Goldstone modes and photonization for higher form symmetries

We discuss generalized global symmetries and their breaking. We extend Goldstone's theorem to higher form symmetries by showing that a perimeter law for an extended p-dimensional defect operator charged under a continuous p-form generalized global symmetry necessarily results in a gapless mode in the spectrum. We also show that a p-form symmetry in a conformal theory in 2 (p + 1) dimensions has a free realization. In four dimensions this means any 1-form symmetry in a CFT4 can be realized by free Maxwell electrodynamics, i.e. the current can be photonized. The theory has infinitely many conserved 0-form charges that are constructed by integrating the symmetry currents against suitable 1-forms. We study these charges by developing a twistor-based formalism that is a 4d analogue of the usual holomorphic complex analysis familiar in CFT2. The charges are shown to obey an algebra with central extension, which is an analogue of the 2d Abelian Kac-Moody algebra for higher form symmetries.