When do non-invertible symmetries forbid a featureless gapped phase?
In plain words
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 statement
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 $d=3 \text{ and } 4$.
What would settle it
A theorem stating the criterion for $d = 3, 4$ with examples where it forces gaplessness or degeneracy.
Status in the literature
Settled in $2\mathrm{D}$ (2018-2023); open in $d \ge 3$.
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