Compare20x20Reps.nb
Loads dWvH.m, OS.m, and nmSG.m, demonstrates their holoraumy and calculates their gadgets, etc. as explained in the paper

$Assumptions = Element[n,Reals]

Import Adinkra.wl

<<Adinkra.wl

BuildDate[Adinkra]

Function Lists

FunctionList[Adinkra]

Importing Data

SetDirectory[NotebookDirectory[]]

Get["dWvH.m"]

Get["nmSG.m"]

Get["OS.m"]

Get["dWvHprime.m"]

Get["nmSGprime.m"]

Get["OSprime.m"]

Reports

Level 0: Preliminary

AdinkraReport[OS]

AdinkraReport[dWvH]

AdinkraReport[nmSG]

AdinkraReport[OSprime]

AdinkraReport[dWvHprime]

AdinkraReport[nmSGprime]

Level 1: V and Vtilde Linear Independence check: all pass

AdinkraReport[OS,1]

AdinkraReport[dWvH,1]

AdinkraReport[nmSG,1]

AdinkraReport[OSprime,1]

AdinkraReport[dWvHprime,1]

AdinkraReport[nmSGprime,1]

Compare prime to unprimed:

L[dWvH]==L[dWvHprime]/.c0prime-> c0//Simplify

R[dWvH]==R[dWvHprime]/.c0prime-> c0//Simplify

V[dWvH]==V[dWvHprime]/.c0prime-> c0//Simplify

Vtilde[dWvH]==Vtilde[dWvHprime]/.c0prime-> c0//Simplify

L[OS]==L[OSprime]/.s0prime-> s0//Simplify

L[nmSG]==L[nmSGprime]/.g0prime-> g0//Simplify

V and Vtilde for all reps

All V and Vtilde for dWvH: all are non-zero

PrintAllV[dWvH]

PrintAllVtilde[dWvH]/.c0->0

All V and Vtilde for OS: all are non-zero

PrintAllV[OS]

PrintAllVtilde[OS]/.s0->0

All V and Vtilde for nmSG: all are non-zero

PrintAllV[nmSG]

PrintAllVtilde[nmSG]/.g0->0

Vpm and Vtildepm for all Reps

All Vpm and Vtildepm for dWvH: all are non-zero

LocalRep = dWvH;

PrintAllVPM[-1][LocalRep]

PrintAllVPM[1][LocalRep]

PrintAllVtildePM[-1][LocalRep]/.c0->0

PrintAllVtildePM[1][LocalRep]/.c0->0

Clear[LocalRep]

All Vpm and Vtildepm for OS: all are non-zero

LocalRep = OS;

PrintAllVPM[-1][LocalRep]

PrintAllVPM[1][LocalRep]

PrintAllVtildePM[-1][LocalRep]/.s0->0

PrintAllVtildePM[1][LocalRep]/.s0->0

Clear[LocalRep]

All Vpm and Vtildepm for nmSG: all are non-zero

LocalRep = nmSG;

PrintAllVPM[-1][LocalRep]

PrintAllVPM[1][LocalRep]

PrintAllVtildePM[-1][LocalRep]/.g0->0

PrintAllVtildePM[1][LocalRep]/.g0->0

Clear[LocalRep]

Eigenvalues: all plus or minus 1 for all reps. nmSG takes a little time to calculate

LocalRep =OS;

(Table[Abs[Eigenvalues[V[LocalRep][[II,JJ]]]]==Abs[{-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1}] && Element[Eigenvalues[V[LocalRep][[II,JJ]]],Reals],{II,1,3},{JJ,II+1,4}]//Flatten)==Table[True,{Index,1,6}]

(Table[Abs[Eigenvalues[Vtilde[LocalRep][[II,JJ]]]]==Abs[{-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1}]&& Element[Eigenvalues[Vtilde[LocalRep][[II,JJ]]],Reals],{II,1,3},{JJ,II+1,4}]//Flatten)==Table[True,{Index,1,6}]

Clear[LocalRep]

Print Things in a Tensor Product basis

Table[ωmatrix[5][slIndex]//MatrixForm,{slIndex,0,24}]

(Subscript[(OverTilde[V]), IJ]) for dWVH for c0=0 and nz[5] = 2 as shown in the paper

