Higher symmetries, anomalies, and crossed squares in lattice gauge theory

TL;DR

This paper explores higher-form symmetries and anomalies in lattice gauge theories using homotopy theory and operator algebras, revealing their algebraic structures as higher groups and crossed squares.

Contribution

It introduces a novel algebraic framework connecting higher symmetries and anomalies to higher groups and crossed n-cubes in lattice gauge theories.

Findings

Higher-form symmetries are described by higher groups.

't Hooft anomalies correspond to restrictions of symmetry transformations.

Crossed squares encode symmetry data in 2D gauge theories.

Abstract

We examine higher-form symmetries of quantum lattice gauge theories through the lens of homotopy theory and operator algebras. We show that in the operator-algebraic approach both higher-form symmetries and 't Hooft anomalies arise from considering restrictions of symmetry transformations to spatial regions. The data of these restrictions are naturally packaged into a higher group. For example, for gauge theories in two spatial dimensions, this information is encoded in a crossed square of groups, which is an algebraic model of a 3-group. In general, we propose that higher groups appear in lattice models and QFT as crossed n-cubes of groups via a nonabelian version of the Cech construction.