added pointer to:

- Alexander Voronov:
*The $E_k$ symmetry of dimensional reductions of M-theory*, talk at*M-Theory and Mathematics 2023*, NYU Abu Dhabi (2023) [web]

Maybe email them personally?

]]>Typos in second paper to pass on:

Diagram label at foot of p.3:

differetnial of map

On p. 61:

]]>we end up with $G_{! 0}$

Sati & Voronov have now split the article on “Mysterious triality” into two, I have updated references accordingly:

Hisham Sati, Alexander Voronov,

*Mysterious Triality and Rational Homotopy Theory*[arXiv:2111.14810]Hisham Sati, Alexander Voronov,

*Mysterious Triality and M-Theory*[arXiv:2212.13968

As in ’Mysterious triality’, a duality between algebraic topology and algebraic geometry in Ben-Zvi’s Electric-Magnetic Duality for Periods and L-functions (slide 15). But then in the former, electric-magnetic duality occurs on both sides of that duality.

]]>Yang-Hui He, is a lecture speaker at your conference.

I did notice :-)

It was funny last Friday in the discussion session we had, because Yang-Hui, who was the last of four hosts on that day, and me kept chatting, until we finally noticed that everyone else had long left the conference chat. (Partly of course due to the ambitious planetary scheduling of the meeting, since we start each day when its already bedtime in Singapore while the US east coast is barely waking up… )

]]>Alright, thanks. Have forwarded this now.

]]>I guess they’ll know about the 120 tritangents, since one of the teachers they dedicate it to is Igor V. Dolgachev who writes on such matters in Classical Algebraic Geometry: a modern view.

But who knows where McKay’s sporadic group connections lead? By the way, the co-author, Yang-Hui He, is a lecture speaker at your conference.

]]>OK.

In your new paper, Mysterious triality, in sec. 6.4 you discuss the 27 lines on a cubic and 28 bitangents on a quartic associated to $E_6$ and $E_7$. I wonder if you know of the work of John McKay on exceptional structures. In

Yang-Hui He, John McKay, Sporadic and Exceptional, (https://arxiv.org/abs/1505.06742),

the authors complete the series by the association of 120 tritangents on a sextic of genus 4 to $E_8$. They also mention del Pezzo surfaces in the context of mysterious duality.

]]>Could you formulate an email message that I could pass on? I am glancing over your comments here only in stolen split-seconds while busy otherwise.

]]>So they see the 27 Lines and 28 bitangents in Sec. 6.4 associated to $E_6$ and $E_7$. How about McKay’s continuation to 120 tritangents on sextic of genus 4 corresponding to $E_8$, as discussed in the reference in #25?

Might be worth passing on if they don’t know of McKay’s work.

]]>added pointer to today’s

- Hisham Sati, Alexander Voronov,
*Mysterious triality*(arXiv:2111.14810)

I guess universal exceptionalism predicts that an important advance in fundamental physics will add novel ideas to the study of exceptional structures, which since they are so thoroughly interrelated, should lead to a cascade of mathematical advances.

]]>While I am not involved in the “mysterious triality”-project, my understanding (which may be outdated) is that the relation between del Pezzo surfaces and U-duality remains as mysterious as it used to be, while the (new) relation between the rational homotopy of iterated cyclifications of the 4-sphere and U-duality (rationally) is now a theorem and is conceptually explained by Hypothesis H.

On the role of $\mathbb{C}P^2$ as a branched double cover of the 4-sphere in view of the latter as coefficients for C-field charge quantization: It’s too early to say anything of substance in public; but one is bound to notice that a singular $\mathbb{Z}/2$-quotient is also at the heart of HW-theory.

]]>So when you, Urs, were telling me about the exceptional features of the $S^4$ appearing in Hypothesis H,

Regarding the 4-sphere: Lots of special aspects of it in low dim topology, like being the base of the quaternionic Hopf fibration, the twistor fibration, etc. Next we’ll bring out its role via the Arnold-Kuiper-Massey theorem,

are we to see the latter

$\mathbb{C}P^2 / \mathrm{O}(1) \simeq S^4$as relating the first spaces in

]]>There must be an explicit relation between the series of del Pezzo surfaces $B_k$, $0\leq k \leq 8$, and the series of iterated cyclic loop spaces $\mathcal{L}^k_c S^4$, $0\leq k \leq 8$. (Voronov’s slides)?

