Goldstone modes and photonization for higher form symmetries

TL;DR

This paper extends Goldstone's theorem to higher form symmetries, showing that certain defect operators imply gapless modes, and demonstrates that 1-form symmetries in 4D conformal theories can be realized by free Maxwell theory, revealing a rich algebraic structure.

Contribution

It generalizes Goldstone's theorem to higher form symmetries and introduces a twistor formalism to analyze conserved charges and their algebra in 4D conformal theories.

Findings

Perimeter law for defect operators implies gapless modes.

1-form symmetries in 4D CFTs can be realized by free Maxwell theory.

Conserved charges obey a Kac-Moody-like algebra with central extension.

Abstract

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 $CFT_4$ can be realized by free Maxwell electrodynamics, i.e. the current can be photonized. The photonized 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 $CFT_2$. 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.