Sigma models as Gross-Neveu models

We review the correspondence between integrable sigma models with complex homogeneous target spaces and chiral bosonic (and possibly mixed bosonic/fermionic) Gross-Neveu models. Mathematically, the latter are models with quiver variety phase spaces, which reduce to the more conventional sigma models in special cases. We discuss the geometry of the models, as well as their trigonometric and elliptic deformations, Ricci flow and the inclusion of fermions.