4D N=4 CV No closure.nb
Demonstrates the Hypermultiplet can not be closed as well as the cross terms unhatted, hatted. There MUST be central charges.

Setting Rep and Importing Adinkra.wl

$RecursionLimit=Infinity

Rep = N4CV;

<<Adinkra.wl

BuildDate[Adinkra]

FunctionList[Adinkra]

FieldLists

FieldList:= list of all fields

FieldListDynamicalBosons = Join[{A[t,x,y,z],B[t,x,y,z]},Table[Am[mu][t,x,y,z],{mu,0,3}],{Ah[t,x,y,z],Bh[t,x,y,z],Ab[t,x,y,z],Bb[t,x,y,z]}]

FieldListAuxiliaryBosons = Join[{F[t,x,y,z],G[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[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}]

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

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]]

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 WF Hypermultiplet (Chiral-Chiral) transformation laws
Untilded, unhatted transformation laws Dia[1][ii][aa]

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

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

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

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

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

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

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

Do[Dia[1][ii][aa][Gh[t,x,y,z]]= I Sum[γ5γ[mu][[aa,bb]]D[Ψh[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][Ab[t,x,y,z]]=  Sum[sigma[1][[ii,jj]]Ψh[jj][aa][t,x,y,z],{jj,1,2}] ,{ii,1,2},{aa,1,4}]

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

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

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

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

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

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

Do[Dia[1][ii][aa][Gb[t,x,y,z]]= -Sum[sigma[2][[ii,jj]]γ5γ[mu][[aa,bb]]D[Ψh[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][Ψh[jj][bb][t,x,y,z]] =I sigma[3][[ii,jj]](Sum[γdown[mu][[aa,bb]]D[Ah[t,x,y,z],Ϛ[mu]],{mu,0,3}] - Cmetric[[aa,bb]]Fh[t,x,y,z]) + KroneckerDelta[ii,jj](Sum[-γ5γdown[mu][[aa,bb]]D[Bh[t,x,y,z],Ϛ[mu]],{mu,0,3}] + γ5down[[aa,bb]]Gh[t,x,y,z]) + I sigma[1][[ii,jj]](Sum[γdown[mu][[aa,bb]]D[Ab[t,x,y,z],Ϛ[mu]],{mu,0,3}] - Cmetric[[aa,bb]]Fb[t,x,y,z]) + I sigma[2][[ii,jj]](Sum[-γ5γdown[mu][[aa,bb]]D[Bb[t,x,y,z],Ϛ[mu]],{mu,0,3}] + γ5down[[aa,bb]]Gb[t,x,y,z]),{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]

Dia[1][1][3][Ψh[1][1][t,x,y,z]]

Vector-Chiral hatted transformation laws Dia[0][ii][aa]

Do[Dia[0][ii][aa][A[t,x,y,z]]=  Sum[W[1][ii,jj]Ψh[jj][aa][t,x,y,z] ,{jj,1,2}],{ii,1,2},{aa,1,4}]

Dia[0][1][1][A[t,x,y,z]]

Dia[0][2][3][A[t,x,y,z]]

Do[Dia[0][ii][aa][B[t,x,y,z]]= I Sum[W[2][ii,jj]γ5[[aa,bb]]Ψh[jj][bb][t,x,y,z],{jj,1,2},{bb,1,4}] ,{ii,1,2},{aa,1,4}]

Dia[0][2][1][B[t,x,y,z]]

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

Dia[0][1][2][F[t,x,y,z]]

Do[Dia[0][ii][aa][G[t,x,y,z]]= I Sum[W[4][ii,jj]γ5γ[mu][[aa,bb]]D[Ψh[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[0][1][2][G[t,x,y,z]]

Do[Dia[0][ii][aa][Am[mu][t,x,y,z]] = Sum[W[5][ii,jj]γstdown[mu][[aa,bb]]Ψh[jj][bb][t,x,y,z],{jj,1,2},{bb,1,4}],{ii,1,2},{aa,1,4},{mu,0,3}]

Dia[0][2][3][Am[1][t,x,y,z]]

Do[Dia[0][ii][aa][d[t,x,y,z]]= I Sum[W[6][ii,jj]γ5γ[mu][[aa,bb]]D[Ψh[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[0][2][2][d[t,x,y,z]]

Do[Dia[0][ii][aa][Ψ[jj][bb][t,x,y,z]] = I W[7][ii,jj]Sum[γdown[mu][[aa,bb]]D[Ah[t,x,y,z],Ϛ[mu]],{mu,0,3}] - I W[8][ii,jj] Cmetric[[aa,bb]]Fh[t,x,y,z] - W[9][ii,jj]Sum[γ5γdown[mu][[aa,bb]]D[Bh[t,x,y,z],Ϛ[mu]],{mu,0,3}] +W[10][ii,jj] γ5down[[aa,bb]]Gh[t,x,y,z]  +  I W[11][ii,jj]Sum[γdown[mu][[aa,bb]]D[Ab[t,x,y,z],Ϛ[mu]],{mu,0,3}] - I W[12][ii,jj] Cmetric[[aa,bb]]Fb[t,x,y,z] - W[13][ii,jj]Sum[γ5γdown[mu][[aa,bb]]D[Bb[t,x,y,z],Ϛ[mu]],{mu,0,3}] +W[14][ii,jj] γ5down[[aa,bb]]Gb[t,x,y,z] ,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}]

Dia[0][1][3][Ψ[1][1][t,x,y,z]]

Do[Dia[0][ii][aa][Ah[t,x,y,z]]=  Sum[X[1][ii,jj]Ψ[jj][aa][t,x,y,z],{jj,1,2}] ,{ii,1,2},{aa,1,4}]

