Volumes of moduli spaces of bordered Klein surfaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918219299684352 |
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| author | Garcia-Failde, Elba Gregori, Paolo Osuga, Kento |
| author_facet | Garcia-Failde, Elba Gregori, Paolo Osuga, Kento |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_21986 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Volumes of moduli spaces of bordered Klein surfaces Garcia-Failde, Elba Gregori, Paolo Osuga, Kento Geometric Topology Mathematical Physics Symplectic Geometry 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. |
| title | Volumes of moduli spaces of bordered Klein surfaces |
| topic | Geometric Topology Mathematical Physics Symplectic Geometry |
| url | https://arxiv.org/abs/2511.21986 |