# Decomposing massless N=8 SUGRA multiplet into multiplets of massless N=4

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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?

This post imported from StackExchange Physics at 2016-01-19 22:07 (UTC), posted by SE-user ryanp16

edited Jan 19, 2016

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