Changeset 4a24b9 in git
- Timestamp:
- Feb 8, 2024, 11:59:57 AM (3 months ago)
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Singular/LIB/tropicalEllipticCovers.lib (modified) (19 diffs)
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Singular/LIB/tropicalEllipticCovers.lib
r1bd321 r4a24b9 32 32 33 33 [BBM] J. Boehm, A. Buchholz, H. Markwig: Tropical mirror symmetry for elliptic curves, 34 34 arXiv:1309.5893 (2013). 35 35 [BGM1] J. Boehm, C. Goldner, H. Markwig: Tropical mirror symmetry in dimension one, 36 36 arXiv:1809.10659 (2018). 37 37 [BGM2] J. Boehm, C. Goldner, H. Markwig, Counts of (tropical) curves in ExP1 and Feynman 38 38 integrals, arXiv:1812.04936 (2018). 39 39 40 40 … … 58 58 of tuples corresponds to a cover of the elliptic curve E. 59 59 CoverMult(graph, list, list, int[, list]) depending on int, computes a list of cover multiplicity corresponding to 60 60 each cover of an elliptic curve having source curve G. 61 61 Tropicalcover(graph, list, list, int[, list]) depending on int, computes a list of matrices such that each matrix in the 62 62 list represents a cover of an elliptic curve E. 63 63 64 64 sinh(poly, int) returns the power series expansion of the hyperbolic sine function given a 65 65 polynomial f up to first n terms. 66 66 Sfunction(poly, int) computes the S-function in the form of (1+X), where X is a taylor series 67 67 expansion of the sinh function upto n terms. 68 68 FloorDiagrams(graph,list,list,list,int) returns a Latex file illustrating floor diagrams, such that each floor 69 69 diagram corresponds to a tropical stable map to ExP1 for a given curled … … 72 72 PropagatorFunction(graph,list,list,int[,list]) depending on int, computes the propagator for the edges of a Feynman graph G. 73 73 FeynmanIntegralo(graph,list,list,int,int[,list]) depending on the second int, computes the coefficient of the Feynman 74 74 integral for a given ordering of the vertices of the graph G. 75 75 FeynmanIntegralO(graph,list,list,int,int[,list]) depending on the second int, computes the coefficients of the sum of Feynman 76 76 integrals (over all orderings of the vertices of the graph G). 77 77 FeynmanIntegralA(graph,list,int,list,int,int[,list]) depending on the third int, computes the coefficients of the sum of Feynman 78 78 integrals for fixed ordering (sum goes over all partitions of the degree of G). 79 79 DrawCovers(graph,list,list,int,int[,list]) depending on int, returns a Latex file illustrating a cover of the elliptic curve E. 80 80 … … 567 567 "USAGE: coverTuple(G,aa,O,t[,l]); G graph, aa list, O list, t int, l list@* 568 568 ASSUME: G is a Feynman graph, aa is a partition of the degree of the graph G, O is a 569 569 given ordering of the vertices of G, t is an integer and l is a list of leaks in G. 570 570 RETURN: list. 571 571 THEORY: Computes a list of tuples (depending on the int t) of the form (a_k, N_k, D_k) for … … 683 683 "USAGE: CoverMult(G,aa,O,t[, l]); G graph, aa list, O list, t int, l list@* 684 684 ASSUME: G is a Feynman graph, aa is a partition of the degree of the graph G, O is a 685 685 given ordering of the vertices of G, l is a list of leaks of G and t is some integer between 0 and 3. 686 686 RETURN: list. 687 687 THEORY: For a given ordering of the vertices of G, this function computes a list of … … 868 868 "USAGE: Tropicalcover(G,aa,O,t[, l]); G graph, aa list, O list, t int, l list@* 869 869 ASSUME: G is a Feynman graph, aa is a partition of the degree of the graph G, O is a 870 870 given ordering of the vertices of G, t is an integer and l is a list of leaks of G. 871 871 RETURN: list of matrices. 872 872 THEORY: For a given ordering of the vertices of G, this function computes a list of … … 1734 1734 RETURN: poly 1735 1735 THEORY: Returns the S-function in the form of (1+X), where X is a taylor series 1736 1736 expansion of the sinh function upto k terms. 1737 1737 1738 1738 KEYWORDS: elliptic curves; S-function … … 2462 2462 "USAGE: FloorDiagrams(G,aa,O,Lw,n); G graph,aa list, O list, Lw list,n int@* 2463 2463 ASSUME: G is a pearl chain of the degree d, aa is a partition of degree d, O is an 2464 2465 2466 2464 ordering of the vertices of G, Lw is a list where the first n terms represent 2465 the leaks in G and the next d2 terms is a list of white pearls (assuming the 2466 first pearl is always white). NOTE: Requires pdflatex to be installed. 