Changeset ca9c14 in git

Timestamp:

Aug 24, 2013, 6:39:15 PM (10 years ago)

Author:

Janko Boehm <boehm@…>

Branches:

(u'jengelh-datetime', 'ceac47cbc86fe4a15902392bdbb9bd2ae0ea02c6')(u'spielwiese', 'f875bbaccd0831e36aaed09ff6adeb3eb45aeb94')

Children:

a57b655308776d26d2a30d610451d0b7955cf233

Parents:

d55b3c24490789ff86cd0aeefd67f1eda2fd8ccf

git-author:

Janko Boehm <boehm@mathematik.uni-kl.de>2013-08-24 18:39:15+02:00

git-committer:

Janko Boehm <boehm@mathematik.uni-kl.de>2013-09-03 19:36:12+02:00

Message:

New version of ellipticCovers.lib

File:

• Singular/LIB/ellipticCovers.lib (modified) (3 diffs)

Singular/LIB/ellipticCovers.lib

@@ -11 +11 @@
 OVERVIEW:
 
+We implement a formula for computing the number of covers of elliptic curves
+which has beed proved by proving mirror symmetry
+for arbitrary genus by tropical methods in [BBM]. A Feynman graph of genus 
+g, which is defined as a trivalent genus g connected graph (with 2g-2 vertices 
+and 3g-3 edges). The branch type a=(a_1,...,a_(3g-3)) of a stable map is the 
+multiplicity of the the edge i over a fixed base point.
+
+Given a Feynman graph and a branch type a, we compute the number 
+N_(Gamma,a) of stable maps from a genus g curve of topological type Gamma
+and branch type a to the elliptic curve by computing a path integral 
+over a rational function (as a residue).
+
+The sum of N_(Gamma,a) over all branch types a of sum d gives the
+Gromov-Witten invariant N_(Gamma,d) of degree d stable maps from a genus g curve
+of topological type Gamma to the elliptic curve.
+
+The sum of N_(Gamma,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.
+
+References:
+
+[BBM] J. Boehm, A. Buchholz, H. Markwig: Tropical mirror symmetry for elliptic curves.
+
 KEYWORDS:
 
+tropical geometry; mirror symmetry; tropical mirror symmetry; Gromov-Witten invariants; elliptic curves; propagator; Feynman graph; path integral
+
 TYPES:
 
+graph
+
 PROCEDURES:
+
+makeGraph(list, list)                     graph from lists of vertices and edges;
+propagator(list, int)                     propagator factor of degree d in the quotient of two variables
+propagator(graph, list)                   propagator for fixed graph and branch type
+computeConstant(number, number)           constant coefficient in the Laurent series expansion of a rational function in a given variable
+evalutateIntegral(number, list)           path integral for a given propagator and ordered sequence of variables
+gromovWitten(number)                      sum of path integrals for a given propagator over all orderings of the variables
+gromovWitten(graph, int)                  list of Gromov Witten invariants for a given graph and all branch types
+
+partitions(int, int)                      partitions of an integer in a fixed number of summands
+permute(list)                             all permutations of a list
+sum(list)                                 sum of the elements of a list
+max(int, int)                             compute the maximum
 
 ";
@@ -217 +257 @@
   }
   if (typeof(P)=="graph"){
+   if (size(#)==1){
      int d =#[1];
+     list pa = partitions(size(P.edges),d);
+     return(gromovWitten(P,list(#[1],1,size(pa))));
+   } else {
+     int d =#[1];
+     int st = #[2];
+     int en = #[3];
      number s =0;
      number p;
@@ -224 +271 @@
      list pa = partitions(size(P.edges),d);
      int ti;
+     int ct=1;
+     print(size(pa));
      for (int j=1; j<=size(pa); j++) {
+      if ((j>=st)&(j<=en)){
        ti=timer;
        //pararg[j]=list(propagator(G,pa[j]));
        re[j]=gromovWitten(propagator(G,pa[j]));
        ti=timer-ti;
-       print(string(j)+" / "+string(size(pa))+"    "+string(pa[j])+"     "+string(re[j])+"     "+string(ti));
+       print(string(j)+" / "+string(size(pa))+"    "+string(pa[j])+"     "+string(re[j])+"      "+string(sum(re))+"     "+string(ti));
+      } else {re[j]=0;}
      }
      //list re = parallelWaitAll("gromovWitten", pararg, list(list(list(2))));
      return(re);
+    }
   }
 }