We introduce and study conformal field theories specified by $W-$algebras commuting with certain set of screening charges. These CFTs possess perturbations which define integrable QFTs. We establish that these QFTs have local and non-local Integrals of Motion and admit the perturbation theory in the weak coupling region. We construct factorized scattering theory which is consistent with non-local Integrals of Motion and perturbation theory. In the strong coupling limit the $S-$matrix of this QFT tends to the scattering matrix of the $O(N)$ sigma model. The perturbation theory, Bethe anzatz technique, renormalization group approach and methods of conformal field theory are applied to show, that the constructed QFTs are dual to integrable deformation of $O(N)$ sigma-models.
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