4DN=4VMTM_offshell.nb
Notebook Author: Kory Stiffler
Off shell Holoraumy Calculations for the 4d, N=4 Vector Tensor multiplet analyzed in 2010.06124
Here “off-shell” refers to the separate N=2 vector and tensor multiplets as these algebras close off-shell, where as the composite N=4 vector-tensor multiplet obviously does not.
Original holoraumy done by Gabriel Hannon and  Rui Xian Siew 
Some corrections and the 1D holoraumy were calculated by KS, this file represents the final check of the holoraumy presented in 2010.06124
The algebra is as in 1812.04236
Appendix F is checked for the cross term algebra  it is also checked the cross term  holoraumy [(Subscript[D, a]^i),(Subscript[(OverTilde[D]), b]^j)]Fermion  in the X and Y bases as well as the algebra {(Subscript[D, a]^i),(Subscript[(OverTilde[D]), b]^j)}Fermion in W-basis.
Appendix H results of the V-tilded 16 color, 16 x 16 holoraumy added 11/14/21

Preliminaries

Importing Adinkra

$RecursionLimit=Infinity

<<Adinkra`

BuildDate[Adinkra]

FieldLists

FieldList:= list of all fields

FieldNameList = {A[t,x,y,z],B[t,x,y,z],F[t,x,y,z],G[t,x,y,z],d[t,x,y,z],Am[mu][t,x,y,z],Ψ[kk][cc][t,x,y,z],At[t,x,y,z],Bt[t,x,y,z],Ft[t,x,y,z],Gt[t,x,y,z],φ[t,x,y,z],Bmn[mu,nu][t,x,y,z],Ψt[kk][cc][t,x,y,z],Ah[t,x,y,z],Ab[t,x,y,z],Bh[t,x,y,z],Bb[t,x,y,z],Fh[t,x,y,z],Fb[t,x,y,z],Gh[t,x,y,z],Gb[t,x,y,z],Ψh[kk,cc][t,x,y,z]};

FieldListDynamicalBosons = Join[{A[t,x,y,z],B[t,x,y,z],At[t,x,y,z],Bt[t,x,y,z],φ[t,x,y,z]},Table[Am[mu][t,x,y,z],{mu,0,3}],Table[Bmn[mu,nu][t,x,y,z],{mu,0,2},{nu,mu+1,3}]//Flatten,{Ah[t,x,y,z],Bh[t,x,y,z],Ab[t,x,y,z],Bb[t,x,y,z]}]/.Bmn[1,3]-> Bmn[3,1]

FieldListAuxiliaryBosons = Join[{F[t,x,y,z],G[t,x,y,z],Ft[t,x,y,z],Gt[t,x,y,z],d[t,x,y,z]},{Fh[t,x,y,z],Gh[t,x,y,z],Fb[t,x,y,z],Gb[t,x,y,z]}]

FieldListFermions = Join[(Table[Ψ[ii][aa][t,x,y,z],{ii,1,2},{aa,1,4}]//Flatten),(Table[Ψt[ii][aa][t,x,y,z],{ii,1,2},{aa,1,4}]//Flatten),(Table[Ψh[ii][aa][t,x,y,z],{ii,1,2},{aa,1,4}]//Flatten)]

FieldList = Join[FieldListDynamicalBosons,FieldListAuxiliaryBosons,FieldListFermions]

FieldListWithDerivatives:= list of all fields and all derivatives of fields, up to 2nd order

FieldListWithDerivatives = Join[FieldList,Table[D[FieldList,Ϛ[mu]],{mu,0,3}]//Flatten,Table[D[FieldList,Ϛ[mu],Ϛ[nu]],{mu,0,3},{nu,0,3}]//Flatten];

FieldProductList := List of all combinations of boson*fermion w/ derivatives on the bosons.

FieldProductsListDynamicalBosons = Table[D[FieldListDynamicalBosons[[ii]],Ϛ[mu],Ϛ[nu]]FieldListFermions[[jhat]],{mu,0,3},{nu,mu,3},{ii,1,Length[FieldListDynamicalBosons]},{jhat,1,Length[FieldListFermions]}]//Flatten;

FieldProductsListAuxiliaryBosons = Table[D[FieldListAuxiliaryBosons[[ii]],Ϛ[mu]]FieldListFermions[[jhat]],{mu,0,3},{ii,1,Length[FieldListAuxiliaryBosons]},{jhat,1,Length[FieldListFermions]}]//Flatten;

FieldProductsList = Join[FieldProductsListDynamicalBosons,FieldProductsListAuxiliaryBosons];

Symmetries, Constraints, and raised indices such as Amup, etc.

Ψup[i][a] := (Ψ^ia)

Do[Ψup[ii][aa][t,x,y,z] = RaiseFermionIndex[Ψ[ii][aa][t,x,y,z]],{ii,1,2},{aa,1,4}]

InverseCmetric//MatrixForm

Table[Ψup[ii][aa][t,x,y,z] ,{ii,1,2},{aa,1,4}]

Ψtup[i][a] := ((OverTilde[Ψ])^ia)

Do[Ψtup[ii][aa][t,x,y,z] = RaiseFermionIndex[Ψt[ii][aa][t,x,y,z]],{ii,1,2},{aa,1,4}]

InverseCmetric//MatrixForm

Table[Ψtup[ii][aa][t,x,y,z] ,{ii,1,2},{aa,1,4}]

Ψhup[i][a] := ((OverHat[Ψ])^ia)

Do[Ψhup[ii][aa][t,x,y,z] = RaiseFermionIndex[Ψh[ii][aa][t,x,y,z]],{ii,1,2},{aa,1,4}]

InverseCmetric//MatrixForm

Table[Ψhup[ii][aa][t,x,y,z] ,{ii,1,2},{aa,1,4}]

The Field strength, covariant and contravariant:  Fmn[μ,ν]:=(Subscript[F, μν]) = (∂_μ)(Subscript[A, ν]) - (∂_ν)(Subscript[A, μ])  ,  Fmnup[μ,ν][t,x,y,z] (F^(μν)) = (∂^μ)(A^ν)-(∂^ν)(A^μ)

Do[Fmn[μ,ν][t,x,y,z] = D[Am[ν][t,x,y,z],Ϛ[μ]] - D[Am[μ][t,x,y,z],Ϛ[ν]], {μ,0,3},{ν,0,3}]

Do[Fmnup[μ,ν][t,x,y,z] =  RaiseSTIndex[FmnHolder[μ,ν][t,x,y,z],UP,UP]/.FmnHolder->Fmn,{μ,0,3},{ν,0,3}]

Table[Fmn[μ,ν][t,x,y,z],{μ,0,3},{ν,0,3}]//MatrixForm

Table[Fmnup[μ,ν][t,x,y,z],{μ,0,3},{ν,0,3}]//MatrixForm

Hard code the indices of (Subscript[B, μν]) so the independent fields are: (Subscript[B, 0i]), (Subscript[B, 12]) , (Subscript[B, 23]) , (Subscript[B, 31])

Bmn[1,0][t,x,y,z]=-Bmn[0,1][t,x,y,z];
Bmn[2,0][t,x,y,z]=-Bmn[0,2][t,x,y,z];
Bmn[3,0][t,x,y,z]=-Bmn[0,3][t,x,y,z];
Bmn[2,1][t,x,y,z]=-Bmn[1,2][t,x,y,z];
Bmn[1,3][t,x,y,z]=-Bmn[3,1][t,x,y,z];
Bmn[3,2][t,x,y,z] =-Bmn[2,3][t,x,y,z];
Do[Bmn[mi,mi][t,x,y,z] = 0,{mi,0,3}];

Bmn[0,1][t]

Bmn[3,3][t,x,y,z]

D[Bmn[3,3][t,x,y,z],z]

Bmn[1,0][t,x,y,z]

Bmnup[μ_,ν_][t,x,y,z]  = (B^(μν))

Do[Bmnup[μ,ν][t,x,y,z] = RaiseSTIndex[BmnHolder[μ,ν][t,x,y,z],UP,UP]/.BmnHolder->Bmn,{μ,0,3},{ν,0,3}]

Bmn[1,0][t,x,y,z]

Bmnup[0,1][t,x,y,z]

Bmnup[1,0][t,x,y,z]

Bmnup[3,0][t,x,y,z]

Bmnup[0,3][t,x,y,z]

D[Bmnup[2,1][t,x,y,z] ,t]

Table[Bmnup[μ,ν][t,x,y,z],{μ,0,3},{ν,0,3}]//MatrixForm

Hamn[ap_,mu_,nu_][t,x,y,z] = (Subscript[H, αμν]) = (1/2)(∂_([α))(Subscript[B, μν]])=(∂_α)(Subscript[B, μν])+(∂_ν)(Subscript[B, αμ])+(∂_μ)(Subscript[B, να])

Do[Hamn[ap,mu,nu][t,x,y,z] = (D[Bmn[mu,nu][t,x,y,z],Ϛ[ap]] + D[Bmn[ap,mu][t,x,y,z],Ϛ[nu]] + D[Bmn[nu,ap][t,x,y,z],Ϛ[mu]] ),{ap,0,3},{mu,0,3},{nu,0,3}];

Hamn[0,1,3][t,x,y,z]

Hamn[0,1,2][t,x,y,z]

Hamn[1,2,3][t,x,y,z]

Hamnup[α_,μ_,ν_][t,x,y,z]  = (H^(αμν))

Do[Hamnup[α,μ,ν][t,x,y,z] = RaiseSTIndex[HamnHolder[α,μ,ν][t,x,y,z],UP,UP,UP]/.HamnHolder->Hamn,{α,0,3},{μ,0,3},{ν,0,3}]

Hamn[1,0,2][t,x,y,z]

Hamnup[1,0,2][t,x,y,z]

Hamnup[1,0,2][t,x,y,z] ==-Hamn[1,0,2][t,x,y,z]

Hamnup[0,1,2][t,x,y,z]

Hamnup[0,1,2][t,x,y,z]==-Hamn[0,1,2][t,x,y,z]

Hamnup[0,1,3][t,x,y,z]==-Hamn[0,1,3][t,x,y,z]

Hamnup[3,1,2][t,x,y,z]==Hamn[3,1,2][t,x,y,z]

Table[Hamnup[0,μ,ν][t,x,y,z],{μ,0,3},{ν,0,3}]//MatrixForm

Check UD[Field_,Ϛ[μ]] := (∂^μ)Field   ,   Lap[Field_] := (∂^μ)(∂_μ)Field defined in SpaceTime2.nb. Doesn’t matter in which order these are taken when strung together.

D[UD[Am[0][t,x,y,z],Ϛ[0]],Ϛ[0]]

Lap[D[Am[1][t,x,y,z],Ϛ[0]]]

UD[D[Lap[Am[2][t,x,y,z]],Ϛ[0]],Ϛ[0]]

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

Lap[Ψ[1][1][t,x,y,z]]

Lap[Ψh[2][1][t,x,y,z]]

Lap[Ψhup[2][1][t,x,y,z]]

Transformation Laws

4D, N=2 Vector transformation laws
Untilded, unhatted transformation laws Dia[1][ii][aa]

Do[Dia[1][ii][aa][A[t,x,y,z]]=  Ψ[ii][aa][t,x,y,z] ,{ii,1,2},{aa,1,4}]

Dia[1][1][1][A[t,x,y,z]]

Dia[1][2][3][A[t,x,y,z]]

Do[Dia[1][ii][aa][B[t,x,y,z]]= I Sum[γ5[[aa,bb]]Ψ[ii][bb][t,x,y,z],{bb,1,4}] ,{ii,1,2},{aa,1,4}]

Dia[1][2][1][B[t,x,y,z]]

Do[Dia[1][ii][aa][F[t,x,y,z]]= Sum[γ[mu][[aa,bb]]D[Ψ[ii][bb][t,x,y,z],Ϛ[mu]],{mu,0,3},{bb,1,4}] ,{ii,1,2},{aa,1,4}]

Dia[1][1][2][F[t,x,y,z]]

Do[Dia[1][ii][aa][G[t,x,y,z]]= I Sum[sigma[3][[ii,jj]]γ5γ[mu][[aa,bb]]D[Ψ[jj][bb][t,x,y,z],Ϛ[mu]],{mu,0,3},{bb,1,4},{jj,1,2}]//Simplify ,{ii,1,2},{aa,1,4}]

Dia[1][1][2][G[t,x,y,z]]

Do[Dia[1][ii][aa][Am[mu][t,x,y,z]] = I Sum[sigma[2][[ii,jj]]γstdown[mu][[aa,bb]]Ψ[jj][bb][t,x,y,z],{jj,1,2},{bb,1,4}],{ii,1,2},{aa,1,4},{mu,0,3}]

Dia[1][2][3][Am[1][t,x,y,z]]

Do[Dia[1][ii][aa][d[t,x,y,z]]= I Sum[sigma[1][[ii,jj]]γ5γ[mu][[aa,bb]]D[Ψ[jj][bb][t,x,y,z],Ϛ[mu]],{mu,0,3},{bb,1,4},{jj,1,2}]//Simplify ,{ii,1,2},{aa,1,4}]

Dia[1][2][2][d[t,x,y,z]]

Do[Dia[1][ii][aa][Ψ[jj][bb][t,x,y,z]] =KroneckerDelta[ii,jj]( Sum[I γdown[mu][[aa,bb]]D[A[t,x,y,z],Ϛ[mu]] -γ5γdown[mu][[aa,bb]]D[B[t,x,y,z],Ϛ[mu]],{mu,0,3}] - I Cmetric[[aa,bb]]F[t,x,y,z]) + sigma[3][[ii,jj]]γ5down[[aa,bb]]G[t,x,y,z] + sigma[1][[ii,jj]]γ5down[[aa,bb]]d[t,x,y,z] +Sum[ (1/4)sigma[2][[ii,jj]]CommuteGammadown[mu,nu][[aa,bb]]Fmn[mu, nu][t,x,y,z],{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]

4D, N=2 Tensor transformation laws
Untilded transformation laws Dia[1][ii][aa]

Do[Dia[1][ii][aa][At[t,x,y,z]]=  Sum[sigma[3][[ii,jj]]Ψt[jj][aa][t,x,y,z],{jj,1,2}] ,{ii,1,2},{aa,1,4}]

Dia[1][1][1][At[t,x,y,z]]

Dia[1][1][3][At[t,x,y,z]]

Do[Dia[1][ii][aa][Bt[t,x,y,z]]= I Sum[γ5[[aa,bb]]Ψt[ii][bb][t,x,y,z],{bb,1,4}] ,{ii,1,2},{aa,1,4}]

Dia[1][2][1][Bt[t,x,y,z]]

Do[Dia[1][ii][aa][Ft[t,x,y,z]]= Sum[γ[mu][[aa,bb]]D[Ψt[ii][bb][t,x,y,z],Ϛ[mu]],{mu,0,3},{bb,1,4}] ,{ii,1,2},{aa,1,4}]

Dia[1][1][2][Ft[t,x,y,z]]

Do[Dia[1][ii][aa][Gt[t,x,y,z]]= I Sum[γ5γ[mu][[aa,bb]]D[Ψt[ii][bb][t,x,y,z],Ϛ[mu]],{mu,0,3},{bb,1,4}]//Simplify ,{ii,1,2},{aa,1,4}]

Dia[1][1][2][G[t,x,y,z]]

Do[Dia[1][ii][aa][φ[t,x,y,z]]= Sum[sigma[1][[ii,jj]]Ψt[jj][aa][t,x,y,z],{jj,1,2}]//Simplify ,{ii,1,2},{aa,1,4}]

Dia[1][1][2][φ[t,x,y,z]]

Do[Dia[1][ii][aa][Bmn[mu,nu][t,x,y,z]] = -I(1/4) Sum[sigma[2][[ii,jj]]CommuteGammastdown[mu,nu][[aa,bb]]Ψt[jj][bb][t,x,y,z],{jj,1,2},{bb,1,4}],{ii,1,2},{aa,1,4},{mu,0,3},{nu,0,3}]

Dia[1][2][3][Bmn[1,2][t,x,y,z]]

Do[Dia[1][ii][aa][Ψt[jj][bb][t,x,y,z]] =KroneckerDelta[ii,jj]( -Sum[γ5γdown[mu][[aa,bb]]D[Bt[t,x,y,z],Ϛ[mu]],{mu,0,3}] - I Cmetric[[aa,bb]]Ft[t,x,y,z] + γ5down[[aa,bb]]Gt[t,x,y,z] ) + I sigma[3][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[At[t,x,y,z],Ϛ[mu]],{mu,0,3}] + I sigma[1][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[φ[t,x,y,z],Ϛ[mu]],{mu,0,3}] -I sigma[2][[ii,jj]] Sum[ϵdownupupup[mu,nu,ap,bt]γ5γdown[mu][[aa,bb]]D[Bmn[ap,bt][t,x,y,z],Ϛ[nu]],{mu,0,3},{nu,0,3},{ap,0,3},{bt,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]

Dia[1][1][3][Ψt[1][1][t,x,y,z]]

Vector-Tensor tilded transformation laws Dia[2][ii][aa]

Do[Dia[2][ii][aa][A[t,x,y,z]]=  Sum[I sigma[2][[ii,jj]]Ψt[jj][aa][t,x,y,z] ,{jj,1,2}],{ii,1,2},{aa,1,4}]

Dia[2][1][1][A[t,x,y,z]]

Dia[2][2][3][A[t,x,y,z]]

Do[Dia[2][ii][aa][B[t,x,y,z]]=  Sum[-sigma[2][[ii,jj]]γ5[[aa,bb]]Ψt[jj][bb][t,x,y,z],{jj,1,2},{bb,1,4}] ,{ii,1,2},{aa,1,4}]

Dia[2][2][1][B[t,x,y,z]]

Do[Dia[2][ii][aa][F[t,x,y,z]]= Sum[I sigma[2][[ii,jj]]γ[mu][[aa,bb]]D[Ψt[jj][bb][t,x,y,z],Ϛ[mu]],{jj,1,2},{mu,0,3},{bb,1,4}] ,{ii,1,2},{aa,1,4}]

Dia[2][1][2][F[t,x,y,z]]

Do[Dia[2][ii][aa][G[t,x,y,z]]= I Sum[sigma[1][[ii,jj]]γ5γ[mu][[aa,bb]]D[Ψt[jj][bb][t,x,y,z],Ϛ[mu]],{mu,0,3},{bb,1,4},{jj,1,2}]//Simplify ,{ii,1,2},{aa,1,4}]

Dia[2][1][2][G[t,x,y,z]]

Do[Dia[2][ii][aa][Am[mu][t,x,y,z]] = Sum[KroneckerDelta[ii,jj]γstdown[mu][[aa,bb]]Ψt[jj][bb][t,x,y,z],{jj,1,2},{bb,1,4}],{ii,1,2},{aa,1,4},{mu,0,3}]

Dia[2][2][3][Am[1][t,x,y,z]]

Do[Dia[2][ii][aa][d[t,x,y,z]]= -I Sum[sigma[3][[ii,jj]]γ5γ[mu][[aa,bb]]D[Ψt[jj][bb][t,x,y,z],Ϛ[mu]],{mu,0,3},{bb,1,4},{jj,1,2}]//Simplify ,{ii,1,2},{aa,1,4}]

