发表机构
School of Mathematics, Southwest Jiaotong University（西南交通大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明中心差分算子可精确可逆地分离曲线计数中的多重覆盖与BPS亏格信息，给出全亏格Gopakumar-Vafa重构的Chebyshev-Fejér鬼变换及Möbius逆，并刻画整数像与局部曲线几何实现。
AI 中文摘要
曲线计数理论中的生成函数混合了两种不同类型的信息：多重覆盖混合了曲线次数，而拓扑弦耦合混合了BPS亏格。我们证明，对于Calabi-Yau三维流形沿有效射线上的全亏格Gopakumar-Vafa展开，单个中心差分算子能以精确且可逆的方式分离这两种效应。记\\[ y=\left(2\sin\frac{\lambda}{2}\right)^2, \\]我们将Bryan-Pandharipande覆盖多项式与相应的Fejér因子结合，得到归一化差分强迫的Chebyshev-Fejér鬼变换。该变换具有显式的算子值Möbius逆。其$y$-adic滤过恰好是BPS亏格滤过：模$y^{G+1}$的强迫等价于亏格至多为$G$的BPS不变量的完整集合。因此，从经典特化得到的亏格零重构是全亏格层级的第一伴随分次块。我们还刻画了该变换的整数像。整数BPS强迫等价于一个两阶段条件：底层鬼坐标的Witt-Dwork型Frobenius同余，加上中心差分算子施加的额外Fejér可除性条件。在素数幂次数下，这给出了包含完整Fejér因子的加强Dwork同余。对于局部曲线，该形式体系具有具体的几何实现。超刚性椭圆曲线给出强迫的精确Dedekind $\eta$表达式。在平衡的高亏格局部曲线理论中，上次数二BPS不变量实现为对称积上平方根自然叠的顶部Chern数。这些结果将中心差分方程与BPS重构、算术鬼结构及局部曲线的几何联系起来。
英文摘要
Generating functions in curve counting theory mix two different kinds of information: multiple covers mix curve degrees, while the topological string coupling mixes BPS genera. We show that a single central difference operator separates these two effects in a precise and invertible way for the all genus Gopakumar-Vafa expansion along an effective ray of a Calabi-Yau threefold. Writing \[ y=\left(2\sin\fracλ{2}\right)^2, \] we combine the Bryan-Pandharipande cover polynomials with the corresponding Fejér factors and obtain a Chebyshev-Fejér ghost transform for the normalized difference forcing. The transform admits an explicit operator-valued Möbius inverse. Its $y$-adic filtration is exactly the BPS genus filtration: the forcing modulo $y^{G+1}$ is equivalent to the complete collection of BPS invariants of genera at most $G$. Thus the genus zero reconstruction from the classical specialization is the first associated graded piece of an all genus hierarchy. We also characterize the integral image of the transform. Integral BPS forcing is equivalent to a two stage condition: Witt-Dwork type Frobenius congruences for the underlying ghost coordinates, together with an additional Fejér divisibility condition imposed by the central difference operator. In prime power degree this gives a strengthened Dwork congruence containing the full Fejér factor. For local curves, the formalism has concrete geometric realizations. A super-rigid elliptic curve gives an exact Dedekind $η$ expression for the forcing. In the balanced higher genus local curve theory, the upper degree two BPS invariants are realized as top Chern numbers on natural stacks of square roots over symmetric products. These results connect central difference equations with BPS reconstruction, arithmetic ghost structures, and the geometry of local curves.
