Merge pull request #81 from singular-gpispace/dev_new

Dev new

docs/make.jl

@@ -42,7 +42,10 @@ makedocs(;
         ],
         "Functions" => "Overview.md",
         #"Quasimodular" =>"Feynman Integral/Feynman.md"
-        "Quasimodular" =>"quasimodular.md"
+        "Quasimodular" =>[
+            "quasimodular.md",
+            "quasimodular_psi.md",
+        ],
     ],
 )
 

docs/src/quasimodular.md

@@ -30,6 +30,13 @@ Consider the Caterpillar 3 graph
 
 ![alt text](img/Cartepillar3.png)
 
+We define first a polynomial ring in one variable $q$.
+
+```jldoctest quasi
+julia> R,q=polynomial_ring(QQ,["q"])
+(Multivariate polynomial ring in 1 variable over QQ, Nemo.QQMPolyRingElem[q])
+```
+
 ```jldoctest quasi
 julia> G = FeynmanGraph([(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)])
@@ -42,29 +49,30 @@ julia> F=FeynmanIntegral(G)
 FeynmanIntegral(FeynmanGraph([(1, 3), (1, 2), (1, 2), (2, 4), (3, 4), (3, 4)]), Dict{Symbol, Dict{Vector{Int64}, Nemo.QQMPolyRingElem}}(), (Multivariate polynomial ring in 14 variables over QQ, Nemo.QQMPolyRingElem[x[1], x[2], x[3], x[4]], Nemo.QQMPolyRingElem[q[1], q[2], q[3], q[4], q[5], q[6]], Nemo.QQMPolyRingElem[z[1], z[2], z[3], z[4]]))
 ```
 
-We compute the  sum of all Feynman Integral of degree up to 6.
+
+We compute the  sum of all Feynman Integral of degree up to $m=\text{number\_of\_monomials}(\text{weightmax}) $
+where weightmax$=6g-6$ where  $g$  is the genus of the graph .
+Here $g=3$ so weightmax$=6g-6=12$
 
 ```jldoctest quasi
-julia> Iq=substitute(feynman_integral_degree_sum(F,6))
-886656*q[1]^12 + 182272*q[1]^10 + 25344*q[1]^8 + 1792*q[1]^6 + 32*q[1]^4
+julia> weightmax=12;
 ```
 
-To express the Feynman Integral Iq in term of Eisenstein series $E_2, E_4, E_6$, we define a polynomial ring in q
-
 ```jldoctest quasi
-julia> R,q=polynomial_ring(QQ,["q"])
-(Multivariate polynomial ring in 1 variable over QQ, Nemo.QQMPolyRingElem[q])
-```
+julia> m = number_of_monomial(weightmax)
+22  
 
+```
+Since $m=22$,  we need to compute the Feynman integral degree sum up to at least degree $m=22$. This results in  Iq=substitute(feynman_integral_degree_sum(F,m)).
+We get Iq as following 
 ```jldoctest quasi
-julia> Iqq=886656*q[1]^12 + 182272*q[1]^10 + 25344*q[1]^8 + 1792*q[1]^6 + 32*q[1]^4
-886656*q^12 + 182272*q^10 + 25344*q^8 + 1792*q^6 + 32*q^4
+julia> Iq=61425005056*q[1]^46+43646419584*q[1]^44+29331341312*q[1]^42+20067375616*q[1]^40 + 12961886976*q[1]^38 + 8490271392*q[1]^36 + 5225373696*q[1]^34 + 3233267712*q[1]^32 + 1875116544*q[1]^30 + 1079026432*q[1]^28 + 577972224*q[1]^26 + 302347264*q[1]^24 + 145337600*q[1]^22 + 66497472*q[1]^20 + 27353088*q[1]^18 + 10246144*q[1]^16 + 3294720*q[1]^14 + 886656*q[1]^12 + 182272*q[1]^10 + 25344*q[1]^8 + 1792*q[1]^6 + 32*q[1]^4;
 ```
 
-we compute  quasimodular form of Iq :
+We can now express the Feynman Integral Iq in term of Eisenstein series $E_2, E_4, E_6$ by call the  quasimodular form of Iq :
 
 ```jldoctest quasi
-julia> quasimodular_form(Iqq,12)
+julia> quasimodularity_form(Iq,weightmax)
 (1//93312, -3*E2^6 + 6*E2^4*E4 + 4*E2^3*E6 - 3*E2^2*E4^2 - 12*E2*E4*E6 + 4*E4^3 + 4*E6^2)
 ```
 
@@ -74,7 +82,7 @@ Consider the Star graph $K_ {1,3}$ .
 
 ![alt text](img/star_graph.png)
 
-We want to find the quasimodular form of the sum of Feynman Integral  $Iq$ up to degree 6.
+We want to find the quasimodular form of the sum of Feynman Integral  $Iq$ up to degree 22.
 
 We define first a polynomial ring in one variable $q$.
 
@@ -86,8 +94,7 @@ julia> R,q=polynomial_ring(QQ,["q"])
 The sum of Feynman Integral up to degree 6 is:
 