Dia[2][2][2][d[t,x,y,z]]

Do[Dia[2][ii][aa][Ψ[jj][bb][t,x,y,z]] = I sigma[2][[ii,jj]](-Sum[γ5γdown[mu][[aa,bb]]D[Bt[t,x,y,z],Ϛ[mu]],{mu,0,3}] -I Cmetric[[aa,bb]]Ft[t,x,y,z] + γ5down[[aa,bb]]Gt[t,x,y,z])  + Sum[I γdown[mu][[aa,bb]]D[(-sigma[1][[ii,jj]]At[t,x,y,z]+ sigma[3][[ii,jj]]φ[t,x,y,z]),Ϛ[mu]],{mu,0,3}] -KroneckerDelta[ii,jj] Sum[ϵdownupupup[mu,nu,ap,bt]γ5γdown[mu][[aa,bb]]D[Bmn[ap,bt][t,x,y,z],Ϛ[nu]],{mu,0,3},{nu,0,3},{ap,0,3},{bt,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]

Dia[2][1][3][Ψ[1][1][t,x,y,z]]

Do[Dia[2][ii][aa][At[t,x,y,z]]=  Sum[-sigma[1][[ii,jj]]Ψ[jj][aa][t,x,y,z],{jj,1,2}] ,{ii,1,2},{aa,1,4}]

Dia[2][1][1][At[t,x,y,z]]

Dia[2][1][3][At[t,x,y,z]]

Do[Dia[2][ii][aa][Bt[t,x,y,z]]= Sum[-sigma[2][[ii,jj]]γ5[[aa,bb]]Ψ[jj][bb][t,x,y,z],{jj,1,2},{bb,1,4}] ,{ii,1,2},{aa,1,4}]

Dia[2][2][1][Bt[t,x,y,z]]

Do[Dia[2][ii][aa][Ft[t,x,y,z]]= Sum[I sigma[2][[ii,jj]]γ[mu][[aa,bb]]D[Ψ[jj][bb][t,x,y,z],Ϛ[mu]],{jj,1,2},{mu,0,3},{bb,1,4}] ,{ii,1,2},{aa,1,4}]

Dia[2][1][2][Ft[t,x,y,z]]

Do[Dia[2][ii][aa][Gt[t,x,y,z]]= Sum[-sigma[2][[ii,jj]]γ5γ[mu][[aa,bb]]D[Ψ[jj][bb][t,x,y,z],Ϛ[mu]],{jj,1,2},{mu,0,3},{bb,1,4}]//Simplify ,{ii,1,2},{aa,1,4}]

Dia[2][1][2][Gt[t,x,y,z]]

Do[Dia[2][ii][aa][φ[t,x,y,z]]= Sum[sigma[3][[ii,jj]]Ψ[jj][aa][t,x,y,z],{jj,1,2}]//Simplify ,{ii,1,2},{aa,1,4}]

Dia[2][1][2][φ[t,x,y,z]]

Do[Dia[2][ii][aa][Bmn[mu,nu][t,x,y,z]] = -(1/4) Sum[KroneckerDelta[ii,jj]CommuteGammastdown[mu,nu][[aa,bb]]Ψ[jj][bb][t,x,y,z],{jj,1,2},{bb,1,4}],{ii,1,2},{aa,1,4},{mu,0,3},{nu,0,3}]

Dia[2][2][3][Bmn[1,2][t,x,y,z]]

Do[Dia[2][ii][aa][Ψt[jj][bb][t,x,y,z]] =I sigma[2][[ii,jj]](Sum[ I γdown[mu][[aa,bb]]D[A[t,x,y,z],Ϛ[mu]] -γ5γdown[mu][[aa,bb]]D[B[t,x,y,z],Ϛ[mu]],{mu,0,3}] - I Cmetric[[aa,bb]]F[t,x,y,z] )+ γ5down[[aa,bb]](sigma[1][[ii,jj]]G[t,x,y,z] -sigma[3][[ii,jj]] d[t,x,y,z]) -Sum[ (I/4)KroneckerDelta[ii,jj]CommuteGammadown[mu,nu][[aa,bb]]Fmn[mu, nu][t,x,y,z],{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]

Dia[2][1][3][Ψt[1][1][t,x,y,z]]

Lagrangian Invariance

Integration by parts commands: Put all derivatives on the bosons

ByParts={ Derivative[0,0,0,1][Ψ[ii_][xp__]][t,x,y,z]efield__[t,x,y,z]-> -  Ψ[ii][xp][t,x,y,z]D[efield[t,x,y,z],z],
Derivative[0,0,1,0][Ψ[ii_][xp__]][t,x,y,z]efield__[t,x,y,z]-> - Ψ[ii][xp][t,x,y,z]D[ efield[t,x,y,z],y],Derivative[0,1,0,0][Ψ[ii_][xp__]][t,x,y,z]efield__[t,x,y,z]-> -  Ψ[ii][xp][t,x,y,z]D[ efield[t,x,y,z],x],Derivative[1,0,0,0][Ψ[ii_][xp__]][t,x,y,z]efield__[t,x,y,z]-> -  Ψ[ii][xp][t,x,y,z]D[ efield[t,x,y,z],t],Derivative[0,0,0,1][Ψt[ii_][xp__]][t,x,y,z]efield__[t,x,y,z]-> -  Ψt[ii][xp][t,x,y,z]D[efield[t,x,y,z],z],
Derivative[0,0,1,0][Ψt[ii_][xp__]][t,x,y,z]efield__[t,x,y,z]-> - Ψt[ii][xp][t,x,y,z]D[ efield[t,x,y,z],y],Derivative[0,1,0,0][Ψt[ii_][xp__]][t,x,y,z]efield__[t,x,y,z]-> -  Ψt[ii][xp][t,x,y,z]D[ efield[t,x,y,z],x],Derivative[1,0,0,0][Ψt[ii_][xp__]][t,x,y,z]efield__[t,x,y,z]-> -  Ψt[ii][xp][t,x,y,z]D[ efield[t,x,y,z],t],Derivative[0,0,0,1][Ψh[ii_][xp__]][t,x,y,z]efield__[t,x,y,z]-> -  Ψh[ii][xp][t,x,y,z]D[efield[t,x,y,z],z],
Derivative[0,0,1,0][Ψh[ii_][xp__]][t,x,y,z]efield__[t,x,y,z]-> - Ψh[ii][xp][t,x,y,z]D[ efield[t,x,y,z],y],Derivative[0,1,0,0][Ψh[ii_][xp__]][t,x,y,z]efield__[t,x,y,z]-> -  Ψh[ii][xp][t,x,y,z]D[ efield[t,x,y,z],x],Derivative[1,0,0,0][Ψh[ii_][xp__]][t,x,y,z]efield__[t,x,y,z]-> -  Ψh[ii][xp][t,x,y,z]D[ efield[t,x,y,z],t]}

ℒ = (Subscript[ℒ, (2VS) ])+(Subscript[ℒ, (2TS)])

Lagrangian Invariance: DL[tt] w/ tt= tilde index (1 = no tilde, 2 = tilde)
(Subscript[D, a]^i)ℒ = 0  = ((Subscript[D, a]^i)A) □A + ((Subscript[D, a]^i)B) □B +F (Subscript[D, a]^i)F +G (Subscript[D, a]^i)G+d (Subscript[D, a]^i)d+ ((Subscript[D, a]^i)(Subscript[A, ν]))(∂_μ)(F^(μν)) - i (Subscript[Ψ, b]^j)((((γ^μ)))^bc)(∂_μ)((Subscript[D, a]^i)(Subscript[Ψ, c]^j)) + ((Subscript[D, a]^i)(OverTilde[A])) □(OverTilde[A]) + ((Subscript[D, a]^i)(OverTilde[B])) □(OverTilde[B]) +(OverTilde[F]) (Subscript[D, a]^i)(OverTilde[F])  +(OverTilde[G])  (Subscript[D, a]^i)(OverTilde[G]) +2((Subscript[D, a]^i)(Subscript[B, να]))(∂_μ)(H^(μνα))+ ((Subscript[D, a]^i)(OverTilde[φ])) □(OverTilde[φ]) - i (Subscript[(OverTilde[Ψ]), b]^j)((((γ^μ)))^bc)(∂_μ)((Subscript[D, a]^i)(Subscript[(OverTilde[Ψ]), c]^j)) 
(Subscript[(OverTilde[D]), a]^i)ℒ = 0  = ((Subscript[(OverTilde[D]), a]^i)A) □A + ((Subscript[(OverTilde[D]), a]^i)B) □B +F (Subscript[(OverTilde[D]), a]^i)F +G (Subscript[(OverTilde[D]), a]^i)G+d (Subscript[(OverTilde[D]), a]^i)d+ ((Subscript[(OverTilde[D]), a]^i)(Subscript[A, ν]))(∂_μ)(F^(μν)) - i (Subscript[Ψ, b]^j)((((γ^μ)))^bc)(∂_μ)((Subscript[(OverTilde[D]), a]^i)(Subscript[Ψ, c]^j)) + ((Subscript[(OverTilde[D]), a]^i)(OverTilde[A])) □(OverTilde[A]) + ((Subscript[(OverTilde[D]), a]^i)(OverTilde[B])) □(OverTilde[B]) +(OverTilde[F]) (Subscript[(OverTilde[D]), a]^i)(OverTilde[F])  +(OverTilde[G])  (Subscript[(OverTilde[D]), a]^i)(OverTilde[G]) +2((Subscript[(OverTilde[D]), a]^i)(Subscript[B, να]))(∂_μ)(H^(μνα))+ ((Subscript[(OverTilde[D]), a]^i)(OverTilde[φ])) □(OverTilde[φ]) - i (Subscript[(OverTilde[Ψ]), b]^j)((((γ^μ)))^bc)(∂_μ)((Subscript[(OverTilde[D]), a]^i)(Subscript[(OverTilde[Ψ]), c]^j))

DLA[tt] = {((Subscript[D, a]^i)A) □A,((Subscript[(OverTilde[D]), a]^i)A) □A}

Do[DLA[tt] = Table[Dia[tt][ii][aa][A[t,x,y,z]]Lap[A[t,x,y,z]],{ii,1,2},{aa,1,4}],{tt,0,2}]

DLB[tt] = {((Subscript[D, a]^i)B) □B,((Subscript[(OverTilde[D]), a]^i)B) □B}

Do[DLB[tt] = Table[Dia[tt][ii][aa][B[t,x,y,z]]Lap[B[t,x,y,z]],{ii,1,2},{aa,1,4}],{tt,0,2}]

DLF[tt] = {((Subscript[D, a]^i)F) F,((Subscript[(OverTilde[D]), a]^i)F) F}

Do[DLF[tt] = Expand[Table[Dia[tt][ii][aa][F[t,x,y,z]]F[t,x,y,z],{ii,1,2},{aa,1,4}]]/.ByParts,{tt,0,2}]

DLG[tt] = {((Subscript[D, a]^i)G) G,((Subscript[(OverTilde[D]), a]^i)G) G}

Do[DLG[tt] =( Table[Dia[tt][ii][aa][G[t,x,y,z]]G[t,x,y,z],{ii,1,2},{aa,1,4}]//Expand)/.ByParts,{tt,0,2}]

DLd[tt] = {((Subscript[D, a]^i)d) d,((Subscript[(OverTilde[D]), a]^i)d) d}

Do[DLd[tt] = Expand[Table[Dia[tt][ii][aa][d[t,x,y,z]]d[t,x,y,z],{ii,1,2},{aa,1,4}]]/.ByParts,{tt,0,2}]

DLAm[tt] = {((Subscript[D, a]^i)(Subscript[A, ν]))(∂_μ)(F^(μν)),((Subscript[(OverTilde[D]), a]^i)(Subscript[A, ν]))(∂_μ)(F^(μν))}

Do[DLAm[tt] = Table[Sum[Dia[tt][ii][aa][Am[nu][t,x,y,z]]D[Fmnup[mu,nu][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4}],{tt,0,2}]

DLPsi[tt] =  {- i (Subscript[(OverTilde[Ψ]), b]^j)((((γ^μ)))^bc)(∂_μ)((Subscript[D, a])(Subscript[(OverTilde[Ψ]), c]^j)),- i (Subscript[(OverTilde[Ψ]), b]^j)((((γ^μ)))^bc)(∂_μ)((Subscript[(OverTilde[D]), a]^i)(Subscript[(OverTilde[Ψ]), c]^j))}

Do[DLPsi[tt] = Table[ - I Sum[Ψ[jj][bb][t,x,y,z]γup[mu][[bb,cc]]D[Dia[tt][ii][aa][Ψ[jj][cc][t,x,y,z]],Ϛ[mu]],{jj,1,2},{bb,1,4},{cc,1,4},{mu,0,3}],{ii,1,2},{aa,1,4}]/.ByParts,{tt,0,2}]

DLPsi[1][[2,2]]

DLAt[tt] = {((Subscript[D, a]^i)At) □At,((Subscript[(OverTilde[D]), a]^i)At) □At}

Do[DLAt[tt] = Table[Dia[tt][ii][aa][At[t,x,y,z]]Lap[At[t,x,y,z]],{ii,1,2},{aa,1,4}],{tt,1,2}]

DLBt[tt] = {((Subscript[D, a]^i)Bt) □Bt,((Subscript[(OverTilde[D]), a]^i)Bt) □Bt}

Do[DLBt[tt] = Table[Dia[tt][ii][aa][Bt[t,x,y,z]]Lap[Bt[t,x,y,z]],{ii,1,2},{aa,1,4}],{tt,1,2}]

DLFt[tt] = {((Subscript[D, a]^i)Ft) Ft,((Subscript[(OverTilde[D]), a]^i)Ft) Ft}

Do[DLFt[tt] = Expand[Table[Dia[tt][ii][aa][Ft[t,x,y,z]]Ft[t,x,y,z],{ii,1,2},{aa,1,4}]]/.ByParts,{tt,1,2}]

DLGt[tt] = {((Subscript[D, a]^i)Gt) Gt,((Subscript[(OverTilde[D]), a]^i)Gt) Gt}

Do[DLGt[tt] = (Table[Dia[tt][ii][aa][Gt[t,x,y,z]]Gt[t,x,y,z],{ii,1,2},{aa,1,4}]//Expand)/.ByParts,{tt,1,2}]

DLφ[tt] = {((Subscript[D, a]^i)φ) □φ,((Subscript[(OverTilde[D]), a]^i)φ) □φ}

Do[DLφ[tt] = Table[Dia[tt][ii][aa][φ[t,x,y,z]]Lap[φ[t,x,y,z]],{ii,1,2},{aa,1,4}],{tt,1,2}]

DLBmn[tt] = {2((Subscript[D, a]^i)(Subscript[B, να]))(∂_μ)(H^(μνα)), 2((Subscript[(OverTilde[D]), a]^i)(Subscript[B, να]))(∂_μ)(H^(μνα))}

Do[DLBmn[tt] = Table[Sum[2 Dia[tt][ii][aa][Bmn[nu,ap][t,x,y,z]]D[Hamnup[mu,nu,ap][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ap,0,3}],{ii,1,2},{aa,1,4}],{tt,1,2}]

DLPsit[tt] =  {- i (Subscript[(OverTilde[Ψ]), b]^j)((((γ^μ)))^bc)(∂_μ)((Subscript[D, a])(Subscript[(OverTilde[Ψ]), c]^j)),- i (Subscript[(OverTilde[Ψ]), b]^j)((((γ^μ)))^bc)(∂_μ)((Subscript[(OverTilde[D]), a]^i)(Subscript[(OverTilde[Ψ]), c]^j))}

Do[DLPsit[tt] = Table[ - I Sum[Ψt[jj][bb][t,x,y,z]γup[mu][[bb,cc]]D[Dia[tt][ii][aa][Ψt[jj][cc][t,x,y,z]],Ϛ[mu]],{jj,1,2},{bb,1,4},{cc,1,4},{mu,0,3}],{ii,1,2},{aa,1,4}]/.ByParts,{tt,1,2}]

DLPsit[1]

Lagrangian invariance

Do[DLTotal[tt]=(DLA[tt] +DLB[tt] +DLF[tt] +DLG[tt] +DLd[tt] +DLAm[tt] + DLPsi[tt] +DLAt[tt] +DLBt[tt] +DLFt[tt] +DLGt[tt] +DLφ[tt] +DLBmn[tt] + DLPsit[tt] ),{tt,1,2}];

DLTotal[1]

Length[FieldListWithDerivatives]

Do[DLFinal[tt] = Collect[DLTotal[tt],{Ψ[_][_][t,x,y,z],Ψt[_][_][t,x,y,z],Derivative[_,_,_,_][A][t,x,y,z],A[t,x,y,z],Derivative[_,_,_,_][B][t,x,y,z],B[t,x,y,z],Derivative[_,_,_,_][F][t,x,y,z],F[t,x,y,z],Derivative[_,_,_,_][G][t,x,y,z],G[t,x,y,z],Derivative[_,_,_,_][Am[_]][t,x,y,z],Am[_][t,x,y,z],Derivative[_,_,_,_][d][t,x,y,z],d[t,x,y,z],Derivative[_,_,_,_][At][t,x,y,z],At[t,x,y,z],Derivative[_,_,_,_][Bt][t,x,y,z],Bt[t,x,y,z],Derivative[_,_,_,_][Ft][t,x,y,z],Ft[t,x,y,z],Derivative[_,_,_,_][Gt][t,x,y,z],Gt[t,x,y,z],Derivative[_,_,_,_][φ][t,x,y,z],φ[t,x,y,z],Derivative[_,_,_,_][Bmn[_,_]][t,x,y,z],Bmn[_,_][t,x,y,z]},Simplify],{tt,1,2}]

Lagrangian invariance: untilded DLFinal[1]

DLFinal[1]

Lagrangian Invariance: tilded

DLFinal[2]

Lagrangian Summary

DLFinal[1]==(DLFinal[2])== {{0,0,0,0},{0,0,0,0}}

Holoraumy

Holoraumy [(Subscript[D, a]^i),(Subscript[D, b]^j)]

DD

SUSYTransformFields[Operator_]:=Table[FieldList[[FieldIndex]]->Operator[FieldList[[FieldIndex]]],{FieldIndex,1,Length[FieldList]}]

SUSYTransformJustB[Operator_]:=Table[FieldList[[FieldIndex]]->Operator[FieldList[[FieldIndex]]],{FieldIndex,1,4}]

Do[SUSYTransform[Dia[tt][ii][aa]] = Join[SUSYTransformFields[Dia[tt][ii][aa]],D[SUSYTransformFields[Dia[tt][ii][aa]],t],D[SUSYTransformFields[Dia[tt][ii][aa]],x],D[SUSYTransformFields[Dia[tt][ii][aa]],y],D[SUSYTransformFields[Dia[tt][ii][aa]],z]],{tt,1,2},{ii,1,2},{aa,1,4}]

DD[LeftOperator_,RightOperator_][Field_]:=RightOperator[Field]/.SUSYTransform[LeftOperator]

AlgebraDDTerms[LeftOperator_,RightOperator_][Field_]:= DD[LeftOperator,RightOperator][Field] + DD[RightOperator,LeftOperator][Field]

