### D.4.6 ellipticcovers_lib

Library:
  ellipticCovers.lib
Purpose:
  Gromov-Witten numbers of elliptic curves

Authors:
  J. Boehm, boehm @ mathematik.uni-kl.de
  A. Buchholz, buchholz @ math.uni-sb.de
  H. Markwig hannah @ math.uni-sb.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 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).

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

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