We have perturbed Wess-Zumino-Witten (WZW) models and also N=(2,2) supersymmetric sigma models on Lie groups by adding a term containing complex structure to their actions. Then, using non-coordinate basis, we have shown that for N=(2,2) supersymmetric sigma models on Lie groups the conditions (from the algebraic point of view) for the preservation of the N=(2,2) supersymmetry impose that the complex structure must be invariant; so only the Abelian Lie algebras admit these deformations preserving the N=(2,2) supersymmetry. Also, we have shown that the perturbed WZW model with this term, using Hermitian (not necessarily invariant) condition, is an integrable model.
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In the light cone coordinates and Lax pair Therefore, we have the monodromy matrix for WZW model as follows: where x is a spectral parameter.
$\xi ^{\pm }=\frac{\tau \pm \sigma }{2}$
, the equations of motion for WZW model are written in the following Lax form:11 $$[\partial _{+}+x\;L_{\mu }\hspace{0.0pt}^{A} X_{A}\partial _{+}x^{\mu }]\psi =0,$$
\begin{equation*}[{\partial }_{-}+\;L_{\nu }\hspace{0.0pt}^{A} X_{A}\partial _{-}x^{\nu }]\psi =0,\end{equation*}
$({\cal A}_{0},{\cal A}_{1})$
are in the following forms: \begin{equation*}{\cal A}_{0}(x)=L_{\mu }\hspace{0.0pt}^{A}X_{A}(\frac{x+1}{2}\partial _{0}x^{\mu }+\frac{x-1}{2}\partial _{1}x^{\mu }),\end{equation*}
\begin{equation*}{\cal A}_{1}(x)=L_{\mu }\hspace{0.0pt}^{A}X_{A}(\frac{x-1}{2}\partial _{0}x^{\mu }+\frac{x+1}{2}\partial _{1}x^{\mu }).\end{equation*}
\begin{equation*}T(x,\tau )=P \exp {\big \lbrace -}X_{A}(\frac{x-1}{2}\int _{0}^{2\pi }L_{\mu }\hspace{0.0pt}^{A}\partial _{0}x^{\mu }d\sigma +\frac{x+1}{2}\int _{0}^{2\pi }L_{\mu }\hspace{0.0pt}^{A}\partial _{1}x^{\mu }d\sigma ){\big \rbrace },\end{equation*}
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