When the moduli space is an orbifold: Spontaneous breaking of continuous non-invertible symmetries

TL;DR

This paper explores theories with non-invertible continuous symmetries, revealing that the vacuum space forms an orbifold and that the spectrum of Nambu-Goldstone bosons varies depending on the smoothness of the vacuum manifold.

Contribution

It provides a detailed analysis of non-invertible symmetries in scalar theories, demonstrating how the vacuum structure affects NG boson spectra and the invertibility of symmetry operators.

Findings

At singular points, NG spectrum is reduced with no single-particle states.

At smooth points, single NG modes exist due to invertible symmetry operators.

Vacuum manifolds can be orbifolds with varying NG excitation spectra.

Abstract

We investigate theories of Nambu-Goldstone bosons where the spontaneously broken continuous symmetry is non-invertible. In such theories, the vacua generically parameterize an orbifold. We study in detail the simplest example of a single free scalar with shift symmetry, modded by reflection symmetry. At singular points of the vacuum manifold, we show that the spectrum of NG excitations is reduced, in particular there are no single-particle states. At the smooth points, on the other hand, single NG modes are present. We show that this is a consequence of the fact that at those points one can construct invertible operators implementing the continuous symmetry on the Hilbert space.