Counting Feynman-like graphs: Quasimodularity and Siegel-Veech weight

TL;DR

This paper proves the quasimodularity of generating functions counting torus covers, using flat surface decompositions and contour integrals, extending previous results and providing an alternative proof approach.

Contribution

It introduces a new proof method for quasimodularity of generating functions related to counting torus covers, generalizing prior work.

Findings

Proves quasimodularity of generating functions for torus covers.

Provides an alternative proof approach based on flat surface decompositions.

Extends previous results to covers with simple ramification.

Abstract

We prove the quasimodularity of generating functions for counting torus covers, with and without Siegel-Veech weight. Our proof is based on analyzing decompositions of flat surfaces into horizontal cylinders. The quasimodularity arise as contour integral of quasi-elliptic functions. It provides an alternative proof of the quasimodularity results of Bloch-Okounkov, Eskin-Okounkov and Chen-Moeller-Zagier, and generalizes the results of Boehm-Bringmann-Buchholz-Markwig for simple ramification covers.