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

## TL;DR

This paper explores the higher-group symmetry structure in a lattice U(1) gauge theory with instanton-sum restrictions, revealing a generalized Green--Schwarz mechanism between certain 1-form and 3-form symmetries.

## Contribution

It demonstrates how higher-group structures emerge in lattice regularized U(1) gauge theories with instanton restrictions, extending the understanding of symmetry structures in lattice gauge theories.

## Key findings

- Higher-group structure identified in lattice U(1) gauge theory.
- Realization of the generalized Green--Schwarz mechanism on the lattice.
- Connection between instanton restrictions and symmetry structures.

## 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.