Topological recursion for symplectic volumes of moduli spaces of curves

The asymptotic Weil-Petersson form and intersection theory on M_{g,n}

“…Therefore, g : 1 N S(NΓ) → Γ is an -GHA where converges to 0 in the N → ∞ limit. It follows that 1 N S(NΓ) converges to Γ in the Gromov-Hausdorff topology.…”

“…On the other hand, if we consider Γ embedded as the spine of 1 N S(NΓ), then the curve γ in the hyperbolic surface consists of geodesic segments along the edges of Γ. By the triangle inequality, the length of such a segment in 1 N S(NΓ) exceeds the length of the corresponding edge in Γ by no more than twice the maximum rib length in 1 N S(NΓ). Summing up over the edges traversed by γ, we obtain the fact that…”

“…Therefore, f : Γ → 1 N S(NΓ) is an -GHA where converges to 0 in the N → ∞ limit. Now define the map g : 1 N S(NΓ) → Γ in the following way. For a point x ∈ 1 N S(NΓ), extend the shortest path from the boundary to x until it meets the spine Γ at g(x).…”

Emergent Geometry of KP Hierarchy

“…It first appeared in the setting of Eynard-Orantin topological recursion [10,9,5,30], establishing [VII] in the above picture. In the approach of [30], we note that the EO topological recursions for the Airy curve is equivalent to the DVV Virasoro constraints.…”

Grothendieck's Dessins d'Enfants in a Web of Dualities. II

“…The original motivation was to understand the mirror symmetry of a point, more precisely, why the n-point functions associated with the Witten-Kontsevich tau-function [12,10] satisfied the Eynard-Orantin topological recursion on the Airy curve y 2 = x, a result proved many times by different approach in e.g. [7,3,13]. The proof by the author in [13] indicates that the Eynard-Orantin topological recursion in this case is equivalent to the Virasoro constraints derived by Dijkgraaf-Verlinde-Verlinde [4], and spectral curve and the Bergman kernel are determined by making sense of genus zero one-point function and the genus zero two-point function respectively.…”