Counting generalized Jenkins-Strebel differentials

TL;DR

This paper develops a combinatorial approach to count lattice Jenkins-Strebel differentials on the Riemann sphere, relating the counts to volumes of moduli space strata and providing explicit formulas and identities.

Contribution

It introduces a new combinatorial framework for counting lattice Jenkins-Strebel differentials using decorated trees and relates these counts to moduli space volume calculations.

Findings

Derived explicit formulas for counting lattice Jenkins-Strebel differentials.

Connected counting problems to volumes of moduli space strata.

Established combinatorial identities related to these counts.

Abstract

We study the combinatorial geometry of "lattice" Jenkins--Strebel differentials with simple zeroes and simple poles on $\mathbb{C}P^1$ and of the corresponding counting functions. Developing the results of M. Kontsevich we evaluate the leading term of the symmetric polynomial counting the number of such "lattice" Jenkins-Strebel differentials having all zeroes on a single singular layer. This allows us to express the number of general "lattice" Jenkins-Strebel differentials as an appropriate weighted sum over decorated trees. The problem of counting Jenkins-Strebel differentials is equivalent to the problem of counting pillowcase covers, which serve as integer points in appropriate local coordinates on strata of moduli spaces of meromorphic quadratic differentials. This allows us to relate our counting problem to calculations of volumes of these strata . A very explicit expression for the volume of any stratum of meromorphic quadratic differentials recently obtained by the authors leads to an interesting combinatorial identity for our sums over trees.