a constant tensor. 
This means that 
the matrix $v^l_k$ is a Killing vector of the tensor $Q_{ij}$ and the Lie algebra 
${\cal{H}}$ is Abelian.
In this case 
equations (\ref{liediff}) is satisfied for any arbitrary 
tensors $\omega^{il}$ and $\Lambda^i$. 
Notice that neither the equations of motion nor the Bianchi identities
have, for this class of sigma models, terms quadratic in the currents
$A^i$ and $\bar A^i$. 
The Lax pair construction for these types of sigma models is given by
\bea
&\left[\dd +\left(\omega^{ik} v^l_k  Q_{jl}\dd X^j+\dd \Lambda^i\right)H_i\,\,,\,\,
\bar\dd +\left(-\omega^{mn} v^r_n  Q_{rs}\bar\dd X^s+\bar\dd \Lambda^m\right)H_m\right]&
\nonumber\\
&=\left\{\omega^{mn}\left[-\dd\left(v^r_n  Q_{rs}\bar\dd X^s\right)-
\bar\dd\left(v^l_n  Q_{jl}\dd X^j\right)\right]
+\left(\dd\bar\dd \Lambda^m-\bar \dd\dd \Lambda^m\right)\right\}H_m \,\,\,.&
\eea
When this curvature vanishes, the term proportional to the arbitrary tensor $\omega^{mn}$
yields the equations of motion while the second term is identically zero. 
We have chosen here to use directly $\dd X^i$ and $\bar \dd X^i$ instead of the
currents $A^i$ and $\bar A^i$.
\par
We conclude that any two-dimensional non-linear sigma model possessing
isometries is classically integrable\footnote{Since the zero curvature condition
leads to $v^i_j{\cal{E}}_i=0$ (and not ${\cal{E}}_i=0$), we require 
the Killing vectors $v^i_j$ to be invertible.}.
In this case, for a given sigma model,
one can find an infinite number of Lax pairs: one for each choice of
$\left(\Lambda^i,\omega^{ij}\right)$. Moreover, one has a Lax pair construction involving 
an arbitrary number of spectral parameters. 
\par
An example of this class is given by a generalisation of the principal
chiral sigma model. We consider the action
\be
S=\int {\rm d} z{\rm d}\bar{z}\,\, \Omega_{kl}\,e^k_i\left(X\right) e^l_j\left(X\right)
\dd X^i \bar\dd X^j
\,\,\,\,,
\label{genchiral}
\ee
where $\Omega_{kl}$ can be any constant matrix (not necessarily the
bilinear form $\eta_{kl}$). The underlying Lie algebra is still 
${\cal{G}}$, defined in the introductory section. 
The Killing vector $v^i_j$ corresponding to the tensor 
$Q_{ij}=\Omega_{kl}\,e^k_i e^l_j$ is given by
$v^i_j=R_j^k\left(e^{-1}\right)^i_k$, where $R^i_j$ is defined by
$g^{-1}T_ig=R^j_i T_j$ and verifies the relations
$\dd_k R^i_j=f^i_{mn}R^m_j e^n_k$ and $f^i_{jk} R^j_m R^k_n=f^j_{mn} R^i_j$.
Here $g\left(X\right)$ is an element in the Lie group
corresponding to the Lie algebra ${\cal{G}}$.
The vielbeins are given by the usual experession 
$e^i_j T_i=\left(g^{-1}\dd_j g\right)^iT_i$.
\par
At this point some remarks are due. Firstly, since 
the  principal chiral sigma model is a special case of the action (\ref{genchiral}),
we have therefore found a new Lax pair formulation for this model. 
