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
本文引入矩阵值有限差分形式，证明BPS重求和Gromov-Witten势的混合差分具有秩一Gram分解，并揭示翻转协变性与对数反常，应用于ADE折叠可区分李代数类型。
AI 中文摘要
我们为三维流形翻转的亏格零 BPS 重求和局部 Gromov-Witten 势引入了一种矩阵值有限差分形式。将中心差分分解为 \\[ \Delta_\eta=\nabla_\eta^2, \qquad \nabla_\eta=T_{h\eta/2}-T_{-h\eta/2}, \\] 我们证明了混合差分 \\( \mathbf H=(\nabla_{\eta_i}\nabla_{\eta_j}F)_{i,j} \\) 允许精确的秩一符号 \\(q\\)-Gram 分解。因此，原始 BPS 类由秩一系数矩阵检测，而矩阵值 Möbius 反演在原始射线上重建所有重数。在简单三维流形翻转下，Gram 强迫在翻转射线之外是协变的，并获得了普适对数反常 \\[ \kappa_C(C^+\otimes C^+)\log r, \qquad \kappa_C=\sum_{d\ge1}d^3n_{dC}. \\] 经过自然重整化后，它变为翻转协变的，而一个经典对数导数恢复了 crepant 变换公式中的三次修正。我们将 \\(\kappa_C\\) 解释为 GV 宽度分布的第三矩：Toda 的非交换宽度是第二矩，而收缩代数 BPS 不变量给出了一个有限上同调宽度序列，其数值实现包含所有这些矩。对于根支撑的 BPS 理论，简单根 Gram 系数重建了带符号的 Cartan 矩阵。应用于标准 ADE 折叠时，该理论区分了 \\(B_n\\) 与 \\(C_n\\)，检测到 \\(F_4\\) 双键和 \\(D_4\\) 三重性，并给出了显式的全局 \\(q\\)-Gram 判别式。
英文摘要
We introduce a matrix-valued finite difference formalism for the genus zero BPS resummed local Gromov-Witten potential of a threefold flop. Factoring the central difference as \[ Δ_η=\nabla_η^2, \qquad \nabla_η=T_{hη/2}-T_{-hη/2}, \] we prove that the mixed differences \( \mathbf H=(\nabla_{η_i}\nabla_{η_j}F)_{i,j} \) admit an exact rank one signed \(q\)-Gram decomposition. Primitive BPS classes are therefore detected by rank one coefficient matrices, and a matrix-valued Möbius inversion reconstructs all multiplicities on a primitive ray. Under a simple threefold flop the Gram forcing is covariant away from the flopped ray and acquires the universal logarithmic anomaly \[ κ_C(C^+\otimes C^+)\log r, \qquad κ_C=\sum_{d\ge1}d^3n_{dC}. \] After a natural renormalization it becomes flop covariant, while one classical logarithmic derivative recovers the cubic correction in the crepant transformation formula. We interpret \(κ_C\) as the third moment of the GV width distribution: Toda's noncommutative width is the second moment, and contraction algebra BPS invariants give a finite cohomological width series whose numerical realization contains all these moments. For root supported BPS theories, simple root Gram coefficients reconstruct the signed Cartan matrix. Applied to the standard ADE foldings, the theory distinguishes \(B_n\) from \(C_n\), detects the \(F_4\) double bond and \(D_4\) triality, and yields explicit global \(q\)-Gram discriminants.
Comments20 pages. Companion to arXiv:2609.10488[math.AG]. Comments welcome
