The Jacobator

We show that the jacobator of in the Killing superalgebra of a d=11 sugra background vanishes.

Explicit Cl(1,10) realisation

Abbreviations,...

Abbreviations and Messages

<<  LinearAlgebra`MatrixManipulation`;
Off[Solve::svars, General::spell1];

N-tuples

\(\(\(diffQ[s_List] := \ Equal[Length[s], \ Length[Union[s]]]\)\(
\)
    \)\)
\(auxtuples[s_List, \ 
        n_Integer] := 
\ \ Union[
\ \ \ \ Union\  /@ 
\ \ \ \ \ \ Select[
\ \ \ \ \ \ \ Flatten[
\ \ \ \ \ \ \ \ \ \(Outer[\ List, \ ##\ ] &\)\  @@ \ 
                Table[s, {n}], \ n - 1], \ diffQ]]\)

tuples[s_List,  n_Integer] := 
        If[2  n  <  Length[s],  
          auxtuples[s, n],  Complement[s, #] &  /@  
            auxtuples[s, Length[s] - n]];

Tensor products of matrices

The pauli matrices

σ[0]  =  IdentityMatrix[2];
σ[1]  =  {{0, 1},  {1, 0}};
σ[2]  =  {{0, -I}, {I, 0}};
σ[3]  =  {{1, 0}, {0, -1}};

Computing tensor products

tp[a_List, b_List]  :=  BlockMatrix[Outer[Times, a, b]]
tp[a__List, b_List, c_List]  :=  tp[a, tp[b, c]]
tp[i_Integer] := σ[i]
\(\(\(tp[i__Integer, j_Integer] := \ 
      tp\  @@ \ \(σ\  /@ \ List[i, j]\)\)\(
\)
    \)\)

Usage: e.g.,  tp[σ[1],σ[2],σ[3],σ[2],σ[1]]

A real representation for the Clifford algebra Cl(1,10)

Cl(0,2)

γ2[1] =  σ[1];  γ2[2] =  σ[
        3];  γ2[3] =  -I σ[2];

Cl(7) from the octonions

A (Subscript[G, 2]) invariant 3-form in (ℝ^7)= Im𝕆

\(ϕ[___, a_, ___, a_, ___]\  = \ 0; \ \  (*\ 
      antisymmetry\ 1\ *) 
ϕ[a___, b_, c_, 
        d___]\  := \ \(-\ ϕ[a, c, b, 
            d]\)\  /; \ \(\(b\)\(>\)\(c\)\(\ \)\( (*\ 
            antisymmetry\ 2\ *) \)\)\)
ϕ[1, 2, 5]  =  1;
ϕ[1, 3, 6]  =  1;
ϕ[1, 4, 7]  =  1;
ϕ[2, 3, 7]  = -1;
ϕ[2, 4, 6]  =  1;
ϕ[3, 4, 5]  = -1;
ϕ[5, 6, 7]  =  1;
ϕ[__]  =  0;

Kronecker δ function

δ[a_, b_]  :=  0  /;  a  !=  b
δ[a_, a_]  =  1 ;

Octonion multiplication

μ[a_, b_, 
        c_] :=  ϕ[a, b, 
          c]  /;  ((a < b < 8)  ||  (b < a < 8))
μ[8, b_, c_] :=  δ[b, c]
μ[a_, 8, c_] :=  δ[a, c]
μ[a_, a_, 8] :=  -1  /;  (a < 8)
μ[__] = 0;

Cl(7) matrices

γ7[
        a_] :=  ( γ7[a]  =  
            Table[μ[a, b, c],  {b, 8},  {c, 8}] )  /;  (a > 
              0  &&  a < 8)

Cl(8)

γ8[
        i_] :=  ( γ8[i]  =  
            tp[γ7[i], γ2[1]] )  /;  ( 
          i < 8  &&  i > 0 )
γ8[8]  =  tp[IdentityMatrix[8], γ2[3]];

γ8[i__Integer] :=  
      Dot  @@  γ8  /@  List[i]

γ9 = γ8[1, 2, 3, 4, 5, 6, 7, 8];

Cl(0,10) ≃ Cl(8) ⊗ Cl(0,2)

Γ[
        i_]  :=  ( Γ[i]  =  
            tp[γ8[i], γ2[3]] )  /;  ( 
          i < 9  &&  i > 0 )
Γ[9]  =  
        tp[IdentityMatrix[16], γ2[1]];
Γ[10]  =  
        tp[IdentityMatrix[16], γ2[2]];

Cl(1,10)

This involves a choice of sign:

Γ[0]  =  
        Dot  @@  Γ  /@  Range[10];

Careful: this is not totally antisymmetric:

Γ[i__Integer] :=  
      Dot  @@  Γ  /@  List[i]

Metric

ϵ[i_] := If[i == 0, 1, -1]

Check of Clifford algebra relations

Union[
      Flatten[Table[Γ[a] . Γ[b] + Γ[
                b] . Γ[a] +  
            2 If[a == b, ϵ[a], 0] IdentityMatrix[
                Length[Γ[0]]], {a, 0, 10}, {b, 0, 
            10}]]]

Spinors and vectors

The vector υ associated to a spinor ζ

The isotropy of the spinor is contained in the isotropy of the vector.  The vector is time-like or null depending on the isotropy subgroup: (Subscript[SU, 5])or (Subscript[Spin, 7])semidirect (ℝ^9), respectively.

