Non-invertible global symmetries and completeness of the spectrum

Abstract: It is commonly believed that there are no global symmetries in quantum gravity, and also that the spectrum of charged states is complete. It is also common lore that these two properties are related or equivalent. I will explore this question in the context of quantum field theory, explicitly showing examples which have an incomplete spectrum, but no ordinary global symmetries. However, these examples have non-invertible topological operators; demanding their absence restores completeness of the spectrum. I will discuss how this equivalence between completeness of the spectrum and absence of non-invertible symmetries can be established rigorously for electric states in gauge theory, for any compact (even disconnected) gauge group, and for twist vortices (codimension-2 objects related to Gukov-Witten operators). I will also briefly comment on the situation for non-compact gauge groups, theories with Chern-Simons terms, and the motivation for the absence of non-invertible topological operators in gravity.

HEP - theorymathematical physics

Audience: researchers in the topic

QFT and Geometry

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