 ```jldoctest form
-julia> Iq= 843264*q[1]^12 + 165888*q[1]^10 + 20736*q[1]^8 + 1152*q[1]^6 
-843264*q^12 + 165888*q^10 + 20736*q^8 + 1152*q^6
+julia> Iq=67007616768*q[1]^46+47407680000*q[1]^44+31871766528*q[1]^42+21710052864*q[1]^40 + 14037577344*q[1]^38 + 9136115712*q[1]^36 + 5629741056*q[1]^34 + 3459760128*q[1]^32 + 2005675776*q[1]^30 + 1145207808*q[1]^28 + 612956160*q[1]^26 + 317058048*q[1]^24 + 152150400*q[1]^22 + 68594688*q[1]^20 + 28035072*q[1]^18 + 10285056*q[1]^16 + 3255552*q[1]^14 + 843264*q[1]^12 + 165888*q[1]^10 + 20736*q[1]^8 + 1152*q[1]^6;
 ```
 
 The quasimodular form of $Iq$  is then:

docs/src/quasimodular_psi.md

@@ -0,0 +1,124 @@
+```@meta
+CurrentModule = GromovWitten
+DocTestSetup = quote
+using GromovWitten
+end
+```
+
+# Quasimodular
+
+In this module, we compute the solution of the system $Ax=b$, where $A$ is a matrix from homogeneous Eisenstein series $E_2, E_4, E_6$ and $b$
+from the Feynman Integral $I(q)$
+The solution is of the form (factor, coefficients) where coefficients is a vector of rationals numbers.
+Given a Feynman Integral
+
+```math
+I(q)=\sum_{n=1}^{d} a_i q^{d},
+```
+
+we compute the coefficients $b_{i,j,k}$ such that
+
+```math
+I(q)=\sum_{\substack{i,j,k \in \mathbb{N}_0 \\ 2i+4j+6k=d}} b_{i,j,k} E_2^i E_4^j E_6^k
+```
+
+## Example
+
+### Graph with vertex contribution
+
+Consider the Graph with vertex contribution
+
+![alt text](img/graph_with_vertex1.png)
+
+```jldoctest quasi
+julia> G = FeynmanGraph( [(1, 2), (2, 3), (3, 1)])
+FeynmanGraph([(1, 2), (2, 3), (3, 1)])
+```
+
+We then define the FeynmanIntegral type.
+
+```jldoctest quasi
+julia> F=FeynmanIntegral(G)
+FeynmanIntegral(FeynmanGraph([(1, 2), (2, 3), (3, 1)]), Dict{Symbol, Dict{Vector{Int64}, Nemo.QQMPolyRingElem}}(), (Multivariate polynomial ring in 9 variables over QQ, Nemo.QQMPolyRingElem[x[1], x[2], x[3]], Nemo.QQMPolyRingElem[q[1], q[2], q[3]], Nemo.QQMPolyRingElem[z[1], z[2], z[3]]))
+```
+
+We compute the  sum of all Feynman Integral of degree up to $m=\text{number\_of\_monomials}(\text{weightmax}) $
+where weightmax$=2(r+\sum_{i=1}^{n} g_i)$ with $r$ the number of edges and $g_i$ satifying  $h^1(\Gamma)+\sum_{i=1}^{n} g_i=g$ .
+Suppose $gg=[1,0,0]$, we have r=3; so
+weightmax$=2(3+1)=8$
+
+```jldoctest quasi
+julia> g =[1,0,0];
+```
+```jldoctest quasi
+julia> weightmax=8;
+```
+
+```jldoctest quasi
+julia> m = number_of_monomial(weightmax)
+10  
+```
+
+
+We computed then the Feynman Integral sum up to the degree $m=10$
+
+```jldoctest quasi
+julia> Iq=substitute(feynman_integral_degree_sum(F, m,g))
+56250*q[1]^20 + 121581//4*q[1]^18 + 18480*q[1]^16 + 8330*q[1]^14 + 4428*q[1]^12 + 3075//2*q[1]^10 + 556*q[1]^8 + 117*q[1]^6 + 15*q[1]^4 + 1//4*q[1]^2
+```
+
+We can now express the Feynman Integral Iq in term of Eisenstein series $E_2, E_4, E_6$ by call the
+
+we compute  quasimodular form of Iq :
+
+```jldoctest quasi
+julia> quasimodularity_form(Iq,weightmax)
+(1//6912, E2^3 + 2*E2^2*E4 - 3*E2*E4 - 4*E2*E6 + 2*E4^2 + 2*E6)
+```
+
+### Graph with loop at the vertex 1
+
+Consider the Graph with loop at the vertex 1 .
+
