Non-Local Conserved Currents and Continuous Non-Invertible Symmetries

TL;DR

This paper develops a generalized Noether's theorem for continuous non-invertible symmetries in 1+1d CFTs, introducing non-local conserved currents and providing new examples in various models, with implications for defect conformal manifolds.

Contribution

It unifies existing constructions of continuous non-invertible symmetries through a generalized Noether's theorem and presents new examples in Wess-Zumino-Witten and minimal models.

Findings

Established a generalized Noether's theorem for non-invertible symmetries.

Identified new continuous non-invertible symmetries in WZW and minimal models.

Constructed new defect conformal manifolds using these symmetries.

Abstract

We embark on a systematic study of continuous non-invertible symmetries, focusing on 1+1d CFTs. We describe a generalized version of Noether's theorem, where continuous non-invertible symmetries are associated to $\textit{non-local}$ conserved currents: point-like operators attached to extended topological defects. The generalized Noether's theorem unifies several constructions of continuous non-invertible symmetries in the literature, and allows us to exhibit many more examples in diverse theories of interest. We first review known examples which are non-intrinsic (i.e., invertible up to gauging), and then describe $\textit{new}$ examples in Wess-Zumino-Witten models and products of minimal models. For some of these new examples, we show that these continuous non-invertible symmetries are intrinsic if we demand that a certain global symmetry is preserved. The continuous non-invertible symmetries in products of minimal models also allow us to construct new examples of defect conformal manifolds in a single copy of a minimal model. Finally, we comment on continuous non-invertible symmetries in higher dimensions.