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).

]]>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.

]]>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. ]

]]>have inserted the previously missing components linear in $\psi$

(but they don’t matter anyway until the proof of the last lemma, which is still not written out here)

]]>have spelled out the proof of the second of the three lemmas (here) that go into the main theorem

(This is where the magic happens – some crazy cancellations.)

]]>have spelled out the proof of the first of the three lemmas (here) that go into the main theorem.

]]>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.

]]>