2467 2467 RETURN: Latex file. 2468 2468 THEORY: If n=0: the procedure generates a Latex file for all the tropical stable maps 2469 2470 2471 2469 to ExP1 and if n=1:the procedure generates the source file that can be executed 2470 in Latex to get all the stable maps of ExP1 (in the form of floor diagrams) for 2471 every curled pearl chain of an elliptic curve E provided similar information. 2472 2472 2473 2473 KEYWORDS: elliptic curves; tropical stable maps; floor diagrams … … 2536 2536 "USAGE: PropagatorFunction(G,aa,O,t[,l]); G graph, aa list, O list, t int, l list@* 2537 2537 ASSUME: G is a graph (a Feynman graph or a pearl chain) of the degree d, aa is a partition 2538 2539 2538 of degree d, O is an ordering of the vertices of G, l is a list representing the 2539 leaky edges of G and t is one of 0,1,2 or 3. 2540 2540 RETURN: list P_t of propagator function for each edge of G as follows: 2541 2541 P_0: Propagator for a Feynman graph G as in [BBM], i.e., a graph without any self- … … 2547 2547 P_3: Propagator for a pearl chain G as in [BGM2], i.e., graph G may have leaks. 2548 2548 THEORY: Computes the numerator and the denominator of the propagator function for every 2549 2549 edge of graph G depending on the integer t. 2550 2550 2551 2551 KEYWORDS: elliptic curves; propagator of a graph … … 2582 2582 proc FeynmanIntegralo(graph G,list aa, list O,list l,int k,int t, list #) 2583 2583 "USAGE: FeynmanIntegralo(G,aa,O,l,k,t[,gg]); G graph, aa list, O list, l list, k int, 2584 2584 t int, gg list@* 2585 2585 ASSUME: G is a graph (a Feynman graph or a pearl chain) of the degree d, aa is a 2586 2587 2588 2586 partition of degree d, O is an ordering of the vertices of G, l is a list 2587 representing the leaky edges of G, k is any integer, gg is a list representing 2588 the genus function and t is one of 0,1,2 or 3. 2589 2589 RETURN: number or list Q_t (depending on k) of Feynman integral for a fixed ordering of 2590 vertices of G, the results are as follows: 2591 @format 2592 Q_0: Feynman integral for a fixed ordering of a Feynman graph G as in [BBM], i.e., 2593 a graph without any self-looping edges, leaks or vertex contributions. 2594 Q_1: Feynman integral for a fixed ordering of a Feynman graph G without vertex 2595 contributions as in [BGM1], i.e. A graph that may have self-looping edges 2596 and leaks. 2597 Q_2: Feynman integral for a fixed ordering of a Feynman graph G with vertex 2598 contributions as in [BGM1] possibly with self-looping edges and leaks. 2599 Q_3: Feynman integral for a fixed ordering of a pearl chain G as in [BGM2], i.e., 2600 graph G may have leaks. 2590 vertices of G, the results are as follows: 2591 Q_0: Feynman integral for a fixed ordering of a Feynman graph G as in [BBM], i.e., 2592 a graph without any self-looping edges, leaks or vertex contributions. 2593 Q_1: Feynman integral for a fixed ordering of a Feynman graph G without vertex 2594 contributions as in [BGM1], i.e. A graph that may have self-looping edges 2595 and leaks. 2596 Q_2: Feynman integral for a fixed ordering of a Feynman graph G with vertex 2597 contributions as in [BGM1] possibly with self-looping edges and leaks. 2598 Q_3: Feynman integral for a fixed ordering of a pearl chain G as in [BGM2], i.e., 2599 graph G may have leaks. 2601 2600 THEORY: If k is zero it returns the coefficient of the Feynman integral for a given 2602 ordering of the vertices of the graph G. Otherwise, returns a list showing the 2603 ordering and the coefficient of the Feynman integral for the corresponding ordering. 2604 2605 2601 ordering of the vertices of the graph G. Otherwise, returns a list showing the 2602 ordering and the coefficient of the Feynman integral for the corresponding ordering. 