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We investigate theories of Nambu-Goldstone bosons where the spontaneouslybroken continuous symmetry is non-invertible. In such theories, the vacuagenerically parameterize an orbifold. We study in detail the simplest exampleof a single free scalar with shift symmetry, modded by reflection symmetry. Atsingular points of the vacuum manifold, we show that the spectrum of NGexcitations is reduced, in particular there are no single-particle states. Atthe smooth points, on the other hand, single NG modes are present. We show thatthis is a consequence of the fact that at those points one can constructinvertible operators implementing the continuous symmetry on the Hilbert space.

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