AlgebraDD[ttl_,ttr_][Field_] := Table[AlgebraDDTerms[Dia[ttl][ii][aa],Dia[ttr][jj][bb]][Field],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]  (* Convention is left D upper index, left D lower index, right D upper index, right D lower index *)

HoloraumyDDTerms[LeftOperator_,RightOperator_][Field_]:= DD[LeftOperator,RightOperator][Field] - DD[RightOperator,LeftOperator][Field]

HoloraumyDD[ttl_,ttr_][Field_]:=Table[HoloraumyDDTerms[Dia[ttl][ii][aa],Dia[ttr][jj][bb]][Field],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]

ClosureTerms[ii_,jj_][aa_,bb_][Field_]:=Sum[2 I KroneckerDelta[ii,jj]γdown[mu][[aa,bb]]D[Field,Ϛ[mu]],{mu,0,3}]

Closure[Field_]:=Table[ClosureTerms[ii,jj][aa,bb][Field],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]

Zeros = Table[0,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

Test it out

HoloraumyDD[1,1][Ψh[1][1][t,x,y,z]]

Closure[A[t,x,y,z]]

HoloraumyDD[1,1][A[t,x,y,z]]

Table[ClosureTerms[ii,jj][aa,bb][Ψ[ll][kk][t,x,y,z]],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{ll,1,2},{kk,1,4}]

To be used in writing holoraumy, 
cV[x][i,j,k,l] :=
V[x][i,j,k,l] := and also
V[x][i,j,k,l]:=
X[x][i,j,k,l]:=and also X-tilde, W, and W-tilde
Y[x][i,j,k,l]:=and also Y-tilde

where alphahat, etc. are different for the different fermions and holoraumies as described in the paper

cV[xx_][ii_,jj_,kk_,ll_]:=αhat[[xx]]sigma[2][[ii,jj]] sigma[2][[kk,ll]] + βhat[[xx]](sigma[1][[ii,jj]]sigma[1][[kk,ll]] + sigma[3][[ii,jj]]sigma[3][[kk,ll]]) + κhat[[xx]]KroneckerDelta[ii,jj]KroneckerDelta[kk,ll]

V[xx_][ii_,jj_,kk_,ll_]:= I Sum[cV[xx][ii,rr,kk,ll]sigma[2][[rr,jj]],{rr,1,2}]

X[xx_][ii_,jj_,kk_,ll_]:=α[[xx]]KroneckerDelta[ii,jj]sigma[2][[kk,ll]] + β[[xx]]sigma[1][[ii,jj]]sigma[3][[kk,ll]] + δ[[xx]]sigma[3][[ii,jj]]sigma[1][[kk,ll]] + κ[[xx]]sigma[2][[ii,jj]]KroneckerDelta[kk,ll]
Y[xx_][ii_,jj_,kk_,ll_]:=α[[xx]]sigma[1][[ii,jj]]sigma[3][[kk,ll]] + β[[xx]]sigma[1][[ii,kk]]sigma[3][[jj,ll]] + δ[[xx]]sigma[1][[jj,kk]]sigma[3][[ii,ll]] + κ[[xx]]sigma[1][[ii,ll]]sigma[3][[jj,kk]]

Set tilde indices: tt1,tt2=1 for untilded, 2 for tilded

tt1 = 1;
tt2 = 1;

4D, N=2 Vector Holoraumy, both untilded [(Subscript[D, a]^i),(Subscript[D, b]^j)]

, FieldNumber = 1

LocalFieldNumber =1 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 KroneckerDelta[ii,jj](-Sum[γ5γdown[mu][[aa,bb]]D[B[t,x,y,z],Ϛ[mu]],{mu,0,3}] - I Cmetric[[aa,bb]]F[t,x,y,z]) +2 sigma[3][[ii,jj]]γ5down[[aa,bb]]G[t,x,y,z] + 2 sigma[1][[ii,jj]]γ5down[[aa,bb]]d[t,x,y,z] + (1/2)sigma[2][[ii,jj]]Sum[CommuteGammadown[mu,nu][[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 2

LocalFieldNumber =2 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 KroneckerDelta[ii,jj](Sum[γ5γdown[mu][[aa,bb]]D[A[t,x,y,z],Ϛ[mu]],{mu,0,3}] +γ5down[[aa,bb]]F[t,x,y,z]) +2 I sigma[3][[ii,jj]]Cmetric[[aa,bb]]G[t,x,y,z] + 2 I sigma[1][[ii,jj]]Cmetric[[aa,bb]]d[t,x,y,z] + (1/2)I sigma[2][[ii,jj]]Sum[(γ5.CommuteGammadown[mu,nu])[[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 3

LocalFieldNumber =3 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 KroneckerDelta[ii,jj](-I Cmetric[[aa,bb]]Lap[A[t,x,y,z]] +γ5down[[aa,bb]]Lap[B[t,x,y,z]]) -2 sigma[3][[ii,jj]]Sum[γ5γdown[mu][[aa,bb]]D[G[t,x,y,z],Ϛ[mu]],{mu,0,3}] -2 sigma[1][[ii,jj]]Sum[γ5γdown[mu][[aa,bb]]D[d[t,x,y,z],Ϛ[mu]],{mu,0,3}] -2 sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]UD[Fmn[mu,nu][t,x,y,z],Ϛ[nu]],{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 4

LocalFieldNumber =4 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 sigma[3][[ii,jj]](γ5down[[aa,bb]]Lap[A[t,x,y,z]] +I Cmetric[[aa,bb]]Lap[B[t,x,y,z]] + Sum[γ5γdown[mu][[aa,bb]]D[F[t,x,y,z],Ϛ[mu]],{mu,0,3}])  +2 sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[d[t,x,y,z],Ϛ[mu]],{mu,0,3}] -2 sigma[1][[ii,jj]]Sum[γ5γdown[mu][[aa,bb]]UD[Fmn[mu,nu][t,x,y,z],Ϛ[nu]],{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 5

LocalFieldNumber =5;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 sigma[1][[ii,jj]](γ5down[[aa,bb]]Lap[A[t,x,y,z]] +I Cmetric[[aa,bb]]Lap[B[t,x,y,z]] + Sum[γ5γdown[mu][[aa,bb]]D[F[t,x,y,z],Ϛ[mu]],{mu,0,3}]) -2 sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[G[t,x,y,z],Ϛ[mu]],{mu,0,3}]+2sigma[3][[ii,jj]]Sum[γ5γdown[mu][[aa,bb]]UD[Fmn[mu,nu][t,x,y,z],Ϛ[nu]],{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 6

LocalFieldNumber =6;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[sigma[2][[ii,jj]](Sum[CommuteGammastdownstupdown[mu,nu][[aa,bb]] D[A[t,x,y,z],Ϛ[nu]] + I (γ5.CommuteGammastdownstupdown[mu,nu])[[aa,bb]] D[B[t,x,y,z],Ϛ[nu]] ,{nu,0,3}] + 2 γstdowndown[mu][[aa,bb]]F[t,x,y,z]) + 2 γ5γstdowndown[mu][[aa,bb]](sigma[1][[ii,jj]]G[t,x,y,z] - sigma[3][[ii,jj]]d[t,x,y,z]) - KroneckerDelta[ii,jj]Sum[ϵdownupupup[mu,nu,ap,bt]γ5γstdowndown[nu][[aa,bb]]Fmn[ap,bt][t,x,y,z],{nu,0,3},{ap,0,3},{bt,0,3}],{mu,0,3},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{mu,0,3}] ==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] //Simplify

FieldNumber = 7

αhat = {1,0,0,0,-1};
βhat = {0,1,0,1,0};
κhat = {0,0,-1,0,0};

LocalFieldNumber =7;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2I Sum[(cV[1][ii,jj,kk,ll] γdown[mu][[aa,bb]]KroneckerDelta[cc,dd] + cV[2][ii,jj,kk,ll] γ5γdown[mu][[aa,bb]]γ5[[cc,dd]])D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}] + I Sum[cV[3][ii,jj,kk,ll] (Cmetric[[aa,bb]]γ[ld][[cc,dd]] + γ5down[[aa,bb]]γ5γ[ld][[cc,dd]]+ Sum[γ5γdown[nu][[aa,bb]]γ5γγstdownstup[nu,ld][[cc,dd]],{nu,0,3}])D[Ψ[ll][dd][t,x,y,z],Ϛ[ld]],{ld,0,3},{ll,1,2},{dd,1,4}]+I Sum[cV[4][ii,jj,kk,ll] (Cmetric[[aa,bb]]γ[ld][[cc,dd]] + γ5down[[aa,bb]]γ5γ[ld][[cc,dd]]-Sum[γ5γdown[nu][[aa,bb]]γ5γγstdownstup[nu,ld][[cc,dd]],{nu,0,3}])D[Ψ[ll][dd][t,x,y,z],Ϛ[ld]],{ld,0,3},{ll,1,2},{dd,1,4}]+ I Sum[cV[5][ii,jj,kk,ll] (Sum[γdown[mu][[aa,bb]] γγstdownstup[mu,ld][[cc,dd]],{mu,0,3}] + (1/8)Sum[CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[ld])[[cc,dd]],{mu,0,3},{nu,0,3}])D[Ψ[ll][dd][t,x,y,z],Ϛ[ld]],{ld,0,3},{ll,1,2},{dd,1,4}],{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{kk,1,2},{cc,1,4}]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber,αhat,βhat,κhat]

4D, N=2 tensor Holoraumy, both untilded [(Subscript[D, a]^i),(Subscript[D, b]^j)]

, FieldNumber = 8

LocalFieldNumber =8 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 sigma[3][[ii,jj]](-Sum[γ5γdown[mu][[aa,bb]]D[Bt[t,x,y,z],Ϛ[mu]],{mu,0,3}] - I Cmetric[[aa,bb]]Ft[t,x,y,z] + γ5down[[aa,bb]]Gt[t,x,y,z])  + 2 sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[φ[t,x,y,z],Ϛ[mu]],{mu,0,3}]+(2/3)sigma[1][[ii,jj]]Sum[ϵdownupupup[mu,nu,ap,bt]γ5γdown[mu][[aa,bb]]Hamn[nu,ap,bt][t,x,y,z],{mu,0,3},{nu,0,3},{ap,0,3},{bt,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 9

LocalFieldNumber =9 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 KroneckerDelta[ii,jj](γ5down[[aa,bb]]Ft[t,x,y,z] + I Cmetric[[aa,bb]]Gt[t,x,y,z]) + 2 Sum[γ5γdown[mu][[aa,bb]]D[sigma[3][[ii,jj]]At[t,x,y,z] + sigma[1][[ii,jj]]φ[t,x,y,z],Ϛ[mu]],{mu,0,3}] -(2/3)sigma[2][[ii,jj]]Sum[ϵdownupupup[mu,nu,ap,bt]γdown[mu][[aa,bb]]Hamn[nu,ap,bt][t,x,y,z],{mu,0,3},{nu,0,3},{ap,0,3},{bt,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 10

LocalFieldNumber =10;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 KroneckerDelta[ii,jj] (γ5down[[aa,bb]] Lap[Bt[t,x,y,z]] - Sum[γ5γdown[mu][[aa,bb]]D[Gt[t,x,y,z],Ϛ[mu]],{mu,0,3}]) - 2 I Cmetric[[aa,bb]]Lap[sigma[3][[ii,jj]]At[t,x,y,z] + sigma[1][[ii,jj]]φ[t,x,y,z]] - I (1/3)sigma[2][[ii,jj]]Sum[ϵdownupupup[ld,nu,ap,bt](γ5.CommuteGammadown[mu,ld])[[aa,bb]]D[Hamn[nu,ap,bt][t,x,y,z],Ϛ[mu]],{ld,0,3},{mu,0,3},{nu,0,3},{ap,0,3},{bt,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 11

LocalFieldNumber =11;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 KroneckerDelta[ii,jj] (I Cmetric[[aa,bb]] Lap[Bt[t,x,y,z]] + Sum[γ5γdown[mu][[aa,bb]]D[Ft[t,x,y,z],Ϛ[mu]],{mu,0,3}]) + 2γ5down[[aa,bb]]Lap[sigma[3][[ii,jj]]At[t,x,y,z] + sigma[1][[ii,jj]]φ[t,x,y,z]] +(1/3)sigma[2][[ii,jj]]Sum[ϵdownupupup[ld,nu,ap,bt](CommuteGammadown[mu,ld])[[aa,bb]]D[Hamn[nu,ap,bt][t,x,y,z],Ϛ[mu]],{ld,0,3},{mu,0,3},{nu,0,3},{ap,0,3},{bt,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber =12

LocalFieldNumber =12 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 sigma[1][[ii,jj]](-Sum[γ5γdown[mu][[aa,bb]]D[Bt[t,x,y,z],Ϛ[mu]],{mu,0,3}] - I Cmetric[[aa,bb]]Ft[t,x,y,z] + γ5down[[aa,bb]]Gt[t,x,y,z])  -2 sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[At[t,x,y,z],Ϛ[mu]],{mu,0,3}]-(2/3) sigma[3][[ii,jj]]Sum[ϵdownupupup[ld,nu,ap,bt]γ5γdown[ld][[aa,bb]]Hamn[nu,ap,bt][t,x,y,z],{ld,0,3},{nu,0,3},{ap,0,3},{bt,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 13

LocalFieldNumber =13;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[sigma[2][[ii,jj]](Sum[ϵdowndowndownup[mu,nu,de,ld]γdown[de][[aa,bb]]D[Bt[t,x,y,z],Ϛ[ld]],{de,0,3},{ld,0,3}] - (1/2)CommuteGammastdowndown[mu,nu][[aa,bb]]Ft[t,x,y,z] - (1/2)I(γ5.CommuteGammastdowndown[mu,nu])[[aa,bb]]Gt[t,x,y,z]) + Sum[ϵdowndowndownup[mu,nu,de,ld]γ5γdown[de][[aa,bb]]D[-sigma[1][[ii,jj]]At[t,x,y,z] + sigma[3][[ii,jj]]φ[t,x,y,z],Ϛ[ld]],{de,0,3},{ld,0,3}]  +(1/3) KroneckerDelta[ii,jj]Sum[(ϵ[kp,ap,bt,mu]γ5γstdowndown[nu][[aa,bb]]-ϵ[kp,ap,bt,nu]γ5γstdowndown[mu][[aa,bb]])Hamnup[kp,ap,bt][t,x,y,z],{kp,0,3},{ap,0,3},{bt,0,3}],{mu,0,3},{nu,0,3},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{mu,0,3},{nu,0,3}] ==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] //Simplify

Clear[LocalFieldNumber]

FieldNumber = 14

αhat = {0,1,1,0,0};
βhat = {0,0,0,0,-1};
κhat = {1,0,0,-1,0};

LocalFieldNumber =14;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2I Sum[cV[1][ii,jj,kk,ll] γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}]   + Sum[I cV[2][ii,jj,kk,ll] CommuteGammadown[mu,ld][[aa,bb]]γstdown[ld][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ld,0,3},{ll,1,2},{dd,1,4}] + I Sum[cV[3][ii,jj,kk,ll] (Sum[γdown[mu][[aa,bb]]γγstdownstup[mu,ld][[cc,dd]],{mu,0,3}] + (1/8)Sum[CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[ld])[[cc,dd]],{mu,0,3},{nu,0,3}])D[Ψt[ll][dd][t,x,y,z],Ϛ[ld]],{ld,0,3},{ll,1,2},{dd,1,4}]+I Sum[cV[4][ii,jj,kk,ll] (Sum[γ5γdown[mu][[aa,bb]]γ5γγstdownstup[mu,ld][[cc,dd]],{mu,0,3}] - Cmetric[[aa,bb]]γ[ld][[cc,dd]]+γ5down[[aa,bb]]γ5γ[ld][[cc,dd]])D[Ψt[ll][dd][t,x,y,z],Ϛ[ld]],{ld,0,3},{ll,1,2},{dd,1,4}]+ I Sum[cV[5][ii,jj,kk,ll] (Sum[γ5γdown[mu][[aa,bb]]γ5γγstdownstup[mu,ld][[cc,dd]],{mu,0,3}] + Cmetric[[aa,bb]]γ[ld][[cc,dd]]-γ5down[[aa,bb]]γ5γ[ld][[cc,dd]])D[Ψt[ll][dd][t,x,y,z],Ϛ[ld]],{ld,0,3},{ll,1,2},{dd,1,4}],{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{kk,1,2},{cc,1,4}]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber,αhat,βhat,κhat]

4D, N=2 Summary, untilded (tt1=tt2=1)
True if Holoraumy is correct

Table[HoloraumyDDResult[tt1,tt2][Index],{Index,1,14}]==Table[True,{Index,1,14}]
tt1
tt2

Holoraumy [(Subscript[(OverTilde[D]), a]^i),(Subscript[(OverTilde[D]), b]^j)]

Set tilde indices: tt1,tt2=1 for untilded, 2 for tilded

tt1 = 2;
tt2 = 2;

4D, N=2 Vector Holoraumy, both tilded [(Subscript[(OverTilde[D]), a]^i),(Subscript[(OverTilde[D]), b]^j)]

, FieldNumber = 1

LocalFieldNumber =1 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[-2 KroneckerDelta[ii,jj](Sum[γ5γdown[mu][[aa,bb]]D[B[t,x,y,z],Ϛ[mu]],{mu,0,3}] + I Cmetric[[aa,bb]]F[t,x,y,z]) +2γ5down[[aa,bb]]( sigma[3][[ii,jj]]G[t,x,y,z] + sigma[1][[ii,jj]]d[t,x,y,z]) -(1/2)sigma[2][[ii,jj]]Sum[CommuteGammadown[mu,nu][[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 2

LocalFieldNumber =2 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 KroneckerDelta[ii,jj](Sum[γ5γdown[mu][[aa,bb]]D[A[t,x,y,z],Ϛ[mu]],{mu,0,3}] +γ5down[[aa,bb]]F[t,x,y,z]) +2 I Cmetric[[aa,bb]](sigma[3][[ii,jj]]G[t,x,y,z] + sigma[1][[ii,jj]]d[t,x,y,z] )- (1/2)I sigma[2][[ii,jj]]Sum[(γ5.CommuteGammadown[mu,nu])[[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 3

LocalFieldNumber =3 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 KroneckerDelta[ii,jj](-I Cmetric[[aa,bb]]Lap[A[t,x,y,z]] +γ5down[[aa,bb]]Lap[B[t,x,y,z]]) -2 Sum[γ5γdown[mu][[aa,bb]]D[sigma[3][[ii,jj]]G[t,x,y,z] + sigma[1][[ii,jj]]d[t,x,y,z],Ϛ[mu]],{mu,0,3}] + 2 sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]UD[Fmn[mu,nu][t,x,y,z],Ϛ[nu]],{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 4

LocalFieldNumber =4 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 sigma[3][[ii,jj]](γ5down[[aa,bb]]Lap[A[t,x,y,z]] +I Cmetric[[aa,bb]]Lap[B[t,x,y,z]] + Sum[γ5γdown[mu][[aa,bb]]D[F[t,x,y,z],Ϛ[mu]],{mu,0,3}])  +2 sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[d[t,x,y,z],Ϛ[mu]],{mu,0,3}] +2 sigma[1][[ii,jj]]Sum[γ5γdown[mu][[aa,bb]]UD[Fmn[mu,nu][t,x,y,z],Ϛ[nu]],{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 5

