Volumes of moduli spaces of bordered Klein surfaces

Abstract

We investigate 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. We derive an explicit formula for the volume of the moduli space of two-bordered real projective planes and of one-bordered Klein bottles --- the latter utilises Norbury's extension of the Mirzakhani--McShane identities to non-orientable surfaces. We then 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. For general topologies, both approaches remain incomplete due to several difficulties, which we discuss in this work. Together with our results, this initiates the search for a geometric recursive structure governing the volumes of moduli spaces of bordered Klein surfaces, whose natural formulation may lie within a suitably extended version of refined topological recursion.