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 CF T4 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 CF T2. 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