标题： 离散黎曼曲面上的周期矩阵和同调拟树 显示英文标题

标题： Period matrices and homological quasi-trees on discrete Riemann surfaces

摘要： 我们研究与闭曲面上胞腔嵌入图相关的离散周期矩阵，类似于黎曼曲面的经典周期矩阵。 通过离散调和1-形式的积分定义，这些周期矩阵已知以圆模式的方式编码离散共形结构。 我们得到了离散周期矩阵的组合解释，其中它的子式对应于某些生成子图上的加权和，我们将这些生成子图称为同调拟树。 此外，我们将周期矩阵与平坦复线丛的拉普拉斯算子的行列式联系起来。 我们推导了泰希米勒空间上威尔-彼得松势的组合类比表达式，表示为同调拟树上的加权和。 最后，我们证明同调拟树的集合构成一个delta-拟阵。 离散周期矩阵在圆平面网络中扮演着响应矩阵类似的角色，从而回答了Richard Kenyon提出的问题。 显示英文摘要

摘要： We study discrete period matrices associated with graphs cellularly embedded on closed surfaces, resembling classical period matrices of Riemann surfaces. Defined via integrals of discrete harmonic 1-forms, these period matrices are known to encode discrete conformal structure in the sense of circle patterns. We obtain a combinatorial interpretation of the discrete period matrix, where its minors correspond to weighted sums over certain spanning subgraphs, which we call homological quasi-trees. Furthermore, we relate the period matrix to the determinant of the Laplacian for a flat complex line bundle. We derive a combinatorial analogue of the Weil-Petersson potential on Teichm\"uller space, expressed as a weighted sum over homological quasi-trees. Finally, we prove that the collection of homological quasi-trees form a delta-matroid. The discrete period matrix plays a role similar to that of the response matrix in circular planar networks, thereby addressing a question posed by Richard Kenyon.