Refined lattice point counting on the moduli space of Klein surfaces
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911669367603200 |
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| author | Chidambaram, Nitin Kumar Garcia-Failde, Elba Giacchetto, Alessandro Osuga, Kento |
| author_facet | Chidambaram, Nitin Kumar Garcia-Failde, Elba Giacchetto, Alessandro Osuga, Kento |
| contents | We introduce the moduli space of metric Möbius graphs, which extend ribbon graphs to the non-orientable world. This space contains both the moduli space of Riemann surfaces and the moduli space of non-orientable Klein surfaces. Each metric Möbius graph is equipped with a measure of non-orientability. We count lattice points in this moduli space, weighted by the measure of non-orientability, and prove a refined version of Norbury's recursion for this count. Taking the limit as the mesh becomes finer, we deduce a recursion for the Euclidean volumes, yielding a refined version of the Witten--Kontsevich recursion. As an application, we give a geometric definition of the refined Euler characteristic of the moduli space and compute it explicitly, thereby answering a question of Goulden, Harer, and Jackson. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09526 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Refined lattice point counting on the moduli space of Klein surfaces Chidambaram, Nitin Kumar Garcia-Failde, Elba Giacchetto, Alessandro Osuga, Kento Algebraic Geometry Mathematical Physics Combinatorics Geometric Topology We introduce the moduli space of metric Möbius graphs, which extend ribbon graphs to the non-orientable world. This space contains both the moduli space of Riemann surfaces and the moduli space of non-orientable Klein surfaces. Each metric Möbius graph is equipped with a measure of non-orientability. We count lattice points in this moduli space, weighted by the measure of non-orientability, and prove a refined version of Norbury's recursion for this count. Taking the limit as the mesh becomes finer, we deduce a recursion for the Euclidean volumes, yielding a refined version of the Witten--Kontsevich recursion. As an application, we give a geometric definition of the refined Euler characteristic of the moduli space and compute it explicitly, thereby answering a question of Goulden, Harer, and Jackson. |
| title | Refined lattice point counting on the moduli space of Klein surfaces |
| topic | Algebraic Geometry Mathematical Physics Combinatorics Geometric Topology |
| url | https://arxiv.org/abs/2605.09526 |