υ[ζ_List] :=  
      Table[ ζ  .  (Γ[0, i]  .  ζ),  {i, 
          0, 10}]

Quartic invariant

This quartic is negative semi-definite.   It is zero on those spinors which have (Subscript[Spin, 7])semidirect (ℝ^9)isotropy, and negative on those spinors which have (Subscript[SU, 5]) isotropy.

The Minkowskian norm of a vector

MNorm[vv_List] := First[vv]^2  -  Rest[vv]  .  Rest[vv]

The quartic of a spinor ζ

Bq[ζ_List] :=  MNorm[υ[ζ]]

Two spinor orbits

ψ[1] =  Table[If[i == 1, 1, 0], {i, 32}]

Bq[ψ[1]]

υ[ψ[1]]

ψ[2] = Table[Random[Integer], {i, 32}]

Bq[ψ[2]]

υ[ψ[2]]

Bispinors,...

Skew-symmetrised products of Γ matrices

Γs[___, i_, ___, i_, ___] = 
        0  Γ[0];

Γs[a___, i_, b___, j_, 
        c___] := -Γs[a, j, b, i, c] /; i > j

Γs[
        a__Integer] := Γs[a] = Γ[
          a]

Check

f[l_List] := Signature[l] (Γ @@ l)

test[a_, b_] := Γs[a, b] - 
        1/2 (Γ[a] . Γ[b] - Γ[
                  b] . Γ[a])

Apply[test, #] & /@ tuples[Range[0, 10], 2] // Flatten // 
      Union

test[a_, b_, c_, d_] := Γs[a, b, c, d] - 
        1/24 (Plus @@ (f /@ Permutations[{a, b, c, d}]))

Apply[test, #] & /@ tuples[Range[0, 10], 4] // Flatten // 
      Union

Bispinors

v[ψ_, 
        i_] := ψ . (Γ[0] . Γ[
              i] . ψ)

ω[ψ_, i_, 
        j_] := ψ  .  (Γ[0] . Γs[i, 
              j] . ψ)

Ω[ψ_, i_, j_, k_, l_, 
        m_] := ψ . (Γ[0] . Γs[i, j, k, l, 
              m] . ψ)

η[ψ_, i_, j_, k_, l_, m_, 
        n_] := ψ . (Γ[0] . Γs[i, j, k, l, 
              m, n] . ψ)

The jacobator

jac1[ψ_, a_, b_, c_, d_] := 
      1/6 (v[ψ, a] Γs[b, c, d] . ψ - 
            v[ψ, b] Γs[a, c, d] . ψ - 
            v[ψ, c] Γs[b, a, d] . ψ - 
            v[ψ, d] Γs[b, c, a] . ψ)

jac2[ψ_, a_, b_, c_, d_] := 
      1/12 Sum[ϵ[m] 
            v[ψ, m] Γs[a, b, c, d, m] . ψ, {m, 0, 
            10}]

jac3[ψ_, a_, b_, c_, d_] := 
      1/6 (ω[ψ, c, 
                d] Γs[a, b] . ψ + ω[ψ, a, 
                b] Γs[c, d] . ψ - ω[ψ, b, 
                d] Γs[a, c] . ψ - ω[ψ, a, 
                c] Γs[b, d] . ψ + ω[ψ, b, 
                c] Γs[a, d] . ψ + ω[ψ, a, 
                d] Γs[b, c] . ψ)

jac4[ψ_, a_, b_, c_, d_] := 
      1/24 Sum[ϵ[m] ϵ[n] η[ψ, m, n, a, b, c, 
              d] Γs[m, n] . ψ, {m, 0, 10}, {n, 0, 
            10}]

eqn[ψ_, a_, b_, c_, d_] :=  
      jac1[ψ, a, b, c, d]  +   jac2[ψ, a, b, c, d] + 
        jac3[ψ, a, b, c, d] + jac4[ψ, a, b, c, d]

equations1 = Table[
              eqn[ψ[1], a, b, c, d], {a, 0, 10}, {b, a + 1, 10}, {c, 
                b + 1, 10}, {d, c + 1, 10}] // Simplify // Flatten // 
        Union

SessionTime[]

equations2 = Table[
              eqn[ψ[2], a, b, c, d], {a, 0, 10}, {b, a + 1, 10}, {c, 
                b + 1, 10}, {d, c + 1, 10}] // Simplify // Flatten // 
        Union

equations = Table[
              eqn[ψ /@ Range[32], a, b, c, d], {a, 0, 10}, {b, a + 1, 
                10}, {c, b + 1, 10}, {d, c + 1, 10}] // Simplify // 
          Flatten // Union

SessionTime[]