Period matrices and homological quasi-trees on discrete Riemann surfaces

TL;DR

This paper explores discrete period matrices on embedded graphs, providing combinatorial interpretations, relating them to Laplacian determinants, and introducing a discrete Weil-Petersson potential, thus advancing discrete conformal geometry understanding.

Contribution

It introduces a combinatorial interpretation of discrete period matrices using homological quasi-trees and relates them to Laplacian determinants and Teichmüller space potentials.

Findings

Minors of the period matrix are expressed as weighted sums over homological quasi-trees.

The period matrix is related to the Laplacian determinant of flat complex line bundles.

A discrete analogue of the Weil-Petersson potential is derived as a sum over homological quasi-trees.

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 Teichm\"uller 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.