My question is about a claim which I suspect is true in some generality, but that I have not seen stated anywhere in the literature, and for which I do not know of any counter-examples. Most of the content of this question arose within a collaboration with Dominic Williamson.
Setup: Consider some theory $\mathcal{T}$ in $D$ spacetime dimensions (could be a QFT or a quantum lattice model) with a non-anomalous finite Abelian $q$-form global symmetry group $A^{(q)}$. Feel free to add whatever other assumptions you would like about $A^{(q)}$. It is well-known that if one gauges such a symmetry, then the gauged theory $\mathcal{T}/A^{(q)}$ enjoys an emergent ``quantum'' $(D-2-q)$-form global symmetry group $\hat{A}^{(D-2-q)}$ with $\hat{A}$ the Pontryagin dual of $A$.
Claim: If $A^{(q)}$ is spontaneously broken in the theory $\mathcal{T}$, then $\hat{A}^{(D-2-q)}$ is preserved in the theory $\mathcal{T}/A^{(q)}$. Conversely, if $A^{(q)}$ is preserved in $\mathcal{T}$, then $\hat{A}^{(D-2-q)}$ is spontaneously broken in $\mathcal{T}/A^{(q)}$. I also have a guess about how this claim generalizes to the case of partial spontaneous symmetry breaking.
Examples: An example of this claim in action is the transverse field Ising model, $$H=-J\sum_{\langle v,v'\rangle} Z_v Z_{v'} -h\sum_v X_v$$ where $X_v,Z_v$ are Pauli operators which act on qubits supported on the sites of a $(D-1)$-dimensional lattice. As the coupling $J/h$ is varied one passes through a spontaneous symmetry breaking transition of its $\mathbb{Z}_2$ global symmetry $U = \prod_v X_v$ . In $D=2$, this happens at $J/h=1$, and gauging the $\mathbb{Z}_2$ global symmetry simply swaps $h$ and $J$ (i.e. gauging recovers the Kramers-Wannier dual), so that the claim is easily seen to be true. In $D=3$, gauging the $\mathbb{Z}_2$ global symmetry results in $\mathbb{Z}_2$ lattice gauge theory, $$H/\mathbb{Z}_2 = -J\sum_e X_e -h\sum_p \prod_{e\in\partial p}Z_e$$ where the first sum is over edges of the lattice, and the second is over plaquettes. (Additionally this model should be supplemented by a Gauss's law constraint of the form $G_v = \prod_{e\in\mathrm{star}(v)} X_v = 1$ for all sites $v$.) This theory famously undergoes a confinement/deconfinement phase transition associated with the spontaneous breaking of its $\mathbb{Z}_2^{(1)}$ one-form symmetry, and it is straightforward to see that when the Ising model is spontaneously broken, $\mathbb{Z}_2$ lattice gauge theory preserves its one-form symmetry, and vice versa. A similar pattern continues to hold for all $D>3$ as well, and I believe there are continuum versions of the statements I've made above.
Another example is (3+1)D $\mathrm{SU}(N)$ Yang-Mills theory which has a $\mathbb{Z}_N^{(1)}$ one-form symmetry which is conjectured to be preserved (as measured by the area law of the fundamental Wilson loop). On the other hand, $\mathrm{SU}(N)/\mathbb{Z}_N^{(1)}$ Yang-Mills theory has a 't Hooft line charged under the emergent $\widehat{\mathbb{Z}}_N^{(1)}$ global symmetry which has a perimeter law, and so the symmetry is spontaneously broken.
My question: Does anyone know if the claim holds in general, and if so, to what extent it has been appreciated in the literature? I suspect that it is possible to prove it by arguing that an order parameter for $A^{(q)}$ in $\mathcal{T}$ maps via the gauging procedure to a disorder parameter for $\hat{A}^{(D-2-q)}$ in $\mathcal{T}/A^{(q)}$, and furthermore that their vacuum expectation values are the same (roughly because discrete gauge theories are topological and don't impact local physics). This would then prove the claim because e.g. a non-zero order parameter indicates spontaneous symmetry breaking, while a non-zero disorder parameter indicates symmetry preserving.
