Volumes of moduli spaces of bordered Klein surfaces

TL;DR

The authors extend Mirzakhani's recursion to moduli spaces of bordered Klein surfaces, including non-orientable cases, by combining Norbury’s top-form with Gendulphe regularisation. They derive explicit volume formulas for the simplest non-orientable topologies and establish a recursion for total volumes V^{ε}_{g,n} that captures ε-dependence and unifies orientable and non-orientable contributions. A refined topological recursion framework is developed, introducing a refined spectral curve and conjectural relations between refined correlators ω_{g,n}^b and the volumes, with concrete evidence in Euler characteristic −1 cases. The work also situates these results within a broader deformation between orientable and non-orientable geometry, connecting to b-Hurwitz-type invariants and opening directions for geometric interpretations of the refinement parameter b.

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 $ε\in\mathbb{R}_{>0}$, 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 $ε$. 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.

Figures (6)

• Figure 1: Diagrammatic representation of the recursion for total volumes.
• Figure 2: The picture on the left shows a two-bordered real-projective plane which contains exactly two 1-sided curves $\alpha$ and $\alpha'$ (red and blue) intersecting exactly once; the third elementary move exchanges them. The picture on the right shows a one-bordered Klein bottle with its unique 2-sided primitive curve (green), a reference 1-sided curve $\alpha_0=\alpha$ (red), and a pair of 1-sided curves $\alpha_1=\alpha'$ and $\alpha_{-1}$ (blue and orange) which intersect each other once but do not intersect $\alpha_0$, and can be obtained respectively by adding/subtracting a Dehn twist along $\gamma$ to $\alpha_0$. The fourth elementary move exchanges the pair $(\alpha,\alpha')$ with $\gamma$, which intersects both exactly once.
• Figure 3: The picture shows the effect of an isotopy $h_t$ of $K$ which moves the second boundary $\beta_2$ around the Möbius strip once, while keeping the boundary (of the Möbius strip) $\beta_1$ fixed. We have $h_0=\mathrm{Id}$, $\left.h_t\right|_{\beta_1}=\mathrm{Id}$ and $h_1=Y^{(1)}$, which is still the identity on $\beta_1$ and reverses the orientation on $\beta_2$.
• Figure 4: In this figure, both surfaces $K$ are obtained by gluing a pair of pants $P_1$, bordered by $\beta_1,\gamma',\gamma"$, to another pair of pants $K\setminus P_1$. Each surface $K$ has topology $(g,n)=(1,2)$; however, the one on the left is orientable, while the one on the right is non-orientable, so they are topologically distinct. This figure illustrates that when $K\setminus P_1$ is connected, gluing $P_1$ regularly along one boundary component $\gamma'$ and antipodally along the other one $\gamma"$, producing a non-orientable handle (right), can produce a different surface than gluing along both $\gamma'$ and $\gamma"$ regularly, producing an orientable handle (left).
• Figure 5: In contrast to Figure \ref{['fig:illustrating-factor-2']}, when $K\setminus P_1$ is disconnected, gluing $P_1$ regularly along one boundary component $\gamma'$ and antipodally along the other one $\gamma"$, or gluing regularly along both $\gamma'$ and $\gamma"$, always produces the same surface.

Theorems & Definitions (61)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Lemma 2.1: Nor07
• Definition 2.2
• Theorem 2.3: Nor07
• Remark 2.4
• Theorem 2.5: Gen17
• Definition 2.6