Pillowcase covers: Counting Feynman-like graphs associated with quadratic differentials

TL;DR

This paper proves the quasimodularity of generating functions for counting pillowcase covers, offering an alternative proof to prior results and introducing new techniques involving half-translation surface decompositions and 2-orbifold Hurwitz numbers.

Contribution

It provides a new proof of quasimodularity for pillowcase cover counts and introduces a quasi-polynomiality result for 2-orbifold Hurwitz numbers with completed cycles.

Findings

Proves quasimodularity of pillowcase cover counting functions

Develops a new method based on surface decompositions

Introduces a quasi-polynomiality result for Hurwitz numbers

Abstract

We prove the quasimodularity of generating functions for counting pillowcase covers, with and without Siegel-Veech weight. Similar to prior work on torus covers, the proof is based on analyzing decompositions of half-translation surfaces into horizontal cylinders. It provides an alternative proof of the quasimodularity results of Eskin-Okounkov and a practical method to compute area Siegel-Veech constants. A main new technical tool is a quasi-polynomiality result for 2-orbifold Hurwitz numbers with completed cycles.