A ceiling inside one family
An arXiv preprint reports a sharp asymptotic ceiling for the topological quantum field theory (TQFT) contribution to torus topological entanglement entropy, or TEE, within the simply connected, untwisted WZW class. As the allowed categorical rank grows, the best simple-WZW value, divided by the square of the cutoff's base-2 logarithm, approaches a fixed coefficient: 7 times the zeta-function value at 3 divided by 4 times pi squared. The statement describes the large-cutoff envelope, while the best theory at each finite cutoff remains unresolved.
That class is defined mathematically by categories C(g,k), built from a finite-dimensional simple Lie algebra g and a positive integral level k. Its categorical rank counts sectors, while D denotes the category's total quantum dimension. The objects compared are categories and asymptotic sequences, not human or laboratory samples.
The analysis first asks which normalized ground state on the torus gives the largest value. For the chosen torus bipartition, the maximum TQFT contribution is twice the natural logarithm of D. It is reached by every definite Abelian-flux state, including the vacuum-flux state. That separates the state-optimization question from the later question of which category gives the largest value under a rank cutoff.
The sequence that reaches it
The asymptotic coefficient is reached by a balanced type-C, or symplectic, sequence in which the rank and level are both indexed by n. For this sequence, the normalized TQFT value approaches the same coefficient when divided by the square of the base-2 logarithm of its sector count. That makes the upper envelope sharp in the WZW class, although the analysis does not name a unique maximizing category at any particular finite cutoff.
Across the classical families, the proportional limits split into two groups. Type A reaches 7 times the zeta-function value at 3 divided by 8 times pi squared, half the coefficient reached by types B, C and D, whose common value is 7 times the zeta-function value at 3 divided by 4 times pi squared. Half filling uniquely maximizes this proportional limit. In type C, equal rank and level give that half-filled case: even Fourier modes vanish and odd modes double.
Other routes through the parameter space do not produce the same quadratic-scale growth. Holding the Lie algebra fixed while increasing the level drives the logarithm of total quantum dimension divided by the square of the logarithm of sector count to zero, including for exceptional algebras. Holding the level fixed while classical Lie rank grows also gives a zero coefficient. The ratio likewise vanishes when both rank and level grow but the level-to-rank ratio tends to zero or infinity.
Why balance matters
To prove a global envelope, the analysis treats four parameter regimes. It combines estimates for the distribution of roots, an inequality linking entropy to a spectrum, rank-level identities, and a final passage from bounds along sequences to the supremum imposed by the cutoff.
The analysis also considers finite semisimple stacking, meaning a finite product of independent WZW factors with their own levels. It finds the same leading coefficient after the same cutoff normalization. Stacking therefore does not improve the asymptotic law within this setting.
The claim has a boundary
The boundary of the claim matters. The result is sharp within the simply connected untwisted WZW class, but it is only a lower bound for the unrestricted modular-category and TQFT problem. Whether other modular categories or other topological theories can produce a larger leading envelope remains open.
Nor does the work settle the winner at any finite cutoff. Sharper subleading expansions and exact sector counts are needed to compare families that are tied at leading order. There is also a microscopic caveat: the reported TEE is the TQFT contribution after nonuniversal boundary effects have been separated. Extracting that contribution from a physical realization requires control of additional constants, and the paper proves no general prescription for WZW realizations.
The work is an arXiv preprint, version 3, dated 8 October 2026, and marked as prepared for submission to JHEP.
Paper data and sources
Original title: Maximal Torus Topological Entanglement Entropy in WZW Theories at Bounded Ground-State Degeneracy
Authors: Ce Shen
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text