Dia[0][1][1][Ah[t,x,y,z]]

Dia[0][1][3][Ah[t,x,y,z]]

Do[Dia[0][ii][aa][Bh[t,x,y,z]]= I Sum[X[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[0][2][1][Bh[t,x,y,z]]

Do[Dia[0][ii][aa][Fh[t,x,y,z]]= Sum[X[3][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[0][1][2][Fh[t,x,y,z]]

Do[Dia[0][ii][aa][Gh[t,x,y,z]]= I Sum[X[4][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[0][1][2][Gh[t,x,y,z]]

Do[Dia[0][ii][aa][Ab[t,x,y,z]]=  Sum[X[5][ii,jj]Ψ[jj][aa][t,x,y,z],{jj,1,2}] ,{ii,1,2},{aa,1,4}]

Dia[0][1][1][Ab[t,x,y,z]]

Dia[0][1][3][Ab[t,x,y,z]]

Do[Dia[0][ii][aa][Bb[t,x,y,z]]= I Sum[X[6][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[0][2][1][Bb[t,x,y,z]]

Do[Dia[0][ii][aa][Fb[t,x,y,z]]= Sum[X[7][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[0][1][2][Fb[t,x,y,z]]

Do[Dia[0][ii][aa][Gb[t,x,y,z]]= I Sum[X[8][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[0][1][2][Gb[t,x,y,z]]

Do[Dia[0][ii][aa][Ψh[jj][bb][t,x,y,z]] =Sum[I X[9][ii,jj] γdown[mu][[aa,bb]]D[A[t,x,y,z],Ϛ[mu]] -X[10][ii,jj]γ5γdown[mu][[aa,bb]]D[B[t,x,y,z],Ϛ[mu]],{mu,0,3}] - I X[11][ii,jj]Cmetric[[aa,bb]]F[t,x,y,z] + X[12][ii,jj]γ5down[[aa,bb]]G[t,x,y,z] + X[13][ii,jj]γ5down[[aa,bb]]d[t,x,y,z] -I (1/4)Sum[ X[14][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[0][1][3][Ψh[1][1][t,x,y,z]]

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

Lagrangian Invariance: DL[tt] w/ tt= tilde index (0 = hat, 1 = no tilde or hat) (just the CC and VS portions)
(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)(OverHat[A])) □(OverHat[A]) + ((Subscript[D, a]^i)(OverHat[B])) □(OverHat[B]) +(OverHat[F]) (Subscript[D, a]^i)(OverHat[F])  +(OverHat[G])  (Subscript[D, a]^i)(OverHat[G]) + ((Subscript[D, a]^i)(Overscript[A, ⌣])) □(Overscript[A, ⌣]) + ((Subscript[D, a]^i)(Overscript[B, ⌣])) □(Overscript[B, ⌣]) +(Overscript[F, ⌣]) (Subscript[D, a]^i)(Overscript[F, ⌣])  +(Overscript[G, ⌣])  (Subscript[D, a]^i)(Overscript[G, ⌣])  - i (Subscript[(OverHat[Ψ]), b]^j)((((γ^μ)))^bc)(∂_μ)((Subscript[D, a]^i)(Subscript[(OverHat[Ψ]), c]^j)) 
(Subscript[(OverHat[D]), a]^i)ℒ = 0  = ((Subscript[(OverHat[D]), a]^i)A) □A + ((Subscript[(OverHat[D]), a]^i)B) □B +F (Subscript[(OverHat[D]), a]^i)F +G (Subscript[(OverHat[D]), a]^i)G+d (Subscript[D, a]^i)d+ ((Subscript[(OverHat[D]), a]^i)(Subscript[A, ν]))(∂_μ)(F^(μν)) - i (Subscript[Ψ, b]^j)((((γ^μ)))^bc)(∂_μ)((Subscript[(OverHat[D]), a]^i)(Subscript[Ψ, c]^j)) + ((Subscript[(OverHat[D]), a]^i)(OverHat[A])) □(OverHat[A]) + ((Subscript[(OverHat[D]), a]^i)(OverHat[B])) □(OverHat[B]) +(OverHat[F]) (Subscript[(OverHat[D]), a]^i)(OverHat[F])  +(OverHat[G])  (Subscript[D, a]^i)(OverHat[G]) + ((Subscript[(OverHat[D]), a]^i)(Overscript[A, ⌣])) □(Overscript[A, ⌣]) + ((Subscript[(OverHat[D]), a]^i)(Overscript[B, ⌣])) □(Overscript[B, ⌣]) +(Overscript[F, ⌣]) (Subscript[(OverHat[D]), a]^i)(Overscript[F, ⌣])  +(Overscript[G, ⌣])  (Subscript[(OverHat[D]), a]^i)(Overscript[G, ⌣])  - i (Subscript[(OverHat[Ψ]), b]^j)((((γ^μ)))^bc)(∂_μ)((Subscript[(OverHat[D]), a]^i)(Subscript[(OverHat[Ψ]), c]^j))

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][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]}

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,0,1}]

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,0,1}]

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,0,1}]

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,0,1}]

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,0,1}]

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,0,1}]

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,0,1}]

DLPsi[1][[2,2]]

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

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

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

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

DLFh[tt] = {((Subscript[D, a]^i)Fh) Fh,((Subscript[(OverTilde[D]), a]^i)Fh) Fh}

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

DLGh[tt] = {((Subscript[D, a]^i)Gh) Gh,((Subscript[(OverTilde[D]), a]^i)Gh) Gh}

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

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

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

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

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

DLFb[tt] = {((Subscript[D, a]^i)Fb) Fb,((Subscript[(OverTilde[D]), a]^i)Fb) Fb}

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

DLGb[tt] = {((Subscript[D, a]^i)Gb) Gb,((Subscript[(OverTilde[D]), a]^i)Gb) Gb}

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

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

Do[DLPsih[tt] = Table[ - I Sum[Ψh[jj][bb][t,x,y,z]γup[mu][[bb,cc]]D[Dia[tt][ii][aa][Ψh[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,0,1}]