+![alt text](img/graph_loop.png)
+
+```jldoctest quasi
+julia> G = FeynmanGraph( [(1, 1),(1, 2), (2, 3), (3, 1)])
+FeynmanGraph([(1, 1), (1, 2), (2, 3), (3, 1)])
+```
+
+We then define the FeynmanIntegral type.
+
+```jldoctest quasi
+julia> F=FeynmanIntegral(G)
+FeynmanIntegral(FeynmanGraph([(1, 1), (1, 2), (2, 3), (3, 1)]), Dict{Symbol, Dict{Vector{Int64}, Nemo.QQMPolyRingElem}}(), (Multivariate polynomial ring in 10 variables over QQ, Nemo.QQMPolyRingElem[x[1], x[2], x[3]], Nemo.QQMPolyRingElem[q[1], q[2], q[3], q[4]], Nemo.QQMPolyRingElem[z[1], z[2], z[3]]))
+```
+
+We compute the  sum of all Feynman Integral of degree up to $m=\text{number\_of\_monomials}(\text{weightmax}) $.
+Here $gg=[0,0,0]$, and r=4; so
+weightmax$=2(4+0)=8$.
+
+```jldoctest quasi
+julia> weightmax=8;
+```
+
+```jldoctest quasi
+julia> m=number_of_monomial(weightmax)
+10
+```
+
+We computed then the Feynman Integral sum up to the degree $m=10$
+
+```jldoctest quasi
+julia> Iq=substitute(feynman_integral_degree_sum(F, m))
+67500*q[1]^20 + 36774*q[1]^18 + 20640*q[1]^16 + 9996*q[1]^14 + 4320*q[1]^12 + 1650*q[1]^10 + 456*q[1]^8 + 90*q[1]^6 + 6*q[1]^4
+```
+
+We can now express the Feynman Integral Iq in term of Eisenstein series $E_2, E_4, E_6$ by call the
+
+we compute  quasimodular form of Iq :
+
+```jldoctest quasi
+julia> quasimodularity_form(Iq,weightmax)
+(1//6912, E2^4 - E2^3 - 3*E2^2*E4 + 3*E2*E4 + 2*E2*E6 - 2*E6)
+```

quasimodular_pcopy.md

@@ -0,0 +1,120 @@
+```@meta
+CurrentModule = GromovWitten
+DocTestSetup = quote
+using GromovWitten
+end
+```
+
+# Quasimodular
+
+In this module, we compute the solution of the system $Ax=b$, where $A$ is a matrix from homogeneous Eisenstein series $E_2, E_4, E_6$ and $b$
+from the Feynman Integral $I(q)$
+The solution is of the form (factor, coefficients) where coefficients is a vector of rationals numbers.
+Given a Feynman Integral
+
+```math
+I(q)=\sum_{n=1}^{d} a_i q^{d},
+```
+
+we compute the coefficients $b_{i,j,k}$ such that
+
+```math
+I(q)=\sum_{\substack{i,j,k \in \mathbb{N}_0 \\ 2i+4j+6k=d}} b_{i,j,k} E_2^i E_4^j E_6^k
+```
+
+## Example
+
+### Graph with vertex contribution
+
+Consider the Graph with vertex contribution
+
+![alt text](img/graph_with_vertex1.png)
+
+```jldoctest quasi
+julia> G = FeynmanGraph( [(1, 2), (2, 3), (3, 1)])
+FeynmanGraph([(1, 2), (2, 3), (3, 1)])
+```
+
+We then define the FeynmanIntegral type.
+
+```jldoctest quasi
+julia> F=FeynmanIntegral(G)
+FeynmanIntegral(FeynmanGraph([(1, 2), (2, 3), (3, 1)]), Dict{Symbol, Dict{Vector{Int64}, Nemo.QQMPolyRingElem}}(), (Multivariate polynomial ring in 9 variables over QQ, Nemo.QQMPolyRingElem[x[1], x[2], x[3]], Nemo.QQMPolyRingElem[q[1], q[2], q[3]], Nemo.QQMPolyRingElem[z[1], z[2], z[3]]))
+```
+
+We compute the  sum of all Feynman Integral of degree up to $m=\text{number\_of\_monomials}(\text{weightmax}) $
+where weightmax$=2(r+\sum_{i=1}^{n} g_i)$ with $r$ the number of edges and $g_i$ satifying  $h^1(\Gamma)+\sum_{i=1}^{n} g_i=g$ .
+Suppose $gg=[1,0,0]$, we have r=3; so
+weightmax$=2(3+1)=8$
+
+```jldoctest quasi
+julia> weightmax=8;
