Enumeration of ramified coverings of the sphere and 2-dimensional gravity

Abstract

Let A be the algebra generated by the power series \sum n^{n-1} q^n/n! and \sum n^n q^n /n! . We prove that many natural generating functions lie in this algebra: those appearing in graph enumeration problems, in the intersection theory of moduli spaces M_{g,n} and in the enumeration of ramified coverings of the sphere. We argue that ramified coverings of the sphere with a large number of sheets provide a model of 2-dimensional gravity. Our results allow us to compute the asymptotic of the number of coverings as the number of sheets goes to infinity. The leading terms of such asymptotics are the values of certain observables in 2-dimensional gravity. We prove that they coincide with the values provided by other models. In particular, we recover a solution of the Painleve I equation and the string solution of the KdV hierarchy.

26 pages, partially repeats math.AG/0403092

References in corpus (5)

• Hurwitz numbers and intersections on moduli spaces of curves
• Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials
• Gromov-Witten theory, Hurwitz numbers, and Matrix models, I
• An algebro-geometric proof of Witten's conjecture
• 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

Cited by in corpus (4)

• An algebro-geometric proof of Witten's conjecture
• Combinatorial solutions to integrable hierarchies
• Masur-Veech volumes and intersection theory: the principal strata of quadratic differentials
• A combinatorial theory of random matrices III: random walks on $\mathfrak{S}(N)$, ramified coverings and the $\mathfrak{S}(\infty )$ Yang-Mills measure