DLPsih[0]

Lagrangian invariance

Do[DLTotal[tt]=(DLA[tt] +DLB[tt] +DLF[tt] +DLG[tt] +DLd[tt] +DLAm[tt] + DLPsi[tt] +DLAh[tt] +DLBh[tt] +DLFh[tt] +DLGh[tt]  +DLAb[tt] +DLBb[tt] +DLFb[tt] +DLGb[tt]+ DLPsih[tt] ),{tt,0,1}];

DLTotal[1]

Do[DLFinal[tt] = Collect[DLTotal[tt],FieldListWithDerivatives,Simplify],{tt,0,1}]

Lagrangian invariance: untilded DLFinal[1]

DLFinal[1]

Lagrangian Parameters

XList = Table[X[Index][ii,jj],{Index,1,14},{ii,1,2},{jj,1,2}]//Flatten

CoefficientEquations[0] = Table[Coefficient[DLFinal[0][[ii,aa]],FieldProductsList[[FieldProductsIndex]]]==0,{ii,1,2},{aa,1,4},{FieldProductsIndex,1,Length[FieldProductsList]}]//Flatten;

LagrangianParameters = Solve[CoefficientEquations[0],XList]//Flatten

Lagrangian Invariance: tilded

DLFinal[0]/.LagrangianParameters

We have 
X[1][i,j] =W[7][i,j] 
X[2][i,j] =W[9][i,j] 
X[3][i,j] =W[8][i,j] 
X[4][i,j]=W[10][i,j] 
X[5][i,j] =W[11][i,j] 
X[6][i,j] =W[13][i,j] 
X[7][i,j] =W[12][i,j]  
X[8][i,j] =W[14][i,j]  
X[9][i,j] =W[1][i,j]  
X[10][i,j] =W[2][i,j]  
X[11][i,j] =W[3][i,j] 
X[12][i,j] =W[4][i,j] 
X[13][i,j] =W[6][i,j] 
X[14][i,j] =W[5][i,j]

XWPairings = {{1,7},{2,9},{3,8},{4,10},{5,11},{6,13},{7,12},{8,14},{9,1},{10,2},{11,3},{12,4},{13,6},{14,5}};

Table[(X[XWPairings[[uvpair,1]]][ii,jj]/.LagrangianParameters)==W[XWPairings[[uvpair,2]]][ii,jj],{uvpair,1,14},{ii,1,2},{jj,1,2}] == Table[True,{uvpair,1,14},{ii,1,2},{jj,1,2}]

Lagrangian Summary

DLFinal[1]==(DLFinal[0]/.LagrangianParameters)=={{0,0,0,0},{0,0,0,0}}

Algebra {(Subscript[D, a]^i),(Subscript[D, b]^j)} and {(Subscript[(OverHat[D]), a]^i),(Subscript[(OverHat[D]), b]^j)}
Neither close on the hypermultiplet, there MUST be central charges

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,0,1},{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

AlgebraDD[tt1,tt2][Ψh[1][1][t,x,y,z]]

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

AlgebraDD[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}]

AllgebraDDResults[tt1,tt2][FieldIndex] = True if the algebra closes fo Field FieldIndex, False if not. tt1,tt2 are tilded indices: , tt1,tt2=0 for hatted, tt1,tt2=1 for untilded and unhatted. set tt1 = tt2 = 1 for the next two unhatted subsections.

tt1 = 1;
tt2 = 1;

4D, N=2 Vector Algebra, both unhatted {(Subscript[D, a]^i),(Subscript[D, b]^j)}

{(Subscript[D, a]^i),(Subscript[D, b]^j)}A= 2 i (Subscript[(δ^ij)((γ^μ)), ab])(∂_μ)A , AlgebraDDResult[1,1][FieldNumber], FieldNumber = 1

AlgebraDDResult[tt1,tt2][1]=(AlgebraDD[tt1,tt2][A[t,x,y,z]] ==Closure[A[t,x,y,z]])//Simplify

{(Subscript[D, a]^i),(Subscript[D, b]^j)}B= 2 i (Subscript[(δ^ij)((γ^μ)), ab])(∂_μ)B , FieldNumber = 2

AlgebraDDResult[tt1,tt2][2]=(AlgebraDD[tt1,tt2][B[t,x,y,z]] ==Closure[B[t,x,y,z]])//Simplify

{(Subscript[D, a]^i),(Subscript[D, b]^j)}F= 2 i (Subscript[(δ^ij)((γ^μ)), ab])(∂_μ)F, FieldNumber = 3

AlgebraDDResult[tt1,tt2][3]=(AlgebraDD[tt1,tt2][F[t,x,y,z]] ==Closure[F[t,x,y,z]])//Simplify

{(Subscript[D, a]^i),(Subscript[D, b]^j)}G= 2 i (Subscript[(δ^ij)((γ^μ)), ab])(∂_μ)G, FieldNumber = 4

AlgebraDDResult[tt1,tt2][4]=(AlgebraDD[tt1,tt2][G[t,x,y,z]] ==Closure[G[t,x,y,z]])//Simplify

{(Subscript[D, a]^i),(Subscript[D, b]^j)}d= 2 i (Subscript[(δ^ij)((γ^μ)), ab])(∂_μ)d, FieldNumber = 5

