Non-Invertible Global Symmetries and Completeness of the Spectrum

TL;DR

This paper explores how non-invertible symmetries ensure the completeness of gauge charge spectra in quantum gravity theories, extending the understanding beyond traditional symmetry concepts.

Contribution

It introduces non-invertible topological operators as a key to restoring the link between symmetries and spectrum completeness in general gauge groups.

Findings

Non-invertible symmetries guarantee spectrum completeness.

Absence of non-invertible operators implies a complete gauge charge spectrum.

Results have implications for the Swampland conjectures and phenomenology.

Abstract

It is widely believed that consistent theories of quantum gravity satisfy two basic kinematic constraints: they are free from any global symmetry, and they contain a complete spectrum of gauge charges. For compact, abelian gauge groups, completeness follows from the absence of a 1-form global symmetry. However, this correspondence breaks down for more general gauge groups, where the breaking of the 1-form symmetry is insufficient to guarantee a complete spectrum. We show that the correspondence may be restored by broadening our notion of symmetry to include non-invertible topological operators, and prove that their absence is sufficient to guarantee a complete spectrum for any compact, possibly disconnected gauge group. In addition, we prove an analogous statement regarding the completeness of twist vortices: codimension-2 objects defined by a discrete holonomy around their worldvolume, such as cosmic strings in four dimensions. We discuss how this correspondence is modified in various, more general contexts, including non-compact gauge groups, Higgsing of gauge theories, and the addition of Chern-Simons terms. Finally, we discuss the implications of our results for the Swampland program, as well as the phenomenological implications of the existence of twist strings.