Volumes of moduli spaces of bordered Klein surfaces

TL;DR

This paper extends Mirzakhani's recursion to compute volumes of moduli spaces of bordered Klein surfaces, including non-orientable cases, by regularising divergent integrals and deriving explicit formulas and recursion relations.

Contribution

It introduces a regularisation approach for non-orientable surfaces' moduli space volumes and derives explicit formulas and recursions using extended identities.

Findings

Derived explicit volume formula for one-bordered Klein bottles.

Established a recursion for volumes of arbitrary topologies.

Connected volume calculations to refined topological recursion.

Abstract

We generalise Mirzakhani's recursion to volumes of moduli spaces of bordered Klein surfaces, which include non-orientable surfaces. On these moduli spaces, the top form introduced by Norbury diverges as the lengths of 1-sided geodesics approach zero. However, when integrated over Gendulphe's regularised moduli space, on which the systole of 1-sided geodesics is bounded below by $\epsilon\in\mathbb{R}_{>0}$, it returns a finite value. Using Norbury's extension of the Mirzakhani--McShane identities to non-orientable surfaces, we derive an explicit formula for the volume of the moduli space of one-bordered Klein bottles, as well as a recursion for arbitrary topologies that fully captures the dependence on Gendulphe's regularisation parameter $\epsilon$. We further relate these results to refined topological recursion, showing that, for a fixed refinement parameter, the volumes of moduli spaces of Klein surfaces with Euler characteristic $-1$ are governed by this procedure, and we conjecture the same holds for general topologies.