### D.15.31 tropicalEllipticCovers_lib

Library:
  tropicalEllipticCovers.lib
Purpose:
  Gromov-Witten numbers of tropical elliptic curves and their covers

Authors:
  J. Boehm, boehm @ mathematik.uni-kl.de
  F. Dastur, dastur.firoozeh @ gmail.com

Overview:
  We implement a formula for computing the number of covers of (abstract) elliptic curves with and without tangency conditions, and for the surface ExP1. For the first case, we have used methods discussed in [BGM1]. For the covers without tangency conditions, we obtain the covers of an elliptic curve using tropical methods in [BBM], and for ExP1, we have used the concepts from [BGM2]. A Feynman graph G of genus g is a trivalent, connected graph (of genus g) with n number of vertices and r number of edges (in case of abstract elliptic curve in [BBM] we have 2g-2 vertices and 3g-3 edges, since we consider only trivalent graphs). A pearl chain P of type (d_2, g) is a genus g, connected graph of degree d_1 such that it has two different types of vertices, say vertices of type w and b (where w represent the white vertices and b the black ones).
  Every edge in P connect vertex of type w to vertex of type b such that the vertices adjacent to each edge are not of the same type. There are precisely d_2 vertices of type w and d_2+g-1 vertices of type b. The vertices of type b should be 2-valent and the vertices of type w can have any valency.
  For each case we compute the respective propagator function from which we later use terms to compute the Gromov-Witten invariants N_(g,d), N_(g,d,Omega) and N_(P,d_1,d_2,g).

References:
  [BBM] J. Boehm, A. Buchholz, H. Markwig: Tropical mirror symmetry for elliptic curves, arXiv:1309.5893 (2013).
  [BGM1] J. Boehm, C. Goldner, H. Markwig: Tropical mirror symmetry in dimension one, arXiv:1809.10659 (2018).
  [BGM2] J. Boehm, C. Goldner, H. Markwig, Counts of (tropical) curves in ExP1 and Feynman integrals, arXiv:1812.04936 (2018).

Types:
  graph,number,list,poly,string

Procedures:

> D.15.31.1 cf_of |  | computes the coefficient of a given monomial g from a polynomial f.
> D.15.31.2 preim |  | returns the entry number of an element x in a list O.

> D.15.31.3 coverTuple

> depending on int, computes a list of tuples of the form (a_k, N_k, D_k) for all k=1,...,r, where r is the number of edges, a_k is the k-th entry of the partition aa of the degree of G, N_k and D_k are the corresponding numerator and denominator (respectively) of the propagator function whose product would return the coefficient of the Feynman integral. Every list of tuples corresponds to a cover of the elliptic curve E.

> D.15.31.4 CoverMult |  | depending on int, computes a list of cover multiplicity corresponding to each cover of an elliptic curve having source curve G.
> D.15.31.5 Tropicalcover |  | depending on int, computes a list of matrices such that each matrix in the list represents a cover of an elliptic curve E.
> D.15.31.6 sinh |  | returns the power series expansion of the hyperbolic sine function given a polynomial f up to first n terms.
> D.15.31.7 Sfunction |  | computes the S-function in the form of (1+X), where X is a taylor series expansion of the sinh function upto n terms.
> D.15.31.8 FloorDiagrams |  | returns a Latex file illustrating floor diagrams, such that each floor diagram corresponds to a tropical stable map to ExP1 for a given curled pearl chain of elliptic curve E.
> D.15.31.9 PropagatorFunction |  | depending on int, computes the propagator for the edges of a Feynman graph G.
> D.15.31.10 FeynmanIntegralo |  | depending on the second int, computes the coefficient of the Feynman integral for a given ordering of the vertices of the graph G.
> D.15.31.11 FeynmanIntegralO |  | depending on the second int, computes the coefficients of the sum of Feynman integrals (over all orderings of the vertices of the graph G).
> D.15.31.12 FeynmanIntegralA |  | depending on the third int, computes the coefficients of the sum of Feynman integrals for fixed ordering (sum goes over all partitions of the degree of G).
> D.15.31.13 DrawCovers |  | depending on int, returns a Latex file illustrating a cover of the elliptic curve E.
> D.15.31.14 TropCovandMaps |  | returns a Latex file that draws all the (cut open) curled pearl chains and its corresponding stable maps (as a floor diagram) given a pearl chain and other information regarding the leaks and white pearls of the pearl chain.

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