We implement a formula for computing the number of covers of elliptic curves.
It has beed obtained by proving mirror symmetry
for arbitrary genus by tropical methods in [BBM]. A Feynman graph of genus
g is a trivalent, connected graph of genus g (with 2g-2 vertices
and 3g-3 edges). The branch type b=(b_1,...,b_(3g-3)) of a stable map is the
multiplicity of the the edge i over a fixed base point.
Given a Feynman graph G and a branch type b, we obtain the number
N_(G,b) of stable maps of branch type b from a genus g curve of topological type G
to the elliptic curve by computing a path integral
over a rational function. The path integral is computed as a residue.
The sum of N_(G,b) over all branch types b of sum d gives N_(G,d)*|Aut(G)|, with the
Gromov-Witten invariant N_(G,d) of degree d stable maps from a genus g curve
of topological type G to the elliptic curve.
The sum of N_(G,d) over all such graphs gives the usual Gromov-Witten invariant N_(g,d)
of degree d stable maps from a genus g curve to the elliptic curve.
The key function computing the numbers N_(G,b) and N_(G,d) is gromovWitten.
References:
[BBM] J. Boehm, A. Buchholz, H. Markwig: Tropical mirror symmetry for elliptic curves, arXiv:1309.5893 (2013).