Obstructions to gapped phases from noninvertible symmetries

Our understanding of quantum symmetry of systems has considerably broadened over the last decade. The idea of symmetry has become intrinsically linked with topology described and algebraically characterized by higher categories. Studying three-dimensional systems with noninvertible symmetries, the authors show here that these symmetries are in general incompatible with a unique gapped ground state. Their results extend the ideas behind the Lieb-Shultz-Mattis theorem to the arena of higher-dimensional field theories invariant under a novel class of symmetries.