AI 中文总结
本文通过高精度Bethe真空数据和符号回归重构ABJM理论有限秩扭曲指标，证明其常数映射为收敛积分，修正数值常数，并揭示指数小扇区的算术结构，为量子引力提供有限N基准。
AI 中文摘要
ABJM理论的拓扑扭曲指标捕捉了磁荷超对称$\mathrm{AdS}_4$黑洞的微观熵。基于姊妹论文arXiv:2608.04107（该文在通用扭曲下重构了有限秩扭曲超势），我们利用高精度Bethe真空数据、物理信息符号回归和整数关系确定了有限秩指标。Hessian矩阵、有效膨胀子和单圈因子编码了壳上扭曲超势之外的信息。这两个可观测量共享相同的移位秩。在声明的一个有限类内，我们从$k$和$k/2$能级的ABJM $S^3$常数映射、扭曲超势的常数映射及初等项重构了指标常数映射。我们证明对于实数$k>0$，它等于一个收敛的单核积分，重新求和了大$k$展开，修正了先前的一个数值常数，并给出了任意亏格下的IIA型系数。我们还证明了arXiv:2608.04107中的Clausen表达式等于一个交错欧拉和与一个绝对收敛积分，为两个有理参数欧拉和族给出了闭式求值。在$k=1,2,4$时，指数抑制扇区表现出算术结构。扭曲超势的重构系数满足一个除数求和公式，并带有一个与权四Eisenstein形式相关的猜想全阶Eichler积分级数。指标系数归结为两个有理生成元；其中一个允许模积公式，该公式重现了每一个可用系数并暗示了一个全阶完备化。在此延拓条件下，模变换确定了最近的对数奇点、收敛半径和首项系数增长。这些结果为量子引力块和体黑洞量子熵提供了有限$N$基准。
英文摘要
The topologically twisted index of ABJM theory captures the microscopic entropy of magnetically charged supersymmetric $\mathrm{AdS}_4$ black holes. Building on the companion Letter arXiv:2608.04107, which reconstructed the finite-rank twisted superpotential at the universal twist, we determine the finite-rank index using high-precision Bethe-vacuum data, physics-informed symbolic regression, and integer relations. The Hessian, effective dilaton, and one-loop factors encode information beyond the on-shell twisted superpotential. The two observables share the same shifted rank. Within a declared finite class, we reconstruct the index constant map from the ABJM $S^3$ constant maps at levels $k$ and $k/2$, the constant map of the twisted superpotential, and elementary terms. We prove that it equals a convergent one-kernel integral for real $k>0$, resumming the large-$k$ expansion, fixing a previously numerical constant, and yielding type-IIA coefficients at arbitrary genus. We also prove that the Clausen expression in arXiv:2608.04107 equals an alternating Euler sum and an absolutely convergent integral, giving closed evaluations for two rational-argument Euler-sum families. At $k=1,2,4$, the exponentially suppressed sectors exhibit arithmetic structure. The reconstructed coefficients of the twisted superpotential obey a divisor-sum formula with a conjectural all-order Eichler-integral series associated with a weight-four Eisenstein form. The index coefficients reduce to two rational generators; one admits modular-product formulas reproducing every available coefficient and suggesting an all-order completion. Conditional on this continuation, modular transformations determine the nearest logarithmic singularity, radius of convergence, and leading coefficient growth. These results provide a finite-$N$ benchmark for quantum gravitational blocks and bulk black hole quantum entropy.
Comments80 pages, 8 figures
