Topological recursion for the Poincaré polynomial of the combinatorial moduli space of curves (Q436238)

The combinatorial model of the moduli space \(\mathcal{M}_{g,n}\) of smooth genus \(g\) algebraic curves with \(n\) marked points is given by metric ribbon graphs. The latter are thickened \(1\)-skeletons of the curve's cell decomposition with a length assigned to each edge. Denoting their moduli space \(RG_{g,n}\) one has an orbifold isomorphism \(RG_{g,n}\simeq\mathcal{M}_{g,n}\times\mathbb{R}^n_+\). It is known that the generating function of the Euler characteristics \(\chi(RG_{g,n})\) is given by the Penner's matrix model. As a generalization, the authors study suitably defined Poincaré polynomials \(F_{g,n}(t_1,\dots,t_n)\) of \(RG_{g,n}\) in this paper. In particular, they derive a recursion of Eynard-Orantin type for them and interpret their top degree terms via the intersection numbers on the Deligne-Mumford stack \(\overline{\mathcal{M}}_{g,n}\).NEWLINENEWLINELet \(N_{g,n}(p)\) denote the number of ribbon graphs in \(RG_{g,n}\) of perimeter \(p\) with integral lengths, these graphs can be interpreted as Grothendieck's dessins d'enfants. It turns out that up to an explicit change of variables \(F_{g,n}\) are the Laplace transforms of \(N_{g,n}\). This is analogous to an earlier result for the Hurwitz numbers, the Laplace transforms plays the role of a mirror map.NEWLINENEWLINEKnown recursions for \(N_{g,n}\) lead to a differential equation for \(F_{g,n}\), which is also recursive in \(2g-2+n\), the absolute value of the Euler characteristic of an \(n\)-punctured surface of genus \(g\). In the stable range \(2g-2+n>0\) this recursion recovers all \(F_{g,n}\) from \(F_{0,3}\) and \(F_{1,1}\). Moreover, \(F_{g,n}\) turns out to be a Laurent polynomial in \(t_1,\dots,t_n\) of degree \(3(2g-2+n)\) with the intersection numbers of \(\psi\)-classes on \(\overline{\mathcal{M}}_{g,n}\) as coefficients in the top degree terms. The above recursion restricts to these terms and recovers the DVV recursion (Virasoro constraint) for the intersection numbers.