Secondly, there is another model based on the Lie algebra ${\cal{G}}=SU\left(2\right)$
and which is known to be classically integrable \cite{cherednik}. This 
model corresponds to taking a diagonal $\Omega_{ij}$ with
$\Omega_{11}=\Omega_{22}\ne \Omega_{33}$. In this case also our Lax pair construction
is different from that of \cite{cherednik}. 
Finally, we should mention that an earlier attempt to study the classical integrability
of the model (\ref{genchiral}) did not produce any new integrable systems \cite{sochen}.
\par
The second example we consider here is described by the action
\be
S=\int {\rm d} z{\rm d}\bar{z}\,\, Q^{ij}\left(\chi\right)\dd \chi_i \bar\dd \chi_j
\,\,\,\,,
\ee
where $Q^{ij}$ is given through its inverse by
\be
Q_{ij}=\eta_{ij} +f^k_{ij}\chi_k\,\,\,.
\ee
This model is the non-Abelian T-dual of the chiral principal sigma model 
\cite{nadual1,nadual2,nadual3,nadual4}.
It is not known whether this model is classically integrable. 
The study of this model sheds, therefore,  some light
on whether duality preserves integrability.
\par
The equations of motion stemming from our action can be cast in the form
\be
\dd \bar J^l - \bar \dd J^l + f^l_{ij} J^i\bar J^j=0\,\,\,,
\ee
where the two currents are given by \cite{amit}
\be
J^l=Q^{il}\dd\chi_i\,\,\,\,\,,\,\,\,\,\,
\bar J^l=-Q^{lj}\bar\dd\chi_j\,\,\,.
\ee
This definition leads to the Bianchi identities ($\dd\bar\dd \chi_i -\bar\dd\dd\chi_i=0$)
\be
\dd \bar J^j + \bar \dd J^j + 
\eta^{jk}f_{kl}^i\chi_i\left(\dd \bar J^l - \bar \dd J^l + f^l_{mn} J^m\bar J^n\right)
=0\,\,\,.
\ee
The Lax pair construction is found through the commutator
\bea
&\left[\dd +\left({1\over 2} +\lambda \pm\sqrt{{1\over 4}+\lambda^2}\right)
J^kT_k\,\,,\,\,
\bar\dd +\left({1\over 2} -\lambda \pm\sqrt{{1\over 4}+\lambda^2}\right) 
\bar J^lT_l\right]&
\nonumber\\
&=\left\{-\lambda\left[
\dd \bar J^j + \bar \dd J^j + 
\eta^{jk}f_{kl}^i\chi_i\left(\dd \bar J^l - \bar \dd J^l + f^l_{mn} J^m\bar J^n\right)\right]
\right. & \nonumber\\
&+\left[\left({1\over 2} \pm\sqrt{{1\over 4}+\lambda^2}\right)\delta^j_l
+\lambda \eta^{jk}f_{kl}^i\chi_i\right]
\left(\dd \bar J^l - \bar \dd J^l + f^l_{mn} J^m\bar J^n\right)
\left.\right\}T_j
\,\,\,.&
\eea
If this commutators is to vanish then the term propotional
$-\lambda$ yields the Bianchi identities while the term proportional to
$\left[\left({1\over 2} \pm\sqrt{{1\over 4}+\lambda^2}\right)\delta^j_l
+\lambda \eta^{jk}f_{kl}^i\chi_i\right]$ leads to the equations of motion.
Therefore, the non-Abelian dual of the principal chiral sigma model is also
classically integrable.
\par
In summary,  we have given in this paper the necessary constraints for a non-linear sigma model to
be classicaly integrable. The equations found are not empty and lead to some unexpected
integrable theories. The geometrical nature of these equations might be employed to
find more solutions.
  



