N!=4.nb

$RecursionLimit=Infinity

Import Adinkra package

<<Adinkra`

BuildDate[Adinkra]

Function Lists

FunctionList[SpaceTime]

FunctionList[GenerateLandR]

FunctionList[AdinkraEssentials]

FunctionList[BasisDecomposition]

FunctionList[BC4Tools]

FunctionList[GraphingTools]

FunctionList[Adinkra]

Introduction to some of these functions

Within FunctionList[SpaceTime]

η[1,1]

Cmetric

Lap[SomeField[t,x,y,z]]

Lap[A[2][t,x,y,z]]

Within FunctionList[AdinkraEssentials]

IndexRange[AdinkraEssentials][ReportLevel]

FunctionList[AdinkraEssentials]

N≠4, Various Representations

3-cube

Generating the Adinkras

Rep = ThreeHypercube;

L[Rep]={IdentityMatrix[4],I SigmaProduct[3,2],-I SigmaProduct[2,0]};

PrintLSigmaProduct[Rep]

AdinkraPreliminaryReportO[Rep]

GenerateAdinkraDataO[Rep];

AdinkraHoloMonoReport[Rep]

AdinkraReport[Rep]

AdinkraFullReport[Rep]

PrintAllGALR[Rep]

PrintAllGARL[Rep]

Holoraumy: 8 distinct elements, Monodromy: 4 distinct elements

PrintAllHoloraumy[Rep]

PrintAllHoloraumy[ThreeHypercube]

PrintAllMonodromy[Rep]

Another 3-cube

Generating the Adinkra

Rep = AnotherThreeHypercube;

L[Rep]={L[ThreeHypercube][[3]],L[ThreeHypercube][[1]],L[ThreeHypercube][[2]]};

PrintLSigmaProduct[Rep]

GenerateAdinkraDataO[Rep];

AdinkraPreliminaryReport[Rep]

AdinkraReport[Rep,0]

AdinkraReport[Rep,1]

AdinkraReport[Rep,2]

AdinkraReport[Rep,3]

AdinkraReport[Rep,4]

AdinkraReport[Rep,5]

AdinkraReport[Rep,6]

AdinkraReport[Rep,7]

AdinkraReport[Rep,8]

AdinkraSummaryReport[Rep]

AdinkraFullReport[Rep]

Holoraumy: 8 distinct elements, Monodromy: 4 distinct elements

PrintAllHoloraumy[Rep]

PrintAllMonodromy[Rep]

PrintHoloraumy[Rep][{1,1,1}]

PrintMonodromy[Rep][{1,1,1}]

Yet Another 3-cube

Generating the Adinkra

Rep = YetAnotherThreeHypercube;

L[Rep]={L[ThreeHypercube][[1]],L[ThreeHypercube][[3]],L[ThreeHypercube][[2]]};

PrintLSigmaProduct[Rep]

GenerateAdinkraDataO[Rep];

AdinkraPreliminaryReport[Rep]

AdinkraReport[Rep]

AdinkraFullReport[Rep]

Holoraumy: 8 distinct elements, Monodromy: 4 distinct elements

PrintAllHoloraumy[Rep]

PrintAllMonodromy[Rep]

The Tesseract or the 4-Hypercube

Generating the Adinkra

Rep = Tesseract;

L[Rep]={IdentityMatrix[8],SigmaMatrixProduct[0,L[ThreeHypercube][[2]]],SigmaMatrixProduct[3,L[ThreeHypercube][[3]]],SigmaMatrixProduct[1,L[ThreeHypercube][[3]]]};

PrintLSigmaProduct[Rep]

GenerateAdinkraDataO[Rep];

AdinkraReport[Rep]

AdinkraFullReport[Rep]

Holoraumy: 16 distinct elements, Monodromy 8 distinct elements

PrintAllHoloraumy[Rep]

PrintAllMonodromy[Rep]

FunctionList[AdinkraEssentials]

ListOfIdenticalMonoOrHolo[Holoraumy,Rep]

ListOfIdenticalMonoOrHolo[Monodromy,Rep]

Another Tesseract or the 4-Hypercube

Generating the Adinkra

Rep = AnotherTesseract;

L[Rep]={L[Tesseract][[4]],L[Tesseract][[1]],L[Tesseract][[2]],L[Tesseract][[3]]};

PrintLSigmaProduct[Rep]

GenerateAdinkraDataO[Rep];

AdinkraReport[Rep]

AdinkraFullReport[Rep]

Holoraumy: 16 distinct elements, Monodromy 8 distinct elements

PrintAllHoloraumy[Rep]

PrintAllMonodromy[Rep]

ListOfIdenticalMonoOrHolo[Holoraumy,Rep]

ListOfIdenticalMonoOrHolo[Monodromy,Rep]

YetAnother Tesseract or the 4-Hypercube

Generating the Adinkra

Rep = YetAnotherTesseract;

L[Rep]={L[Tesseract][[2]],L[Tesseract][[1]],L[Tesseract][[4]],L[Tesseract][[3]]};

PrintLSigmaProduct[Rep]

GenerateAdinkraDataO[Rep];

AdinkraReport[Rep]

