A Mirzakhani recursion for non-orientable surfaces

TL;DR

This paper extends Mirzakhani's recursion to non-orientable surfaces, connecting it to matrix models with orthogonal symmetry and enabling volume computations with crosscap size cutoffs.

Contribution

It generalizes Mirzakhani's recursion to non-orientable surfaces and links it to matrix integral loop equations, addressing divergences in the process.

Findings

Recursion for non-orientable surface volumes is derived.

Integral kernels correspond to orthogonal matrix model loop equations.

Regularized volumes are computed with a crosscap size cutoff.

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.