Flag manifold sigma models

This review is dedicated to two-dimensional sigma models with flag manifold target spaces, which are generalizations of the familiar $CP^{n-1}$ and Grassmannian models. They naturally arise in description continuum limits spin chains, their phase structure sensitive values topological angles, determined by representations spins chain. Gapless phases can certain cases be explained presence discrete 't Hooft anomalies theory. We also discuss integrable models, provide a generalization theory symmetric spaces. These as well deformations, have an alternative equivalent formulation bosonic Gross-Neveu proves useful for demonstrating that deformed geometries solutions renormalization group (Ricci flow) equations, analysis describing potential couplings fermions.