+```
+
+```jldoctest quasi
+julia> m = number_of_monomial(weightmax)
+10  
+
+```
+
+
+We computed then the Feynman Integral sum up to the degree $m=10$
+
+```jldoctest quasi
+julia> Iq=substitute(feynman_integral_degree_sum(F, m,g))
+56250*q[1]^20 + 121581//4*q[1]^18 + 18480*q[1]^16 + 8330*q[1]^14 + 4428*q[1]^12 + 3075//2*q[1]^10 + 556*q[1]^8 + 117*q[1]^6 + 15*q[1]^4 + 1//4*q[1]^2
+```
+
+We can now express the Feynman Integral Iq in term of Eisenstein series $E_2, E_4, E_6$ by call the
+
+we compute  quasimodular form of Iq :
+
+```jldoctest quasi
+julia> quasimodularity_form(Iq,weightmax)
+(1//6912, E2^3 + 2*E2^2*E4 - 3*E2*E4 - 4*E2*E6 + 2*E4^2 + 2*E6)
+```
+
+### Graph with loop at the vertex 1
+
+Consider the Graph with loop at the vertex 1 .
+
+![alt text](img/graph_loop.png)
+
+```jldoctest quasi
+julia> G = FeynmanGraph( [(1, 1),(1, 2), (2, 3), (3, 1)])
+FeynmanGraph( [(1, 2), (2, 3), (3, 1)])
+```
+
+We then define the FeynmanIntegral type.
+
+```jldoctest quasi
+julia> F=FeynmanIntegral(G)
+FeynmanIntegral(FeynmanGraph( [(1, 1),(1, 2), (2, 3), (3, 1)]), Dict{Symbol, Dict{Vector{Int64}, Nemo.QQMPolyRingElem}}(), (Multivariate polynomial ring in 10 variables over QQ, Nemo.QQMPolyRingElem[x[1], x[2], x[3]], Nemo.QQMPolyRingElem[q[1], q[2], q[3], q[4]], Nemo.QQMPolyRingElem[z[1], z[2],z[3]]))
+```
+
+We compute the  sum of all Feynman Integral of degree up to $m=\text{number\_of\_monomials}(\text{weightmax}) $.
+Here $gg=[0,0,0]$, and r=4; so
+weightmax$=2(4+0)=8$.
+
+```jldoctest quasi
+julia> weightmax=8;
+julia> m=number_of_monomial(weightmax)
+10
+```
+
+We computed then the Feynman Integral sum up to the degree $m=10$
+
+```jldoctest quasi
+julia> Iq=substitute(feynman_integral_degree_sum(F, m))
+67500*q[1]^20 + 36774*q[1]^18 + 20640*q[1]^16 + 9996*q[1]^14 + 4320*q[1]^12 + 1650*q[1]^10 + 456*q[1]^8 + 90*q[1]^6 + 6*q[1]^4
+```
+
+We can now express the Feynman Integral Iq in term of Eisenstein series $E_2, E_4, E_6$ by call the
+
+we compute  quasimodular form of Iq :
+
+```jldoctest quasi
+julia> quasimodularity_form(Iq,weightmax)
+(1//6912, E2^4 - E2^3 - 3*E2^2*E4 + 3*E2*E4 + 2*E2*E6 - 2*E6)
+```
+

src/GromovWitten.jl

@@ -6,6 +6,7 @@ module GromovWitten
 using Nemo
 
 import Combinatorics: permutations
+export number_of_monomial
 export next_partition, combination
 export count_member
 export cache_integral_result
@@ -59,7 +60,6 @@ export substitute
 export sum_of_coeff
 export sum_of_divisor_powers
 export vector_to_monomial
-export number_of_monomials
 include("graph.jl")
 include("coeftermV.jl")
 include("coefterm.jl")

src/graph.jl

@@ -69,3 +69,21 @@ end
 function feynman_integral(G::FeynmanGraph)
     return FeynmanIntegral(G)
 end
+
+
+function number_of_monomial(weightmax)
+    w = [2, 4, 6]
+    count = 0
+    for e2 in 0:weightmax
+        for e4 in 0:weightmax
+            for e6 in 0:weightmax
+                degree = w[1] * e2 + w[2] * e4 + w[3] * e6
+                if 0 < degree <= weightmax
+                    count += 1
+                end
+            end
+        end
+    end
+
+    return count
+end