First-order GLSM construction in sigma models

Sigma models are a class of 2D field theories that play a crucial role in various branches of modern theoretical and mathematical physics. However, studying these models is challenging due to the highly nonlinear nature of their Lagrangians. Certain methods, such as the background field method, can be used, but they have their limitations. An alternative method, recently proposed, is the "first-order GLSM formulation" (or "Gross-Neveu formalism"). In this approach, one casts these models as gauge theories with a finite number of interactions using the idea of symplectic reduction. I will illustrate the ideas using the simplest example, the $\mathbb{CP}^n$ sigma model. I will then explain how these ideas can be extended to other target spaces such as complex Grassmannians.