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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).
We now have analogous analysis also for the 3-form flux on M5-branes inside 11d SuGra backgrounds:
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