Sigma models as Gross-Neveu models

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