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

Goujard, Elise; Möller, Martin

doi:10.2140/agt.2020.20.2451

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

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Goujard, Elise
Max Planck Institute for Mathematics, Max Planck Society;

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Möller, Martin
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Goujard, E., & Möller, M. (2020). Pillowcase covers: counting Feynman-like graphs associated with quadratic differentials. Algebraic & Geometric Topology, 20(5), 2451-2510. doi:10.2140/agt.2020.20.2451.

Cite as: https://hdl.handle.net/21.11116/0000-0007-A385-A

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.