\smallskip
\smallskip
\smallskip
\bigskip

\noindent $\underline{\hbox{\bf Acknowledgments}}$: I would like to thank the members of the 
Division of Theoretical Physics at the University of Liverpool for their hospitality. 
I would like also to thank P\'eter Forg\'acs, Ian Jack , Tim Jones and Max Niedermaier for very
useful discussions.
This research is supported by the CNRS.

\smallskip
\smallskip   


\begin{thebibliography}{99}

\bibitem{pohlmeyer}
K. Pohlmeyer, {\it Integrable Hamiltonian systems and interactions through quadratic constraints},
Commun. Math. Phys. {\bf 46} (1976) 207.
\bibitem{polyakov}
A. M. Polyakov and P. B. Wiegmann, {\it Theory of non-Abelian Goldstone bosons},
Phys. Lett. {\bf B131} (1983) 121.

\bibitem{klimcik1}
 C. Klim$\check{c}$\'{\i}k and P. $\check{S}$evera, {\it Dual non-Abelian duality and the Drinfeld double},
Phys. Lett. {\bf B351} (1995) 455, hep-th/9502122.

\bibitem{klimcik2}
C. Klim$\check{c}$\'{\i}k, {\it Poisson-Lie T-duality}, Nucl. Phys. Proc. Suppl. {\bf 46}
(1996) 116, hep-th/9509095.

\bibitem{balog1}
J. Balog, P. Forg\'acs,  Z. Horv\'ath and L. Palla, {\it A new family of $SU(2)$ symmetric integrable
sigma models}, Phys. Lett. {\bf B324} (1994) 403, hep-th/9307030.

\bibitem{evans}
J. M. Evans and T. J. Hollowood, {\it Integrable theories that are asymptotically CFT},
Nucl. Phys. {\bf B438} (1995) 469, hep-th/9407113.

\bibitem{balog2}
J. Balog, P. Forg\'acs and L. Palla, {\it A two-dimensional integrable axionic 
$\sigma$-model and T-duality}, Phys. Lett. {\bf B484} (2000) 367, hep-th/0004180. 

\bibitem{cherednik}
I. V. Cherednik, {\it Relativistically invariant quasiclassical limits of integrable
two-dimensional quantum models}, Theor. Math. Phys. {\bf 47} (1981) 422.

\bibitem{sochen} 
N. Sochen, {\it Integrable generalized principal chiral models},
Phys. Lett. {\bf B391} (1997) 374,  hep-th/9607009 .


\bibitem{nadual1}
B. E. Fridling and A. Jevicki, {\it Dual representations and ultraviolet divergences in non-linear sigma
models}, Phys. Lett. {\bf B134} (1984) 70.


\bibitem{nadual2}
E. S. Fradkin and  A. A. Tseytlin, {\it Quantum equivalence of dual field theories},
Annals Phys. {\bf 162} (1985) 31.

\bibitem{nadual3}
X.  C. de la Ossa and  F. Quevedo, {\it Duality symmetries from non-Abelian 
isometries in string theories}, Nucl. Phys. {\bf B403} (1993) 377, hep-th/9210021.

\bibitem{nadual4}
E. \'Alvarez, L. \'Alvarez-Gaum\'e and Y. Lozano, {\it On non-Abelian Duality},
Nucl. Phys. {\bf B424} (1994) 155, hep-th/9403155.

\bibitem{amit}
A. Giveon and M. Ro$\check{c}$ek, {\it On non-Abelian duality},
Nucl. Phys. {\bf B421} (1994) 173, hep-th/9308154.


