Volume 20, issue 5 (2020)

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

Elise Goujard and Martin Möller

Algebraic & Geometric Topology 20 (2020) 2451–2510

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 and Okounkov and a practical method to compute area Siegel–Veech constants.

A main new technical tool is a quasipolynomiality result for 2–orbifold Hurwitz numbers with completed cycles.

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flat surfaces, covers, Feynman graphs, quasimodular forms
Mathematical Subject Classification 2010
Primary: 11F11, 32G15
Secondary: 14H30, 14N10, 30F30, 81T18
Received: 17 October 2018
Revised: 29 August 2019
Accepted: 7 January 2020
Published: 4 November 2020
Elise Goujard
Institut de Mathématiques de Bordeaux
Université de Bordeaux
Martin Möller
Institut für Mathematik
Goethe-Universität Frankfurt
Frankfurt am Main