Higher-group structure in lattice Abelian gauge theory under instanton-sum modification

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 ${\mathbb{Z}}_q$ $1$-form and ${\mathbb{Z}}_{pq}$ $3$-form symmetries. Here, ${\mathbb{Z}}_q$ is a subgroup of the center of~$U(1)$. Following the recent study of the lattice construction of the $U(1)\mathrm{/}{\mathbb{Z}}_q$ principal bundle, we examine how such a structure is realized on the basis of lattice regularization.