\end{thebibliography}

\end{document}







 In this case the equations of motion are 
in the form of an Abelian curvature $\dd\left(K^{kl}Q_{lj}\bar\dd X^j\right)
+\bar\dd\left(K^{kl}Q_{il}\bar\dd X^i\right)=0$.
Hence the Lax pair construction follows 
the same steps as in the case of the principal chiral sigma nodel (with the obvious replacements
for the currents involved).
\par
The case of interest is therefore when $K^{kl}$ is not a Killing vector
of $Q_{ij}$ (namely, $K^{kl}\dd_lQ_{ij} + Q_{lj}\dd_iK^{kl} +Q_{il}\dd_j K^{kl}\ne 0$).
Let us now find the conditions under which the equations of motion take 
the form of a zero curvature condition. 
This curvature is certainly non-Abelian. This is because whatever currents
$I^i$ and $\bar I^i$ we choose to express the equations of motion, they must be proportional  
to $\dd X^j$ and $\bar\dd X^j$, respectively. Since our equations of motion contain 
terms involving $\dd X^i \bar \dd X^j$, they can only come from a combination of the
product $I^i\bar I^j$. This is why we must look for a non-Abelian curvature whose
vanishing yields the equations of motion.
\par
We will first include a non-Abelian structure in the previous Lie algebra (\ref{Halgebra}). 
We take this to be given by
\be
\left[H_i\,\,,\,\,H_j\right]=h^k_{ij}h_k\,\,,\,\,
\left[J_i\,\,,\,\,J_j\right]=0\,\,,\,\,
\left[H_i\,\,,\,\,J_j\right]=0\,\,\,,
\ee
where $h^k_{ij}$ are some structure constants.
\par
The two terms $\dd\left(K^{kl}Q_{lj}\bar\dd X^j\right)$ and 
$\bar\dd\left(K^{kl}Q_{il}\bar\dd X^i\right)$ in the equations of motion suggest
the examination of the following commutator
\bea
&&\left[\dd + \left(K^{kl}Q_{il}\bar\dd X^i\right)H_k\,\,,\,\,
\dd + \left(-K^{mn}Q_{nj}\bar\dd X^j\right)H_m\right]
\nonumber\\
&&=
\left\{-K^{kl}K^{mn}Q_{il}Q_{nj}f^p_{km}\dd X^i\bar\dd X^j 
- \dd\left(K^{pl}Q_{lj}\bar\dd X^j\right)
- \bar\dd\left(K^{pl}Q_{il}\bar\dd X^i\right)\right\}H_p\,\,\,.
\eea
In order for the equations of motion to correspond to the vanishing of this curvature,
one must impose the conditions
\be
K^{pl}\dd_lQ_{ij} + Q_{lj}\dd_iK^{pl} +Q_{il}\dd_j K^{pl}
=-K^{kl}K^{mn}Q_{il}Q_{nj}h^p_{km} \,\,\,.
\label{PL}
\ee
These are precisely the constraints needed for a Poisson-Lie duality to occur.
\par
Provided that the condition (\ref{PL}) holds, the parametric Lax pair formulation
is then as follows
\bea
\left[\dd + \left(K^{kl}Q_{il}\bar\dd X^i\right)H_k +\left(\alpha^k_i\dd X^i\right)J_k\,\,,\,\,
\dd + \left(-K^{mn}Q_{nj}\bar\dd X^j\right)H_m +\left(\alpha^m_j\bar\dd X^j\right)J_m\right]
\nonumber\\
=
\left\{-K^{kl}K^{mn}Q_{il}Q_{nj}f^p_{km}\dd X^i\bar\dd X^j 
- \dd\left(K^{pl}Q_{lj}\bar\dd X^j\right)
- \bar\dd\left(K^{pl}Q_{il}\bar\dd X^i\right)\right\}H_p
+\alpha^k_i\left\{\dd\bar\dd X^i-\bar\dd \dd X^i\right\}J_k
\,\,\,.
\eea



\end{document}


\smallskip
\smallskip
\smallskip
\bigskip

\noindent $\underline{\hbox{\bf Acknowledgments}}$: I would like to thank the members of the 
Division of Theoretical Physics at the University of Liverpool for their hospitality.  
This research is supported by the CNRS.

\smallskip
\smallskip   


\begin{thebibliography}{99}


\end{thebibliography}
















\documentstyle[prd,aps,epsf,eqsecnum]{revtex}
\begin{document}
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\title{CLASSICAL INTEGRABILITY OF TWO DIMENSIONAL NON LINEAR SIGMA MODELS}



\author{N. Mohammedi \footnote{En d\'el\'egation au CNRS.}$^{,}$\footnote{Permanent address: 
Laboratoire de Math\'ematiques et Physique Th\'eorique, 
Universit\'e Fran\c{c}ois Rabelais,
Facult\'e des Sciences et Techniques,  
Parc de Grandmont, F-37200 Tours, France.}$^{,}$\footnote{E-mail: 
nouri@celfi.phys.univ-tours.fr}}



\address{\textit{
Division of Theoretical Physics, 
Department of Mathematical Sciences,\\ 
Chadwick Building, 
The University of Liverpool, \\
Liverpool, 
L69 3BX, 
England, U.K.\\ }}

\maketitle

\vspace{5mm}

\begin{abstract}
Write something nice !. 
 
