Refined lattice point counting on the moduli space of Klein surfaces

Nitin Kumar Chidambaram; Elba Garcia-Failde; Alessandro Giacchetto; Kento Osuga

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arXiv · 2605.09526

Refined lattice point counting on the moduli space of Klein surfaces

Nitin Kumar Chidambaram·Elba Garcia-Failde·Alessandro Giacchetto·Kento Osuga

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Abstract

We introduce the moduli space of metric M\"obius 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 M\"obius 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.

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BibTeX RIS

Nitin Kumar Chidambaram, Elba Garcia-Failde, Alessandro Giacchetto, Kento Osuga. 2026-05-10. Refined lattice point counting on the moduli space of Klein surfaces. https://arxiv.org/abs/2605.09526

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