A comparison between statistics and 't Hooft anomaly
Abstract
Anomalies of generalized symmetries are closely related to the statistics of topological excitations. Truncating a symmetry transformation to a finite region produces an operator that creates symmetry defects on its boundary; we call this a symmetry patch operator. Such operators resemble hopping operators that create topological excitations at the boundary of their support, but this geometric similarity does not justify identifying the two. For finite Abelian groups in one spatial dimension, we prove that if the same family of local operators defines both valid hopping operators and compatible symmetry patches, the type-III component of the anomaly is −2 times that of the statistics. We establish a one-to-one correspondence between statistics and anomaly by treating hopping operators as symmetric operators that commute with the global symmetry, following the description of symmetry through the algebra of local symmetric operators. We explain this correspondence through gauging by coupling boundary matter to a Dijkgraaf-Witten gauge field in one higher spatial dimension. Hopping operators act on the gauge-invariant degrees of freedom. After fixing the background gauge field, the residual gauge transformations become the global symmetry, and the gauge invariance of hopping operators becomes the requirement that they commute with this symmetry.
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BibTeX
@misc{xue2026comparison,
title = {{A comparison between statistics and 't Hooft anomaly}},
author = {Hanyu Xue},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.20813},
eprint = {2609.20813},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20813},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A comparison between statistics and 't Hooft anomaly
AU - Hanyu Xue
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2609.20813
UR - https://arxiv.org/abs/2609.20813
AB - Anomalies of generalized symmetries are closely related to the statistics of topological excitations. Truncating a symmetry transformation to a finite region produces an operator that creates symmetry defects on its boundary; we call this a symmetry patch operator. Such operators resemble hopping operators that create topological excitations at the boundary of their support, but this geometric similarity does not justify identifying the two. For finite Abelian groups in one spatial dimension, we prove that if the same family of local operators defines both valid hopping operators and compatible symmetry patches, the type-III component of the anomaly is −2 times that of the statistics. We establish a one-to-one correspondence between statistics and anomaly by treating hopping operators as symmetric operators that commute with the global symmetry, following the description of symmetry through the algebra of local symmetric operators. We explain this correspondence through gauging by coupling boundary matter to a Dijkgraaf-Witten gauge field in one higher spatial dimension. Hopping operators act on the gauge-invariant degrees of freedom. After fixing the background gauge field, the residual gauge transformations become the global symmetry, and the gauge invariance of hopping operators becomes the requirement that they commute with this symmetry.
ER -