    
\end{abstract}


\smallskip
\smallskip

\centerline{April  2002}


%\tableofcontents

  
%\newpage 


















\section{Introduction}



As it is well-known, a two-dimensional theory is classically integrable if its equations
of motion can be represented as a zero curvature relation. That is, a  
Lax pair $\left({\cal{A}}_0,{\cal{A}}_1\right)$ can be found such that the commutator
\be
\left[\dd_0+{\cal A}_0\left(\lambda\right)\,\,,\,\,
\dd_1+{\cal A}_1\left(\lambda\right)\right] =0
\ee
yields the equations of motion of the model under consideration for all values 
of the spectral parameter $\lambda$.
\par
The classical example of this setting is a class of non-linear sigma 
models represented by the following pair of equations
\bea
&&\dd \bar I +\bar\dd I=0 \nonumber\\
&&  \dd \bar I - \bar\dd I +\left[I\,\,,\,\,\bar I\right]=0\,\,\,,
\eea
where $I_\mu$ is a current valued in the Lie algebra of a Lie group ${\cal{G}}$ such that
$I_\mu =I_\mu^a T_a$ and $\left[T_a\,\,,\,\,T_b\right]=f^c_{ab}T_c$.
The Lax pair for this model is given by
\bea
\left[\dd +{1\over 1+\lambda}I\,\,,\,\,
\bar\dd +{1\over 1-\lambda} \bar I\right]
={\lambda\over 1-\lambda^2}\left(\dd \bar I +\bar\dd I\right)
+{1\over 1-\lambda^2}\left(\dd \bar I - \bar\dd I +\left[I\,\,,\,\,\bar I\right]\right)
\,\,\,.
\eea
\end{document}


