Supersymmetric Grassmannian Sigma Models in Gross-Neveu Formalism

We revisit the classical aspects of $\mathcal{N}=(2,2)$ supersymmetric sigma models with Hermitian symmetric target spaces, using the so-called Gross-Neveu (first-order GLSM) formalism. We reformulate these models for complex Grassmannians in terms of simple supersymmetric Lagrangians with polynomial interactions. For maximal isotropic Grassmannians we propose two types of equivalent Lagrangians, which make either supersymmetry or the geometry of target space manifest. These reformulations can be seen as current-current deformations of curved $\beta \gamma$ systems. The $\mathsf{CP}^{1}$ supersymmetric sigma model is our prototypical example.