AlgebraDDResult[tt1,tt2][5]=(AlgebraDD[tt1,tt2][d[t,x,y,z]] ==Closure[d[t,x,y,z]])//Simplify

{(Subscript[D, a]^i),(Subscript[D, b]^j)}(Subscript[Ψ, c]^k)= 2 i (Subscript[(δ^ij)((γ^μ)), ab])(∂_μ)(Subscript[Ψ, c]^k), FieldNumber = 6

AlgebraDDResult[tt1,tt2][6]=Table[AlgebraDD[tt1,tt2][Ψ[kk][cc][t,x,y,z]],{kk,1,2},{cc,1,4}] ==Table[Closure[Ψ[kk][cc][t,x,y,z]],{kk,1,2},{cc,1,4}]//Simplify

{(Subscript[D, a]^i),(Subscript[D, b]^j)}(Subscript[A, μ])= 2 i (Subscript[(δ^ij)((γ^ν)), ab])(∂_ν)(Subscript[A, μ]) -(∂_μ)(Subscript[λ, ab]^ij) , FieldNumber = 7
(Subscript[λ, ab]^ij)= 2 i (Subscript[(δ^ij)((γ^ν)), ab])(Subscript[A, ν]) - 2 i ((((σ^2)))^ij)( i (Subscript[C, ab])A - (Subscript[((γ^5)), ab])B)

λ[tt1,tt2][ii_,jj_][aa_,bb_][t,x,y,z] := 2 I KroneckerDelta[ii,jj]Sum[γdown[nu][[aa,bb]]Am[nu][t,x,y,z],{nu,0,3}] - 2I sigma[2][[ii,jj]](I Cmetric[[aa,bb]] A[t,x,y,z] - γ5down[[aa,bb]]B[t,x,y,z])

λTable[tt1,tt2][t,x,y,z] = Table[λ[tt1,tt2][ii,jj][aa,bb][t,x,y,z],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

AlgebraDDResult[tt1,tt2][7]=(Table[AlgebraDD[tt1,tt2][Am[mu][t,x,y,z]],{mu,0,3}] ==Table[Closure[Am[mu][t,x,y,z]]- D[λTable[tt1,tt2][t,x,y,z],Ϛ[mu]],{mu,0,3}] )//Simplify

4D, N=2 Chiral-Chiral, both unhatted {(Subscript[D, a]^i),(Subscript[D, b]^j)}

, FieldNumber = 8

AlgebraDDResult[tt1,tt2][8]=(AlgebraDD[tt1,tt2][Ah[t,x,y,z]] ==Closure[Ah[t,x,y,z]] + Table[I sigma[2][[ii,jj]] 2 I Cmetric[[aa,bb]]Fb[t,x,y,z],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}])//Simplify

, FieldNumber = 9

AlgebraDDResult[tt1,tt2][9]=(AlgebraDD[tt1,tt2][Ab[t,x,y,z]] ==Closure[Ab[t,x,y,z]] - Table[I sigma[2][[ii,jj]] 2 I Cmetric[[aa,bb]]Fh[t,x,y,z],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}])//Simplify

, FieldNumber = 10

AlgebraDDResult[tt1,tt2][10]=(AlgebraDD[tt1,tt2][Bh[t,x,y,z]] ==Closure[Bh[t,x,y,z]] +  Table[I sigma[2][[ii,jj]] 2 I Cmetric[[aa,bb]]Gb[t,x,y,z],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}])//Simplify

, FieldNumber = 11

AlgebraDDResult[tt1,tt2][11]=(AlgebraDD[tt1,tt2][Bb[t,x,y,z]] ==Closure[Bb[t,x,y,z]] -  Table[I sigma[2][[ii,jj]] 2 I Cmetric[[aa,bb]]Gh[t,x,y,z],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}])//Simplify

, FieldNumber = 12

AlgebraDDResult[tt1,tt2][12]=(AlgebraDD[tt1,tt2][Fh[t,x,y,z]] ==Closure[Fh[t,x,y,z]] + Table[I sigma[2][[ii,jj]]2 I Cmetric[[aa,bb]]Lap[Ab[t,x,y,z]],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}])//Simplify

, FieldNumber = 13

AlgebraDDResult[tt1,tt2][13]=(AlgebraDD[tt1,tt2][Fb[t,x,y,z]] ==Closure[Fb[t,x,y,z]] - Table[I sigma[2][[ii,jj]]2 I Cmetric[[aa,bb]]Lap[Ah[t,x,y,z]],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}])//Simplify

, FieldNumber = 14

AlgebraDDResult[tt1,tt2][14]=(AlgebraDD[tt1,tt2][Gh[t,x,y,z]] ==Closure[Gh[t,x,y,z]] + Table[I sigma[2][[ii,jj]]2 I Cmetric[[aa,bb]]Lap[Bb[t,x,y,z]],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}])//Simplify

, FieldNumber = 15

AlgebraDDResult[tt1,tt2][15]=(AlgebraDD[tt1,tt2][Gb[t,x,y,z]] ==Closure[Gb[t,x,y,z]] - Table[I sigma[2][[ii,jj]]2 I Cmetric[[aa,bb]]Lap[Bh[t,x,y,z]],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}])//Simplify

, FieldNumber = 16

Closure[Ψh[1][2][t,x,y,z]][[2,4,2,4]]

Table[Closure[Ψh[kk][cc][t,x,y,z]],{kk,1,2},{cc,1,4}][[1,2,2,4,2,4]]

