Defining 2-group symmetries on lattices and extracting their anomalies
Abstract
Ordinary $0$-form symmetries G can mix with $1$-form symmetries Π2 to form a $2$-group 𝒢. We give a concrete lattice definition for a class of finite $2$-group symmetries in terms of local relations among symmetry operators. These relations identify the Postnikov class [ρ]∈ H3(BG,Π2) that characterizes the mixing. We develop a method to extract anomalies of these symmetries in one spatial dimension by restricting symmetry operations to an interval and comparing their composition near its endpoints. We also construct explicit qubit symmetry actions in d=1,2,3 spatial dimensions for a $2$-group with G=Π2=ℤ2. These actions share the same nontrivial Postnikov class but include both anomaly-free and anomalous realizations. Their cohomological anomalies are classified by Hd+2(B𝒢,U(1)), which is ℤ2 for d=1,2 and ℤ2×ℤ2 for d=3, and our constructions realize every class in these dimensions. Our results connect higher-group symmetry and its anomalies to concrete lattice operations, providing a microscopic framework for studying anomalous symmetries.
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BibTeX
@misc{feng2026defining,
title = {{Defining 2-group symmetries on lattices and extracting their anomalies}},
author = {Yitao Feng and Yu-An Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37745},
eprint = {2610.37745},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37745},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Defining 2-group symmetries on lattices and extracting their anomalies
AU - Yitao Feng
AU - Yu-An Chen
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37745
UR - https://arxiv.org/abs/2610.37745
AB - Ordinary $0$-form symmetries G can mix with $1$-form symmetries Π_2 to form a $2$-group 𝒢. We give a concrete lattice definition for a class of finite $2$-group symmetries in terms of local relations among symmetry operators. These relations identify the Postnikov class [ρ]∈ H^3(BG,Π_2) that characterizes the mixing. We develop a method to extract anomalies of these symmetries in one spatial dimension by restricting symmetry operations to an interval and comparing their composition near its endpoints. We also construct explicit qubit symmetry actions in d=1,2,3 spatial dimensions for a $2$-group with G=Π_2=ℤ_2. These actions share the same nontrivial Postnikov class but include both anomaly-free and anomalous realizations. Their cohomological anomalies are classified by H^d+2(B𝒢,U(1)), which is ℤ_2 for d=1,2 and ℤ_2×ℤ_2 for d=3, and our constructions realize every class in these dimensions. Our results connect higher-group symmetry and its anomalies to concrete lattice operations, providing a microscopic framework for studying anomalous symmetries.
ER -