Superdeformed $$\mathbb{C}\mathbb{P}$$ σ-Model Equivalence

We find the novel supersymmetric deformation of the $$\mathbb{C}{{\mathbb{P}}^{1}}$$ σ-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 β function. Provided approach is useful in finding conformal limits and establishes relation between chiral (GN) and sigma model (geometric) descriptions, which is explicitly demonstrated for the case of $$\mathbb{R} \times {{S}^{1}}$$ /Super-Thirring models. We also make remarks on its emergence in $$\mathcal{N} = 2$$ Liouville and 4-dim Chern–Simons theory.