Torus Berry Data Determine All-Genus Abelian Topological Orders
Abstract
We show that for Abelian Chern-Simons topological orders, torus Berry matrices determine the all-genus extended TQFT. We identify the topological part of the projective Berry holonomy under metric deformations with the mapping-class-group representation of the Abelian Chern-Simons TQFT and prove that the normalized torus data reconstruct its finite quadratic module (G,q). Recent work showed that (G,q) classifies the extended theory up to symmetric monoidal natural isomorphism, and that the associated Abelian defect extension is determined by the pointed modular category C(G,q) arising from the same finite quadratic-module data. Therefore genus-one Berry data determine not only the all-genus bulk theory but also its associated defect structures without choosing a K-matrix presentation. We also prove that, for normalized character row errors δ<21.96%, nearest-row decoding recovers the Abelian fusion algebra independently of the number of anyons. The result applies to Abelian fractional quantum Hall and spin-liquid phases described by even-lattice Chern-Simons theories.
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BibTeX
@misc{galviz2026torus,
title = {{Torus Berry Data Determine All-Genus Abelian Topological Orders}},
author = {Daniel Galviz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.20330},
eprint = {2608.20330},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.20330},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Torus Berry Data Determine All-Genus Abelian Topological Orders
AU - Daniel Galviz
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2608.20330
UR - https://arxiv.org/abs/2608.20330
AB - We show that for Abelian Chern-Simons topological orders, torus Berry matrices determine the all-genus extended TQFT. We identify the topological part of the projective Berry holonomy under metric deformations with the mapping-class-group representation of the Abelian Chern-Simons TQFT and prove that the normalized torus data reconstruct its finite quadratic module (G,q). Recent work showed that (G,q) classifies the extended theory up to symmetric monoidal natural isomorphism, and that the associated Abelian defect extension is determined by the pointed modular category C(G,q) arising from the same finite quadratic-module data. Therefore genus-one Berry data determine not only the all-genus bulk theory but also its associated defect structures without choosing a K-matrix presentation. We also prove that, for normalized character row errors δ<21.96%, nearest-row decoding recovers the Abelian fusion algebra independently of the number of anyons. The result applies to Abelian fractional quantum Hall and spin-liquid phases described by even-lattice Chern-Simons theories.
ER -