\be
S=\int {\rm d} z{\rm d}\bar{z}\,\, \eta_{kl}\,e^k_i\left(X\right) e^l_j\left(X\right)
\dd X^i \bar\dd X^j
\,\,\,\,,
\ee
where $\eta_{kl}$ is an invertible invariant bilinear form of the Lie algebra
$\left[T_i\,\,,\,\,T_j\right]=f_{ij}^kT_k$ such that $\eta_{ij}f^j_{kl} +\eta_{kj}f^j_{il}=0$.
The vielbeins $e^k_i$ satisfy the 
Cartan-Maurer relation 
\be
\dd_i e^k_j -\dd_j e^k_i + f^k_{mn} e^m_i e^n_j=0\,\,\,.  
\ee
The equations of motion of the principal chiral sigma model can be cast in the form
\be
\dd \left(e^k_i\bar\dd X^i\right) + \bar\dd \left(e^k_i\dd X^i\right)=0
\,\,\,.
\ee
This expression is reached after using the properties of $\eta_{ij}$ and $e^k_i$.
The Lax pair for this model is given by
\bea
&\left[\dd +{1\over 1+\lambda}\left(e^k_i\dd X^i\right)T_k\,\,,\,\,
\bar\dd +{1\over 1-\lambda} \left(-e^l_j\bar \dd X^j\right)T_l\right]&
\nonumber\\
&={\lambda\over 1-\lambda^2}
\left\{\dd \left(e^a_i\bar\dd X^i\right) + \bar\dd \left(e^a_i\dd X^i\right)\right\}T_a
+{1\over 1-\lambda^2}\left\{\dd\left(e^k_i\bar\dd X^i\right)T_k
-\bar\dd\left(e^k_i\dd X^i\right)T_k +
\left[\left(e^k_i\bar\dd X^i\right)T_k\,\,,\,\,\left(e^l_i\bar\dd X^i\right)T_k\right]
\right\}
\,\,\,.&
\eea
If this commutator vanishes for all values of $\lambda$ then the term proportional to
${\lambda\over 1-\lambda^2}$ yields the equations of motion of the principal chiral sigma model.
The second term is identically zero due to the Cartan-Maurer relations between the vielbeins
$e^k_i$.
\par
In summary, the equations of motion are completely specified in terms of the currents
\be
I^j=e^j_i\dd X^i\,\,\,\,,\,\,\,\,\bar I^j_i=e^j_i \bar\dd X^i \,\,\,.
\ee
These currents satisfy the Bianchi identities $\dd \bar I^i -\bar \dd I^i +f^i_{jk}I^j\bar I^k=0$.
A convenient way of interpreting the Bianchi identities stems from the expressions
\be
\dd X^i=E^i_j I^j\,\,\,\,,\,\,\,\,
\bar\dd X^i=E^i_j \bar I^j \,\,\,\,,
\ee
where $E^i_j$ are the inverses of the vielbeins $e^j_i$ (that is $E^i_j e^k_i=\delta^k_j$, $E^i_k e^k_j=\delta^i_j$).
They satisfy the relations $E^i_m\dd_iE^j_n-E^i_n\dd_iE^j_m=f^i_{mn}E^j_i$.
The Bianchi identities are nothing else than the integrability conditions of these 
last relations. Namely,
\be
\dd \bar \dd X^i - \bar\dd \dd X^i=0\,\,\,.
\ee
This remark will be of use in what follows.
\par
We would like to give here another Lax pair formulation for the principal chiral sigma model.
We will relay on some observations from the above review of this model. 
A general feature of the Lax pair formulation is that the zero curvature relation yields 
always two expressions: One of these is the equations of motion while the other corresponds
to some Bianchi identities. The most important remark, however, is that the equations of motion of the principal 
chiral sigma model are 
already in the form of a zero curvature condition:
$\dd \left(e^j_i\bar\dd X^i\right) + \bar\dd \left(e^j_i\dd X^i\right)=0$. This 
curvature is Abelian in nature (there are no quadratic terms in the currents).
The other observation is that the Bianchi identities are simply
$\dd \bar \dd X^i - \bar\dd \dd X^i=0$ when the currents $I^i$ and $\bar I^i$ are introduced through
$\dd X^i=E^i_j I^j$ and $\bar\dd X^i=E^i_j \bar I^j$, respectively. 
\par
We start from the Abelian Lie algebra as given by the commutation relations
\be
\left[H_i\,\,,\,\,H_j\right]=0\,\,,\,\,
\left[J_i\,\,,\,\,J_j\right]=0\,\,,\,\,
\left[H_i\,\,,\,\,J_j\right]=0\,\,\,.
\label{Halgebra}
\ee
The above remarks suggest the following Lax pair formulation for the principal chiral sigma
model
\bea
&& \left[\dd +\left(e^k_i\dd X^i\right)H_k + \left(\alpha^k_i\dd X^i\right)J_k
\,\,,\,\,
\bar\dd + \left(-e^l_j\bar \dd X^j\right)H_l+ \left(\alpha^l_j\dd X^j\right)J_l\right]
\nonumber\\
&& = \left\{\dd \left(e^k_i\bar\dd X^i\right) + \bar\dd \left(e^k_i\dd X^i\right)\right\}H_k
+\alpha^k_i\left\{\dd \bar \dd X^i - \bar\dd \dd X^i\right\}J_k\,\,\,.
\eea
Here $\alpha^k_i$ is an invertible constant tensor which plays the role of the spectral parameter.
When expressed in terms of the currents $I^j=e^j_i\dd X^i$ and $\bar I^j=e^j_i\bar\dd X^i$, the commutator
takes the form
\bea
&& \left[\dd + I^i H_i + \left(\alpha^k_i E^i_l I^l\right)J_k
\,\,,\,\,
\bar\dd - \bar I^j H_j +\left(\alpha^m_j E^j_n\bar I^nJ_m\right)\right]
\nonumber\\
&& = \left\{\dd \bar I^i + \bar\dd I^i \right\}H_i
+\alpha^j_i E^i_k\left\{ \dd \bar I^k_i -\bar \dd I^k +f^k_{mn}I^m\bar I^n\right\}J_j\,\,\,.
\eea 
To reach this form, we have used the chain rule to express $\dd E^i_k=\left(\dd_j E^i_k\right)E^j_l I^l$
and $\bar\dd E^i_k=\left(\dd_j E^i_k\right)E^j_l \bar I^l$ together with
$E^i_m\dd_iE^j_n-E^i_n\dd_iE^j_m=f^i_{mn}E^j_i$.



