Period matrices and homological quasi-trees on discrete Riemann surfaces

TL;DR

The paper develops a fully combinatorial framework for the discrete period matrix of graphs embedded on closed surfaces by expressing minors of the matrix as weighted sums over homological quasi-trees and by relating the matrix to the determinant of a flat $\\mathbb{C}$-bundle Laplacian. It proves a duality between $k$-quasi-trees in a graph and $(2g-k)$-quasi-trees in the dual, and it connects the Hessian of the bundle-Laplacian determinant to the period matrix, providing a discrete analogue of Weil-Petersson geometry. A normalized-period-matrix perspective links the discrete theory to the classical Teichmüller world and yields a combinatorial interpretation of the Weil-Petersson potential in terms of $g$-quasi-trees. The work also places homological quasi-trees into a delta-matroidal framework, clarifying their structural and matroidal properties and answering a question of Kenyon about extending planar results to closed surfaces. Together, these results integrate discrete conformal geometry, topological graph theory, and matroid theory to advance the understanding of discrete Riemann surfaces.

Abstract

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 are expressed as 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 the Teichmüller space, expressed as a weighted sum over homological quasi-trees. Finally, we study the collection of homological quasi-trees from a (delta-)matroidal perspective. 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.

Figures (4)

• Figure 1: On the left-hand side: a graph (represented by a white vertex and four solid edges) and its dual (represented by a black vertex and four dotted edges), both embedded on the double torus. On the right-hand side: a subgraph of the original graph.
• Figure 2: A graph $G$ cellularly embedded on a genus-$2$ surface.
• Figure 3: A $1$-quasi-tree $T_1$ (thick edges) on the genus-2 surface. It contains a unique cycle that is homologically non-trivial on the surface. The cycle consists of the edges marked with a transverse stroke.
• Figure 4: A $2$-quasi-tree $T_2$ (thick edges) on the genus-2 surface. It is the union of $T_1$ together with the edge labeled by $26$. The added edge forms a new cycle, consisting of the edges marked with a transverse stroke. This new cycle, together with the cycle from $T_1$, forms a basis of $H_1(T_2;\mathbb{Z})$.

Theorems & Definitions (44)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1: GrossTuckerMondello2009
• Definition 2.2
• Lemma 2.3