A Mirzakhani recursion for non-orientable surfaces

Douglas Stanford

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arXiv · 2303.04049

A Mirzakhani recursion for non-orientable surfaces

Douglas Stanford

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Abstract

We review Mirzakhani's recursion for the volumes of moduli spaces of orientable surfaces, using a perspective that generalizes to non-orientable surfaces. The non-orientable version leads to divergences when the recursion is iterated, from regions in moduli space with small crosscaps. However, the integral kernels of the recursion are well-defined and they map precisely onto the loop equations for a matrix integral with orthogonal symmetry class and classical density of eigenvalues proportional to $\sinh(2\pi\sqrt{E})$ for $E>0$. The recursion can be used to compute regularized volumes with a cutoff on the minimal size of a crosscap.

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Douglas Stanford. 2023-03-07. A Mirzakhani recursion for non-orientable surfaces. https://arxiv.org/abs/2303.04049

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