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
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ξ±=τ±σ2
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and Lax pair
$({\cal A}_{0},{\cal A}_{1})$
(A0,A1)
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Therefore, we have the monodromy matrix for WZW model as follows:
where x is a spectral parameter.
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