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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 24th 2024
   • (edited Mar 6th 2024)
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarting 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](https://ncatlab.org/nlab/show/D'Auria-Fre+formulation+of+supergravity#CastellaniDAuriaFre). I am going to spell out at least parts of the proof, with some attention to the prefactors. <a href="https://ncatlab.org/nlab/revision/11d+SuGra+from+super+C-field+flux+quantization+--+section/1">v1</a>, <a href="https://ncatlab.org/nlab/show/11d+SuGra+from+super+C-field+flux+quantization+--+section">current</a>

   starting a stand-alone Section-entry (to be !includeed 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.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 25th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave spelled out the proof of the first of the three lemmas ([here](https://ncatlab.org/nlab/show/11d+SuGra+from+super+C-field+flux+quantization+--+section#SuperBianchiIdentityOnG4InComponents)) that go into the main theorem. <a href="https://ncatlab.org/nlab/revision/diff/11d+SuGra+from+super+C-field+flux+quantization+--+section/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/11d+SuGra+from+super+C-field+flux+quantization+--+section/4">v4</a>, <a href="https://ncatlab.org/nlab/show/11d+SuGra+from+super+C-field+flux+quantization+--+section">current</a>

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

   diff, v4, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 25th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave spelled out the proof of the second of the three lemmas ([here](https://ncatlab.org/nlab/show/11d+SuGra+from+super+C-field+flux+quantization+--+section#SuperBianchiIdentityOnG7InComponents)) that go into the main theorem (This is where the magic happens -- some crazy cancellations.) <a href="https://ncatlab.org/nlab/revision/diff/11d+SuGra+from+super+C-field+flux+quantization+--+section/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/11d+SuGra+from+super+C-field+flux+quantization+--+section/6">v6</a>, <a href="https://ncatlab.org/nlab/show/11d+SuGra+from+super+C-field+flux+quantization+--+section">current</a>

   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.)

   diff, v6, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeFeb 27th 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave 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) <a href="https://ncatlab.org/nlab/revision/diff/11d+SuGra+from+super+C-field+flux+quantization+--+section/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/11d+SuGra+from+super+C-field+flux+quantization+--+section/10">v10</a>, <a href="https://ncatlab.org/nlab/show/11d+SuGra+from+super+C-field+flux+quantization+--+section">current</a>

   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)

   diff, v10, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 7th 2024
   • (edited Mar 10th 2024)
   • PermaLink
   Author: Urs
   Format: MarkdownItexRegarding 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. ]

   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. ]

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMar 22nd 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe detailed computation/proof is now spelled out in section 3 at *[[schreiber:Flux Quantization on 11d Superspace|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.

   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.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeApr 9th 2024
   • (edited Apr 9th 2024)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWas still absorbed with filling a gap ([[schreiber:Flux Quantization on 11d Superspace|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).

   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).