AdinkraFullReport[Rep]

Holoraumy: 16 distinct elements, Monodromy 8 distinct elements

PrintAllHoloraumy[Rep]

PrintAllMonodromy[Rep]

ListOfIdenticalMonoOrHolo[Holoraumy,Rep]

ListOfIdenticalMonoOrHolo[Monodromy,Rep]

(4,1) Adinkra

Generating the Adinkra

Rep = 41;

L[Rep] = {L[ThreeHypercube][[1]],L[ThreeHypercube][[2]],L[ThreeHypercube][[3]],I SigmaProduct[1,2]};

PrintLSigmaProduct[Rep]

GenerateAdinkraDataO[Rep];

AdinkraReport[Rep]

AdinkraFullReport[Rep]

Holoraumy for (4,1):  only 8 distinct elements, Monodromy 4 distinct elements

PrintAllHoloraumy[Rep]

PrintAllMonodromy[Rep]

5-Hypercube

Generating the Adinkra

Rep = FiveHypercube;

L[Rep]={IdentityMatrix[(2^4)],SigmaMatrixProduct[0,L[Tesseract][[2]]],SigmaMatrixProduct[0,L[Tesseract][[3]]],SigmaMatrixProduct[3,L[Tesseract][[4]]],SigmaMatrixProduct[1,L[Tesseract][[4]]]};

PrintLSigmaProduct[Rep]

GenerateAdinkraDataO[Rep];

AdinkraReport[Rep]

AdinkraFullReport[Rep]

Another 5-Hypercube

Generating the Adinkra

Rep = AnotherFiveHypercube;

L[Rep]={L[FiveHypercube][[5]],L[FiveHypercube][[1]],L[FiveHypercube][[2]],L[FiveHypercube][[3]],L[FiveHypercube][[4]]};

PrintLSigmaProduct[Rep]

GenerateAdinkraDataO[Rep];

AdinkraReport[Rep]

AdinkraFullReport[Rep]

Yet Another 5-Hypercube

Generating the Adinkra

Rep = YetAnotherFiveHypercube;

L[Rep]={L[FiveHypercube][[5]],L[FiveHypercube][[4]],L[FiveHypercube][[1]],L[FiveHypercube][[2]],L[FiveHypercube][[3]]};

PrintLSigmaProduct[Rep]

GenerateAdinkraDataO[Rep];

AdinkraReport[Rep]

AdinkraFullReport[Rep]

(5,1) Adinkra

Generating the Adinkra

Rep = 51;

L[Rep] = {SigmaMatrixProduct[0,L[41][[1]]],SigmaMatrixProduct[0,L[41][[2]]],SigmaMatrixProduct[0,L[41][[3]]],SigmaMatrixProduct[3,L[41][[4]]],SigmaMatrixProduct[1,L[41][[4]]]};

PrintLSigmaProduct[Rep]

GenerateAdinkraDataO[Rep];

Reports

AdinkraReport[Rep]

AdinkraFullReport[Rep]

Another (5,1) Adinkra

Generating the Adinkra

Rep = Another51;

L[Rep] = {L[51][[1]],L[51][[2]],L[51][[3]],L[51][[5]],L[51][[4]]};

PrintLSigmaProduct[Rep]

GenerateAdinkraDataO[Rep];

Reports

AdinkraReport[Rep]

AdinkraFullReport[Rep]

Yet Another (5,1) Adinkra

Generating the Adinkra

Rep = YetAnother51;

L[Rep] = {L[51][[5]],L[51][[1]],L[51][[2]],L[51][[3]],L[51][[4]]};

PrintLSigmaProduct[Rep]

GenerateAdinkraDataO[Rep];

Reports

AdinkraReport[Rep]

AdinkraFullReport[Rep]

Gadgets

Gadget[51,Another51]

Gadget[51,YetAnother51]

Gadget[Another51,YetAnother51]

Gadget[ThreeHypercube,AnotherThreeHypercube]

Gadget[ThreeHypercube,YetAnotherThreeHypercube]

Gadget[AnotherThreeHypercube,YetAnotherThreeHypercube]

Gadget[FiveHypercube,AnotherFiveHypercube]

Gadget[FiveHypercube,YetAnotherFiveHypercube]

Gadget[AnotherFiveHypercube,YetAnotherFiveHypercube]

Gadget[Tesseract,AnotherTesseract]

Gadget[Tesseract,YetAnotherTesseract]

Gadget[AnotherTesseract,YetAnotherTesseract]

Print Functions: choose Rep to be anything from the previous sections

GraphAdinkra[41]
PrintAllL[41]

Rep = 41

PrintL[Rep][1]

PrintAllL[Rep]

PrintR[Rep][2]

PrintAllR[Rep]

PrintAllV[Rep]

Table[PrintV[Rep][Ii,Ji],{Ii,1,4},{Ji,1,4}] ==Table[V[Rep][[Ii,Ji]]//MatrixForm,{Ii,1,4},{Ji,1,4}]

PrintAllVtildePM[1][Rep]

PrintAllHoloraumytilde[Rep][{1,1,1,1}]