LocalFieldNumber =5;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 sigma[1][[ii,jj]](γ5down[[aa,bb]]Lap[A[t,x,y,z]] +I Cmetric[[aa,bb]]Lap[B[t,x,y,z]] + Sum[γ5γdown[mu][[aa,bb]]D[F[t,x,y,z],Ϛ[mu]],{mu,0,3}]) -2 sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[G[t,x,y,z],Ϛ[mu]],{mu,0,3}]-2sigma[3][[ii,jj]]Sum[γ5γdown[mu][[aa,bb]]UD[Fmn[mu,nu][t,x,y,z],Ϛ[nu]],{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 6

LocalFieldNumber =6;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[-sigma[2][[ii,jj]](Sum[CommuteGammastdownstupdown[mu,nu][[aa,bb]] D[A[t,x,y,z],Ϛ[nu]] + I (γ5.CommuteGammastdownstupdown[mu,nu])[[aa,bb]] D[B[t,x,y,z],Ϛ[nu]] ,{nu,0,3}] + 2 γstdowndown[mu][[aa,bb]]F[t,x,y,z]) - 2 γ5γstdowndown[mu][[aa,bb]](sigma[1][[ii,jj]]G[t,x,y,z] - sigma[3][[ii,jj]]d[t,x,y,z]) - KroneckerDelta[ii,jj]Sum[ϵdownupupup[mu,ld,kp,de]γ5γstdowndown[de][[aa,bb]]Fmn[ld,kp][t,x,y,z],{ld,0,3},{kp,0,3},{de,0,3}],{mu,0,3},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{mu,0,3}] ==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] //Simplify

FieldNumber = 7

αhat = {0,1,1,0,0};
βhat = {0,0,0,0,1};
κhat = {1,0,0,-1,0};

LocalFieldNumber =7;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[I Sum[(2cV[1][ii,jj,kk,ll] γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]  + cV[2][ii,jj,kk,ll] Sum[CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]],{nu,0,3}])D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}] + I Sum[cV[3][ii,jj,kk,ll] (Sum[γdown[mu][[aa,bb]]γγstdownstup[mu,ld][[cc,dd]],{mu,0,3}] + (1/8)Sum[CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[ld])[[cc,dd]],{mu,0,3},{nu,0,3}])D[Ψ[ll][dd][t,x,y,z],Ϛ[ld]],{ld,0,3},{ll,1,2},{dd,1,4}]+I Sum[ cV[4][ii,jj,kk,ll] (Sum[γ5γdown[mu][[aa,bb]]γ5γγstdownstup[mu,ld][[cc,dd]],{mu,0,3}] - Cmetric[[aa,bb]]γ[ld][[cc,dd]]+γ5down[[aa,bb]]γ5γ[ld][[cc,dd]])D[Ψ[ll][dd][t,x,y,z],Ϛ[ld]],{ld,0,3},{ll,1,2},{dd,1,4}]+ I Sum[cV[5][ii,jj,kk,ll] (Sum[γ5γdown[mu][[aa,bb]]γ5γγstdownstup[mu,ld][[cc,dd]],{mu,0,3}] + Cmetric[[aa,bb]]γ[ld][[cc,dd]]-γ5down[[aa,bb]]γ5γ[ld][[cc,dd]])D[Ψ[ll][dd][t,x,y,z],Ϛ[ld]],{ld,0,3},{ll,1,2},{dd,1,4}],{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{kk,1,2},{cc,1,4}]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber,αhat,βhat,κhat]

4D, N=2 tensor Holoraumy, both tilded [(Subscript[(OverTilde[D]), a]^i),(Subscript[(OverTilde[D]), b]^j)]

, FieldNumber = 8

LocalFieldNumber =8 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 sigma[3][[ii,jj]](Sum[γ5γdown[mu][[aa,bb]]D[Bt[t,x,y,z],Ϛ[mu]],{mu,0,3}] + I Cmetric[[aa,bb]]Ft[t,x,y,z] - γ5down[[aa,bb]]Gt[t,x,y,z])  + 2 sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[φ[t,x,y,z],Ϛ[mu]],{mu,0,3}]+(2/3)sigma[1][[ii,jj]]Sum[ϵdownupupup[mu,nu,ap,bt]γ5γdown[mu][[aa,bb]]Hamn[nu,ap,bt][t,x,y,z],{mu,0,3},{nu,0,3},{ap,0,3},{bt,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 9

LocalFieldNumber =9 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 KroneckerDelta[ii,jj](γ5down[[aa,bb]]Ft[t,x,y,z] + I Cmetric[[aa,bb]]Gt[t,x,y,z]) -2 Sum[γ5γdown[mu][[aa,bb]]D[sigma[3][[ii,jj]]At[t,x,y,z] + sigma[1][[ii,jj]]φ[t,x,y,z],Ϛ[mu]],{mu,0,3}] +(2/3)sigma[2][[ii,jj]]Sum[ϵdownupupup[mu,nu,ap,bt]γdown[mu][[aa,bb]]Hamn[nu,ap,bt][t,x,y,z],{mu,0,3},{nu,0,3},{ap,0,3},{bt,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 10

LocalFieldNumber =10;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 KroneckerDelta[ii,jj] (γ5down[[aa,bb]] Lap[Bt[t,x,y,z]] - Sum[γ5γdown[mu][[aa,bb]]D[Gt[t,x,y,z],Ϛ[mu]],{mu,0,3}]) + 2 I Cmetric[[aa,bb]]Lap[sigma[3][[ii,jj]]At[t,x,y,z] + sigma[1][[ii,jj]]φ[t,x,y,z]] +I  (1/3) sigma[2][[ii,jj]]Sum[ϵdownupupup[mu,nu,ap,bt](γ5.CommuteGammadown[ld,mu])[[aa,bb]]D[Hamn[nu,ap,bt][t,x,y,z],Ϛ[ld]],{ld,0,3},{mu,0,3},{nu,0,3},{ap,0,3},{bt,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 11

LocalFieldNumber =11;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 KroneckerDelta[ii,jj] (I Cmetric[[aa,bb]] Lap[Bt[t,x,y,z]] + Sum[γ5γdown[mu][[aa,bb]]D[Ft[t,x,y,z],Ϛ[mu]],{mu,0,3}]) -2γ5down[[aa,bb]]Lap[sigma[3][[ii,jj]]At[t,x,y,z] + sigma[1][[ii,jj]]φ[t,x,y,z]]-(1/3)sigma[2][[ii,jj]]Sum[ϵdownupupup[mu,nu,ap,bt](CommuteGammadown[ld,mu])[[aa,bb]]D[Hamn[nu,ap,bt][t,x,y,z],Ϛ[ld]],{ld,0,3},{mu,0,3},{nu,0,3},{ap,0,3},{bt,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber =12

LocalFieldNumber =12 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 sigma[1][[ii,jj]](Sum[γ5γdown[mu][[aa,bb]]D[Bt[t,x,y,z],Ϛ[mu]],{mu,0,3}] + I Cmetric[[aa,bb]]Ft[t,x,y,z] - γ5down[[aa,bb]]Gt[t,x,y,z])  -2 sigma[2][[ii,jj]]Sum[γdown[mu][[aa,bb]]D[At[t,x,y,z],Ϛ[mu]],{mu,0,3}]-(2/3) sigma[3][[ii,jj]]Sum[ϵdownupupup[mu,nu,ap,bt]γ5γdown[mu][[aa,bb]]Hamn[nu,ap,bt][t,x,y,z],{mu,0,3},{nu,0,3},{ap,0,3},{bt,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 13

LocalFieldNumber =13;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[sigma[2][[ii,jj]](Sum[ϵdowndownupup[mu,nu,ld,de]γstdowndown[de][[aa,bb]]D[Bt[t,x,y,z],Ϛ[ld]],{de,0,3},{ld,0,3}] +(1/2)CommuteGammastdowndown[mu,nu][[aa,bb]]Ft[t,x,y,z] + (1/2)I(γ5.CommuteGammastdowndown[mu,nu])[[aa,bb]]Gt[t,x,y,z]) + Sum[ϵdowndownupup[mu,nu,ld,de]γ5γstdowndown[de][[aa,bb]]D[sigma[1][[ii,jj]]At[t,x,y,z] -  sigma[3][[ii,jj]]φ[t,x,y,z],Ϛ[ld]],{de,0,3},{ld,0,3}]  +(1/3) KroneckerDelta[ii,jj]Sum[(ϵ[kp,ap,bt,mu]γ5γstdowndown[nu][[aa,bb]]-ϵ[kp,ap,bt,nu]γ5γstdowndown[mu][[aa,bb]])Hamnup[kp,ap,bt][t,x,y,z],{kp,0,3},{ap,0,3},{bt,0,3}],{mu,0,3},{nu,0,3},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{mu,0,3},{nu,0,3}] ==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] //Simplify

Clear[LocalFieldNumber]

FieldNumber = 14

αhat = {1,0,-1,0,0};
βhat = {0,-1,0,0,1};
κhat = {0,0,0,-1,0};

LocalFieldNumber =14;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2I Sum[(cV[1][ii,jj,kk,ll] γdown[mu][[aa,bb]]KroneckerDelta[cc,dd] +cV[2][ii,jj,kk,ll] γ5γdown[mu][[aa,bb]]γ5[[cc,dd]])D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}] + I Sum[ cV[3][ii,jj,kk,ll] (Sum[γdown[mu][[aa,bb]] γγstdownstup[mu,ld][[cc,dd]],{mu,0,3}] + (1/8)Sum[CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[ld])[[cc,dd]],{mu,0,3},{nu,0,3}])D[Ψt[ll][dd][t,x,y,z],Ϛ[ld]],{ld,0,3},{ll,1,2},{dd,1,4}]+I Sum[cV[4][ii,jj,kk,ll] (Sum[γ5γdown[mu][[aa,bb]]γ5γγstdownstup[mu,ld][[cc,dd]],{mu,0,3}]+Cmetric[[aa,bb]]γ[ld][[cc,dd]] + γ5down[[aa,bb]]γ5γ[ld][[cc,dd]])D[Ψt[ll][dd][t,x,y,z],Ϛ[ld]],{ld,0,3},{ll,1,2},{dd,1,4}]+ I Sum[cV[5][ii,jj,kk,ll] (Sum[γ5γdown[mu][[aa,bb]]γ5γγstdownstup[mu,ld][[cc,dd]],{mu,0,3}]-Cmetric[[aa,bb]]γ[ld][[cc,dd]] - γ5down[[aa,bb]]γ5γ[ld][[cc,dd]])D[Ψt[ll][dd][t,x,y,z],Ϛ[ld]],{ld,0,3},{ll,1,2},{dd,1,4}],{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{kk,1,2},{cc,1,4}]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber,αhat,βhat,κhat]

4D, N=2 Summary, tilded (tt1=tt2=2)
True if Holoraumy is correct

Table[HoloraumyDDResult[tt1,tt2][Index],{Index,1,14}]==Table[True,{Index,1,14}]
tt1
tt2

Holoraumy [(Subscript[D, a]^i),(Subscript[(OverTilde[D]), b]^j)]

Set tilde indices: tt1,tt2=1 for untilded, 2 for tilded

tt1 = 1;
tt2 = 2;

4D, N=2 Vector Holoraumy, one tilded, one untilded [(Subscript[D, a]^i),(Subscript[(OverTilde[D]), b]^j)]

, FieldNumber = 1

LocalFieldNumber =1 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 sigma[2][[ii,jj]](I Sum[γ5γdown[mu][[aa,bb]]D[Bt[t,x,y,z],Ϛ[mu]],{mu,0,3}] -  Cmetric[[aa,bb]]Ft[t,x,y,z] -I γ5down[[aa,bb]]Gt[t,x,y,z])  -(2/3)KroneckerDelta[ii,jj]Sum[ϵdownupupup[mu,nu,ap,bt]γ5γdown[mu][[aa,bb]]Hamn[nu,ap,bt][t,x,y,z],{mu,0,3},{nu,0,3},{ap,0,3},{bt,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 2

LocalFieldNumber =2 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 sigma[2][[ii,jj]](-I γ5down[[aa,bb]]Ft[t,x,y,z] + Cmetric[[aa,bb]]Gt[t,x,y,z]) +2 Sum[γ5γdown[mu][[aa,bb]]D[-sigma[1][[ii,jj]]At[t,x,y,z] + sigma[3][[ii,jj]]φ[t,x,y,z],Ϛ[mu]],{mu,0,3}] ,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 3

LocalFieldNumber =3;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 I sigma[2][[ii,jj]] (-γ5down[[aa,bb]] Lap[Bt[t,x,y,z]] + Sum[γ5γdown[mu][[aa,bb]]D[Gt[t,x,y,z],Ϛ[mu]],{mu,0,3}]) + 2 I Cmetric[[aa,bb]]Lap[sigma[1][[ii,jj]]At[t,x,y,z] - sigma[3][[ii,jj]]φ[t,x,y,z]] ,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 4

LocalFieldNumber =4;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 I sigma[1][[ii,jj]]  Sum[γdown[mu][[aa,bb]]D[Gt[t,x,y,z],Ϛ[mu]],{mu,0,3}] +2 KroneckerDelta[ii,jj]γ5down[[aa,bb]] Lap[φ[t,x,y,z]],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 5

LocalFieldNumber =5;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[-2 I sigma[3][[ii,jj]]  Sum[γdown[mu][[aa,bb]]D[Gt[t,x,y,z],Ϛ[mu]],{mu,0,3}] -2 KroneckerDelta[ii,jj]γ5down[[aa,bb]] Lap[At[t,x,y,z]],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 6

LocalFieldNumber =6;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 KroneckerDelta[ii,jj](γ5down[[aa,bb]]D[Bt[t,x,y,z],Ϛ[mu]]- γ5γstdowndown[mu][[aa,bb]]Gt[t,x,y,z]) + I Sum[CommuteGammastdowndown[mu,nu][[aa,bb]]UD[sigma[3][[ii,jj]]At[t,x,y,z] + sigma[1][[ii,jj]]φ[t,x,y,z],Ϛ[nu]],{nu,0,3}] + I (2/3)sigma[2][[ii,jj]]Sum[ϵdownupupup[mu,nu,ap,bt]γ5down[[aa,bb]]Hamn[nu,ap,bt][t,x,y,z],{nu,0,3},{ap,0,3},{bt,0,3}],{mu,0,3},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{mu,0,3}] ==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] //Simplify

, FieldNumber = 7

αhat = {1/4,-1/8,3/4,-(1/4),-(1/32),-1/4,-1/4,1/2};
βhat = {1/4,-1/8,-1/4,-(1/4),-(1/32),1/4,1/4,0};
κhat = {-1/4,1/8,-3/4,0,0,1/2,-1/2,-(1/4)};

LocalFieldNumber =7;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 I Sum[V[1][ii,jj,kk,ll]γdown[mu][[aa,bb]]D[Ψt[ll][cc][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2}]+2I Sum[( V[2][ii,jj,kk,ll]Sum[CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]],{nu,0,3}] + V[3][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]])D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}] + 2 I Sum[(V[4][ii,jj,kk,ll]Sum[γstdowndown[nu][[aa,bb]]γγ[nu,ld][[cc,dd]],{nu,0,3}] + V[5][ii,jj,kk,ll]Sum[CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[ld])[[cc,dd]],{mu,0,3},{nu,0,3}])D[Ψt[ll][dd][t,x,y,z],Ϛ[ld]],{ld,0,3},{ll,1,2},{dd,1,4}] + 2 I Sum[( V[6][ii,jj,kk,ll]Sum[γ5γstdowndown[nu][[aa,bb]]γ5γγ[nu,ld][[cc,dd]],{nu,0,3}] +V[7][ii,jj,kk,ll]γ5down[[aa,bb]]γ5γ[ld][[cc,dd]])D[Ψt[ll][dd][t,x,y,z],Ϛ[ld]],{ld,0,3},{ll,1,2},{dd,1,4}]+ 2 I Sum[V[8][ii,jj,kk,ll]Cmetric[[aa,bb]]γ[ld][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[ld]],{ld,0,3},{ll,1,2},{dd,1,4}],{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{kk,1,2},{cc,1,4}]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber,αhat,βhat,κhat]

, FieldNumber = 8

LocalFieldNumber =8 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 sigma[1][[ii,jj]](Sum[γ5γdown[mu][[aa,bb]]D[B[t,x,y,z],Ϛ[mu]],{mu,0,3}] + I Cmetric[[aa,bb]]F[t,x,y,z]) -2 KroneckerDelta[ii,jj]γ5down[[aa,bb]]d[t,x,y,z] +I(1/2)sigma[3][[ii,jj]]Sum[CommuteGammadown[mu,nu][[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 9

LocalFieldNumber =9 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[-2 I sigma[2][[ii,jj]](Sum[γ5γdown[mu][[aa,bb]]D[A[t,x,y,z],Ϛ[mu]],{mu,0,3}] +γ5down[[aa,bb]]F[t,x,y,z]) ,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 10

LocalFieldNumber =10 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[-2 sigma[2][[ii,jj]](Cmetric[[aa,bb]]Lap[A[t,x,y,z]] +I γ5down[[aa,bb]]Lap[B[t,x,y,z]]),{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 11

LocalFieldNumber =11;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 sigma[2][[ii,jj]](-I γ5down[[aa,bb]]Lap[A[t,x,y,z]] +Cmetric[[aa,bb]]Lap[B[t,x,y,z]] -I Sum[γ5γdown[mu][[aa,bb]]D[F[t,x,y,z],Ϛ[mu]],{mu,0,3}])  +2 I Sum[γdown[mu][[aa,bb]]D[(-sigma[1][[ii,jj]]G[t,x,y,z]+sigma[3][[ii,jj]]d[t,x,y,z]),Ϛ[mu]],{mu,0,3}] -(1/2)KroneckerDelta[ii,jj]Sum[(γ5γ[ld].CommuteGammadown[mu,nu])[[aa,bb]]D[Fmn[mu,nu][t,x,y,z],Ϛ[ld]],{ld,0,3},{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 12

LocalFieldNumber =12 ;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[-2 sigma[3][[ii,jj]](Sum[γ5γdown[mu][[aa,bb]]D[B[t,x,y,z],Ϛ[mu]],{mu,0,3}] + I Cmetric[[aa,bb]]F[t,x,y,z]) +2 KroneckerDelta[ii,jj]γ5down[[aa,bb]]G[t,x,y,z] +I(1/2)sigma[1][[ii,jj]]Sum[CommuteGammadown[mu,nu][[aa,bb]]Fmn[mu,nu][t,x,y,z],{mu,0,3},{nu,0,3}],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber]

, FieldNumber = 13

LocalFieldNumber =13;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[KroneckerDelta[ii,jj]Sum[ϵdowndownupup[mu,nu,de,ld]γ5γstdowndown[de][[aa,bb]]D[A[t,x,y,z],Ϛ[ld]],{de,0,3},{ld,0,3}] + KroneckerDelta[ii,jj](γ5γstdowndown[mu][[aa,bb]]D[B[t,x,y,z],Ϛ[nu]]-γ5γstdowndown[nu][[aa,bb]]D[B[t,x,y,z],Ϛ[mu]]) - Cmetric[[aa,bb]]sigma[2][[ii,jj]]Fmn[mu,nu][t,x,y,z]+ (I/2)sigma[2][[ii,jj]]Sum[ϵdowndownupup[mu,nu,ro,sg]γ5down[[aa,bb]]Fmn[ro,sg][t,x,y,z],{ro,0,3},{sg,0,3}],{mu,0,3},{nu,0,3},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{mu,0,3},{nu,0,3}] ==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] //Simplify

