On First-Order GLSM for Sigma Models

We review some recent developments in 1st order GLSM construction, or so-called Gross–Neveu formalism for sigma models. We recall the general idea behind this framework and describe a 1st order GLSM data from which the general generalized Gross–Neveu model can be constructed, using $$\mathbb{C}{{\mathbb{P}}^{{n - 1}}}$$ sigma model as simplest example. Then, we formulate a general statement that answers the question of which generalized Gross–Neveu models are equivalent to sigma models on compact homogeneous Hermitian spaces of classical groups endowed with the normal metrics.