Study of gapped phases of 4d gauge theories using temporal gauging of the ℤN 1-form symmetry

Abstract

Abstract To study gapped phases of 4d gauge theories, we introduce the temporal gauging of ℤN 1-form symmetry in 4d quantum field theories (QFTs), thereby defining effective 3d QFTs with $$ \overset{\sim }{\mathbb{Z}} $$ ℤ ~ N × ℤN 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 4d QFT, the $$ \overset{\sim }{\mathbb{Z}} $$ ℤ ~ N × ℤN symmetry must be spontaneously broken to an order-N subgroup H, and we can classify the 4d gapped phases by specifying H. This establishes the 1-to-1 correspondence between the two classification schemes for gapped phases of 4d gauge theories: one is the conventional Wilson-’t Hooft classification, and the other is the modern classification using the spontaneous breaking of 4d 1-form symmetry enriched with symmetry-protected topological states.