Holoraumytilde[Rep][[15]]//MatrixForm

PrintZetaGen[Rep][4]

PrintAllZetaGen[Rep]

PrintAllHoloraumy[Rep][{1,1,1,1}]==PrintAllHoloraumy[Rep][[15]]==(-ZetaGen[Rep][[2]].ZetaGen[Rep][[3]].ZetaGen[Rep][[4]]//MatrixForm)

WordW[{0,0,0,1}]

PrintAllHoloraumytilde[Rep]

PrintAllMonodromytilde[Rep]

12x12 decomposition

L[Test12] = {(({{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0}})),(({{0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1}})),(({{0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0}})),(({{0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0}}))};

AdinkraPreliminaryReportO[Test12] (* Report to check to see if it is satisfies the garden algebra,etc. *)

GenerateAdinkraDataO[Test12] (* Generating the following data *)

AdinkraReport[Test12] (* Report on the generated data *)

PrintL[Test12][1]

PrintR[Test12][2]

PrintVtilde[Test12][1,2]

12 x 12 basis elements  (Subscript[c, 0])I + (Subscript[c, ai])(Subscript[ω, a]^3)⊗(Subscript[α, i]) + (Subscript[c, ai])(Subscript[ω, a]^3)⊗(Subscript[β, i])+...
nz[3] is a normalization that can be chose later

Table[ωmatrix[3][slnindex]//MatrixForm,{slnindex,0,8}]

Table[αmatrix[ihat]//MatrixForm,{ihat,1,3}]

Table[βmatrix[ihat]//MatrixForm,{ihat,1,3}]

Table[PrintSigmaProduct[αmatrix[ihat]],{ihat,1,3}]

Table[PrintSigmaProduct[βmatrix[ihat]],{ihat,1,3}]

expanding in terms of 12 x 12 basis elements  (Subscript[c, 0])I + (Subscript[c, ai])(Subscript[ω, a]^3)⊗(Subscript[α, i]) + (Subscript[c, ai])(Subscript[ω, a]^3)⊗(Subscript[β, i])+...

GenerateCoeffs[Test12] (* Generating basis coefficients *)

PrintLBasis[Test12][1]

PrintLBasis[Test12][2]

PrintLBasis[Test12][3]

PrintLBasis[Test12][4]

PrintGALRBasis[Test12][1,1]

ωmatrix[3][0]

PrintGALRBasis[Test12][1,2]

PrintGALRBasis[Test12][1,1]

PrintGALRBasis[Test12][2,3]

PrintGALRBasis[Test12][4,4]

PrintVtildeBasis[Test12][1,2]

PrintVtildeBasis[Test12][1,3]

PrintVtildeBasis[Test12][1,4]

PrintVtildeBasis[Test12][2,3]

PrintVtildeBasis[Test12][2,4]

PrintVtildeBasis[Test12][3,4]

Looks like nz[3] = 1 works pretty well for Vtilde:

Table[PrintVtildeBasis[Test12][Ii,Ji],{Ii,1,NColors[Test12]-1},{Ji,Ii+1,NColors[Test12]}]/.nz[3]->1//Flatten//MatrixForm

Table[ωmatrix[3][slnindex]//MatrixForm,{slnindex,0,8}]/.nz[3]->1

PrintVtildeBasis[Test12][1,2]/.nz[3]->1

Checking that the code is working right.

PrintVtilde[Test12][1,2]

Vtilde[Test12][[1,2]]//MatrixForm

PrintAllVtildePM[1][Test12]

ωmatrix[3][3]/.nz[3]->1//MatrixForm

-βmatrix[3]//MatrixForm

αmatrix[2]//MatrixForm

((Outer[Times,{{1,0,0},{0,0,0},{0,0,0}},αmatrix[2]]+Outer[Times,{{0,0,0},{0,1,0},{0,0,0}},βmatrix[3]]+Outer[Times,{{0,0,0},{0,0,0},{0,0,-1}},βmatrix[3]])//ArrayFlatten)==Vtilde[Test12][[1,2]]

Print Tests

PrintL[Test12][1]

PrintAllL[Test12]

PrintR[Test12][2]

PrintAllR[Test12]

PrintAllGALR[Test12]

PrintAllGARL[Test12]

Table[PrintV[Test12][Ii,Ji],{Ii,1,4},{Ji,1,4}] ==Table[V[Test12][[Ii,Ji]]//MatrixForm,{Ii,1,4},{Ji,1,4}]

PrintAllVtildePM[1][Test12]

PrintHoloraumytilde[Test12][{1,1,1,1}]

Holoraumytilde[Test12][[15]]//MatrixForm

PrintZetaGen[Test12][4]

PrintAllZetaGen[Test12]

PrintHoloraumytilde[Test12][{1,1,1,0}]==PrintAllHoloraumytilde[Test12][[14]]==(-ZetatildeGen[Test12][[2]].ZetatildeGen[Test12][[3]]//MatrixForm)

WordW[{0,0,0,1}]

PrintAllHoloraumytilde[Test12]

PrintAllMonodromytilde[Test12]

sl(5) basis

(ωmatrix[5][8]/.nz[5]->2)*12//MatrixForm