AlgebraDDResult[tt1,tt2][16]=(Table[AlgebraDD[tt1,tt2][Ψh[kk][cc][t,x,y,z]],{kk,1,2},{cc,1,4}] ==Table[Closure[Ψh[kk][cc][t,x,y,z]],{kk,1,2},{cc,1,4}] - Table[sigma[2][[ii,jj]]Sum[sigma[2][[kk,rr]]2I Cmetric[[aa,bb]]γ[mu][[cc,dd]]D[Ψh[rr][dd][t,x,y,z],Ϛ[mu]],{mu,0,3},{dd,1,4},{rr,1,2}],{kk,1,2},{cc,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}])//Simplify

4D, N=2 Summary, unhatted

Table[AlgebraDDResult[tt1,tt2][Index],{Index,1,16}]==Table[True,{Index,1,16}]
tt1
tt2

tt1 = 0, tt2 = 0 for both hatted D’s

Clear[tt1,tt2];
tt1 = 0;
tt2 = 0;

4D, N=2 Vector Algebra, both hatted {(Subscript[(OverHat[D]), a]^i),(Subscript[(OverHat[D]), b]^j)}

Write ((((Subscript[W, n])))^ij) = (((Subscript[cw, nμ])((l^μ)))^ij) where (l^μ)=(I,(σ^1),i (σ^2),(σ^3)) the generators of ℓ(2,R)

lgen[0] = sigma[0];
lgen[1] = sigma[1];
lgen[2] = I sigma[2];
lgen[3] = sigma[3];

Wtolgen= Table[W[nn_][ii,jj]-> Sum[cw[nn][mu]lgen[mu][[ii,jj]],{mu,0,3}],{ii,1,2},{jj,1,2}]//Flatten

W[2][1,2]/.Wtolgen

cwList = Table[cw[nn][mu],{nn,1,12},{mu,0,3}]//Flatten

{(Subscript[(OverHat[D]), a]^i),(Subscript[(OverHat[D]), b]^j)}A= 2 i (Subscript[(δ^ij)((γ^μ)), ab])(∂_μ)A , FieldNumber = 1

LocalField = A[t,x,y,z];
LocalFieldNumber = 1;

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=(Collect[(AlgebraDD[tt1,tt2][LocalField] -Closure[LocalField])/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify]);

AlgebraDDEqs[tt1,tt2][LocalFieldNumber] = DeleteCases[Table[(Coefficient[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber][[ii,aa,jj,bb]],FieldListWithDerivatives[[FieldListIndex]]])==0,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{FieldListIndex,1,Length[FieldListWithDerivatives]}]//Simplify//Flatten,True]//DeleteDuplicates;

Length[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]]

Clear[LocalField,LocalFieldNumber]

{(Subscript[(OverHat[D]), a]^i),(Subscript[(OverHat[D]), b]^j)}B= 2 i (Subscript[(δ^ij)((γ^μ)), ab])(∂_μ)B , FieldNumber = 2

LocalField = B[t,x,y,z];
LocalFieldNumber = 2;

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=(Collect[(AlgebraDD[tt1,tt2][LocalField] -Closure[LocalField])/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify]);

AlgebraDDEqs[tt1,tt2][LocalFieldNumber] = DeleteCases[Table[(Coefficient[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber][[ii,aa,jj,bb]],FieldListWithDerivatives[[FieldListIndex]]])==0,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{FieldListIndex,1,Length[FieldListWithDerivatives]}]//Simplify//Flatten,True]//DeleteDuplicates;

Length[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]]

Clear[LocalField,LocalFieldNumber]

{(Subscript[(OverHat[D]), a]^i),(Subscript[(OverHat[D]), b]^j)}F= 2 i (Subscript[(δ^ij)((γ^μ)), ab])(∂_μ)F , FieldNumber = 3

LocalField = F[t,x,y,z];
LocalFieldNumber = 3;

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=(Collect[(AlgebraDD[tt1,tt2][LocalField] -Closure[LocalField])/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify]);

AlgebraDDEqs[tt1,tt2][LocalFieldNumber] = DeleteCases[Table[(Coefficient[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber][[ii,aa,jj,bb]],FieldListWithDerivatives[[FieldListIndex]]])==0,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{FieldListIndex,1,Length[FieldListWithDerivatives]}]//Simplify//Flatten,True]//DeleteDuplicates;

Length[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]]

Clear[LocalField,LocalFieldNumber]

{(Subscript[(OverHat[D]), a]^i),(Subscript[(OverHat[D]), b]^j)}G= 2 i (Subscript[(δ^ij)((γ^μ)), ab])(∂_μ)G , FieldNumber = 4

LocalField = G[t,x,y,z];
LocalFieldNumber = 4;

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=(Collect[(AlgebraDD[tt1,tt2][LocalField] -Closure[LocalField])/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify]);

AlgebraDDEqs[tt1,tt2][LocalFieldNumber] = DeleteCases[Table[(Coefficient[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber][[ii,aa,jj,bb]],FieldListWithDerivatives[[FieldListIndex]]])==0,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{FieldListIndex,1,Length[FieldListWithDerivatives]}]//Simplify//Flatten,True]//DeleteDuplicates;

Length[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]]

Clear[LocalField,LocalFieldNumber]

{(Subscript[(OverHat[D]), a]^i),(Subscript[(OverHat[D]), b]^j)}d= 2 i (Subscript[(δ^ij)((γ^μ)), ab])(∂_μ)d , FieldNumber = 5

LocalField = d[t,x,y,z];
LocalFieldNumber = 5;

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=(Collect[(AlgebraDD[tt1,tt2][LocalField] -Closure[LocalField])/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify]);

