  Quantum systems in 3+1-dimensions that are invariant under gauging a one-form
symmetry enjoy novel non-invertible duality symmetries encoded by topological
defects. These symmetries are renormalization group invariants which constrain
dynamics. We show that such non-invertible symmetries often forbid a
symmetry-preserving vacuum state with a gapped spectrum. In particular, we
prove that a self-dual theory with $\mathbb{Z}_{N}^{(1)}$ one-form symmetry is
gapless or spontaneously breaks the self-duality symmetry unless $N=k^{2}\ell$
where $-1$ is a quadratic residue modulo $\ell$. We also extend these results
to non-invertible symmetries arising from invariance under more general gauging
operations including e.g. triality symmetries. Along the way, we discover how
duality defects in symmetry protected topological phases have a hidden
time-reversal symmetry that organizes their basic properties. These
non-invertible symmetries are realized in lattice gauge theories, which serve
to illustrate our results.