I see Hisham is speaking on this next week, Mysterious triality in M-theory:

Our approach allows for extending both mysterious duality and triality to the Kac-Moody case, $k \geq 9$, and for physical and topological interpretations of several prominent statements in algebraic geometry, including the famous 27 lines on a cubic.

I’m reminded of a talk by John McKay I attended which linked the 27 lines via a triplet to other triplets $E_6, E_7, E_8$, sporadic finite simple groups,… He wrote it up in

- Yang-Hui He, John McKay,
*Sporadic and Exceptional*, (arXiv:1505.06742),

where he links things with del Pezzo surfaces, part of the mysterious duality.

The co-author has a paper

- Yang-Hui He,
*The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning*, (arXiv:1812.02893),

which mentions the mysterious duality on p. 213. And earlier p. 24

- Yang-Hui He,
*Calabi-Yau Geometries: Algorithms, Databases, and Physics*, (arXiv:1308.0186),

This Mckay-esque curiosity was further explored [84], wherein the re-markable observation that (3.22) resembles the structure of M-theory compactification was made.

But no time for this today.

]]>Since the rational cohomology of 4-spere encodes the rational M2/M5-brane charges, it’s by application of iterated cyclification to the 4-sphere that the rational brane charges for all toroidal compactifications of type IIA string theory appear, which is the situation for which U-duality has been observed.

But once we understand U-duality as an effect on iterated cyclic loop spaces, this way, it should be interesting to check if other iterated cyclic loop spaces have their own special symmetry groups. I’d expect so, but I don’t know.

]]>To return to #14,

To keep that in mind as Sati & Voronov discuss the ordinary (here: rational) cohomology of the $n$-fold cyclic loop space (here: of the 4-sphere, but one could consider this more generally).

There is an evident pattern here, related in maths to chromatic red-shift, transchromatic characters etc., now identified in physics with double dimensional reduction of brane charges on tori. It seems however that the appearance of U-duality groups of these toroidal double dimensional reductions is something whose analog on the pure maths side has not been recognized before.

Is the idea here that forming $n$-fold cyclic loop spaces in general will give rise to extra U-duality symmetry, or is it that their application to the 4-sphere in particular is especially important for the role the latter plays in cohomotopic brane charges, and this is where U-duality will occur?

]]>Thanks! That helps.

]]>Cyclification is about *both* (I have just uploaded an extended diagram here to bring this out more clearly). But the left base change to the point is just to isolate the homtopy quotient. What makes cyclification tick is that this is combined with right base change from the point.

And it’s the fact that base change from a pointed object classifies principal fibrations which makes these play a special role in this story. Reduction/oxidation over non-principal; $F$-fiber bundles can’t have such a neat fundamental formulation by just some $\infty$-base change. (Or at least I don’t see an evident modification).

]]>Hmm, why is oxidation/reduction about base change and adjoints for $\ast \to B G$, but the general convariance discussion concerns base change for $B G \to \ast$?

]]>Maybe that means that it needs to be generalized (one would want to try $[F,X] \sslash Aut(F)$)

So the kind of thing at general covariance

$[\Sigma//Diff(\Sigma),\; \mathbf{Fields}] \simeq [\Sigma,\; \mathbf{Fields}]//Diff(\Sigma) \,.$ ]]>So what do you do if you want to compactify on something that isn’t a (∞\infty-)group?

Good question. The $Ext \dashv Cyc$-adjunction won’t apply. Maybe that means that it needs to be generalized (one would want to try $[F,X] \sslash Aut(F)$). Or maybe that means reduction/compactification ultimately makes sense only on groupal fibers. This is reminiscent of the curious fact that of all possible KK-compactifications, it’s the only toroidal ones that dominate all the discussion of U-dualities. Maybe it’s because these are the simplest, and everything beyond will only be yet more out of reach. Or maybe it means that gorup structure such as on tori is necessary for the U-duality story.

Perhaps it deserves its own page.

Yes, maybe, though the page “double dimensional reduction” seems like a canonical place to host it.

]]>OK, that would be good to include as an example. So what do you do if you want to compactify on something that isn’t a ($\infty$-)group?

We have the oxidation-reduction adjunction in a few places: base change, dependent product, double dimensional reduction, geometry of physics – fundamental super p-branes. Perhaps it deserves its own page.

]]>