AlgebraDDEqs[tt1,tt2][LocalFieldNumber] = DeleteCases[Table[(Coefficient[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber][[ii,aa,jj,bb]],FieldListWithDerivatives[[FieldListIndex]]])==0,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{FieldListIndex,1,Length[FieldListWithDerivatives]}]//Simplify//Flatten,True]//DeleteDuplicates;

Length[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]]

Clear[LocalField,LocalFieldNumber]

{(Subscript[(OverHat[D]), a]^i),(Subscript[(OverHat[D]), b]^j)}(Subscript[A, μ])= 2 i (Subscript[(δ^ij)((γ^ν)), ab])(∂_ν)(Subscript[A, μ]) -(∂_μ)(Subscript[(OverHat[(OverHat[λ])]), ab]^ij)  , 
FieldNumber = 7
(Subscript[(OverHat[(OverHat[λ])]), ab]^ij)= 2 i (((a1δ)^ij) + a2 ((((σ^1)))^ij)+ a3 ((((σ^3)))^ij) )(Subscript[((γ^ν)), ab])(Subscript[A, ν]) - 2 i ((((σ^2)))^ij)( i a4 (Subscript[C, ab])A - a5 (Subscript[((γ^5)), ab])B)

LocalFieldNumber = 7;

λ[tt1,tt2][ii_,jj_][aa_,bb_][t,x,y,z] := 2 I (a1 KroneckerDelta[ii,jj] + a2 sigma[1][[ii,jj]] + a3 sigma[3][[ii,jj]])Sum[γdown[nu][[aa,bb]]Am[nu][t,x,y,z],{nu,0,3}] - 2I sigma[2][[ii,jj]](I a4 Cmetric[[aa,bb]] A[t,x,y,z] -a5 γ5down[[aa,bb]]B[t,x,y,z])

λTable[tt1,tt2][t,x,y,z] = Table[λ[tt1,tt2][ii,jj][aa,bb][t,x,y,z],{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4}];

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=Collect[(Table[AlgebraDD[tt1,tt2][Am[mu][t,x,y,z]],{mu,0,3}] -Table[Closure[Am[mu][t,x,y,z]]- D[λTable[tt1,tt2][t,x,y,z],Ϛ[mu]],{mu,0,3}] )/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify];

Dimensions[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]]

AlgebraDDEqs[tt1,tt2][LocalFieldNumber] = DeleteCases[Table[(Coefficient[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber][[mu,ii,aa,jj,bb]],FieldListWithDerivatives[[FieldListIndex]]])==0,{mu,1,4},{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{FieldListIndex,1,Length[FieldListWithDerivatives]}]//Simplify//Flatten,True]//DeleteDuplicates;

Length[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]]

Clear[LocalFieldNumber]

Solving Equations, solution is stored in FinalSolutions[tt1,tt2][ss]
Solution Set ss= 1 for this the Vector Algebra

Clear[ss];
ss = 1;

BosonNumbersList[tt1,tt2][ss] = {1,2,3,4,5,7};
FieldNumbersList[tt1,tt2][ss] = {1,2,3,4,5,7,16};

BosonAlgebraDDEqs[tt1,tt2][ss]=Table[AlgebraDDEqs[tt1,tt2][BosonNumbersList[tt1,tt2][ss][[Index]]],{Index,1,Length[BosonNumbersList[tt1,tt2][ss]]}]//Flatten//Simplify//DeleteDuplicates;

Dimensions[BosonAlgebraDDEqs[tt1,tt2][ss]]

BosonSolutions[tt1,tt2][ss] = Solve[BosonAlgebraDDEqs[tt1,tt2][ss],Join[cwList,{a1,a2,a3,a4,a5}]];

Dimensions[BosonSolutions[tt1,tt2][ss]]

{(Subscript[(OverHat[D]), a]^i),(Subscript[(OverHat[D]), b]^j)}(Subscript[(OverHat[Ψ]), c]^k)= 2 i (Subscript[(δ^ij)((γ^μ)), ab])(∂_μ)(Subscript[(OverHat[Ψ]), c]^k)  , FieldNumber = 16 should, is satisfied BosonSolutions[tt1,tt2][ss]

LocalFieldNumber = 16;

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=Collect[(Table[AlgebraDD[tt1,tt2][Ψh[kk][cc][t,x,y,z]],{kk,1,2},{cc,1,4}] -Table[Closure[Ψh[kk][cc][t,x,y,z]],{kk,1,2},{cc,1,4}])/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify];

AlgebraDDEqs[tt1,tt2][LocalFieldNumber]=(AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber] == 0 * AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]);

AlgebraDDEqs[tt1,tt2][LocalFieldNumber][[1,1,1,1,1,1]] (* Sanity check that it is not trivially satisfied, it is not *)

ClosureResult[tt1,tt2][ss]=Table[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]/.BosonSolutions[tt1,tt2][ss][[SolnNum]]//Simplify,{SolnNum,1,Length[BosonSolutions[tt1,tt2][ss]]}] == Table[True,{SolnNum,1,Length[BosonSolutions[tt1,tt2][ss]]}]

tt2
tt2

Clear[LocalFieldNumber]

Final Check on Closure, worked once, takes a few minutes so commented out

(*tt1
tt2*)

(*Table[(AlgebraDDMinusClosure[tt1,tt2][FieldNumbersList[tt1,tt2][ss][[FieldIndex]]]/.BosonSolutions[tt1,tt2][ss][[SolnNumber]]//Simplify) == 0*AlgebraDDMinusClosure[tt1,tt2][FieldNumbersList[tt1,tt2][ss][[FieldIndex]]],{FieldIndex,1,Length[FieldNumbersList[tt1,tt2][ss]]},{SolnNumber,1,Length[BosonSolutions[tt1,tt2][ss]]}]==Table[True,{FieldIndex,1,Length[FieldNumbersList[tt1,tt2][ss]]},{SolnNumber,1,Length[BosonSolutions[tt1,tt2][ss]]}]*)

Closure summary of N=2 Vector hat-hat fermion using all solutions that close the bosons:

ClosureResult[tt1,tt2][ss]

4D, N=2 Chiral-Chiral Algebra, both hatted {(Subscript[(OverHat[D]), a]^i),(Subscript[(OverHat[D]), b]^j)}, no solution even for {(Subscript[(OverHat[D]), a]^i),(Subscript[(OverHat[D]), b]^j)}Ah= 2 i (Subscript[(δ^ij)((γ^μ)), ab])(∂_μ)Ah, There MUST be central charges.