Clear[LocalFieldNumber]

, FieldNumber = 14

αhat = {1/4,-1/4,-1/8,-1/32,-3/4,1/4,1/4,-1/2};
βhat = {-1/4,1/4,1/8,1/32,-1/4,1/4,1/4,0};
κhat = {1/4,0,-1/8,0,-3/4,1/2,-1/2,-1/4};

LocalFieldNumber =14;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHS[tt1,tt2][LocalFieldNumber] = Table[2 I Sum[V[1][ii,jj,kk,ll]γdown[mu][[aa,bb]]D[Ψ[ll][cc][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2}]+2I Sum[V[2][ii,jj,kk,ll]Sum[γstdowndown[nu][[aa,bb]]γγ[nu,mu][[cc,dd]],{nu,0,3}] D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}] +2I Sum[V[3][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]] D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}] +2I Sum[V[4][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]](CommuteGammastdown[mu,nu].γ[ld])[[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[ld]],{mu,0,3},{nu,0,3},{ld,0,3},{ll,1,2},{dd,1,4}] +2I Sum[V[5][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}] +2I Sum[ V[6][ii,jj,kk,ll]γ5γstdowndown[nu][[aa,bb]]γ5γγ[nu,mu][[cc,dd]] D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+2 I Sum[V[7][ii,jj,kk,ll]γ5down[[aa,bb]]γ5γ[mu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}] +2 I  Sum[V[8][ii,jj,kk,ll]Cmetric[[aa,bb]]γ[mu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}],{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResult[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{kk,1,2},{cc,1,4}]==HoloraumyDDRHS[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber,αhat,βhat,κhat]

4D, N=2 Summary, one untiled, one tilded (tt1=1, tt2=2)
True if Holoraumy is correct

Table[HoloraumyDDResult[tt1,tt2][Index],{Index,1,14}]==Table[True,{Index,1,14}]
tt1
tt2

Checking antisymmetry

Scalars

Table[HoloraumyDD[tindex,tindex][ScalarFieldNumber]==-Table[HoloraumyDD[tindex,tindex][ScalarFieldNumber][[jj,bb,ii,aa]],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}],{tindex,1,2},{ScalarFieldNumber,{1,2,3,4,5,8,9,10,11,12}}]

Dimensions[HoloraumyDD[1,1][FieldNameList[[7]]]]

TableHoloraumyDD for fermions, vectors and tensors

Fermions: TableHoloraumyDD[t1,t2][FermionFieldNumber][[kk,cc,ii,aa,jj,bb]]:= [(Subscript[(OverTilde[D]), a]^i),(Subscript[(OverTilde[D]), b]^j)](Subscript[(OverTilde[Ψ]), c]^k) for t1 = t2 = 2, FermionFieldNumber = 14 and similar for other D’s and the other fermion

Do[TableHoloraumyDD[t1,t2][FermionFieldNumber] =Table[HoloraumyDD[t1,t2][FieldNameList[[FermionFieldNumber]]],{kk,1,2},{cc,1,4}],{t1,1,2},{t2,1,2},{FermionFieldNumber,{7,14}}]

TableHoloraumyDD[1,1][7]==HoloraumyDDRHS[1,1][7]//Simplify

TableHoloraumyDD[1,1][7]==-Table[TableHoloraumyDD[1,1][7][[kk,cc,jj,bb,ii,aa]],{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]

Table[TableHoloraumyDD[tindex,tindex][FermionFieldNumber]==-Table[TableHoloraumyDD[tindex,tindex][FermionFieldNumber][[kk,cc,jj,bb,ii,aa]],{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]//Simplify,{tindex,1,2},{FermionFieldNumber,{7,14}}]

Vector: TableHoloraumyDD[t1,t2][6][[mu,ii,aa,jj,bb]]:=[(Subscript[(OverTilde[D]), a]^i),(Subscript[(OverTilde[D]), b]^j)](Subscript[A, μ]) for t1 = t2 = 2, similar for t1,t2 = other values . This makes mu be from 1 to 4 instead of 0 to 3 as a table index

Do[TableHoloraumyDD[t1,t2][6] =Table[HoloraumyDD[t1,t2][FieldNameList[[6]]],{mu,0,3}],{t1,1,2},{t2,1,2}]

Table[TableHoloraumyDD[tindex,tindex][6]==-Table[TableHoloraumyDD[tindex,tindex][6][[mu,jj,bb,ii,aa]],{mu,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]//Simplify,{tindex,1,2}]

Tensor: TableHoloraumyDD[t1,t2][6][[mu,nu,ii,aa,jj,bb]]:=[(Subscript[(OverTilde[D]), a]^i),(Subscript[(OverTilde[D]), b]^j)](Subscript[B, μν]) for t1 = t2 = 2, similar for t1,t2 = other values . This makes mu, nu be from 1 to 4 instead of 0 to 3 as a table index

Do[TableHoloraumyDD[t1,t2][13] =Table[HoloraumyDD[t1,t2][FieldNameList[[13]]],{mu,0,3},{nu,0,3}],{t1,1,2},{t2,1,2}]

Table[TableHoloraumyDD[tindex,tindex][13]==-Table[TableHoloraumyDD[tindex,tindex][13][[mu,nu,jj,bb,ii,aa]],{mu,1,4},{nu,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]//Simplify,{tindex,1,2}]

Appendix F check for cross term   [(Subscript[D, a]^i),(Subscript[(OverTilde[D]), b]^j)]Fermion  in X and Y bases and {(Subscript[D, a]^i),(Subscript[(OverTilde[D]), b]^j)}Fermion  in W basis.

Set tilde indices: tt1,tt2=1 for untilded, 2 for tilded

tt1 = 1;
tt2 = 2;

, FieldNumber = 7

α = {0,1/4,0,-(1/4),-1,1/4,1/2,-1};
β = {0,-I 1/4,0,I 1/4,0,I 1/4,I 1/2,0};
δ={0,I/4,0,-I/4,0,-I/4,-I/2,0};
κ = {1/2,0,-1/4,0,1/2,-1/2,1,1/2};

LocalFieldNumber =7;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHSX[tt1,tt2][LocalFieldNumber] = Table[ Sum[X[1][ii,jj,kk,ll]γdown[mu][[aa,bb]]D[Ψt[ll][cc][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2}]+Sum[X[2][ii,jj,kk,ll]γstdowndown[nu][[aa,bb]]CommuteGamma[nu,mu][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[3][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[4][ii,jj,kk,ll](γ5.CommuteGammadown[mu,nu])[[aa,bb]]γ5γstdown[nu][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}] +Sum[ X[5][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[6][ii,jj,kk,ll]γ5γstdowndown[nu][[aa,bb]](γ5.CommuteGamma[nu,mu])[[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[7][ii,jj,kk,ll]γ5down[[aa,bb]]γ5γ[mu][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[8][ii,jj,kk,ll]Cmetric[[aa,bb]]γ[mu][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}] ,{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResultX[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{kk,1,2},{cc,1,4}]==HoloraumyDDRHSX[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber,α,β,δ,κ]

, FieldNumber = 4

α={a1,a2,a3,a4,a5,a6,a7,a8};
β={I b1,I b2,I b3,I b4,I b5,I b6,I b7,I b8};
δ={I d1,I d2,I d3,I d4,I d5,I d6,I d7,I d8};
κ={k1,k2,k3,k4,k5,k6,k7,k8};

LocalFieldNumber =14;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHSX[tt1,tt2][LocalFieldNumber] = Table[Sum[X[1][ii,jj,kk,ll]γdown[mu][[aa,bb]]D[Ψ[ll][cc][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2}]+Sum[X[2][ii,jj,kk,ll]γstdowndown[nu][[aa,bb]]CommuteGamma[nu,mu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+ Sum[ X[3][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+ Sum[ X[4][ii,jj,kk,ll](γ5.CommuteGammadown[mu,nu])[[aa,bb]]γ5γstdown[nu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}] + Sum[ X[5][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}]+ Sum[ X[6][ii,jj,kk,ll]γ5γstdowndown[nu][[aa,bb]](γ5.CommuteGamma[nu,mu])[[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[7][ii,jj,kk,ll]γ5down[[aa,bb]]γ5γ[mu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[8][ii,jj,kk,ll]Cmetric[[aa,bb]]γ[mu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}] ,{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResultSubtractX[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{kk,1,2},{cc,1,4}]-HoloraumyDDRHSX[tt1,tt2][LocalFieldNumber]//Simplify;

HoloraumyDDEquationsX[tt1,tt2][LocalFieldNumber]=Table[DeleteCases[Table[Coefficient[HoloraumyDDResultSubtractX[tt1,tt2][LocalFieldNumber][[kk,cc,ii,aa,jj,bb]],Derivative[{1,0,0,0}[[derselect]],{0,1,0,0}[[derselect]],{0,0,1,0}[[derselect]],{0,0,0,1}[[derselect]]][Ψ[ll][dd]][t,x,y,z]]==0,{ll,1,2},{dd,1,4},{derselect,1,4}]//Flatten,True],{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]//Simplify//Flatten//DeleteDuplicates

HoloraumyDDSolutionX[tt1,tt2][LocalFieldNumber]=Solve[HoloraumyDDEquationsX[tt1,tt2][LocalFieldNumber],{a1,a2,a3,a4,a5,a6,a7,a8,b1,b2,b3,b4,b5,b6,b7,b8,d1,d2,d3,d4,d5,d6,d7,d8,k1,k2,k3,k4,k5,k6,k7,k8}]//Flatten

α =α//.HoloraumyDDSolutionX[tt1,tt2][LocalFieldNumber]

β=β//.HoloraumyDDSolutionX[tt1,tt2][LocalFieldNumber]

δ=δ//.HoloraumyDDSolutionX[tt1,tt2][LocalFieldNumber]

κ=κ//.HoloraumyDDSolutionX[tt1,tt2][LocalFieldNumber]

HoloraumyDDRHSFinalX[tt1,tt2][LocalFieldNumber] = Table[Sum[X[1][ii,jj,kk,ll]γdown[mu][[aa,bb]]D[Ψ[ll][cc][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2}]+Sum[X[2][ii,jj,kk,ll]γstdowndown[nu][[aa,bb]]CommuteGamma[nu,mu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+ Sum[ X[3][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+ Sum[ X[4][ii,jj,kk,ll](γ5.CommuteGammadown[mu,nu])[[aa,bb]]γ5γstdown[nu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}] + Sum[ X[5][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}]+ Sum[ X[6][ii,jj,kk,ll]γ5γstdowndown[nu][[aa,bb]](γ5.CommuteGamma[nu,mu])[[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[7][ii,jj,kk,ll]γ5down[[aa,bb]]γ5γ[mu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[8][ii,jj,kk,ll]Cmetric[[aa,bb]]γ[mu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}] ,{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResultX[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{kk,1,2},{cc,1,4}]==HoloraumyDDRHSFinalX[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber,α,β,δ,κ]

Same thing but in Y-basis:

, FieldNumber = 7

α = {0,0,-1/4,(1/4),0,1/4,1/2,0};
β = {1/4,- 1/8,1/8,-1/8,-1/4,-1/8,3/4,-1/4};
δ={-1/4,1/8,1/8,-1/8,-1/4,1/8,-3/4,-1/4};
κ = {0,0,0,0,1/2,-1/4,-1/2,1/2};

LocalFieldNumber =7;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHSY[tt1,tt2][LocalFieldNumber] = Table[ 2I Sum[Y[1][ii,jj,kk,ll]γdown[mu][[aa,bb]]D[Ψt[ll][cc][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2}]+2I Sum[Y[2][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+2I Sum[Y[3][ii,jj,kk,ll]γstdowndown[nu][[aa,bb]]CommuteGamma[nu,mu][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+2I Sum[Y[4][ii,jj,kk,ll](γ5.CommuteGammadown[mu,nu])[[aa,bb]]γ5γstdown[nu][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}] +2 I Sum[Y[5][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}]+2 I Sum[Y[6][ii,jj,kk,ll]γ5γstdowndown[nu][[aa,bb]](γ5.CommuteGamma[nu,mu])[[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+2 I Sum[Y[7][ii,jj,kk,ll]γ5down[[aa,bb]]γ5γ[mu][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}]+2 I Sum[Y[8][ii,jj,kk,ll]Cmetric[[aa,bb]]γ[mu][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}] ,{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResultY[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{kk,1,2},{cc,1,4}]==HoloraumyDDRHSY[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber,α,β,δ,κ]

, FieldNumber = 14

α = {0,0,1/4,-(1/4),0,1/4,1/2,0};
β = {-1/4,1/8,1/8,-1/8,3/4,-3/8,1/4,3/4};
δ={1/4,-1/8,-1/8,1/8,-1/4,1/8,-3/4,-1/4};
κ = {0,0,-1/4,1/4,-1/2,0,0,-1/2};

LocalFieldNumber =14;
FieldNameList[[LocalFieldNumber]]

HoloraumyDDRHSY[tt1,tt2][LocalFieldNumber] = Table[ 2I Sum[Y[1][ii,jj,kk,ll]γdown[mu][[aa,bb]]D[Ψ[ll][cc][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2}]+2I Sum[Y[2][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+2I Sum[Y[3][ii,jj,kk,ll]γstdowndown[nu][[aa,bb]]CommuteGamma[nu,mu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+2I Sum[Y[4][ii,jj,kk,ll](γ5.CommuteGammadown[mu,nu])[[aa,bb]]γ5γstdown[nu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}] +2 I Sum[Y[5][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}]+2 I Sum[Y[6][ii,jj,kk,ll]γ5γstdowndown[nu][[aa,bb]](γ5.CommuteGamma[nu,mu])[[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+2 I Sum[Y[7][ii,jj,kk,ll]γ5down[[aa,bb]]γ5γ[mu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}]+2 I Sum[Y[8][ii,jj,kk,ll]Cmetric[[aa,bb]]γ[mu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}] ,{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

HoloraumyDDResultY[tt1,tt2][LocalFieldNumber]=Table[HoloraumyDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{kk,1,2},{cc,1,4}]==HoloraumyDDRHSY[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber,α,β,δ,κ]

Algebra in W-basis: recall W is coded as the function X[xx][ii,jj,kk,ll], the same used for X above

, FieldNumber = 7

α = {-1/2,0,1/4,0,-1/2,-1/2,1,-1/2};
β = {-3 I/2,0,-I/4,0,-I/2,0,0,-I/2};
δ={3 I/2,0,I/4,0,I/2,0,0,I/2};
κ = {0,-1/4,0,1/4,-1,-1/4,-1/2,-1};

LocalFieldNumber =7;
FieldNameList[[LocalFieldNumber]]

AlgebraDDRHSW[tt1,tt2][LocalFieldNumber] = Table[ Sum[X[1][ii,jj,kk,ll]γdown[mu][[aa,bb]]D[Ψt[ll][cc][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2}]+Sum[X[2][ii,jj,kk,ll]γstdowndown[nu][[aa,bb]]CommuteGamma[nu,mu][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[3][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[4][ii,jj,kk,ll](γ5.CommuteGammadown[mu,nu])[[aa,bb]]γ5γstdown[nu][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}] +Sum[ X[5][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[6][ii,jj,kk,ll]γ5γstdowndown[nu][[aa,bb]](γ5.CommuteGamma[nu,mu])[[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[7][ii,jj,kk,ll]γ5down[[aa,bb]]γ5γ[mu][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[8][ii,jj,kk,ll]Cmetric[[aa,bb]]γ[mu][[cc,dd]]D[Ψt[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}] ,{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

AlgebraDDResultW[tt1,tt2][LocalFieldNumber]=Table[AlgebraDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{kk,1,2},{cc,1,4}]==AlgebraDDRHSW[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber,α,β,δ,κ]

, FieldNumber = 14

α = {1/2,0,-1/4,0,-1/2,-1/2,1,-1/2};
β = {-3 I/2,0,-I/4,0,I/2,0,0,I/2};
δ={3 I/2,0,I/4,0,-I/2,0,0,-I/2};
κ = {0,-1/4,0,1/4,1,1/4,1/2,1};

LocalFieldNumber =14;
FieldNameList[[LocalFieldNumber]]

AlgebraDDRHSW[tt1,tt2][LocalFieldNumber] = Table[ Sum[X[1][ii,jj,kk,ll]γdown[mu][[aa,bb]]D[Ψ[ll][cc][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2}]+Sum[X[2][ii,jj,kk,ll]γstdowndown[nu][[aa,bb]]CommuteGamma[nu,mu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[3][ii,jj,kk,ll]CommuteGammadown[mu,nu][[aa,bb]]γstdown[nu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[4][ii,jj,kk,ll](γ5.CommuteGammadown[mu,nu])[[aa,bb]]γ5γstdown[nu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}] +Sum[ X[5][ii,jj,kk,ll]γ5γdown[mu][[aa,bb]]γ5[[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[6][ii,jj,kk,ll]γ5γstdowndown[nu][[aa,bb]](γ5.CommuteGamma[nu,mu])[[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{nu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[7][ii,jj,kk,ll]γ5down[[aa,bb]]γ5γ[mu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}]+Sum[ X[8][ii,jj,kk,ll]Cmetric[[aa,bb]]γ[mu][[cc,dd]]D[Ψ[ll][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{ll,1,2},{dd,1,4}] ,{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

AlgebraDDResultW[tt1,tt2][LocalFieldNumber]=Table[AlgebraDD[tt1,tt2][FieldNameList[[LocalFieldNumber]]],{kk,1,2},{cc,1,4}]==AlgebraDDRHSW[tt1,tt2][LocalFieldNumber]//Simplify

Clear[LocalFieldNumber,α,β,δ,κ]

Holoraumy Final Check, true if all Holoraumy are correct

Table[HoloraumyDDResult[1,1][Index],{Index,1,14}]==Table[HoloraumyDDResult[2,2][Index],{Index,1,14}]==Table[HoloraumyDDResult[1,2][Index],{Index,1,14}]==Table[True,{Index,1,14}]

Appendix F final Check. True if correct

Table[HoloraumyDDResultX[1,2][Index],{Index,{7,14}}]==Table[HoloraumyDDResultY[1,2][Index],{Index,{7,14}}]==Table[AlgebraDDResultW[1,2][Index],{Index,{7,14}}]=={True,True}

Adinkra reduction to 1D and gadget comparison with the VC multiplet in 1402.5765

Adinkra Reduction and Psi replacement commands

AdinkraReduceJustFermions = {Ψ[ii0_][aa0_][t,x,y,z]-> Ψ[ii0][aa0][t],D[Ψ[ii0_][aa0_][t,x,y,z],t]->D[Ψ[ii0][aa0][t],t] ,D[Ψ[ii0_][aa0_][t,x,y,z],x]->0 ,D[Ψ[ii0_][aa0_][t,x,y,z],y]->0,D[Ψ[ii0_][aa0_][t,x,y,z],z]->0  ,Ψt[ii0_][aa0_][t,x,y,z]-> Ψt[ii0][aa0][t],D[Ψt[ii0_][aa0_][t,x,y,z],t]->D[Ψt[ii0][aa0][t],t] ,D[Ψt[ii0_][aa0_][t,x,y,z],x]->0 ,D[Ψt[ii0_][aa0_][t,x,y,z],y]->0,D[Ψt[ii0_][aa0_][t,x,y,z],z]->0}

