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To study gapped phases of $4$d gauge theories, we introduce the temporalgauging of $\mathbb{Z}_N$ $1$-form symmetry in $4$d quantum field theories(QFTs), thereby defining effective $3$d QFTs with$\widetilde{\mathbb{Z}}_N\times \mathbb{Z}_N$ $1$-form symmetry. In this way,spatial fundamental Wilson and 't Hooft loops are simultaneously genuine lineoperators. Assuming a mass gap and Lorentz invariant vacuum of the $4$d QFT,the $\widetilde{\mathbb{Z}}_N\times \mathbb{Z}_N$ symmetry must bespontaneously broken to an order-$N$ subgroup $H$, and we can classify the $4$dgapped phases by specifying $H$. This establishes the $1$-to-$1$ correspondencebetween the two classification schemes for gapped phases of $4$d gaugetheories: One is the conventional Wilson-'t Hooft classification, and the otheris the modern classification using the spontaneous breaking of $4$d $1$-formsymmetry enriched with symmetry-protected topological states.

Study of gapped phases of 4d gauge theories using temporal gauging of the $\mathbb{Z}_N$ 1-form symmetry