{(Subscript[(OverHat[D]), a]^i),(Subscript[(OverHat[D]), b]^j)}Ah= 2 i (Subscript[(δ^ij)((γ^μ)), ab])(∂_μ)Ah , FieldNumber = 8
16 Equations all can not be simultaneoulsy solved, 6 can and possibly up to 12 though it crashes the kernel.

LocalField = Ah[t,x,y,z];
LocalFieldNumber = 8;

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=(Collect[(AlgebraDD[tt1,tt2][LocalField] -Closure[LocalField])/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify]);

AlgebraDDEqs[tt1,tt2][LocalFieldNumber] = DeleteCases[Table[(Coefficient[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber][[ii,aa,jj,bb]],FieldListWithDerivatives[[FieldListIndex]]])==0,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{FieldListIndex,1,Length[FieldListWithDerivatives]}]//Simplify//Flatten,True]//DeleteDuplicates;

Length[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]]

Solve[AlgebraDDEqs[tt1,tt2][LocalFieldNumber],cwList]

Solve[Table[AlgebraDDEqs[tt1,tt2][LocalFieldNumber][[EqNo]],{EqNo,1,6}],cwList]

(*Solve[Table[AlgebraDDEqs[tt1,tt2][LocalFieldNumber][[EqNo]],{EqNo,1,7}],cwList]*) (* Takes at least 20 minutes, never been run to completion *)

(*Solve[Table[AlgebraDDEqs[tt1,tt2][LocalFieldNumber][[EqNo]],{EqNo,1,8}],cwList]*) (* Takes at least 8 minutes, never been run to completion *)

(*Solve[Table[AlgebraDDEqs[tt1,tt2][LocalFieldNumber][[EqNo]],{EqNo,1,12}],cwList]*) (* This kills the kernel *)

Solve[Table[AlgebraDDEqs[tt1,tt2][LocalFieldNumber][[EqNo]],{EqNo,1,13}],cwList]

Clear[LocalField,LocalFieldNumber]

Algebra {(Subscript[D, a]^i),(Subscript[(OverHat[D]), b]^j)}= 0 up to gauge transformations. There is no solution, even without imposing the unhatted results. There MUST be central charges.

tt1 = 1, tt2 = 2

Clear[tt1,tt2,ss]
tt1 = 0
tt2 = 1

{(Subscript[D, a]^i),(Subscript[(OverHat[D]), b]^j)}Ah= 0 , FieldNumber = 8

LocalField = Ah[t,x,y,z];
LocalFieldNumber = 8;

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=(Collect[(AlgebraDD[tt1,tt2][LocalField])/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify]);

AlgebraDDEqs[tt1,tt2][LocalFieldNumber] = DeleteCases[Table[(Coefficient[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber][[ii,aa,jj,bb]],FieldListWithDerivatives[[FieldListIndex]]])==0,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{FieldListIndex,1,Length[FieldListWithDerivatives]}]//Simplify//Flatten,True]//DeleteDuplicates;

Length[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]]

Solve[AlgebraDDEqs[tt1,tt2][LocalFieldNumber],cwList]

Clear[LocalField,LocalFieldNumber]

{(Subscript[D, a]^i),(Subscript[(OverHat[D]), b]^j)}Ab= 0 , FieldNumber = 9

LocalField = Ab[t,x,y,z];
LocalFieldNumber = 9;

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=(Collect[(AlgebraDD[tt1,tt2][LocalField])/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify]);

AlgebraDDEqs[tt1,tt2][LocalFieldNumber] = DeleteCases[Table[(Coefficient[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber][[ii,aa,jj,bb]],FieldListWithDerivatives[[FieldListIndex]]])==0,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{FieldListIndex,1,Length[FieldListWithDerivatives]}]//Simplify//Flatten,True]//DeleteDuplicates;

Length[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]]

Solve[AlgebraDDEqs[tt1,tt2][LocalFieldNumber],cwList]

Clear[LocalField,LocalFieldNumber]

{(Subscript[D, a]^i),(Subscript[(OverHat[D]), b]^j)}Bh= 0 , FieldNumber = 10

LocalField = Bh[t,x,y,z];
LocalFieldNumber = 10;

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=(Collect[(AlgebraDD[tt1,tt2][LocalField])/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify]);

AlgebraDDEqs[tt1,tt2][LocalFieldNumber] = DeleteCases[Table[(Coefficient[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber][[ii,aa,jj,bb]],FieldListWithDerivatives[[FieldListIndex]]])==0,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{FieldListIndex,1,Length[FieldListWithDerivatives]}]//Simplify//Flatten,True]//DeleteDuplicates;

Length[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]]

Solve[AlgebraDDEqs[tt1,tt2][LocalFieldNumber],cwList]

Clear[LocalField,LocalFieldNumber]