PrintVtildeBasis[dWvH][1,2]/.c0->0/.nz[5]->2

PrintVtildeBasis[dWvH][1,3]/.c0->0/.nz[5]->2

PrintVtildeBasis[dWvH][1,4]/.c0->0/.nz[5]->2

PrintVtildeBasis[dWvH][2,3]/.c0->0/.nz[5]->2

PrintVtildeBasis[dWvH][2,4]/.c0->0/.nz[5]->2

PrintVtildeBasis[dWvH][2,4]/.c0->0/.nz[5]->2

Gadgets

Check normalization, all equal 5

Gadget[nmSG,nmSG]==Gadget[OS,OS] ==Gadget[dWvH,dWvH] ==5

Gadget[dWvH,OS]//Simplify

Gadget[dWvH,nmSG] //Simplify

Gadget[OS,nmSG]//Simplify

Concise forms of Gadget[dWvH, OS], Gadget[dWvH,nmSG], and Gadget[OS,nmSG]

GadgetdWvHOSHazelFactorization = ((s0 - 1 - c0)(1-c0-3 (c0^3)+(1+9 (c0^2)) s0-9 c0 (s0^2)+3 (s0^3))+5);

Gadget[dWvH,OS]== GadgetdWvHOSHazelFactorization//Simplify

GadgetdWvHnmSGHazelFactorization = ((-1)/(4n))(3 (((3n+1))^2)(((c0 + g0))^4) + 6 (3n +1)(7n+1)(((c0+g0))^3)+2(9n+1)(13n+3)(((c0+g0))^2)+2 (97 (n^2)+22n + 1)(c0+g0)) - (1/(12 n))(181(n^2)+26n+1)

Gadget[dWvH,nmSG]==GadgetdWvHnmSGHazelFactorization//Simplify

GadgetOSnmSGHazelFactorization = (1/(16n))(-9 (n+1)(3n+1)(((s0 + g0))^4) + 12 (n -1)(3n+1)(((s0+g0))^3)+2(153 (n^2)+52n - 5)(((s0+g0))^2)+4 (91 (n^2)+30 n - 1)(s0+g0)) + (1/(48 n(3n+1)))(1077(n^3)+531 (n^2)+59 n - 3)

Gadget[OS,nmSG]==GadgetOSnmSGHazelFactorization//Simplify

Self-gadgets: prime w/ unprimed

SelfGadgetPaperForm[dWvH] = 5+9/2 (c0-c0prime)^4+2 (c0-c0prime)^2

SelfGadgetPaperForm[OS] =  5+ 21/8 (s0-s0prime)^4-3 (s0-s0prime)^2

SelfGadgetPaperForm[nmSG] =5+(g0-g0prime)^2(3 n+1)^2/(2n^2)(9/16 (9 n^2+2 n+1) (g0-g0prime)^2-n )

Gadget[dWvH,dWvHprime] == SelfGadgetPaperForm[dWvH]//Simplify

Gadget[OS,OSprime] == SelfGadgetPaperForm[OS]//Simplify

Gadget[nmSG,nmSGprime] == SelfGadgetPaperForm[nmSG]//Simplify

Minimum values

FindMinimum[SelfGadgetPaperForm[dWvH]/.c0-> c0prime +x,x]

FindMinimum[SelfGadgetPaperForm[OS]/.s0-> s0prime +x,x]

InputForm[%]

N[29/7]

FindMinimum[SelfGadgetPaperForm[nmSG]/.g0-> g0prime +x,{x,n}]

SelfGadgetPaperForm[nmSG]/.g0-> g0prime+1/3/.n-> 1/3

When Equals Five

Solve[(SelfGadgetPaperForm[OS]/.s0-> s0prime + x)==5,x]

Solve[(SelfGadgetPaperForm[nmSG]/.g0-> g0prime + x)==5,{x,n}]//Simplify

rpm[x_] :=((8 - 9 x^2 +pms 2 (Sqrt[2])(Sqrt[8-18 (x^2)-81 (x^4)]))/(81 (x^2))) ;

(SelfGadgetPaperForm[nmSG]/.n-> rpm[g0- g0prime]/.pms->1)==5//Simplify

(SelfGadgetPaperForm[nmSG]/.n-> rpm[g0- g0prime]/.pms->-1)==5//Simplify

Parallel Cases (Gadget = 5)

dWvH and OS can be made parallel, gadget to five with s0 = 1 + c0 and so and c0 satisfy a third order equation:

MGM5Soln = Solve[Gadget[dWvH,OS]== 5, {c0,s0},Reals]//Simplify

Solutions for g0 with conditions on n for dWvH, nmSG parallel

dWvHnmSGsoln5 = Solve[Gadget[dWvH,nmSG]== 5, {c0,g0}, Reals] // Simplify

Solutions for g0 with conditions on n for OS, nmSG parallel

OSnmSGsoln5 = Solve[Gadget[OS,nmSG]== 5, {c0,g0}, Reals] // Simplify

Setting c0 = -g0  leads to parallel dWvH,nmSG when  n=-0.463211 or n=-0.0119273

dWvHnmSGsoln5 = Solve[(Gadget[dWvH,nmSG]/.c0->-g0)== 5, {n}, Reals] // Simplify

dWvHnmSGsoln5//N

Setting s0 = -g0  leads to parallel OS,nmSG when n=-0.321012,n=-0.0169016, or n=0.513401

