Generalized symmetries and spontaneous symmetry breaking

The fundamental concept of global symmetry in quantum field theory has been generalized in many different directions in the last few years. Some of these generalizations include: higher form symmetries that only act on operators supported on manifolds of dimension greater than 0; higher group symmetries, where the associativity law for the composition of symmetries is modified by the higher form terms; non-invertible symmetries, where groups are replaced by more general mathematical structures such as fusion or n-categories. Most of the properties of the standard global symmetries admit an analogue in these generalized settings: they lead to selection rules for correlation functions, they can be spontaneously broken, they can be gauged, they admit 't Hooft anomalies. In this thesis, after a description of some these generalized definitions of symmetries, the focus will be on spontaneous symmetry breaking. By an extension of Goldstone theorem, it can be proven that spontaneous breaking of a p-form symmetry implies the existence of massless p-form excitation in the spectrum (Goldstone boson); the standard case corresponds to p=0, where the boson is a scalar. A similar results also holds for non-invertible symmetries. Finally, an argument will be presented showing that, for p>0, the mass of a p-form Goldstone mode remains exactly massless even if the spontaneously broken p-form symmetry is only an emergent IR symmetry, i.e. that the mass does not receive corrections due to UV perturbations. This phenomenon represents a major qualitative difference between higher (p>0) form and the standard (p=0) symmetries.