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starting a stand-alone Section-entry (to be !include
ed as a section into D=11 supergravity and into D’Auria-Fré formulation of supergravity)
So far it contains lead-in and statement of the result, in mild but suggestive paraphrase of CDF91, §III.8.5.
I am going to spell out at least parts of the proof, with some attention to the prefactors.
Regarding the last remaining lemma, I am preparing to type out in this nLab page the remaining discussion of the torsion Bianchi identity and the gravitino Bianchi identity up to order $\psi^2$ —showing that to this order this is equivalent to the gravitino eom.
(a derivation which, incidentally, takes several seconds on my computer to verify in Mathematica — I wonder if the original authors really computed this by hand).
But first a question, in case anyone knowledgeable is reading:
Is there any discussion in the literature of the gravitino Bianchi at order $\psi^3$?
There is a non-trivial Fierz identity to verify here, namely that (or whether)
$- \tfrac{1}{6} \tfrac{1}{3!} \Gamma_{[a_1 a_2 a_3}\psi \, \big( \overline{\psi} \,\Gamma_{a_4}\, \psi \big) \;-\; \tfrac{1}{12} \tfrac{1}{4!} \Gamma_{b a_1 \cdots a_4}\psi \, \big( \overline{\psi} \,\Gamma^b\, \psi \big) \;+\; \tfrac{1}{4 \cdot 6} \Gamma_{[a_1 a_2}\psi \, \big( \overline{\psi} \,\Gamma_{a_3 a_4]}\, \psi \big) \;+\; \tfrac{1}{4 \cdot 6} \tfrac{1}{24} \Gamma^{b_1 b_2} \psi \, \big( \overline{\psi} \,\Gamma_{b_1 b_2 a_1 \cdots a_4}\, \psi \big) \;=\; 0 \,.$Is this discussed anywhere in existing literature?
[edit: I have now managed to prove that this expression indeed vanishes. ]
The detailed computation/proof is now spelled out in section 3 at Flux Quantization on 11d Superspace, complete with computer algebra checks.
I will try to bring in more of the details into the entry here, but probably not before next month.
Was still absorbed with filling a gap (here):
Now also the $(\psi^0)$-component of the gravitino Bianchi is checked (to not imply any further conditions beyond the Rarita-Schwinger equation, once the $\psi^2$-component is satisfied).
On my notebook computer the relevant Clifford algebra takes $\gt 10 min$ to verify, and thus two orders of magnitude more than all the other checks combined.
Also, from talking to the experts I am not getting the impression that this was previously checked (certainly no indication of the need or the way to check this is in the literature).
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