# Symmetry-Enriched Quantum Spin Liquids in $(3+1)d$

## TL;DR

This paper classifies symmetry-enriched quantum spin liquids in (3+1)d using higher-form symmetries, providing a systematic framework for understanding their phases, anomalies, and dualities.

## Contribution

It introduces a classification scheme for (3+1)d symmetry-enriched phases based on higher-form symmetries and their couplings, extending previous (2+1)d results.

## Key findings

- Classified (3+1)d symmetry-enriched phases using higher-form symmetries.
- Identified a systematic method for constructing symmetry protected topological phases.
- Discovered a tension with a conjectured duality in (3+1)d $SU(2)$ gauge theory.

## Abstract

We use the intrinsic one-form and two-form global symmetries of (3+1)$d$ bosonic field theories to classify quantum phases enriched by ordinary ($0$-form) global symmetry. Different symmetry-enriched phases correspond to different ways of coupling the theory to the background gauge field of the ordinary symmetry. The input of the classification is the higher-form symmetries and a permutation action of the $0$-form symmetry on the lines and surfaces of the theory. From these data we classify the couplings to the background gauge field by the 0-form symmetry defects constructed from the higher-form symmetry defects. For trivial two-form symmetry the classification coincides with the classification for symmetry fractionalizations in $(2+1)d$. We also provide a systematic method to obtain the symmetry protected topological phases that can be absorbed by the coupling, and we give the relative 't Hooft anomaly for different couplings. We discuss several examples including the gapless pure $U(1)$ gauge theory and the gapped Abelian finite group gauge theory. As an application, we discover a tension with a conjectured duality in $(3+1)d$ for $SU(2)$ gauge theory with two adjoint Weyl fermions.