The Non-linear σ Model

According to the Goldstone theorem, systems with spontaneous symmetry breaking have one massless mode for every broken generator. As a consequence, these Nambu-Goldstone boson (NGB) modes are the natural degrees of freedom of the system at low energies. In this chapter we will illustrate how the non-linear σmodel (NLSM) provides a general and systematic description of the low energy NGB dynamics as well as how their effective Lagrangian can be obtained through a beautiful mathematical picture based on the geometry of homogeneous spaces.