Goldstone modes and photonization for higher form symmetries

TL;DR

The paper extends Goldstone's theorem to higher-form symmetries, showing that a spontaneously broken p-form symmetry entails a gapless Goldstone mode and, in 4d, that a conserved 1-form symmetry is photonized into free Maxwell dynamics. It demonstrates conformal constraints that force a free realization of the current when d = 2(p+1), leading to a photonization picture with J = dB and a free 2-form/gauge-field description. A covariant twistor formalism is developed to represent the infinite set of conserved charges and their central-extended algebra, providing a higher-dimensional analogue of the Abelian Kac-Moody structure and connecting to soft photon theorems. The results offer a framework to study higher-form symmetries in 4d CFTs, with potential applications to asymptotic symmetries, emergent gauge theories, and broader extensions to non-Abelian theories and gravity.

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.

Figures (1)

• Figure 1: Geometry for Goldstone theorem: $C$ is an infinite $p$-plane that intersects $B_{d-p}$ at a single point; $S^{d-p-1}$ is the boundary of $B_{d-p}$.