PsiTo16 = {Ψ[1][aa0_]-> Function[{t},I Ψ16[8+aa0][t]],Ψ[2][aa0_]-> Function[{t},I Ψ16[12+aa0][t]],Ψt[1][aa0_]-> Function[{t},I Ψ16[aa0][t]],Ψt[2][aa0_]-> Function[{t},I Ψ16[4+aa0][t]]}

AdinkraReduce = {At->Function[{t,x,y,z},At[t]],Bt->Function[{t,x,y,z},Bt[t]],Ft->Function[{t,x,y,z},IntFt'[t]],Gt->Function[{t,x,y,z},IntGt'[t]],φ->Function[{t,x,y,z},φ[t]],Bmn[mu_,nu_]->Function[{t,x,y,z},1/2TwoBmn[mu,nu][t]],A->Function[{t,x,y,z},A[t]],B->Function[{t,x,y,z},B[t]],F->Function[{t,x,y,z},IntF'[t]],G->Function[{t,x,y,z},IntG'[t]],Am[nu_]-> Function[{t,x,y,z},Am[nu][t]],d->Function[{t,x,y,z},Intd'[t]],Ψ[ii_][aa_]->Function[{t,x,y,z},Ψ[ii][aa][t]],Ψt[ii_][aa_]->Function[{t,x,y,z},Ψt[ii][aa][t]]}

TemporalGauge = {TwoBmn[0,nu_]->Function[{t},0],TwoBmn[mu_,0]->Function[{t},0],Am[0]->Function[{t},0]}

Dimensionally reduced transformation laws to create L and R matrices, then generate holoraumy


Dimensional reduction utilizing Adinkra.m package, requires defining DColor and Phi and Psi as above

Phi, Psi

Phi[t] = {At[t],Bt[t],IntFt[t],IntGt[t],φ[t],TwoBmn[1,2][t],TwoBmn[2,3][t],TwoBmn[3,1][t],A[t],B[t],IntF[t],IntG[t],Am[1][t],Am[2][t],Am[3][t],Intd[t]};
Psi[t]=-I Join[Table[Ψt[1][aa][t],{aa,1,4}],Table[Ψt[2][aa][t],{aa,1,4}],Table[Ψ[1][aa][t],{aa,1,4}],Table[Ψ[2][aa][t],{aa,1,4}]];

DColor

At

LocalField=At;
DColor[LocalField[t]]=Join[Table[Dia[1][1][aa][LocalField[t,x,y,z]],{aa,1,4}],Table[Dia[1][2][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][1][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][2][aa][LocalField[t,x,y,z]],{aa,1,4}]]/.AdinkraReduce//.TemporalGauge
Clear[LocalField]

Bt

LocalField=Bt;
DColor[LocalField[t]]=Join[Table[Dia[1][1][aa][LocalField[t,x,y,z]],{aa,1,4}],Table[Dia[1][2][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][1][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][2][aa][LocalField[t,x,y,z]],{aa,1,4}]]/.AdinkraReduce//.TemporalGauge
Clear[LocalField]

Ft

LocalField=Ft;
IntLocalField = IntFt;
DColor[IntLocalField[t]]=Integrate[Join[Table[Dia[1][1][aa][LocalField[t,x,y,z]],{aa,1,4}],Table[Dia[1][2][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][1][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][2][aa][LocalField[t,x,y,z]],{aa,1,4}]]/.AdinkraReduce//.TemporalGauge,t]//Simplify
Clear[LocalField,IntLocalField]

Gt

LocalField=Gt;
IntLocalField = IntGt;
DColor[IntLocalField[t]]=Integrate[Join[Table[Dia[1][1][aa][LocalField[t,x,y,z]],{aa,1,4}],Table[Dia[1][2][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][1][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][2][aa][LocalField[t,x,y,z]],{aa,1,4}]]/.AdinkraReduce//.TemporalGauge,t]//Simplify
Clear[LocalField,IntLocalField]

φ

LocalField=φ;
DColor[LocalField[t]]=Join[Table[Dia[1][1][aa][LocalField[t,x,y,z]],{aa,1,4}],Table[Dia[1][2][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][1][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][2][aa][LocalField[t,x,y,z]],{aa,1,4}]]/.AdinkraReduce//.TemporalGauge
Clear[LocalField]

(Subscript[B, μν])

LocalField=Bmn[mu,nu];
LocalFieldPhi=TwoBmn[mu,nu];
Do[DColor[LocalFieldPhi[t]]=2*Join[Table[Dia[1][1][aa][LocalField[t,x,y,z]],{aa,1,4}],Table[Dia[1][2][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][1][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][2][aa][LocalField[t,x,y,z]],{aa,1,4}]]/.AdinkraReduce//.TemporalGauge,{mu,0,3},{nu,0,3}]
Clear[LocalField,LocalFieldPhi]

DColor[TwoBmn[1,2][t]]

Ψt

LocalField=Ψt[ii][bb];
Do[DColor[-I LocalField[t]]=-I Join[Table[Dia[1][1][aa][LocalField[t,x,y,z]],{aa,1,4}],Table[Dia[1][2][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][1][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][2][aa][LocalField[t,x,y,z]],{aa,1,4}]]/.AdinkraReduce//.TemporalGauge,{ii,1,2},{bb,1,4}]
Clear[LocalField]

DColor[-I Ψt[1][3][t]]

A

LocalField=A;
DColor[LocalField[t]]=Join[Table[Dia[1][1][aa][LocalField[t,x,y,z]],{aa,1,4}],Table[Dia[1][2][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][1][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][2][aa][LocalField[t,x,y,z]],{aa,1,4}]]/.AdinkraReduce//.TemporalGauge
Clear[LocalField]

B

LocalField=B;
DColor[LocalField[t]]=Join[Table[Dia[1][1][aa][LocalField[t,x,y,z]],{aa,1,4}],Table[Dia[1][2][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][1][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][2][aa][LocalField[t,x,y,z]],{aa,1,4}]]/.AdinkraReduce//.TemporalGauge
Clear[LocalField]

F

LocalField=F;
IntLocalField = IntF;
DColor[IntLocalField[t]]=Integrate[Join[Table[Dia[1][1][aa][LocalField[t,x,y,z]],{aa,1,4}],Table[Dia[1][2][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][1][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][2][aa][LocalField[t,x,y,z]],{aa,1,4}]]/.AdinkraReduce//.TemporalGauge,t]//Simplify
Clear[LocalField,IntLocalField]

G

LocalField=G;
IntLocalField = IntG;
DColor[IntLocalField[t]]=Integrate[Join[Table[Dia[1][1][aa][LocalField[t,x,y,z]],{aa,1,4}],Table[Dia[1][2][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][1][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][2][aa][LocalField[t,x,y,z]],{aa,1,4}]]/.AdinkraReduce//.TemporalGauge,t]//Simplify
Clear[LocalField,IntLocalField]

(Subscript[A, μ])

LocalField=Am[mu];
Do[DColor[LocalField[t]]=Join[Table[Dia[1][1][aa][LocalField[t,x,y,z]],{aa,1,4}],Table[Dia[1][2][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][1][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][2][aa][LocalField[t,x,y,z]],{aa,1,4}]]/.AdinkraReduce//.TemporalGauge,{mu,0,3}]
Clear[LocalField]

DColor[Am[2][t]]

d

LocalField=d;
IntLocalField = Intd;
DColor[IntLocalField[t]]=Integrate[Join[Table[Dia[1][1][aa][LocalField[t,x,y,z]],{aa,1,4}],Table[Dia[1][2][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][1][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][2][aa][LocalField[t,x,y,z]],{aa,1,4}]]/.AdinkraReduce//.TemporalGauge,t]//Simplify
Clear[LocalField,IntLocalField]

Ψ

LocalField=Ψ[ii][bb];
Do[DColor[-I LocalField[t]]=-I Join[Table[Dia[1][1][aa][LocalField[t,x,y,z]],{aa,1,4}],Table[Dia[1][2][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][1][aa][LocalField[t,x,y,z]],{aa,1,4}],-Table[Dia[2][2][aa][LocalField[t,x,y,z]],{aa,1,4}]]/.AdinkraReduce//.TemporalGauge,{ii,1,2},{bb,1,4}]
Clear[LocalField]

DColor[-I Ψ[1][3][t]]

Generate Adinkra Data, neither Vtilde nor V satisy so16 algebra, neither Vtilde nor V are linearly independent

GenerateLandR[DColor,Phi[t],Psi[t],N4VT]

PrintAllL[N4VT]

AdinkraPreliminaryReport[N4VT]

GenerateAdinkraData[N4VT]

AdinkraReport[N4VT,5]

soNTest[VtildesoN[N4VT]]

soNTest[VsoN[N4VT]]

Importing SM-I, II, III, IV data

SetDirectory[NotebookDirectory[]]

Get["SMI.m"]

Get["SMII.m"]

Get["SMIII.m"]

Get["SMIV.m"]

AdinkraReport[SMI,5]

AdinkraReport[SMII,5]

AdinkraReport[SMIII,5]

AdinkraReport[SMIV,5]

Comparing N4VT with SMi, TM
Looks like top to bottom, the L’s are (up to signs) SMI, TM, SMI, SMII
Also, R = L^T = L^-1, so just need to determine the 16 L’s and we’re good
Define Sign Matrix using Isaac’s BooleanFactors
VierZ[ii][jj]:= Vierrergruppe element[i] with the all rows zeroed except the jjth row

Preliminaries and Load L[TM]

WordToBoolean[{p1_,p2_,p3_,p4_}]:=p1*(2^0)+p2*(2^1)+p3*(2^2)+p4*(2^3)

Do[BooleanToWord[WordToBoolean[{p1,p2,p3,p4}]]={p1,p2,p3,p4},{p1,0,1},{p2,0,1},{p3,0,1},{p4,0,1}];

BooleanToWord[15]

SignMatrix4[BooleanFactor_] := {{(((-1))^(BooleanToWord[BooleanFactor][[1]])),0,0,0},{0,(((-1))^(BooleanToWord[BooleanFactor][[2]])),0,0},{0,0,(((-1))^(BooleanToWord[BooleanFactor][[3]])),0},{0,0,0,(((-1))^(BooleanToWord[BooleanFactor][[4]]))}}

Vier[[1]]

SelectRow[1] ={{1,0,0,0},{0,0,0,0},{0,0,0,0},{0,0,0,0}};
SelectRow[2] ={{0,0,0,0},{0,1,0,0},{0,0,0,0},{0,0,0,0}};
SelectRow[3] ={{0,0,0,0},{0,0,0,0},{0,0,1,0},{0,0,0,0}};
SelectRow[4] ={{0,0,0,0},{0,0,0,0},{0,0,0,0},{0,0,0,1}};

Do[VierZ[ii][jj]=SelectRow[jj].Vier[[ii]],{ii,1,4},{jj,1,4}]

VierZ[4][4]//MatrixForm

SignMatrix4[13]

L[TM]=L[553212];

PrintAllL[TM]//TeXForm

PrintAllL[TM]

I=1, same Vierergruppe and SignMatrix structure as 2,3,4

LocalColor =1;
{PrintL[SMI][LocalColor],PrintL[SMII][LocalColor],PrintL[TM][LocalColor]}

PrintL[N4VT][LocalColor]

LTest[N4VT][LocalColor]= Outer[Times,VierZ[1][1],L[SMI][[LocalColor]]]+Outer[Times,VierZ[1][2],L[TM][[LocalColor]]]+Outer[Times,VierZ[1][3],L[SMI][[LocalColor]]]+Outer[Times,VierZ[1][4],L[SMII][[LocalColor]]]//ArrayFlatten;
LTest[N4VT][LocalColor]==L[N4VT][[LocalColor]]

Clear[LocalColor]

I=2

LocalColor =2;
{PrintL[SMI][LocalColor],PrintL[SMII][LocalColor],PrintL[TM][LocalColor]}

PrintL[N4VT][LocalColor]

LTest[N4VT][LocalColor]= Outer[Times,VierZ[1][1],L[SMI][[LocalColor]]]+Outer[Times,VierZ[1][2],L[TM][[LocalColor]]]+Outer[Times,VierZ[1][3],L[SMI][[LocalColor]]]+Outer[Times,VierZ[1][4],L[SMII][[LocalColor]]]//ArrayFlatten;
LTest[N4VT][LocalColor]==L[N4VT][[LocalColor]]

Clear[LocalColor]

I=3

LocalColor =3;
{PrintL[SMI][LocalColor],PrintL[SMII][LocalColor],PrintL[TM][LocalColor]}

PrintL[N4VT][LocalColor]

LTest[N4VT][LocalColor]= Outer[Times,VierZ[1][1],L[SMI][[LocalColor]]]+Outer[Times,VierZ[1][2],L[TM][[LocalColor]]]+Outer[Times,VierZ[1][3],L[SMI][[LocalColor]]]+Outer[Times,VierZ[1][4],L[SMII][[LocalColor]]]//ArrayFlatten;
LTest[N4VT][LocalColor]==L[N4VT][[LocalColor]]

Clear[LocalColor]

I=4

LocalColor =4;
{PrintL[SMI][LocalColor],PrintL[SMII][LocalColor],PrintL[TM][LocalColor]}

PrintL[N4VT][LocalColor]

LTest[N4VT][LocalColor]= Outer[Times,VierZ[1][1],L[SMI][[LocalColor]]]+Outer[Times,VierZ[1][2],L[TM][[LocalColor]]]+Outer[Times,VierZ[1][3],L[SMI][[LocalColor]]]+Outer[Times,VierZ[1][4],L[SMII][[LocalColor]]]//ArrayFlatten;
LTest[N4VT][LocalColor]==L[N4VT][[LocalColor]]

Clear[LocalColor]

I=5, same Vierergruppe and SignMatrix structure as 6,7,8

LocalSMColor =1;
SetNo = 2;  (* which group of 4 colors out of 16: set 1 is 1-4, set 2 is 5-8, set 3 is 9-12, set 4 is 13-16 *)
LocalColor = LocalSMColor + 4*(SetNo-1);
LocalVierNum = 2;
{PrintL[SMI][LocalSMColor],PrintL[SMII][LocalSMColor],PrintL[TM][LocalSMColor]}

PrintL[N4VT][LocalColor]

LTest[N4VT][LocalColor]= Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[1].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[14].L[TM][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[8].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[7].L[SMII][[LocalSMColor]]]//ArrayFlatten;
LTest[N4VT][LocalColor]==L[N4VT][[LocalColor]]

Clear[LocalColor,LocalSMColor,LocalVierNum,SetNo]

I=6

LocalSMColor =2;
SetNo = 2;  (* which group of 4 colors out of 16: set 1 is 1-4, set 2 is 5-8, set 3 is 9-12, set 4 is 13-16 *)
LocalColor = LocalSMColor + 4*(SetNo-1);
LocalVierNum = 2;
{PrintL[SMI][LocalSMColor],PrintL[SMII][LocalSMColor],PrintL[TM][LocalSMColor]}

PrintL[N4VT][LocalColor]

LTest[N4VT][LocalColor]= Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[1].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[14].L[TM][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[8].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[7].L[SMII][[LocalSMColor]]]//ArrayFlatten;
LTest[N4VT][LocalColor]==L[N4VT][[LocalColor]]
LocalColor

Clear[LocalColor,LocalSMColor,LocalVierNum,SetNo]

I=7

LocalSMColor =3;
SetNo = 2;  (* which group of 4 colors out of 16: set 1 is 1-4, set 2 is 5-8, set 3 is 9-12, set 4 is 13-16 *)
LocalColor = LocalSMColor + 4*(SetNo-1);
LocalVierNum = 2;
{PrintL[SMI][LocalSMColor],PrintL[SMII][LocalSMColor],PrintL[TM][LocalSMColor]}

PrintL[N4VT][LocalColor]

LTest[N4VT][LocalColor]= Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[1].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[14].L[TM][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[8].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[7].L[SMII][[LocalSMColor]]]//ArrayFlatten;
LTest[N4VT][LocalColor]==L[N4VT][[LocalColor]]
LocalColor

Clear[LocalColor,LocalSMColor,LocalVierNum,SetNo]

I=8

LocalSMColor =4;
SetNo = 2;  (* which group of 4 colors out of 16: set 1 is 1-4, set 2 is 5-8, set 3 is 9-12, set 4 is 13-16 *)
LocalColor = LocalSMColor + 4*(SetNo-1);
LocalVierNum = 2;
{PrintL[SMI][LocalSMColor],PrintL[SMII][LocalSMColor],PrintL[TM][LocalSMColor]}

PrintL[N4VT][LocalColor]

LTest[N4VT][LocalColor]= Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[1].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[14].L[TM][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[8].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[7].L[SMII][[LocalSMColor]]]//ArrayFlatten;
LTest[N4VT][LocalColor]==L[N4VT][[LocalColor]]
LocalColor

Clear[LocalColor,LocalSMColor,LocalVierNum,SetNo]

I=9

LocalSMColor =1;
SetNo = 3;  (* which group of 4 colors out of 16: set 1 is 1-4, set 2 is 5-8, set 3 is 9-12, set 4 is 13-16 *)
LocalColor = LocalSMColor + 4*(SetNo-1)
LocalVierNum = 4;
{PrintL[SMI][LocalSMColor],PrintL[SMII][LocalSMColor],PrintL[TM][LocalSMColor]}

PrintL[N4VT][LocalColor]

LTest[N4VT][LocalColor]= Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[14].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[15].L[TM][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[15].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[7].L[SMII][[LocalSMColor]]]//ArrayFlatten;
LTest[N4VT][LocalColor]==L[N4VT][[LocalColor]]
LocalColor

Clear[LocalColor,LocalSMColor,LocalVierNum,SetNo]

Find same Vierergruppe and SignMatrix form for I=9,10,11,12

SetNo = 3;  (* which group of 4 colors out of 16: set 1 is 1-4, set 2 is 5-8, set 3 is 9-12, set 4 is 13-16 *)
LocalColor[LocalSMColor_]:= LocalSMColor + 4*(SetNo-1)
LocalVierNum = 4;

Do[{LTest[N4VT][LocalColor[LocalSMColor]]= Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[14].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[15].L[TM][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[15].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[7].L[SMII][[LocalSMColor]]]//ArrayFlatten,
Print[{LTest[N4VT][LocalColor[LocalSMColor]]==L[N4VT][[LocalColor[LocalSMColor]]],LocalColor[LocalSMColor]}]},{LocalSMColor,1,4}]

Clear[SetNo,LocalVierNum,LocalColor]

I=13

LocalSMColor =1;
SetNo = 4;  (* which group of 4 colors out of 16: set 1 is 1-4, set 2 is 5-8, set 3 is 9-12, set 4 is 13-16 *)
LocalColor = LocalSMColor + 4*(SetNo-1)
LocalVierNum = 3;
{PrintL[SMI][LocalSMColor],PrintL[SMII][LocalSMColor],PrintL[TM][LocalSMColor]}

