A q-analogue of Mirzakhani's recursion for Weil-Petersson volumes
Norman Do, Paul Norbury
TL;DR
This work defines $q$-analogues of Mirzakhani's recursion for Weil–Petersson volumes and of the Stanford–Witten recursion for super Weil–Petersson volumes. It constructs $q$-kernels $D_q,R_q$ from $H_q$, and their super counterparts from $\widehat{H}_q$, to produce polynomials $V_{g,n}(L_1,\ldots,L_n)$ and $\widehat{V}_{g,n}^{(m)}(L_1,\ldots,L_n)$ with coefficients in $\mathbb{Q}[[q]]$ (and $\zeta_q^{odd}$ in the super case). The main results prove that a suitable rescaled $q\to1$ limit recovers the classical Weil–Petersson and super Weil–Petersson volumes, thereby linking the $q$-deformations to Mirzakhani's and Stanford–Witten's recursions; the coefficients are expressible as polynomials in $q$-zeta values $\zeta_q(2k)$ and $\zeta_q^{odd}(2k)$. The work also relates these $q$-polynomials to Okuyama's quasi-polynomials from a double-scaled SYK matrix model, suggesting broader connections to quantum and discrete geometry and potential root-of-unity phenomena via Habiro-type structures.
Abstract
We define q-analogues of Mirzakhani's recursion for Weil-Petersson volumes and the Stanford-Witten recursion for super Weil-Petersson volumes. Okuyama recently introduced a q-deformation of the Gaussian Hermitian matrix model which produces quasi-polynomials that recover the Weil-Petersson volumes via a rescaled q to 1 limit. The q-deformations of the Weil-Petersson volumes produced here agree with the top degree terms of Okuyama's quasi-polynomials and suggest a variation of Okuyama's methods to the super setting.