Student Workshop Integrability 2023

We consider the reformulation the sigma-models as a generalized Gross-Neveu models. In fact, this representation crucially simplifies the analisys of quantum aspects of sigma-models. We show, as simplest application, that in this formulation the derivation of the $\beta$-function of the sigma-models reduces to ordinary calculation of Feynman diagrams (as in $\phi^4$ model), while in the usual formulation one need more complicated tools (background field method, for example). We construct the Gross-Neveu representation for the sigma models, which target spaces are $\mathbb{CP}^n$ and Grassmannians (include orthogonal and symplectic one). We also consider the connection of this representation with algebraic and complex geometry, theory of nilpotent orbits and integrability (potentially, on Riemann surfaces).