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

Paul Norbury

Published 2009-05-26, updated 2011-02-07Version 2

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.