A Goldstone theorem for continuous non-invertible symmetries

TL;DR

This work extends Goldstone's theorem to continuous non-invertible symmetries arising from Adler-Bell-Jackiw anomalies by constructing a topological defect with a defect-local current and an auxiliary defect scalar, enabling a Euclidean-path-integral proof that a charged operator vev implies a bulk gapless mode. The authors derive a derivative-based Ward identity, present an axion-like effective theory, and discuss generalizations to higher-form non-invertible symmetries and nontrivial flux sectors. They apply the framework to string theory, showing that massless bulk fields can be viewed as Goldstone modes of spontaneously broken non-invertible axial symmetries and analyzing Page-charge-type defect operators for D-branes, including robustness against screening. The results offer a symmetry-based mechanism for massless modes beyond invertible symmetries and hint at connections to emergent higher-form structures in gravity.

Abstract

We study systems with an Adler-Bell-Jackiw anomaly in terms of non-invertible symmetry. We present a new kind of non-invertible charge defect where a key role is played by a local current operator localized on the defect. The charge defects are now labeled by elements of a continuous $U(1)$. We use this construction to prove an analogue of Goldstone's theorem for such non-invertible symmetries. We comment on possible applications to string theory.

Figures (1)

• Figure 1: Charge operator defined on an $S^{3}$ of radius $R$ wrapping operator ${{\mathcal{O}}}(0)$ at the origin.