Superdeformed $\mathbb{CP}$ $\sigma$-model equivalence

We find the novel class of the supersymmetric deformation of the $\mathbb{CP}^{1}$ $\sigma$-model and its equivalence with the generalised chiral Gross-Neveu. This construction allows the use of field-theoretic techniques and particularly the study of renormalisability and $\beta$-function. Provided approach is useful in finding conformal limits and establishes relation between chiral (GN) and sigma model description (geometric), which is explicitly demonstrated for the case of $\mathbb{R} \times S^{1}$/Super-Thirring models. We also provide discussion on its emergence in $\mathcal{N}=2$ Liouville and 4-dim Chern-Simons theory.