A Mirzakhani recursion for non-orientable surfaces
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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2023
- DOI:
- 10.48550/arXiv.2303.04049
- arXiv:
- arXiv:2303.04049
- Bibcode:
- 2023arXiv230304049S
- Keywords:
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- High Energy Physics - Theory;
- Mathematics - Geometric Topology
- E-Print:
- 15 pages, v2: minor correction