\section{Generalisation}


We would like to study the generalisation of the previous discussion to
any two-dimensional non-linear sigma model.
We start from the action
\be
S=\int {\rm d} z{\rm d}\bar{z}\,\, Q_{ij}\left(X\right)\dd X^i \bar\dd X^j
\,\,\,\,.
\ee
The tensor $Q_{ij}$ has symmetric and anti-symmetric parts.
The equations of motion resulting from the variation of this action are given by
\be
\dd _lQ_{ij}\dd X^i \bar\dd X^j - \dd\left(Q_{lj}\bar\dd X^j\right)
- \bar\dd\left(Q_{il}\bar\dd X^i\right)=0\,\,\,.
\ee
By contacting the equations of motion with an invertible matrix $K^{kl}\left(X\right)$, one gets
\be
\left(K^{kl}\dd_lQ_{ij} + Q_{lj}\dd_iK^{kl} +Q_{il}\dd_j K^{kl}\right)
\dd X^i\bar\dd X^j - \dd\left(K^{kl}Q_{lj}\bar\dd X^j\right)
- \bar\dd\left(K^{kl}Q_{il}\bar\dd X^i\right)=0\,\,\,.
\ee
We notice that if $K^{kl}$ is a Killing vector of the tensor $Q_{ij}$ then the
term proportional to $\dd X^i\bar\dd X^j$ vanishes. This takes place when the 
action has the global symmetry $X^i\longrightarrow X^i +\epsilon_j K^{ji}$, where
$\epsilon_j$ is a constant. In this case the equations of motion are 
in the form of an Abelian curvature $\dd\left(K^{kl}Q_{lj}\bar\dd X^j\right)
+\bar\dd\left(K^{kl}Q_{il}\bar\dd X^i\right)=0$.
Hence the Lax pair construction follows 
the same steps as in the case of the principal chiral sigma nodel (with the obvious replacements
for the currents involved).
\par
The case of interest is therefore when $K^{kl}$ is not a Killing vector
of $Q_{ij}$ (namely, $K^{kl}\dd_lQ_{ij} + Q_{lj}\dd_iK^{kl} +Q_{il}\dd_j K^{kl}\ne 0$).
Let us now find the conditions under which the equations of motion take 
the form of a zero curvature condition. 
This curvature is certainly non-Abelian. This is because whatever currents
$I^i$ and $\bar I^i$ we choose to express the equations of motion, they must be proportional  
to $\dd X^j$ and $\bar\dd X^j$, respectively. Since our equations of motion contain 
terms involving $\dd X^i \bar \dd X^j$, they can only come from a combination of the
product $I^i\bar I^j$. This is why we must look for a non-Abelian curvature whose
vanishing yields the equations of motion.
\par
We will first include a non-Abelian structure in the previous Lie algebra (\ref{Halgebra}). 
We take this to be given by
\be
\left[H_i\,\,,\,\,H_j\right]=h^k_{ij}h_k\,\,,\,\,
\left[J_i\,\,,\,\,J_j\right]=0\,\,,\,\,
\left[H_i\,\,,\,\,J_j\right]=0\,\,\,,
\ee
where $h^k_{ij}$ are some structure constants.
\par
The two terms $\dd\left(K^{kl}Q_{lj}\bar\dd X^j\right)$ and 
$\bar\dd\left(K^{kl}Q_{il}\bar\dd X^i\right)$ in the equations of motion suggest
the examination of the following commutator
\bea
&&\left[\dd + \left(K^{kl}Q_{il}\bar\dd X^i\right)H_k\,\,,\,\,
\dd + \left(-K^{mn}Q_{nj}\bar\dd X^j\right)H_m\right]
\nonumber\\
&&=
\left\{-K^{kl}K^{mn}Q_{il}Q_{nj}f^p_{km}\dd X^i\bar\dd X^j 
- \dd\left(K^{pl}Q_{lj}\bar\dd X^j\right)
- \bar\dd\left(K^{pl}Q_{il}\bar\dd X^i\right)\right\}H_p\,\,\,.
\eea
In order for the equations of motion to correspond to the vanishing of this curvature,
one must impose the conditions
\be
K^{pl}\dd_lQ_{ij} + Q_{lj}\dd_iK^{pl} +Q_{il}\dd_j K^{pl}
=-K^{kl}K^{mn}Q_{il}Q_{nj}h^p_{km} \,\,\,.
\label{PL}
\ee
These are precisely the constraints needed for a Poisson-Lie duality to occur.
\par
Provided that the condition (\ref{PL}) holds, the parametric Lax pair formulation
is then as follows
\bea
\left[\dd + \left(K^{kl}Q_{il}\bar\dd X^i\right)H_k +\left(\alpha^k_i\dd X^i\right)J_k\,\,,\,\,
\dd + \left(-K^{mn}Q_{nj}\bar\dd X^j\right)H_m +\left(\alpha^m_j\bar\dd X^j\right)J_m\right]
\nonumber\\
=
\left\{-K^{kl}K^{mn}Q_{il}Q_{nj}f^p_{km}\dd X^i\bar\dd X^j 
- \dd\left(K^{pl}Q_{lj}\bar\dd X^j\right)
- \bar\dd\left(K^{pl}Q_{il}\bar\dd X^i\right)\right\}H_p
+\alpha^k_i\left\{\dd\bar\dd X^i-\bar\dd \dd X^i\right\}J_k
\,\,\,.
\eea



