We introduce the moduli space of metric Möbius 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 Möbius 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.

DOI： 10.48550/arXiv.2605.09526

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arXiv

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Other Link： https://arxiv.org/pdf/2605.09526v1

Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1007/s00208-026-03418-4

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arXiv

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Other Link： https://arxiv.org/pdf/2412.17502v2

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.

DOI： 10.48550/arXiv.2511.21986

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arXiv

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Other Link： https://arxiv.org/pdf/2511.21986v1

We investigate a supersymmetric generalisation of topological recursion from two perspectives: algebraic and geometric. The algebraic side concerns a recursive structure encoded in modules of a super Virasoro algebra, and the geometric counterpart is what we call superconformal topological recursion defined on a super Riemann surface. Superconformal topological recursion indicates that odd holomorphic one-forms on a super Riemann surface are related to zero modes of copies of the Clifford algebras, and it also provides a tool to study deformation of non-split super Riemann surfaces, e.g. certain families of super Riemann surfaces carrying odd parameters. On a super Riemann surface over a non-reduced base, the formalism is recursive not only in terms of pants-decomposition but also in terms of odd parameters in a suitable sense.

DOI： 10.48550/arXiv.2511.17320

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arXiv

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Other Link： https://arxiv.org/pdf/2511.17320v1

Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.1093/imrn/rnaf116

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arXiv

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Other Link： https://arxiv.org/pdf/2311.17046v1