Refined lattice point counting on the moduli space of Klein surfaces

TL;DR

This paper extends lattice point counting to the moduli space of Klein surfaces using metric M"obius graphs, deriving refined recursions for volumes and Euler characteristics, bridging orientable and non-orientable cases.

Contribution

It introduces metric M"obius graphs to generalize moduli spaces, establishing refined counting and recursion formulas for non-orientable surfaces, and provides explicit geometric Euler characteristic calculations.

Findings

Derived a refined recursion for lattice point counts in the moduli space.

Established a limit process leading to a refined Witten--Kontsevich recursion.

Explicitly computed the refined Euler characteristic of the moduli space.

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.