{(Subscript[D, a]^i),(Subscript[(OverHat[D]), b]^j)}Bb= 0 , FieldNumber = 11

LocalField = Bb[t,x,y,z];
LocalFieldNumber = 11;

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=(Collect[(AlgebraDD[tt1,tt2][LocalField])/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify]);

AlgebraDDEqs[tt1,tt2][LocalFieldNumber] = DeleteCases[Table[(Coefficient[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber][[ii,aa,jj,bb]],FieldListWithDerivatives[[FieldListIndex]]])==0,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{FieldListIndex,1,Length[FieldListWithDerivatives]}]//Simplify//Flatten,True]//DeleteDuplicates;

Length[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]]

Solve[AlgebraDDEqs[tt1,tt2][LocalFieldNumber],cwList]

Clear[LocalField,LocalFieldNumber]

{(Subscript[D, a]^i),(Subscript[(OverHat[D]), b]^j)}Fh= 0 , FieldNumber = 12

LocalField = Fh[t,x,y,z];
LocalFieldNumber = 12;

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=(Collect[(AlgebraDD[tt1,tt2][LocalField])/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify]);

AlgebraDDEqs[tt1,tt2][LocalFieldNumber] = DeleteCases[Table[(Coefficient[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber][[ii,aa,jj,bb]],FieldListWithDerivatives[[FieldListIndex]]])==0,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{FieldListIndex,1,Length[FieldListWithDerivatives]}]//Simplify//Flatten,True]//DeleteDuplicates;

Length[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]]

Solve[AlgebraDDEqs[tt1,tt2][LocalFieldNumber],cwList]

Clear[LocalField,LocalFieldNumber]

{(Subscript[D, a]^i),(Subscript[(OverHat[D]), b]^j)}Fb= 0 , FieldNumber = 13

LocalField = Fb[t,x,y,z];
LocalFieldNumber = 13;

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=(Collect[(AlgebraDD[tt1,tt2][LocalField])/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify]);

AlgebraDDEqs[tt1,tt2][LocalFieldNumber] = DeleteCases[Table[(Coefficient[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber][[ii,aa,jj,bb]],FieldListWithDerivatives[[FieldListIndex]]])==0,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{FieldListIndex,1,Length[FieldListWithDerivatives]}]//Simplify//Flatten,True]//DeleteDuplicates;

Length[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]]

Solve[AlgebraDDEqs[tt1,tt2][LocalFieldNumber],cwList]

Clear[LocalField,LocalFieldNumber]

{(Subscript[D, a]^i),(Subscript[(OverHat[D]), b]^j)}Gh= 0 , FieldNumber = 14

LocalField = Gh[t,x,y,z];
LocalFieldNumber = 14;

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=(Collect[(AlgebraDD[tt1,tt2][LocalField])/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify]);

AlgebraDDEqs[tt1,tt2][LocalFieldNumber] = DeleteCases[Table[(Coefficient[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber][[ii,aa,jj,bb]],FieldListWithDerivatives[[FieldListIndex]]])==0,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{FieldListIndex,1,Length[FieldListWithDerivatives]}]//Simplify//Flatten,True]//DeleteDuplicates;

Length[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]]

Solve[AlgebraDDEqs[tt1,tt2][LocalFieldNumber],cwList]

Clear[LocalField,LocalFieldNumber]

{(Subscript[D, a]^i),(Subscript[(OverHat[D]), b]^j)}Gb= 0 , FieldNumber = 15

LocalField = Gb[t,x,y,z];
LocalFieldNumber = 15;

AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber]=(Collect[(AlgebraDD[tt1,tt2][LocalField])/.LagrangianParameters/.Wtolgen,FieldListWithDerivatives,Simplify]);

AlgebraDDEqs[tt1,tt2][LocalFieldNumber] = DeleteCases[Table[(Coefficient[AlgebraDDMinusClosure[tt1,tt2][LocalFieldNumber][[ii,aa,jj,bb]],FieldListWithDerivatives[[FieldListIndex]]])==0,{ii,1,2},{aa,1,4},{jj,1,2},{bb,1,4},{FieldListIndex,1,Length[FieldListWithDerivatives]}]//Simplify//Flatten,True]//DeleteDuplicates;

Length[AlgebraDDEqs[tt1,tt2][LocalFieldNumber]]

Solve[AlgebraDDEqs[tt1,tt2][LocalFieldNumber],cwList]

Clear[LocalField,LocalFieldNumber]

Solving Equations, solution is stored in FinalSolutions[tt1,tt2][ss]
Solution Set ss= 2 for the chiral-chiral algebra.
Even for the first 48 equations, the only solution is for there to be no SUSY transformations for the vector multiplet. For the full set of 192 equations, there is no solution at all.

Clear[ss];
ss = 2;

BosonNumbersList[tt1,tt2][ss] = {8,9,10,11,12,13,14,15};
FieldNumbersList[tt1,tt2][ss] = {8,9,10,11,12,13,14,15,6};

BosonAlgebraDDEqs[tt1,tt2][ss]=Table[AlgebraDDEqs[tt1,tt2][BosonNumbersList[tt1,tt2][ss][[Index]]],{Index,1,Length[BosonNumbersList[tt1,tt2][ss]]}]//Flatten//Simplify//DeleteDuplicates;

Dimensions[BosonAlgebraDDEqs[tt1,tt2][ss]]

BosonSolutions[tt1,tt2][ss] = Solve[BosonAlgebraDDEqs[tt1,tt2][ss],cwList]

Dimensions[BosonSolutions[tt1,tt2][ss]]

Solve[Table[BosonAlgebraDDEqs[tt1,tt2][ss][[SolnNo]],{SolnNo,1,48}],cwList]