On polynomials counting essentially irreducible maps

TL;DR

This paper generalizes the enumeration of maps on genus-$g$ surfaces by introducing a class of essentially $2b$-irreducible maps, deriving symmetric polynomial formulas, and establishing string and dilaton equations for their counts.

Contribution

It extends known map enumeration results to a broader class of essentially $2b$-irreducible maps with polynomial dependence on $b$ and derives related recursive equations.

Findings

Enumeration given by symmetric polynomials in face degrees

Polynomials satisfy generalized string and dilaton equations

Results hold for genus $g \\leq 1$ with unique determination

Abstract

We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work of Norbury that, if one disallows vertices of degree one, the enumeration of such maps is related to the counting of lattice point in the moduli space of genus-$g$ curves with $n$ labeled points and is given by a symmetric polynomial $N_{g,n}(\ell_1,\ldots,\ell_n)$ in the face degrees $2\ell_1, \ldots, 2\ell_n$. We generalize this by restricting to genus-$g$ maps that are essentially $2b$-irreducible for $b\geq 0$, which loosely speaking means that they are not allowed to possess contractible cycles of length less than $2b$ and each such cycle of length $2b$ is required to bound a face of degree $2b$. The enumeration of such maps is shown to be again given by a symmetric polynomial $\hat{N}_{g,n}^{(b)}(\ell_1,\ldots,\ell_n)$ in the face degrees with a polynomial dependence on $b$. These polynomials satisfy (generalized) string and dilaton equations, which for $g\leq 1$ uniquely determine them. The proofs rely heavily on a substitution approach by Bouttier and Guitter and the enumeration of planar maps on genus-$g$ surfaces.