Period matrices and homological quasi-trees on discrete Riemann surfaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915769149816832 |
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| author | Lam, Wai Yeung Lo, On-Hei Solomon Yuen, Chi Ho |
| author_facet | Lam, Wai Yeung Lo, On-Hei Solomon Yuen, Chi Ho |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_02317 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Period matrices and homological quasi-trees on discrete Riemann surfaces Lam, Wai Yeung Lo, On-Hei Solomon Yuen, Chi Ho Complex Variables Mathematical Physics Combinatorics Geometric Topology 53A70, 52C26, 30F60, 05B35, 05C10 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. |
| title | Period matrices and homological quasi-trees on discrete Riemann surfaces |
| topic | Complex Variables Mathematical Physics Combinatorics Geometric Topology 53A70, 52C26, 30F60, 05B35, 05C10 |
| url | https://arxiv.org/abs/2506.02317 |