PrintL[N4VT][LocalColor]

LTest[N4VT][LocalColor]= Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[0].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[14].L[TM][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[8].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[15].L[SMII][[LocalSMColor]]]//ArrayFlatten;
LTest[N4VT][LocalColor]==L[N4VT][[LocalColor]]
LocalColor

Clear[LocalColor,LocalSMColor,LocalVierNum,SetNo]

Find same Vierergruppe and SignMatrix form for I=13,14,15,16

SetNo = 4;  (* which group of 4 colors out of 16: set 1 is 1-4, set 2 is 5-8, set 3 is 9-12, set 4 is 13-16 *)
LocalColor[LocalSMColor_]:= LocalSMColor + 4*(SetNo-1)
LocalVierNum = 3;

Do[{LTest[N4VT][LocalColor[LocalSMColor]]= Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[0].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[14].L[TM][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[8].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[15].L[SMII][[LocalSMColor]]]//ArrayFlatten,
Print[{LTest[N4VT][LocalColor[LocalSMColor]]==L[N4VT][[LocalColor[LocalSMColor]]],LocalColor[LocalSMColor]}]},{LocalSMColor,1,4}]

Clear[SetNo,LocalVierNum,LocalColor]

Dimensionally reducing 4D holoraumy to 1D holoraumy V-tilde, show that this is identical to first dimensionally reducing the transformation laws and then generating 1D holoraumy V-tilde as above

Initiate the 16 x 16 x 16 table:
HoloraumyDD16[I_,J_,khat_]:=

HoloraumyDD16 = Table[0,{II,1,16},{JJ,1,16},{khat,1,16}];

HoloraumyDD16[I,J,khat] defined off of
HoloraumyDDRHS[tileft,tiright][FieldNumber][[k,c,i,a,j,b]]
ti = 1 no tilde, ti =2 tilde
Field number = 7 no tilded fermion
Field numbrer = 14 tilded fermion
Indices identifications:
I= 8(tileft - 1)+ 4(ii-1) + aa
J = 8 (tiright -1)+4 (jj-1) + bb
khat = (8/7)(14 - FieldNumber) + 4 (kk-1) + cc
-i and SignDti factor below is because HoloraumyDD16:= [D,D]Ψ16 = [D,D](-i) SignDti[tiLeftD]SignDti[tiRightD] Ψ4D = -i SignDti[tiLeftD]SignDti[tiRightD]TableHoloraumyDD[tiLeftD,tiRightD]

IorJindex[ti_,ii_,aa_]:=8(ti-1)+4(ii-1)+aa
khatindex[FieldNumber_,kk_,cc_]:=(8/7)(14-FieldNumber)+4(kk-1)+cc
SignDti[1]=1; 
SignDti[2]=-1;  (* tilded D have minus sign in their 16 color embedding, see Eq. (6.6) in the manuscript. *)

IorJindex[1,2,3]
SignDti[2]

Do[HoloraumyDD16[[IorJindex[tileft,ii,aa],IorJindex[tiright,jj,bb],khatindex[FieldNumber,kk,cc]]] = (-I)SignDti[tileft]SignDti[tiright]TableHoloraumyDD[tileft,tiright][FieldNumber][[kk,cc,ii,aa,jj,bb]]/.AdinkraReduce/.PsiTo16,{tileft ,1,2},{tiright,1,2},{FieldNumber,{7,14}},{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]

HoloraumyDD16[[1,2]]

Make the Vtilde[N4VTFrom4DHoloRed] (modified from LTable function in Adinkra.m) and compare wtih Vtilde[N4VT]. These are identical

VtildeTerms[N4VTFrom4DHoloRed][II_,JJ_][khat_,lhat_]:= Coefficient[HoloraumyDD16[[II,JJ,khat]], (Ψ16[lhat])'[t]]/2
Vtilde[N4VTFrom4DHoloRed]=Table[VtildeTerms[N4VTFrom4DHoloRed][II,JJ][khat,lhat],{II,1,16},{JJ,1,16},{khat,1,16},{lhat,1,16}];

Vtilde[N4VTFrom4DHoloRed]==Vtilde[N4VT]//Simplify

VT Summary

AdinkraPreliminaryReport[N4VT]

Table[LTest[N4VT][ColorIndex],{ColorIndex,1,16}]==L[N4VT]

Encoding the VC L and R matrices as in 1402.5765

S4Perm[PermCode]:= e.g. S4Perm[12,34] = (12)(34) = permutations of the rows
S4PermJim[PermCode]:=Inverse[S4Perm] (Jim’s convention is to permute the columns, not the rows)

S4Perm[0]=BC4Perm[0,0,0,0];
S4Perm[12,34]=BC4Perm[0,0,0,1234];
S4Perm[13,24]=BC4Perm[0,0,0,1324];
S4Perm[14,23]=BC4Perm[0,0,0,1423];
S4Perm[12]=BC4Perm[0,0,12,0];
S4Perm[34]=BC4Perm[0,0,12,1234];
S4Perm[1324]=BC4Perm[0,0,12,1324];
S4Perm[1423]=BC4Perm[0,0,12,1423];
S4Perm[13]=BC4Perm[0,0,13,0];
S4Perm[1234]=BC4Perm[0,0,13,1234];
S4Perm[24]=BC4Perm[0,0,13,1324];
S4Perm[1432]=BC4Perm[0,0,13,1423];
S4Perm[23]=BC4Perm[0,0,23,0];
S4Perm[1342]=BC4Perm[0,0,23,1234];
S4Perm[1243]=BC4Perm[0,0,23,1324];
S4Perm[14]=BC4Perm[0,0,23,1423];
S4Perm[123]=BC4Perm[0,0,123,0];
S4Perm[134]=BC4Perm[0,0,123,1234];
S4Perm[243]=BC4Perm[0,0,123,1324];
S4Perm[142]=BC4Perm[0,0,123,1423];
S4Perm[132]=BC4Perm[0,0,132,0];
S4Perm[234]=BC4Perm[0,0,132,1234];
S4Perm[124]=BC4Perm[0,0,132,1324];
S4Perm[143]=BC4Perm[0,0,132,1423];
S4PermMatrixForm[aa_,bb_]:=S4Perm[aa,bb]//MatrixForm
S4PermMatrixForm[aa_]:=S4Perm[aa]//MatrixForm
S4PermJim[PermCode_]:=Inverse[S4Perm[PermCode]]
S4PermJim[PermCode1_,PermCode2_]:=Inverse[S4Perm[PermCode1,PermCode2]]
S4PermJimMatrixForm[PermCode_]:=S4PermJim[PermCode]//MatrixForm
S4PermJimMatrixForm[PermCode1_,PermCode2_]:=S4PermJim[PermCode1,PermCode2]//MatrixForm

S4PermMatrixForm[143]

S4PermJimMatrixForm[143]

Encoding in the permutation notation of 1402.5765

L,R matrices for N=4 VC multiplet,
I=1

LTermsMatrixForm[Rep_][Color_]:=LTerms[Rep][Color]//MatrixForm
RTermsMatrixForm[Rep_][Color_]:=RTerms[Rep][Color]//MatrixForm

LocalColor = 1;
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[1][1],SignMatrix4[10].S4PermJim[243]]+Outer[Times,VierZ[1][2],SignMatrix4[10].S4PermJim[243]]+Outer[Times,VierZ[1][3],SignMatrix4[10].S4PermJim[243]]+Outer[Times,VierZ[1][4],SignMatrix4[10].S4PermJim[1243]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,VierZ[1][1],SignMatrix4[12].S4PermJim[234]]+Outer[Times,VierZ[1][2],SignMatrix4[12].S4PermJim[234]]+Outer[Times,VierZ[1][3],SignMatrix4[12].S4PermJim[234]]+Outer[Times,VierZ[1][4],SignMatrix4[12].S4PermJim[1342]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor]

I=2, the Vierz[1][4] term is corrected here from a typo in 1402.5765

LocalColor = 2;
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[1][1],SignMatrix4[12].S4PermJim[123]]+Outer[Times,VierZ[1][2],SignMatrix4[12].S4PermJim[123]]+Outer[Times,VierZ[1][3],SignMatrix4[12].S4PermJim[123]]+Outer[Times,VierZ[1][4],SignMatrix4[12].S4PermJim[23]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,VierZ[1][1],SignMatrix4[9].S4PermJim[132]]+Outer[Times,VierZ[1][2],SignMatrix4[9].S4PermJim[132]]+Outer[Times,VierZ[1][3],SignMatrix4[9].S4PermJim[132]]+Outer[Times,VierZ[1][4],SignMatrix4[10].S4PermJim[23]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor]

Vierz[1][4] term for I=2 in question directly from the transformation laws Eq. (4.2) in 1402.5765: it is indeed (Subscript[(12), b])(23) as corrected above instead of (Subscript[(4), b])(23) that was apparently a typo in 1402.5765

Sum[γstdown[1][[2,bb]]λ[bb],{bb,1,4}]

Sum[γstdown[2][[2,bb]]λ[bb],{bb,1,4}]

Sum[γstdown[3][[2,bb]]λ[bb],{bb,1,4}]

Sum[I γ5γ[0][[2,bb]]λ[bb],{bb,1,4}]

I=3

LocalColor = 3;
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[1][1],SignMatrix4[6].S4PermJim[134]]+Outer[Times,VierZ[1][2],SignMatrix4[6].S4PermJim[134]]+Outer[Times,VierZ[1][3],SignMatrix4[6].S4PermJim[134]]+Outer[Times,VierZ[1][4],SignMatrix4[0].S4PermJim[14]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,VierZ[1][1],SignMatrix4[10].S4PermJim[143]]+Outer[Times,VierZ[1][2],SignMatrix4[10].S4PermJim[143]]+Outer[Times,VierZ[1][3],SignMatrix4[10].S4PermJim[143]]+Outer[Times,VierZ[1][4],SignMatrix4[0].S4PermJim[14]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor]

I=4

LocalColor = 4;
LocalVierNum = 1;
LChiralPerm = 142;
LVectorPerm = 1342;
LBoolean = {0,0,0,6};
RChiralPerm = 124;
RVectorPerm = 1243;
RBoolean = {0,0,0,9};
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[LBoolean[[1]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[LBoolean[[2]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[LBoolean[[3]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[LBoolean[[4]]].S4PermJim[LVectorPerm]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,VierZ[1][1],SignMatrix4[RBoolean[[1]]].S4PermJim[RChiralPerm]]+Outer[Times,VierZ[1][2],SignMatrix4[RBoolean[[2]]].S4PermJim[RChiralPerm]]+Outer[Times,VierZ[1][3],SignMatrix4[RBoolean[[3]]].S4PermJim[RChiralPerm]]+Outer[Times,VierZ[1][4],SignMatrix4[RBoolean[[4]]].S4PermJim[RVectorPerm]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor,LocalVierNum,LChiralPerm,LVectorPerm,LBoolean,RChiralPerm,RVectorPerm,RBoolean]

GA satisfied for IJ = 1,2,3,4 subset after including correction to I=2 as above

Table[LTerms[N4VC][II].RTerms[N4VC][JJ]+LTerms[N4VC][JJ].RTerms[N4VC][II]==2KroneckerDelta[II,JJ]IdentityMatrix[16],{II,1,4},{JJ,1,4}]//MatrixForm
Table[RTerms[N4VC][II].LTerms[N4VC][JJ]+RTerms[N4VC][JJ].LTerms[N4VC][II]==2KroneckerDelta[II,JJ]IdentityMatrix[16],{II,1,4},{JJ,1,4}]//MatrixForm

I=5,6,7,8: (Subscript[L, I]^([3]))
First I = 5

LocalColor = 5;
LocalVierNum = 2;
LChiralPerm = 243;
LVectorPerm = 1243;
LBoolean = {15,0,2,13};
RChiralPerm = 234;
RVectorPerm = 1342;
RBoolean = {15,0,8,7};
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[LBoolean[[1]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[LBoolean[[2]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[LBoolean[[3]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[LBoolean[[4]]].S4PermJim[LVectorPerm]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,Transpose[VierZ[LocalVierNum][1]],SignMatrix4[RBoolean[[1]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][2]],SignMatrix4[RBoolean[[2]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][3]],SignMatrix4[RBoolean[[3]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][4]],SignMatrix4[RBoolean[[4]]].S4PermJim[RVectorPerm]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor,LocalVierNum,LChiralPerm,LVectorPerm,LBoolean,RChiralPerm,RVectorPerm,RBoolean]

I = 6

LocalColor = 6;
LocalVierNum = 2;
LChiralPerm = 123;
LVectorPerm = 23;
LBoolean = {9,6,4,11};
RChiralPerm = 132;
RVectorPerm = 23;
RBoolean = {10,5,1,13};
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[LBoolean[[1]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[LBoolean[[2]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[LBoolean[[3]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[LBoolean[[4]]].S4PermJim[LVectorPerm]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,Transpose[VierZ[LocalVierNum][1]],SignMatrix4[RBoolean[[1]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][2]],SignMatrix4[RBoolean[[2]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][3]],SignMatrix4[RBoolean[[3]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][4]],SignMatrix4[RBoolean[[4]]].S4PermJim[RVectorPerm]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LocalColor
RTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor,LocalVierNum,LChiralPerm,LVectorPerm,LBoolean,RChiralPerm,RVectorPerm,RBoolean]

I = 7

LocalColor = 7;
LocalVierNum = 2;
LChiralPerm = 134;
LVectorPerm = 14;
LBoolean = {3,12,14,7};
RChiralPerm = 143;
RVectorPerm = 14;
RBoolean = {6,9,11,14};
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[LBoolean[[1]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[LBoolean[[2]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[LBoolean[[3]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[LBoolean[[4]]].S4PermJim[LVectorPerm]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,Transpose[VierZ[LocalVierNum][1]],SignMatrix4[RBoolean[[1]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][2]],SignMatrix4[RBoolean[[2]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][3]],SignMatrix4[RBoolean[[3]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][4]],SignMatrix4[RBoolean[[4]]].S4PermJim[RVectorPerm]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LocalColor
RTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor,LocalVierNum,LChiralPerm,LVectorPerm,LBoolean,RChiralPerm,RVectorPerm,RBoolean]

I = 8

LocalColor = 8;
LocalVierNum = 2;
LChiralPerm = 142;
LVectorPerm = 1342;
LBoolean = {5,10,8,1};
RChiralPerm = 124;
RVectorPerm = 1243;
RBoolean = {12,3,2,4};
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[LBoolean[[1]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[LBoolean[[2]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[LBoolean[[3]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[LBoolean[[4]]].S4PermJim[LVectorPerm]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,Transpose[VierZ[LocalVierNum][1]],SignMatrix4[RBoolean[[1]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][2]],SignMatrix4[RBoolean[[2]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][3]],SignMatrix4[RBoolean[[3]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][4]],SignMatrix4[RBoolean[[4]]].S4PermJim[RVectorPerm]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LocalColor
RTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor,LocalVierNum,LChiralPerm,LVectorPerm,LBoolean,RChiralPerm,RVectorPerm,RBoolean]

GA satisfied for IJ = 5,6,7,8 subset

Table[LTerms[N4VC][II].RTerms[N4VC][JJ]+LTerms[N4VC][JJ].RTerms[N4VC][II]==2KroneckerDelta[II,JJ]IdentityMatrix[16],{II,5,8},{JJ,5,8}]//MatrixForm
Table[RTerms[N4VC][II].LTerms[N4VC][JJ]+RTerms[N4VC][JJ].LTerms[N4VC][II]==2KroneckerDelta[II,JJ]IdentityMatrix[16],{II,5,8},{JJ,5,8}]//MatrixForm

I=9,10,11,12: (Subscript[L, I]^([1]))
First I = 9

LocalColor = 9;
LocalVierNum = 4;
LChiralPerm = 243;
LVectorPerm = 1243;
LBoolean = {2,15,0,13};
RChiralPerm = 234;
RVectorPerm = 1342;
RBoolean = {8,15,0,7};
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[LBoolean[[1]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[LBoolean[[2]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[LBoolean[[3]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[LBoolean[[4]]].S4PermJim[LVectorPerm]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,Transpose[VierZ[LocalVierNum][1]],SignMatrix4[RBoolean[[1]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][2]],SignMatrix4[RBoolean[[2]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][3]],SignMatrix4[RBoolean[[3]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][4]],SignMatrix4[RBoolean[[4]]].S4PermJim[RVectorPerm]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LocalColor
RTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor,LocalVierNum,LChiralPerm,LVectorPerm,LBoolean,RChiralPerm,RVectorPerm,RBoolean]

I = 10

LocalColor = 10;
LocalVierNum = 4;
LChiralPerm = 123;
LVectorPerm = 23;
LBoolean = {4,9,6,11};
RChiralPerm = 132;
RVectorPerm = 23;
RBoolean = {1,10,5,13};
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[LBoolean[[1]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[LBoolean[[2]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[LBoolean[[3]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[LBoolean[[4]]].S4PermJim[LVectorPerm]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,Transpose[VierZ[LocalVierNum][1]],SignMatrix4[RBoolean[[1]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][2]],SignMatrix4[RBoolean[[2]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][3]],SignMatrix4[RBoolean[[3]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][4]],SignMatrix4[RBoolean[[4]]].S4PermJim[RVectorPerm]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LocalColor
RTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor,LocalVierNum,LChiralPerm,LVectorPerm,LBoolean,RChiralPerm,RVectorPerm,RBoolean]

I = 11

LocalColor = 11;
LocalVierNum = 4;
LChiralPerm = 134;
LVectorPerm = 14;
LBoolean = {14,3,12,7};
RChiralPerm = 143;
RVectorPerm = 14;
RBoolean = {11,6,9,14};
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[LBoolean[[1]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[LBoolean[[2]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[LBoolean[[3]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[LBoolean[[4]]].S4PermJim[LVectorPerm]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,Transpose[VierZ[LocalVierNum][1]],SignMatrix4[RBoolean[[1]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][2]],SignMatrix4[RBoolean[[2]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][3]],SignMatrix4[RBoolean[[3]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][4]],SignMatrix4[RBoolean[[4]]].S4PermJim[RVectorPerm]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LocalColor
RTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor,LocalVierNum,LChiralPerm,LVectorPerm,LBoolean,RChiralPerm,RVectorPerm,RBoolean]

I = 12

LocalColor = 12;
LocalVierNum = 4;
LChiralPerm = 142;
LVectorPerm = 1342;
LBoolean = {8,5,10,1};
RChiralPerm = 124;
RVectorPerm = 1243;
RBoolean = {2,12,3,4};
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[LBoolean[[1]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[LBoolean[[2]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[LBoolean[[3]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[LBoolean[[4]]].S4PermJim[LVectorPerm]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,Transpose[VierZ[LocalVierNum][1]],SignMatrix4[RBoolean[[1]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][2]],SignMatrix4[RBoolean[[2]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][3]],SignMatrix4[RBoolean[[3]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][4]],SignMatrix4[RBoolean[[4]]].S4PermJim[RVectorPerm]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LocalColor
RTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor,LocalVierNum,LChiralPerm,LVectorPerm,LBoolean,RChiralPerm,RVectorPerm,RBoolean]

