Maximal Total Quantum Dimension at Bounded Rank in Dijkgraaf-Witten Theories
Abstract
We investigate the maximal total quantum dimension at a bounded categorical rank cutoff R in three-dimensional Dijkgraaf--Witten theories. This optimization problem physically governs the maximal state-optimized torus topological entanglement entropy at a fixed torus ground-state degeneracy, while providing a quantitative constraint on the search space required to classify these topological order phases. We analyze how topological twists (3-cocycles) and gauge-group structures independently govern this trade-off. For elementary-Abelian gauge groups at a fixed prime p, we prove that introducing cocycle twists elevates the maximum total quantum dimension from square-root growth Θp(R1/2) to linear growth Θp(R) in the rank cutoff, driven by the geometric protection of cyclic holonomies. For untwisted theories, we establish sharp envelopes for symmetric and alternating gauge groups, and derive group-theoretic bounds across almost-simple, solvable, and radical-free families. In particular, almost-simple groups and their direct products obey a quadratic-logarithmic upper envelope: log𝒟 = O((log R)2). Finally, we compare these scales against WZW models and discuss the status of the general quadratic-logarithmic envelope conjecture.
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BibTeX
@misc{shen2026maximal,
title = {{Maximal Total Quantum Dimension at Bounded Rank in Dijkgraaf-Witten Theories}},
author = {Ce Shen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26914},
eprint = {2609.26914},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26914},
}
RIS
TY - UNPB
TI - Maximal Total Quantum Dimension at Bounded Rank in Dijkgraaf-Witten Theories
AU - Ce Shen
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26914
UR - https://arxiv.org/abs/2609.26914
AB - We investigate the maximal total quantum dimension at a bounded categorical rank cutoff R in three-dimensional Dijkgraaf--Witten theories. This optimization problem physically governs the maximal state-optimized torus topological entanglement entropy at a fixed torus ground-state degeneracy, while providing a quantitative constraint on the search space required to classify these topological order phases. We analyze how topological twists (3-cocycles) and gauge-group structures independently govern this trade-off. For elementary-Abelian gauge groups at a fixed prime p, we prove that introducing cocycle twists elevates the maximum total quantum dimension from square-root growth Θ_p(R^1/2) to linear growth Θ_p(R) in the rank cutoff, driven by the geometric protection of cyclic holonomies. For untwisted theories, we establish sharp envelopes for symmetric and alternating gauge groups, and derive group-theoretic bounds across almost-simple, solvable, and radical-free families. In particular, almost-simple groups and their direct products obey a quadratic-logarithmic upper envelope: log𝒟 = O((log R)^2). Finally, we compare these scales against WZW models and discuss the status of the general quadratic-logarithmic envelope conjecture.
ER -