AI 中文总结
通过构造高维随机游走和拉普拉斯离散化的反例，提出量子Clebsch–Gordan多面体重构定理，证明其可唯一恢复带色三价图，并应用于辛几何与镜像对称，给出非阿贝尔Torelli定理。
AI 中文摘要
我们构造了不同的高维零均值有限范围格点随机游走，使得它们在所有步数下具有两两相等的返回概率。同样的例子提供了标准拉普拉斯算子离散化的两两不同形状，且其密度状态函数两两相等。主要贡献是一个重构定理：对带色三价图，可关联一个量子Clebsch–Gordan多面体，且该关联是一个完全函子，特别是从该多面体可以唯一恢复原始图。这些多面体作为特征簇（曲线上的秩2向量丛模空间）的环面退化矩多面体出现，从而得到一个组合非阿贝尔Torelli定理。在辛几何中，该重构定理意味着与这些退化相关的奇特征簇上的单调拉格朗日环面两两非哈密顿同伦。这些结果及其部分应用源于对向量丛模空间的镜像对称性研究，以及相关的图势能突变中的Laurent现象。
英文摘要
We construct distinct high-dimensional mean-zero finite range lattice random walks having pairwise-equal return probabilities for all step counts. The same examples provide pairwise-distinct shapes of discretizations of the standard Laplacian with pairwise-equal density state functions. The main contribution is a reconstruction theorem: to a colored trivalent graph one associates a quantum Clebsch--Gordan polytope, and this association is a full functor, in particular from the polytope one can uniquely recover the original graph. These polytopes appear as moment polytopes of toric degenerations of character varieties (moduli spaces of rank-2 bundles on curves), yielding a combinatorial non-abelian Torelli theorem. In symplectic geometry, the reconstruction theorem implies that monotone Lagrangian tori on odd character varieties associated with these degenerations are pairwise non-Hamiltonian isotopic. These results, and some of the applications, arose from the study of mirror symmetry for moduli spaces of vector bundles, and of the related Laurent phenomenon for mutations of graph potentials.
Comments25 pages, all comments welcome
