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

Goujard, Elise¹ ; Möller, Martin²

¹Institut de Mathématiques de Bordeaux, Université de Bordeaux, Talence, France
²Institut für Mathematik, Goethe-Universität Frankfurt, Frankfurt am Main, Germany

Algebraic and Geometric Topology, Tome 20 (2020) no. 5, pp. 2451-2510

Résumé

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.

Détail

DOI : 10.2140/agt.2020.20.2451

Classification : 11F11, 32G15, 14H30, 14N10, 30F30, 81T18
Keywords: flat surfaces, covers, Feynman graphs, quasimodular forms

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