An algebra of power series arising in the intersection theory of moduli spaces of curves and in the enumeration of ramified coverings of the sphere

Abstract

A bracket is a function that assigns a number to each monomial in variables τ_0, τ_1, ... We show that any bracket satisfying the string and the dilaton relations gives rise to a power series lying in the algebra A generated by the series \sum n^{n-1} q^n/n! and \sum n^n q^n /n! . As a consequence, various series from A appear in the intersection theory of moduli spaces of curves. A connection between the counting of ramified coverings of the sphere and the intersection theory on moduli spaces allows us to prove that some natural generating functions enumerating the ramified coverings lie, yet again, in A. As an application, one can find the asymptotic of the number of such coverings as the number of sheets tends to infinity. We believe that the leading terms of the asymptotics like that correspond to observables in 2-dimensional gravity.