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Perturbed Wess-Zumino-Witten models and N=(2,2) supersymmetric sigma models on Lie groups with complex structure

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.