GA satisfied for IJ = 9,10,11,12 subset

Table[LTerms[N4VC][II].RTerms[N4VC][JJ]+LTerms[N4VC][JJ].RTerms[N4VC][II]==2KroneckerDelta[II,JJ]IdentityMatrix[16],{II,9,12},{JJ,9,12}]//MatrixForm
Table[RTerms[N4VC][II].LTerms[N4VC][JJ]+RTerms[N4VC][JJ].LTerms[N4VC][II]==2KroneckerDelta[II,JJ]IdentityMatrix[16],{II,9,12},{JJ,9,12}]//MatrixForm

I=13,14,15,16: (Subscript[L, I]^([2]))
First I = 13, correcting a typo here from 1402.5765: should be (1243) instead of (1234) for the VierZ[3][4] term

LocalColor = 13;
LocalVierNum = 3;
LChiralPerm = 243;
LVectorPerm = 1243;
LBoolean = {0,2,15,13};
RChiralPerm = 234;
RVectorPerm = 1342;
RBoolean = {0,8,15,7};
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[LBoolean[[1]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[LBoolean[[2]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[LBoolean[[3]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[LBoolean[[4]]].S4PermJim[LVectorPerm]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,Transpose[VierZ[LocalVierNum][1]],SignMatrix4[RBoolean[[1]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][2]],SignMatrix4[RBoolean[[2]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][3]],SignMatrix4[RBoolean[[3]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][4]],SignMatrix4[RBoolean[[4]]].S4PermJim[RVectorPerm]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LocalColor
RTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor,LocalVierNum,LChiralPerm,LVectorPerm,LBoolean,RChiralPerm,RVectorPerm,RBoolean]

Vierz[3][4] term for I=13 in question directly from the transformation laws Eq. (4.2) in 1402.5765: it is indeed (Subscript[(13), b])(1243) as corrected above instead of (Subscript[(13), b])(1234) that was apparently a typo in 1402.5765

Sum[-γstdown[1][[1,bb]]ψ[2][bb],{bb,1,4}]

Sum[-γstdown[2][[1,bb]]ψ[2][bb],{bb,1,4}]

Sum[-γstdown[3][[1,bb]]ψ[2][bb],{bb,1,4}]

Sum[I γ5γ[0][[1,bb]]ψ[2][bb],{bb,1,4}]

I = 14

LocalColor = 14;
LocalVierNum = 3;
LChiralPerm = 123;
LVectorPerm = 23;
LBoolean = {6,4,9,11};
RChiralPerm = 132;
RVectorPerm = 23;
RBoolean = {5,1,10,13};
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[LBoolean[[1]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[LBoolean[[2]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[LBoolean[[3]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[LBoolean[[4]]].S4PermJim[LVectorPerm]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,Transpose[VierZ[LocalVierNum][1]],SignMatrix4[RBoolean[[1]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][2]],SignMatrix4[RBoolean[[2]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][3]],SignMatrix4[RBoolean[[3]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][4]],SignMatrix4[RBoolean[[4]]].S4PermJim[RVectorPerm]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LocalColor
RTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor,LocalVierNum,LChiralPerm,LVectorPerm,LBoolean,RChiralPerm,RVectorPerm,RBoolean]

I = 15

LocalColor = 15;
LocalVierNum = 3;
LChiralPerm = 134;
LVectorPerm = 14;
LBoolean = {12,14,3,7};
RChiralPerm = 143;
RVectorPerm = 14;
RBoolean = {9,11,6,14};
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[LBoolean[[1]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[LBoolean[[2]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[LBoolean[[3]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[LBoolean[[4]]].S4PermJim[LVectorPerm]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,Transpose[VierZ[LocalVierNum][1]],SignMatrix4[RBoolean[[1]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][2]],SignMatrix4[RBoolean[[2]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][3]],SignMatrix4[RBoolean[[3]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][4]],SignMatrix4[RBoolean[[4]]].S4PermJim[RVectorPerm]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LocalColor
RTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor,LocalVierNum,LChiralPerm,LVectorPerm,LBoolean,RChiralPerm,RVectorPerm,RBoolean]

I = 16

LocalColor = 16;
LocalVierNum = 3;
LChiralPerm = 142;
LVectorPerm = 1342;
LBoolean = {10,8,5,1};
RChiralPerm = 124;
RVectorPerm = 1243;
RBoolean = {3,2,12,4};
LTerms[N4VC][LocalColor]=Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[LBoolean[[1]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[LBoolean[[2]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[LBoolean[[3]]].S4PermJim[LChiralPerm]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[LBoolean[[4]]].S4PermJim[LVectorPerm]]//ArrayFlatten;
RTerms[N4VC][LocalColor]=Outer[Times,Transpose[VierZ[LocalVierNum][1]],SignMatrix4[RBoolean[[1]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][2]],SignMatrix4[RBoolean[[2]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][3]],SignMatrix4[RBoolean[[3]]].S4PermJim[RChiralPerm]]+Outer[Times,Transpose[VierZ[LocalVierNum][4]],SignMatrix4[RBoolean[[4]]].S4PermJim[RVectorPerm]]//ArrayFlatten;
LTerms[N4VC][LocalColor]==Inverse[RTerms[N4VC][LocalColor]] ==Transpose[RTerms[N4VC][LocalColor]] 
LocalColor
RTermsMatrixForm[N4VC][LocalColor]
Clear[LocalColor,LocalVierNum,LChiralPerm,LVectorPerm,LBoolean,RChiralPerm,RVectorPerm,RBoolean]

GA satisfied for IJ = 13,14,15,16 subset with above change for (Subscript[L, 13]) implemented

Table[LTerms[N4VC][II].RTerms[N4VC][JJ]+LTerms[N4VC][JJ].RTerms[N4VC][II]==2KroneckerDelta[II,JJ]IdentityMatrix[16],{II,13,16},{JJ,13,16}]//MatrixForm
Table[RTerms[N4VC][II].LTerms[N4VC][JJ]+RTerms[N4VC][JJ].LTerms[N4VC][II]==2KroneckerDelta[II,JJ]IdentityMatrix[16],{II,13,16},{JJ,13,16}]//MatrixForm

Save into L[N4VC] and generate Adinkra Data
In summary: orthogonal-trace condition satisfied for L and R. The V and Vtilde are NOT all linearly independent, and do not satisfy so(16)

L[N4VC]= Table[LTerms[N4VC][II],{II,1,16}];
R[N4VC]=Table[RTerms[N4VC][JJ],{JJ,1,16}];

L[N4VC][[16]]==LTerms[N4VC][16]

AdinkraPreliminaryReport[N4VC]

GenerateAdinkraData[N4VC]

AdinkraReport[N4VC,5]

soNTest[VtildesoN[N4VC]]

soNTest[VsoN[N4VC]]

Encoding the VC L and R matrices from 1402.5765 in terms of SM-i

I = 1,2,3,4

{PrintL[SMI][1],PrintL[SMII][1]}

PrintL[N4VC][1]

SetNo = 1;  (* which group of 4 colors out of 16: set 1 is 1-4, set 2 is 5-8, set 3 is 9-12, set 4 is 13-16 *)
LocalColor[LocalSMColor_]:= LocalSMColor + 4*(SetNo-1)
LocalVierNum = 1;
SignBoolean = {0,0,0,0};

Do[{LTest[N4VC][LocalColor[LocalSMColor]]= Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[SignBoolean[[1]]].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[SignBoolean[[2]]].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[SignBoolean[[3]]].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[SignBoolean[[4]]].L[SMII][[LocalSMColor]]]//ArrayFlatten,
Print[{LTest[N4VC][LocalColor[LocalSMColor]]==L[N4VC][[LocalColor[LocalSMColor]]],LocalColor[LocalSMColor]}]},{LocalSMColor,1,4}]

LTest[N4VC][2]//MatrixForm

PrintL[N4VC][2]

Clear[SetNo,LocalVierNum,SignBoolean]

I = 5,6,7,8

{PrintL[SMI][1],PrintL[SMII][1]}

PrintL[N4VC][5]

SetNo = 2;  (* which group of 4 colors out of 16: set 1 is 1-4, set 2 is 5-8, set 3 is 9-12, set 4 is 13-16 *)
LocalVierNum = 2;
SignBoolean = {5,10,8,7};

Do[{LTest[N4VC][LocalColor[LocalSMColor]]= Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[SignBoolean[[1]]].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[SignBoolean[[2]]].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[SignBoolean[[3]]].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[SignBoolean[[4]]].L[SMII][[LocalSMColor]]]//ArrayFlatten,
Print[{LTest[N4VC][LocalColor[LocalSMColor]]==L[N4VC][[LocalColor[LocalSMColor]]],LocalColor[LocalSMColor]}]},{LocalSMColor,1,4}]

LTest[N4VC][8]//MatrixForm

PrintL[N4VC][8]

Clear[SetNo,LocalVierNum,SignBoolean]

I = 9,10,11,12

{PrintL[SMI][1],PrintL[SMII][1]}

PrintL[N4VC][9]

SetNo = 3;  (* which group of 4 colors out of 16: set 1 is 1-4, set 2 is 5-8, set 3 is 9-12, set 4 is 13-16 *)
LocalVierNum = 4;
SignBoolean = {8,5,10,7};

Do[{LTest[N4VC][LocalColor[LocalSMColor]]= Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[SignBoolean[[1]]].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[SignBoolean[[2]]].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[SignBoolean[[3]]].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[SignBoolean[[4]]].L[SMII][[LocalSMColor]]]//ArrayFlatten,
Print[{LTest[N4VC][LocalColor[LocalSMColor]]==L[N4VC][[LocalColor[LocalSMColor]]],LocalColor[LocalSMColor]}]},{LocalSMColor,1,4}]

LTest[N4VC][11]//MatrixForm

PrintL[N4VC][11]

Clear[SetNo,LocalVierNum,SignBoolean]

I = 13,14,15,16

{PrintL[SMI][1],PrintL[SMII][1]}

PrintL[N4VC][13]

SetNo = 4;  (* which group of 4 colors out of 16: set 1 is 1-4, set 2 is 5-8, set 3 is 9-12, set 4 is 13-16 *)
LocalVierNum = 3;
SignBoolean = {10,8,5,7};

Do[{LTest[N4VC][LocalColor[LocalSMColor]]= Outer[Times,VierZ[LocalVierNum][1],SignMatrix4[SignBoolean[[1]]].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][2],SignMatrix4[SignBoolean[[2]]].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][3],SignMatrix4[SignBoolean[[3]]].L[SMI][[LocalSMColor]]]+Outer[Times,VierZ[LocalVierNum][4],SignMatrix4[SignBoolean[[4]]].L[SMII][[LocalSMColor]]]//ArrayFlatten,
Print[{LTest[N4VC][LocalColor[LocalSMColor]]==L[N4VC][[LocalColor[LocalSMColor]]],LocalColor[LocalSMColor]}]},{LocalSMColor,1,4}]

LTest[N4VC][14]//MatrixForm

PrintL[N4VC][14]

Clear[SetNo,LocalVierNum,SignBoolean]

VC Summary

AdinkraPreliminaryReport[N4VC]

Table[LTest[N4VC][ColorIndex],{ColorIndex,1,16}]==L[N4VC]

Gadget between N4VT and N4VC

dmin[4]

dmin[16]

Gadget[N4VT,N4VC]

Gadget[N4VT,N4VT]

Gadget[N4VC,N4VC]

thetaRadians = ArcCos[Gadget[N4VT,N4VC]/(Sqrt[Gadget[N4VC,N4VC]]*Sqrt[Gadget[N4VT,N4VT]])]

ThetaDegrees = thetaRadians*(180/π)//N

so(8) subalgebras for VT and VC

Singular for V-tilde corresponding to the D,D-tilde cross terms (I = 1,8 with J = 9,16 and antisymmetric in IJ of course)

SingularList[N4VT] ={};
SingularList[N4VC]={};

Do[If[Det[Vtilde[N4VT][[II,JJ]]]==0, SingularList[N4VT] = Join[SingularList[N4VT],{{II,JJ}}]],{II,1,15},{JJ,II+1,16}]
Do[If[Det[Vtilde[N4VC][[II,JJ]]]==0, SingularList[N4VC] = Join[SingularList[N4VC],{{II,JJ}}]],{II,1,15},{JJ,II+1,16}]

SingularList[N4VT]

SingularList[N4VC]

Find algebra for non-singular, 8 color sub-holoraumy:
Vtilde[N4VTN8a] := 16 x 16 matrices for the first 8 colors (I,J = 1,8)
Vtilde[N4VTN8b] := 16 x 16 matrices for the second 8 colors (I,J = 9,16)
Each satisfy so(8): [(Subscript[M, IJ]),(Subscript[M, KL])] = i ((Subscript[δ, I[L])(Subscript[M, K]J])-(Subscript[δ, J[L])(Subscript[M, K]I]))  
with M = 1/2 Vtilde
so [(Subscript[(OverTilde[V]), IJ]),(Subscript[(OverTilde[V]), KL])] =2 i ((Subscript[δ, I[L])(Subscript[(OverTilde[V]), K]J])-(Subscript[δ, J[L])(Subscript[(OverTilde[V]), K]I]))

Vtilde[N4VTN8a] = Table[Vtilde[N4VT][[II,JJ,khat,lhat]],{II,1,8},{JJ,1,8},{khat,1,16},{lhat,1,16}];
Vtilde[N4VTN8b] = Table[Vtilde[N4VT][[II,JJ,khat,lhat]],{II,9,16},{JJ,9,16},{khat,1,16},{lhat,1,16}];

LinearlyIndependent[Vtilde[N4VTN8a]]

LinearlyIndependent[Vtilde[N4VTN8b]]

soNTest[1/2Vtilde[N4VTN8a]]

soNTest[1/2Vtilde[N4VTN8b]]

Check algebra 8 color subsets of N4VC with 16 x 16 matrices
Vtilde[N4VCN8a] := 16 x 16 matrices for the first 8 colors (I,J = 1,8)
Vtilde[N4VCN8b] := 16 x 16 matrices for the second 8 colors (I,J = 9,16)
Vtilde[N4VCN8c] := 16 x 16 matrices for  (I,J = 5,6,7,8,13,14,15,16)
Vtilde[N4VCN8c] := 16 x 16 matrices for  (I,J = 1,2,3,4,13,14,15,16)
None satisfy so(8): [(Subscript[M, IJ]),(Subscript[M, KL])] = i ((Subscript[δ, I[L])(Subscript[M, K]J])-(Subscript[δ, J[L])(Subscript[M, K]I]))  
with M = 1/2 Vtilde
so [(Subscript[(OverTilde[V]), IJ]),(Subscript[(OverTilde[V]), KL])] =2 i ((Subscript[δ, I[L])(Subscript[(OverTilde[V]), K]J])-(Subscript[δ, J[L])(Subscript[(OverTilde[V]), K]I]))

Vtilde[N4VCN8a] = Table[Vtilde[N4VC][[II,JJ,khat,lhat]],{II,1,8},{JJ,1,8},{khat,1,16},{lhat,1,16}];
Vtilde[N4VCN8b] = Table[Vtilde[N4VC][[II,JJ,khat,lhat]],{II,9,16},{JJ,9,16},{khat,1,16},{lhat,1,16}];
Vtilde[N4VCN8c] = Table[Vtilde[N4VC][[II,JJ,khat,lhat]],{II,{5,6,7,8,13,14,15,16}},{JJ,{5,6,7,8,13,14,15,16}},{khat,1,16},{lhat,1,16}];
Vtilde[N4VCN8d] = Table[Vtilde[N4VC][[II,JJ,khat,lhat]],{II,{1,2,3,4,13,14,15,16}},{JJ,{1,2,3,4,13,14,15,16}},{khat,1,16},{lhat,1,16}];

LinearlyIndependent[Vtilde[N4VCN8a]]

LinearlyIndependent[Vtilde[N4VCN8b]]

LinearlyIndependent[Vtilde[N4VCN8c]]

LinearlyIndependent[Vtilde[N4VCN8d]]

soNTest[1/2Vtilde[N4VCN8a]]

soNTest[1/2Vtilde[N4VCN8b]]

soNTest[1/2Vtilde[N4VCN8c]]

soNTest[1/2Vtilde[N4VCN8d]]

Check algebra 8 color subsets of N4VC with 8 x 8 matrices
Vtilde[N4VCN8d8a] := 8 x 8 matrices for the set of colors I,J= 1,2,3,4,5,6,7,8, and fields  khat, lhat = 9,10,11,12,13,14,15,16
Vtilde[N4VCN8d8b] := 8 x 8 matrices for the set of colors I,J= 1,2,3,4,9,10,11,12, and fields  khat, lhat = 1,2,3,4,13,14,15,16
Vtilde[N4VCN8d8c] := 8 x 8 matrices for the set of colors I,J= 1,2,3,4,13,14,15,16, and fields  khat, lhat = 5,6,7,8,13,14,15,16
All satisfy so(8): [(Subscript[M, IJ]),(Subscript[M, KL])] = i ((Subscript[δ, I[L])(Subscript[M, K]J])-(Subscript[δ, J[L])(Subscript[M, K]I]))  
with M = 1/2 Vtilde
so [(Subscript[(OverTilde[V]), IJ]),(Subscript[(OverTilde[V]), KL])] =2 i ((Subscript[δ, I[L])(Subscript[(OverTilde[V]), K]J])-(Subscript[δ, J[L])(Subscript[(OverTilde[V]), K]I]))  
Not surprising as these all are images of 4D, N=2 8x8 vector-chiral systems

Vtilde[N4VCN8d8a] = Table[Vtilde[N4VC][[II,JJ,khat,lhat]],{II,1,8},{JJ,1,8},{khat,9,16},{lhat,9,16}];
Vtilde[N4VCN8d8b] = Table[Vtilde[N4VC][[II,JJ,khat,lhat]],{II,{1,2,3,4,9,10,11,12}},{JJ,{1,2,3,4,9,10,11,12}},{khat,{1,2,3,4,13,14,15,16}},{lhat,{1,2,3,4,13,14,15,16}}];
Vtilde[N4VCN8d8c] = Table[Vtilde[N4VC][[II,JJ,khat,lhat]],{II,{1,2,3,4,13,14,15,16}},{JJ,{1,2,3,4,13,14,15,16}},{khat,{5,6,7,8,13,14,15,16}},{lhat,{5,6,7,8,13,14,15,16}}];

LinearlyIndependent[Vtilde[N4VCN8d8a]]

LinearlyIndependent[Vtilde[N4VCN8d8b]]

LinearlyIndependent[Vtilde[N4VCN8d8c]]

soNTest[1/2Vtilde[N4VCN8d8a]]

soNTest[1/2Vtilde[N4VCN8d8b]]

soNTest[1/2Vtilde[N4VCN8d8c]]

Save L, R, Vtilde, etc. Ran once and saved, so commented out here.

SetDirectory[NotebookDirectory[]]
DeleteFile["N44DOffShellAdinkraData.m"]
LoadString = "L, R, GALR, GARL, V, Vtilde, VPM, VtildePM, ncis, ntrans, chi0 for Reps N4VT, N4VC, SMI, SMII, SMIII, and SMIV are loaded"
Save["N44DOffShellAdinkraData.m",{L, R, GALR, GARL, V, Vtilde, VPM,VtildePM, ncis, ntrans, chi0,  LoadString}]