Cut-and-join operators for higher Weil-Petersson volumes
Alexander Alexandrov
TL;DR
This work develops a cut-and-join operator framework for generating functions of intersection numbers of $\psi$, $\kappa$, and $\Theta$ classes on $\overline{\mathcal M}_{g,n}$, including higher Weil–Petersson volumes. It shows that the generating functions $\tau_\alpha(\mathbf t, \mathbf s)$ are produced by a Virasoro-conjugated, cubic cut-and-join operator: $\tau_\alpha(\mathbf t, \mathbf s)=\exp\big(\hbar\widehat W_\alpha(\mathbf s)\big)\cdot 1$, with $\widehat W_\alpha(\mathbf s)$ obtained from a Virasoro group action and expressible as residues in terms of the odd current $\widehat{J}(z)$ and $\widehat{L}^o(z)$. This yields an algebraic topological recursion $\tau_\alpha^{(p)}(\mathbf t, \mathbf s)=\frac{1}{p}\widehat W_\alpha(\mathbf s)\cdot\tau_\alpha^{(p-1)}(\mathbf t, \mathbf s)$, enabling recursive computation of all intersection numbers and related Weil–Petersson volumes. The approach unifies several known recursions (e.g., Eynard–Orantin, Mirzakhani, Stanford–Witten) in the appropriate parameter regimes and connects to JT gravity and its supergravity counterparts, with explicit formulas for volumes in hyperbolic and super hyperbolic settings. Key explicit structures involve translations induced by $\mathbf s$, Laguerre-type integral representations for $f_\alpha(z)$, and detailed coefficient formulas expressed via Euler numbers and Bernoulli polynomials.
Abstract
In this paper, we construct the cut-and-join operator description for the generating functions of all intersection numbers of $ψ$, $κ$, and $Θ$ classes on the moduli spaces $\overline{\mathcal M}_{g,n}$. The cut-and-join operators define an algebraic version of topological recursion. This recursion allows us to compute all these intersection numbers recursively. For the specific values of parameters, the generating functions describe the volumes of moduli spaces of (super) hyperbolic Riemann surfaces with geodesic boundaries, which are also related to the Jackiw-Teitelboim (JT) (super)gravity.