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

Abstract

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 $U(1)/{\mathbb{Z}}_q$ principal bundle, we examine how such a structure is realized on the basis of lattice regularization.

Comments: 16 pages, to appear in EPJC. In Sect. 2, our discussions in continuum theory were falsely attributed to the periodicity of the compact scalar and have been corrected. These modifications do not modify any conclusion in lattice theory