2606 2603 KEYWORDS: elliptic curves; Feynman integral of a graph for a fixed ordering 2607 2604 EXAMPLE: example FeynmanIntegralo; shows an example … … 2645 2642 ASSUME: G is a graph (a Feynman graph or a pearl chain) of the degree d, aa is a partition 2646 2643 of degree d, l is a list representing the leaky edges of G, k is any integer, 2647 2644 gg is a list representing the genus function and t is one of 0,1,2 or 3. 2648 2645 RETURN: number or list Q_t (depending on k) of Feynman integral for all possible 2649 orderings of vertices of G, the results are as follows: 2650 @format 2646 orderings of vertices of G, the results are as follows: 2651 2647 Q_0: Feynman integral for all orderings of a Feynman graph G as in [BBM], i.e., a 2652 2648 graph without any self-looping edges, leaks or vertex contributions. … … 2658 2654 may have leaks. 2659 2655 THEORY: If k is zero it returns the coefficient which is a sum of Feynman integrals over 2660 all orderings of the vertices of the graph G. Otherwise, returns a list showing 2661 the ordering, the coefficient of the Feynman integral for the corresponding 2662 ordering and the sum. 2663 2656 all orderings of the vertices of the graph G. Otherwise, returns a list showing 2657 the ordering, the coefficient of the Feynman integral for the corresponding 2658 ordering and the sum. 2664 2659 KEYWORDS: elliptic curves; sum of Feynman integrals 2665 2660 EXAMPLE: example FeynmanIntegralO; shows an example … … 2695 2690 one of 0,1,2 or 3. 2696 2691 RETURN: number or list Q_t (depending on k) of Feynman integral for a fixed ordering of 2697 vertices of G, the results are as follows: 2698 @format 2692 vertices of G, the results are as follows: 2699 2693 Q_0: sum of Feynman integrals (over all partitions of d) for a fixed ordering of a 2700 2694 Feynman graph G as in [BBM], i.e. a graph without any self-looping edges, leaks … … 2712 2706 the Feynman integral is computed, the respective coefficient of the Feynman 2713 2707 integral for the given ordering and the sum over all partitions. 2714 2715 2716 2708 KEYWORDS: elliptic curves; Gromov-Witten invariants;Feynman integrals 2717 2709 EXAMPLE: example FeynmanIntegralA; shows an example … … 2743 2735 "USAGE: DrawCovers(G,aa,O,t[, lw]); G graph, aa list, O list, t int,n int and lw list@* 2744 2736 ASSUME: G is a Feynman graph/pearl chain, aa is a partition of the degree of the graph G, 2745 2746 2747 2737 O is a given ordering of the vertices of G, t is an integer between 0 and 3 and 2738 lw is a list where the first n entries represent the leaks in G and the remaining 2739 terms tells us about the white pearls of G. NOTE: Requires pdflatex to be installed. 2748 2740 RETURN: if n=0: C_t is a Latex file generating all the covers of an elliptic curve and 2749 2741 if n=1: C_t is the source file that can be executed in Latex to get the covers 2750 2742 of an elliptic curve (given an ordering O and a partition aa). 2751 2743 The results for C_t are as follows: 2752 @format2753 2744 C_0: returns a Latex file drawing all simply ramified covers of an elliptic curve 2754 2745 given a fixed ordering of the vertices of G and a fixed partition of its degree. … … 2761 2752 C_3: returns a Latex file drawing all curled pearl chains of an elliptic curve given 2762 2753 a fixed ordering of the vertices of G and a fixed partition of its degree. 2763 2764 2754 THEORY: For a given ordering of the vertices of G, this function computes Latex file 2765 2755 drawing all the covers of an elliptic curve E. If G is a pearl chain then this … … 2769 2759 we get results for graphs w/o vertex contributions respectively w/ vertex 2770 2760 contributions as in [BGM1], and for t=3 we get the results for pearl chains as in [BGM2]) 2771 2772 2773 2761 KEYWORDS: elliptic curve draw covers 2774 2762 EXAMPLE: example DrawCovers; shows an example … … 2850 2838 "USAGE: TropCovandMaps(G,aa,O,Lw,n); G graph, aa list, O list, Lw list, n int@* 2851 2839 ASSUME: G is a pearl chain, aa is a partition of the degree of the graph G, O is a given 2852 2840 ordering of the vertices of G, Lw is a list of leaks and white vertices in G. 2853 2841 NOTE: Requires pdflatex to be installed. 2854 2842 RETURN: Depending on n this procedure returns either a pdf or a latex file for the curled
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