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 - y2 = 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-y2 = 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. © 2012 American Mathematical Society.