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

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 arises from representing the generating series as a contour integral of quasi-elliptic functions. This provides an alternative proof of the quasimodularity results of Bloch–Okounkov, Eskin–Okounkov and Chen–Möller–Zagier, and generalizes the results of Böhm–Bringmann–Buchholz–Markwig for simple ramification covers.