Algorithms for Gromov-Witten Invariants of Elliptic Curves

Authors: Janko Böhm, Firoozeh Dastur, Alain Hoffmann, Hannah Markwig and Ali Traore

Interested readers are referred to this article for more details. See also this article.

using Pkg                            
Pkg.add(url = "https://github.com/singular-gpispace/GromovWitten", rev="d7af772"; io=devnull)
using GromovWitten

julia> using GromovWitten

julia> G =  feynman_graph([(1,2),(1,2),(1,2)]);

julia> G = feynman_graph([(1,2),(1,2),(2,4),(1,3),(3,4),(3,4)]);

julia> G = feynman_graph([(1,2),(1,2),(1,3),(2,4),(3,4),(3,5),(4,6),(5,6),(5,6)]);

julia> G = feynman_graph([(1,2),(1,3),(1,4),(2,4),(2,3),(3,4)]);

julia> using GromovWitten

julia> G = feynman_graph([(1, 3),(1, 2),(1, 2),(2, 4),(3, 4),(3, 4)])
FeynmanGraph([(1, 3), (1, 2), (1, 2), (2, 4), (3, 4), (3, 4)])

julia> F = feynman_integral(G);

julia> a = [0, 2, 1, 0, 0, 1];

julia> feynman_integral_branch_type(F, a)
256*q[2]^4*q[3]^2*q[6]^2

julia> feynman_integral_degree(F, 3)
288*q[1]^6 + 32*q[1]^4*q[2]^2 + 32*q[1]^4*q[3]^2 + 32*q[1]^4*q[5]^2 + 32*q[1]^4*q[6]^2 + 8*q[1]^2*q[2]^2*q[5]^2 + 8*q[1]^2*q[2]^2*q[6]^2 + 8*q[1]^2*q[3]^2*q[5]^2 + 8*q[1]^2*q[3]^2*q[6]^2 + 24*q[2]^6 + 152*q[2]^4*q[3]^2 + 8*q[2]^4*q[5]^2 + 8*q[2]^4*q[6]^2 + 152*q[2]^2*q[3]^4 + 32*q[2]^2*q[3]^2*q[5]^2 + 32*q[2]^2*q[3]^2*q[6]^2 + 32*q[2]^2*q[4]^4 + 8*q[2]^2*q[4]^2*q[5]^2 + 8*q[2]^2*q[4]^2*q[6]^2 + 8*q[2]^2*q[5]^4 + 32*q[2]^2*q[5]^2*q[6]^2 + 8*q[2]^2*q[6]^4 + 24*q[3]^6 + 8*q[3]^4*q[5]^2 + 8*q[3]^4*q[6]^2 + 32*q[3]^2*q[4]^4 + 8*q[3]^2*q[4]^2*q[5]^2 + 8*q[3]^2*q[4]^2*q[6]^2 + 8*q[3]^2*q[5]^4 + 32*q[3]^2*q[5]^2*q[6]^2 + 8*q[3]^2*q[6]^4 + 288*q[4]^6 + 32*q[4]^4*q[5]^2 + 32*q[4]^4*q[6]^2 + 24*q[5]^6 + 152*q[5]^4*q[6]^2 + 152*q[5]^2*q[6]^4 + 24*q[6]^6

julia> f = feynman_integral_degree_sum(F, 3)
288*q[1]^6 + 32*q[1]^4*q[2]^2 + 32*q[1]^4*q[3]^2 + 32*q[1]^4*q[5]^2 + 32*q[1]^4*q[6]^2 + 8*q[1]^4 + 8*q[1]^2*q[2]^2*q[5]^2 + 8*q[1]^2*q[2]^2*q[6]^2 + 8*q[1]^2*q[3]^2*q[5]^2 + 8*q[1]^2*q[3]^2*q[6]^2 + 24*q[2]^6 + 152*q[2]^4*q[3]^2 + 8*q[2]^4*q[5]^2 + 8*q[2]^4*q[6]^2 + 152*q[2]^2*q[3]^4 + 32*q[2]^2*q[3]^2*q[5]^2 + 32*q[2]^2*q[3]^2*q[6]^2 + 8*q[2]^2*q[3]^2 + 32*q[2]^2*q[4]^4 + 8*q[2]^2*q[4]^2*q[5]^2 + 8*q[2]^2*q[4]^2*q[6]^2 + 8*q[2]^2*q[5]^4 + 32*q[2]^2*q[5]^2*q[6]^2 + 8*q[2]^2*q[6]^4 + 24*q[3]^6 + 8*q[3]^4*q[5]^2 + 8*q[3]^4*q[6]^2 + 32*q[3]^2*q[4]^4 + 8*q[3]^2*q[4]^2*q[5]^2 + 8*q[3]^2*q[4]^2*q[6]^2 + 8*q[3]^2*q[5]^4 + 32*q[3]^2*q[5]^2*q[6]^2 + 8*q[3]^2*q[6]^4 + 288*q[4]^6 + 32*q[4]^4*q[5]^2 + 32*q[4]^4*q[6]^2 + 8*q[4]^4 + 24*q[5]^6 + 152*q[5]^4*q[6]^2 + 152*q[5]^2*q[6]^4 + 8*q[5]^2*q[6]^2 + 24*q[6]^6

julia> substitute(f)
1792*q[1]^6 + 32*q[1]^4