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

Abstract

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

23 pages, 2 figures; (v2) refs added, fixed typos

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• Center-vortex semiclassics with non-minimal 't Hooft fluxes on ${\mathbb{R}}^2\times T^2$ and center stabilization at large $N$
• Tensor renormalization group approach to critical phenomena via symmetry-twisted partition functions