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Since it was mentioned by Urs on g+, I thought I’d start mysterious duality. Maybe not a great name when someone discovers how it works (as someone claims to have done here).
Thanks. And thanks for the pointer to Alastair King’s talk. Though it seems there is no details on his claim available. (?)
I started del Pezzo surface too.
Is it that we don’t hear so much now of the mysterious duality because it was subsumed into a larger picture? If the duality concerns second cohomology, then
It turns out that the full cohomology of these [Del Pezzo] surfaces spans the root lattice of a Borcherds superalgebra.
Is it that we don’t hear so much now of the mysterious duality because it was subsumed into a larger picture?
My impression is that a problem is that the “mysterious duality” (more a mysterious dictionary) has remained mysterious: it is not clear why or where the del Pezzo surfaces should show up.
The fact that S-duality of IIB is supposed to match to the exchange symmetry of the two Cartesian factors in $\mathbb{P}^1 \times \mathbb{P}^1$ might suggest that the del Pezzo is to be thought of as some kind of complexification of the F-theory elliptic curve fiber, because that, too, geometrizes S-duality by exchange of its two canonical coordinates. Such an identification would help, but maybe it’s just a superficial analogy.
I spent an idle hour yesterday evening rooting about and encountered:
An exchange between you and Motl on your old blog about (2,1)-strings.
A talk by Bernard Julia which mentions a wide range of interconnected concepts, such as Kervaire spheres and the magic triangle, and a further duality
CJLP noted enlargment of U- to V-duality: spin mixing (super)algebra with p-form coefficients. Multiplying supergenerators of degree –p such that total degree vanishes. V recognized as (Super)Borcherds
with a hint of a W-duality.
Mention of ’oxidation chains’, with non-unique inverses to dimension reduction sounding ’bouquet’-like, and a missing preprint, Julia B and Paulot L 2004 Superalgebras of oxidation chains Preprint ULB-TH/04-09.
Re #5
might suggest that the del Pezzo is to be thought of as some kind of complexification of the F-theory elliptic curve fiber
I did come across someone relating del Pezzo surfaces to F-theoretic compactification (in French) here:
on considere des modeles dans lesquels nous pouvons definir une limite locale de la theorie-F. Cette restriction a pour but justement de decoupler la gravite, ne laissant par consequent quelques interactions de jauge a quatre dimensions. Ceci se realiserait si on impose que le volume des dimensions transversales des 4-cycles enroules des 7-branes soit arbitrairement grand…Les 4-cycles, qui sont contractables, doivent etre des varietes de K ̈ahler a courbure positive. Ils sont class ́es entierement et sont donnes par des varietes dites varietes de del Pezzo (ou surfaces de del Pezzo)
Sasha and Hisham are in the process of explaining the mysterious duality (or maybe rather the U-duality pattern) by Hypothesis H:
Interesting. So there’s some kind of iterated oxidization process happening on the loop space side.
Looking above to #6, I see there was talk of oxidation and indeed ’oxidation chains’. But the link to that talk by Bernard Julia doesn’t work any longer.
I would say that at face value this is about iterated reduction, but of course by their adjunction, you may regard it the other way around, too.
We had previously shown that the single cyclification of the 4-sphere, regarded as the coefficient space of 4-Cohomotopy, yields, rationally, the double dimensional reduction of M-brane charges to type IIA D-brane charges.
This suggests that iterating this (double dimensional-) reduction procedure yields the brane charges of higher toroidal compactifications of M-theory down to smaller spacetime dimensions, at least rationally.
By the U-duality-conjecture these should transform under groups in the infamous E-series. And it seems that can indeed be seen, rationally, from the rational cohomology of the iterated cyclic loop spaces of the 4-sphere.
Here the reduction/oxidation adjunction guarantees that these doubly-dimensionally reduced brane charges are still equivalent to their “oxiation” back to the full M-brane charges up in 11d. But their symmetries/automorphisms increase upon reduction, and decrease again upon oxidation.
Are there any hints for the conjecture on slide 14 on an explicit relation between del Pezzo surfaces and iterated cyclic loop spaces?
I suppose the Arnold-Kuiper-Massey theorem points to a connection between the starting points of the sequences.
On the del Pezzo aspect I am not in position to comment. While I know that this has been a motivation for the project, I don’t know about results beyond those on slide 13. But I am not part of this project and wouldn’t know either way.
However, I find the result (on that slide 13) on U-duality (which is widely understood/expected to be a central aspect of string/M-theory) – potentially more relevant than the relation to del Pezzo surfaces (which remain a curious side remark).
It’ll be interesting to see the unifying effect over tracts of mathematics as the consequences of Hypothesis H are unfolded.
Besides Hypothesis H itself, the other mathematical aspect to take note of here is the identification of physicist’s “doble dimensional reduction” with the construction of cyclic loop spaces/cylic loop stacks as a right adjoint to (Kaluza-Klein-)extension.
This runs deeper than has been made fully manifest. It’s also behind the discussion by Ganter & Rezk & Huan of Tate-elliptic cohomology as either the K-theory of the cyclic loop stack (they never quite say it that way, but that’s what it is) or, in the next step, as (this has not $n$Lab representation yet, but see Spong’s thesis) the ordinary cohomology of the doubly cyclic double loop space.
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.
Sounds like a good mathematical prediction.
Has anyone looked at oxidation/reduction with respect to other groups such as $SU(2)$?
Yes, that yields the spherical T-duality discussed here (in super-rational geometry).
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.
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.
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
$[\Sigma//Diff(\Sigma),\; \mathbf{Fields}] \simeq [\Sigma,\; \mathbf{Fields}]//Diff(\Sigma) \,.$Hmm, why is oxidation/reduction about base change and adjoints for $\ast \to B G$, but the general convariance discussion concerns base change for $B G \to \ast$?
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).
Thanks! That helps.
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?
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.
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
where he links things with del Pezzo surfaces, part of the mysterious duality.
The co-author has a paper
which mentions the mysterious duality on p. 213. And earlier p. 24
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.
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)?
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.
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.
added pointer to today’s
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.
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.
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.
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.
Alright, thanks. Have forwarded this now.
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… )
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