OSnmSGsoln5 = Solve[(Gadget[OS,nmSG]/.s0->-g0)== 5, {n}, Reals] // Simplify

OSnmSGsoln5//N

Some interesting solutions regarding orthogonality

dWvH and OS are necessarily NOT orthogonal:

Solve[Gadget[dWvH,OS]== 0, {c0,s0},Reals]//Simplify

dWvH and nmSG are orthogonal for a couple interesting choices of g0 and n

Solve[Gadget[dWvH,nmSG]== 0, {c0,g0}, Reals]//Simplify

dWvHnmSGsoln = Solve[Gadget[dWvH,nmSG]== 0, {c0,g0,n}, Reals]//Simplify

OS and nmSG can me made orthogonal within ranges of n, but not arbitrary n
There are 4 such solutions

OSnmSGOrthogonal = Solve[(Gadget[OS,nmSG])==0, {s0,c0}, Reals]

Dimensions[OSnmSGOrthogonal]

dWvH and nmSG, and OS and nmSG can be made orthogonal with the following choices (others exist, but these are the simplest)

DoubleZeroChoice = {g0-> -s0 -2/3,c0-> s0+1,n-> -1/9};

Gadget[dWvH,nmSG]==Gadget[OS,nmSG]==0/.DoubleZeroChoice//Simplify

with the obvious choice c0 = - g0, dWvH and nmSG are only orthogonal for imaginary n (no real solutions)

Solve[((Gadget[dWvH,nmSG]/.c0->-g0)//Simplify)==0,n]

Solve[((Gadget[dWvH,nmSG]/.c0->-g0)//Simplify)==0,n,Reals]

with the obvious choice s0 = - g0, OS and nmSG are only orthogonal for a single real solution n=0.0373449

Solve[((Gadget[OS,nmSG]/.s0->-g0)//Simplify)==0,n]

%//N

For the cases n = -1 (extends to N=2 SUGRA in a tower of higher spin) , -1/3 (msG), and n= 0 (newSG) we have

Divergent cases are for n =0, both  Gadget[dWvH,nmSG] and Gadget[OS,nmSG] and Gadget[OS,nmSG] for n=-1/3

Limit[Gadget[dWvH,nmSG],n-> 0]

Limit[Gadget[OS,nmSG],n->0]

Limit[Gadget[OS,nmSG],n-> -1/3]

GadgetOSnmSGSets0  =Gadget[OS,nmSG]/.s0-> -g0//Simplify

GadgetdWvHnmSGSetc0  =Gadget[dWvH,nmSG]/.c0-> -g0//Simplify

?Plot

Plot[GadgetdWvHnmSGSetc0 ,{n,-1.5,0.5},AxesLabel->Automatic,PlotLabel->"𝒢(dWvH,nmSG) for (Subscript[c, 0])=-(Subscript[g, 0])"]

Plot[GadgetOSnmSGSets0,{n,-1.5,0.5},AxesLabel->Automatic,PlotLabel->"𝒢(OS,nmSG) for (Subscript[s, 0])=-(Subscript[g, 0])"]

Convergent cases for Gadget[dWvH,nmSG] exist for n = -1 and n = -1/3 and  Gadget[OS,nmSG] for n = -1

dWvH,nmSG

Gadget[dWvH,nmSG], n = -1

Collect[Gadget[dWvH,nmSG]/.n-> -1/.c0-> c0plusg0 - g0,{c0plusg0},Simplify]/.c0plusg0-> c0+g0/.c0-> (Subscript[c, 0])/.g0-> (Subscript[g, 0])

TeXForm[%]

Gadget[dWvH,nmSG]/.n-> -1/.c0-> -g0//Simplify

Gadget[dWvH,nmSG], n = -1/3

Collect[Gadget[dWvH,nmSG]/.n-> -1/3/.c0-> c0plusg0 - g0,{c0plusg0},Simplify]/.c0plusg0-> c0+g0/.c0-> (Subscript[c, 0])/.g0-> (Subscript[g, 0])

TeXForm[%]

Gadget[dWvH,nmSG]/.n-> -1/3/.c0-> -g0//Simplify

Gadget[OS,nmSG], n = -1

Collect[Gadget[OS,nmSG]/.n-> -1/.s0-> s0plusg0 - g0,{c0plusg0},Simplify]/.s0plusg0->s0+g0/.s0-> (Subscript[s, 0])/.g0-> (Subscript[g, 0])

TeXForm[%]

Gadget[OS,nmSG]/.n-> -1/.s0-> -g0//Simplify