2025/06/02 by Lam, Wai Yeung, Lo, On-Hei Solomon, Yuen, Chi Ho
#05B35 #05C10 #30F60 #52C26 #53A70 #Combinatorics (math.CO) #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2506.02317
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 Teichmüller 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.