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

We prove that the Eynard-Orantin symplectic invariants of the curve x y − y 2 = 1 xy-y^2=1 are the orbifold Euler characteristics of the moduli spaces of genus g g curves. We do this by associating to the Eynard-Orantin invariants of x y − y 2 = 1 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.