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PreprintCC BY 4.0arXiv

Large gaps and BTZ entropy in modular spectra with positive integer degeneracies

  • Chi-Ming Chang
  • Reiko Liu
  • Wen-Jie Ma
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Abstract

We construct modular-invariant torus partition functions with a unique vacuum, discrete energy levels and positive integer degeneracies by recursively repairing the Maloney--Witten--Keller modular completion of the Virasoro vacuum. Exact modular repairs and finite moment matching discretize successive spectral bands while preserving earlier levels and controlling convergence. For cL=cR=c and a=(c−1)/12, the construction realizes primary dimension gaps Δ1=(1+κ)a for sufficiently small fixed κ>0, and Δ1=a+δ for any fixed δ≥0, at every sufficiently large real c. Every nonvacuum primary satisfies h,h̅≥(c−1)/24. In the fixed-δ family, spectra can be chosen whose densities of states, smoothed with a fixed nonnegative normalized smooth kernel of compact support, match the correspondingly smoothed prediction of a single perturbative BTZ saddle through every fixed finite order in 1/c. The count includes all spins and Virasoro descendants. Its logarithm reproduces the Bekenstein--Hawking entropy and its corrections at every fixed positive E/c, where E=Δ−c/12. This includes 0<E<c/12, where thermal AdS dominates the canonical ensemble.

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Cite

BibTeXRIS
BibTeX
@misc{chang2026large,
  title = {{Large gaps and BTZ entropy in modular spectra with positive integer degeneracies}},
  author = {Chi-Ming Chang and Reiko Liu and Wen-Jie Ma},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.39930},
  eprint = {2609.39930},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39930},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Large gaps and BTZ entropy in modular spectra with positive integer degeneracies
AU  - Chi-Ming Chang
AU  - Reiko Liu
AU  - Wen-Jie Ma
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.39930
UR  - https://arxiv.org/abs/2609.39930
AB  - We construct modular-invariant torus partition functions with a unique vacuum, discrete energy levels and positive integer degeneracies by recursively repairing the Maloney--Witten--Keller modular completion of the Virasoro vacuum. Exact modular repairs and finite moment matching discretize successive spectral bands while preserving earlier levels and controlling convergence. For c_L=c_R=c and a=(c−1)/12, the construction realizes primary dimension gaps Δ_1=(1+κ)a for sufficiently small fixed κ>0, and Δ_1=a+δ for any fixed δ≥0, at every sufficiently large real c. Every nonvacuum primary satisfies h,h̅≥(c−1)/24. In the fixed-δ family, spectra can be chosen whose densities of states, smoothed with a fixed nonnegative normalized smooth kernel of compact support, match the correspondingly smoothed prediction of a single perturbative BTZ saddle through every fixed finite order in 1/c. The count includes all spins and Virasoro descendants. Its logarithm reproduces the Bekenstein--Hawking entropy and its corrections at every fixed positive E/c, where E=Δ−c/12. This includes 0<E<c/12, where thermal AdS dominates the canonical ensemble.
ER  -