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

Elise Goujard and Martin Möller

Algebraic & Geometric Topology 20 (2020) 2451–2510

DOI: 10.2140/agt.2020.20.2451

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 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.

Keywords

flat surfaces, covers, Feynman graphs, quasimodular forms

Mathematical Subject Classification 2010

Primary: 11F11, 32G15

Secondary: 14H30, 14N10, 30F30, 81T18

References

Publication

Received: 17 October 2018

Revised: 29 August 2019

Accepted: 7 January 2020

Published: 4 November 2020

Authors

Elise Goujard
Institut de Mathématiques de Bordeaux Université de Bordeaux Talence France

Martin Möller
Institut für Mathematik Goethe-Universität Frankfurt Frankfurt am Main Germany

Volume 20, issue 5 (2020)