We explore topological defects in the 4-dimensional pure $\mathbb {Z}_2$ lattice gauge theory. This theory has 1-form $\mathbb {Z}_{2}$ center symmetry as well as the Kramers-Wannier-Wegner (KWW) duality. We construct the KWW duality topological defects in the similar way to that constructed by Aasen, Mong, Fendley [1] for the 2-dimensional Ising model. These duality defects turn out to be non-invertible. We also construct the 1-form $\mathbb {Z}_{2}$ symmetry defects as well as the junctions among KWW duality defects and 1-form $\mathbb {Z}_{2}$ center symmetry defects. The crossing relations among these defects are derived. The expectation values of some configurations of these topological defects are calculated by using these crossing relations.

We explore topological defects in the 4-dimensional pure Z 2 lattice gauge theory. This theory has 1-form Z 2 center symmetry as well as the Kramers-Wannier-Wegner (KWW) duality. We construct the KWW duality topological defects in the similar way to that constructed by Aasen, Mong, Fendley [1] for the 2-dimensional Ising model. These duality defects turn out to be non-invertible. We also construct the 1-form Z 2 symmetry defects as well as the junctions among KWW duality defects and 1-form Z 2 center symmetry defects. The crossing relations among these defects are derived. The expectation values of some configurations of these topological defects are calculated by using these crossing relations.

We consider the U(1) gauge theory on a four-dimensional torus, where the instanton number is restricted to an integral multiple of p. This theory possesses the nontrivial higher-group structure, which can be regarded as a generalization of the Green–Schwarz mechanism, between $$\mathbb {Z}_q$$ Z q 1-form and $$\mathbb {Z}_p$$ Z p 3-form symmetries. Here, $$\mathbb {Z}_q$$ Z q is a subgroup of the center of U(1). Following the recent study of the lattice construction of the $$U(1)/\mathbb {Z}_q$$ U ( 1 ) / Z q principal bundle, we examine how such a structure is realized on the basis of lattice regularization.

No abstract