update Quasimodularity

singular-gpispace · Oct 30, 2024 · 128a2fd · 128a2fd
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diff --git a/Project.toml b/Project.toml
@@ -6,5 +6,4 @@ version = "0.1.1"
 [deps]
 Combinatorics = "861a8166-3701-5b0c-9a16-15d98fcdc6aa"
 Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
-Revise = "295af30f-e4ad-537b-8983-00126c2a3abe"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
diff --git a/docs/make.jl b/docs/make.jl
@@ -44,8 +44,8 @@ makedocs(;
         #"Quasimodular" =>"Feynman Integral/Feynman.md"
         "Quasimodular" =>[
             "quasimodular.md",
-            "quasimodular_psi.md"
-        ]
+            "quasimodular_psi.md",
+        ],
     ],
 )
 

diff --git a/docs/src/quasimodular.md b/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:

diff --git a/docs/src/quasimodular_psi.md b/docs/src/quasimodular_psi.md
@@ -32,27 +32,34 @@ Consider the Graph with vertex contribution
 
 ```jldoctest quasi
 julia> G = FeynmanGraph( [(1, 2), (2, 3), (3, 1)])
-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]]))
+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;
-julia> m=number_of_monomials(weightmax)
-10
 ```
 
+```jldoctest quasi
+julia> m = number_of_monomial(weightmax)
+10  
+```
+
+
 We computed then the Feynman Integral sum up to the degree $m=10$
 
 ```jldoctest quasi
@@ -77,14 +84,14 @@ Consider the Graph with loop at the vertex 1 .
 
 ```jldoctest quasi
 julia> G = FeynmanGraph( [(1, 1),(1, 2), (2, 3), (3, 1)])
-FeynmanGraph( [(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]]))
+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}) $.
@@ -93,7 +100,10 @@ weightmax$=2(4+0)=8$.
 
 ```jldoctest quasi
 julia> weightmax=8;
-julia> m=number_of_monomials(weightmax)
+```
+
+```jldoctest quasi
+julia> m=number_of_monomial(weightmax)
 10
 ```
 
@@ -112,4 +122,3 @@ we compute  quasimodular form of Iq :
 julia> quasimodularity_form(Iq,weightmax)
 (1//6912, E2^4 - E2^3 - 3*E2^2*E4 + 3*E2*E4 + 2*E2*E6 - 2*E6)
 ```
-
diff --git a/quasimodular_pcopy.md b/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)
+```
+
diff --git a/src/GromovWitten.jl b/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")

diff --git a/src/graph.jl b/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
diff --git a/src/quasimodular.jl b/src/quasimodular.jl
@@ -1,21 +1,4 @@
 
-function number_of_monomials(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
-
 #=function filter_term(pols::Union{QQMPolyRingElem, Int64}, variables::Vector{QQMPolyRingElem}, power::Vector{Int64})
     T = parent(variables[1])
     if typeof(pols) <: Integer

diff --git a/test/quasimod.jl b/test/quasimod.jl
@@ -1,7 +1,7 @@
 @testset "quasimod.jl" begin
     w = [2, 4, 6]
     d = 12
-    @test number_of_monomials( d) == 22
+    @test number_of_monomial( d) == 22
     import Nemo: QQFieldElem
     S, (E2, E4, E6) = polynomial_ring(QQ, ["E2", "E4", "E6"])
     R, q = polynomial_ring(QQ, ["q"])
@@ -114,7 +114,7 @@
 
         F = FeynmanIntegral(ve)
         weightmax=12
-        m=number_of_monomials(6)
+        m=number_of_monomial(6)
         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
         result=quasimodular_form(Iq,weightmax)
         @test result == expected_result
@@ -126,7 +126,7 @@
 
         F = FeynmanIntegral(ve)
         weightmax=6
-        m=number_of_monomials(6)
+        m=number_of_monomial(6)
         Iq=substitute(feynman_integral_degree_sum(F, m))
         result=quasimodularity_form(Iq,weightmax)
         @test result == expected_result

diff --git a/z_install.sh b/z_install.sh
@@ -13,6 +13,13 @@ ve = [(1, 2), (1, 2), (1, 2)]
 
 F = FeynmanIntegral(ve)
 weightmax=6
-m=number_of_monomials(6)
+m=number_of_monomial(6)
  Iq=substitute(feynman_integral_degree_sum(F, m))
- quasimodular_form(Iq,weightmax)
+ quasimodularity_form(Iq,weightmax)
+
+ $ julia --project=docs/
+pkg> instantiate
+pkg> dev .
+pkg> up
+julia> include("docs/make.jl")
+julia> include("test/runtests.jl")