大间隙与正整数简并度模谱中的BTZ熵
Large gaps and BTZ entropy in modular spectra with positive integer degeneracies
- Yau Mathematical Sciences Center (YMSC), Tsinghua University（清华大学丘成桐数学科学中心）
- Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京国际数学研究中心）
- Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）
- Fudan Center for Mathematics and Interdisciplinary Study, Fudan University（复旦大学数学与交叉学科研究中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过递归修复模完备化构造模不变配分函数，实现大主维数间隙和正整数简并度，并证明其态密度对数在固定阶重现BTZ熵及修正。
AI中文摘要：
我们通过递归修复Virasoro真空的Maloney-Witten-Keller模完备化，构造了具有唯一真空、离散能级和正整数简并度的模不变环面配分函数。精确的模修复和有限矩匹配使连续谱带离散化，同时保留早期能级并控制收敛性。对于$c_L=c_R=c$且$a=(c-1)/12$，该构造在足够大的实数$c$下，对于足够小的固定$\kappa>0$实现主维数间隙$\Delta_1=(1+\kappa)a$，对于任意固定$\delta\ge0$实现$\Delta_1=a+\delta$。每个非真空主态满足$h,\bar h\ge(c-1)/24$。在固定$\delta$族中，可以选择谱，使其态密度（用固定非负归一化紧支撑光滑核平滑后）在$1/c$的每个固定有限阶上与单个微扰BTZ鞍点的相应平滑预测匹配。计数包括所有自旋和Virasoro后代。其对数在$E=\Delta-c/12$的每个固定正$E/c$处重现Bekenstein-Hawking熵及其修正。这包括$0<E<c/12$，其中热AdS在正则系综中占主导。
英文摘要：
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 $δ\ge0$, at every sufficiently large real $c$. Every nonvacuum primary satisfies $h,\bar h\ge(c-1)/24$. 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.
