The only massless $N=8$ SUGRA multiplet is given by

$(g_{\mu\nu},\psi_\mu^{\Sigma},A_\mu^{[\Sigma\Pi]},\chi_{\alpha}^{[\Sigma\Pi\Lambda]} ,\phi^{[\Sigma\Pi\Lambda\Omega]})$

where the greek upper indicies run from 1 to 8, which correspong from left to right as

  • 1 graviton
  • 8 gravitinos
  • 28 vectors
  • 56 fermions
  • 70 scalars.

I want to decompose this multiplet into multiplets of massless N=4 SUGRA, which has the following multiplets

  • Graviton multiplet; 1 graviton, 4 gravitinos, 6 vectors, 4 fermions, 1 scalar
  • Gravitino multiplet; 1 gravitino, 4 vectors, 7 fermions, 4 scalars
  • Vector multiplet; 1 vector, 4 fermions, 6 scalars.

The best I can do is try to write the N=8 Multiplet as (in terms of N=4 multiplets)

1 graviton multiplet + 4 gravitino multiplets + 6 vector multiplets.

but this still leaves me with 13 scalars left over. What am I doing wrong?


Found the issue. The scalars in the N=8 gravity multiplet are real scalars, not complex. Same for the vector multiplet.

Thus the N=8 gravity multiplet can be decomposed as 1 copy of the N=4 gravity multiplet, 4 copies of the N=4 gravitino multiplet and 6 copies of the N=4 vector multiplet.


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