\end{document}


\smallskip
\smallskip
\smallskip
\bigskip

\noindent $\underline{\hbox{\bf Acknowledgments}}$: I would like to thank the members of the 
Division of Theoretical Physics at the University of Liverpool for their hospitality.  
This research is supported by the CNRS.

\smallskip
\smallskip   


\begin{thebibliography}{99}


\end{thebibliography}




Annika Grosse <annikagrosse@web.de>  Nikdirt@aol.com  cmarzban@ou.edu                                       
                                          marzban@ou.edu Eholzer <eholzer@debitel.net>  
                                          Susanne K�ffers <susanne_koeffers@hotmail.com>













hep-th0206179.extracted_bib
\begin{thebibliography}{99}

\bibitem{pohlmeyer}
K. Pohlmeyer, {\it Integrable Hamiltonian systems and interactions through quadratic constraints},
Commun. Math. Phys. {\bf 46} (1976) 207.
\bibitem{polyakov}
A. M. Polyakov and P. B. Wiegmann, {\it Theory of non-Abelian Goldstone bosons},
Phys. Lett. {\bf B131} (1983) 121.

\bibitem{klimcik1}
 C. Klim$\check{c}$\'{\i}k and P. $\check{S}$evera, {\it Dual non-Abelian duality and the Drinfeld double},
Phys. Lett. {\bf B351} (1995) 455, hep-th/9502122.

\bibitem{klimcik2}
C. Klim$\check{c}$\'{\i}k, {\it Poisson-Lie T-duality}, Nucl. Phys. Proc. Suppl. {\bf 46}
(1996) 116, hep-th/9509095.

\bibitem{balog1}
J. Balog, P. Forg\'acs,  Z. Horv\'ath and L. Palla, {\it A new family of $SU(2)$ symmetric integrable
sigma models}, Phys. Lett. {\bf B324} (1994) 403, hep-th/9307030.

\bibitem{evans}
J. M. Evans and T. J. Hollowood, {\it Integrable theories that are asymptotically CFT},
Nucl. Phys. {\bf B438} (1995) 469, hep-th/9407113.

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