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Title · changed: Revision 1: When do non-invertible symmetries forbid a featureless gapped phase; Revision 2: When do non-invertible symmetries forbid a featureless gapped phase?
Plain: Some symmetries force a theory to have massless particles or a degenerate vacuum. A test telling which non-invertible symmetries do this exists only in two dimensions.
Precise: For a QFT with a finite non-invertible symmetry $C$, find a computable criterion on $C$ that decides whether a trivially gapped, symmetric, non-degenerate phase is possible. In 2D the criterion is the existence of a fiber functor for the fusion category $C$; find the analogue in $d=3 \text{ and } 4$.
Settled by: A theorem stating the criterion for $d = 3, 4$ with examples where it forces gaplessness or degeneracy.
Kind: well-posed
Answer type: classification
References: (empty)
Literature status: partially-resolved
Status note: Settled in $2\mathrm{D}$ (2018-2023); open in $d \ge 3$.
Posed since: (empty)

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