String and dilaton equations for counting lattice points in the moduli space of curves

Abstract

We prove that the Eynard-Orantin symplectic invariants of the curve xy-y^2=1 are the orbifold Euler characteristics of the moduli spaces of genus g curves. We do this by associating to the Eynard-Orantin invariants of xy-y^2=1 a problem of enumerating covers of the two-sphere branched over three points. This viewpoint produces new recursion relations---string and dilaton equations---between the quasi-polynomials that enumerate such covers.

23 pages

References in corpus (4)

• A matrix model for simple Hurwitz numbers, and topological recursion
• Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
• Local Mirror Symmetry for the Topological Vertex
• Counting lattice points in compactified moduli spaces of curves

Cited by in corpus (3)

• A generalized topological recursion for arbitrary ramification
• Pruned Hurwitz numbers
• The Laplace transform, mirror symmetry, and the topological recursion of Eynard-Orantin