Year: 2026
Type: preprint
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 $Θ_p(R^{1/2})$ 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: $\log\mathcal{D} = O((\log R)^2)$. Finally, we compare these scales against WZW models and discuss the status of the general quadratic-logarithmic envelope conjecture.
Source: arXiv (Cornell University)
Author: Shen, Ce
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214174655