
D.4.6 ellipticcovers_lib
 Library:
 ellipticCovers.lib
 Purpose:
 GromovWitten numbers of elliptic curves
 Authors:
 J. Boehm, boehm @ mathematik.unikl.de
A. Buchholz, buchholz @ math.unisb.de
H. Markwig hannah @ math.unisb.de
 Overview:
 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 2g2 vertices
and 3g3 edges). The branch type b=(b_1,...,b_(3g3)) 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
GromovWitten 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 GromovWitten 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).
 Types:
 graph
Procedures:
D.4.6.1 makeGraph   generate a graph from a list of vertices and a lsit of edges 
D.4.6.2 printGraph   print procedure for graphs 
D.4.6.3 propagator   propagator factor of degree d in the quotient of two variables, or propagator for fixed graph and branch type 
D.4.6.4 computeConstant   constant coefficient in the Laurent series expansion of a rational function in a given variable 
D.4.6.5 evalutateIntegral   path integral for a given propagator and ordered sequence of variables 
D.4.6.6 gromovWitten   sum of path integrals for a given propagator over all orderings of the variables, or Gromov Witten invariant for a given graph and a fixed branch type, or list of Gromov Witten invariants for a given graph and all branch types 
D.4.6.7 computeGromovWitten   compute the Gromov Witten invariants for a given graph and some branch types generatingFunction (graph, int) multivariate generating function for the Gromov Witten invariants of a graph up to fixed degree 
D.4.6.8 partitions   partitions of an integer into a fixed number of summands 
D.4.6.9 permute   all permutations of a list 
D.4.6.10 lsum   sum of the elements of a list 
