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stub for conformal structure
Have been adding references on (super-)conformal geometry here and there. For instance I gave Nahm’s Supersymmetries and their Representations a category:references-page and added links to it from various relevant pages.
added more references to conformal geometry, started stubs for things like Fefferman-Graham ambient construction, tractor bundle, conformal compactification, but nothing much there yet.
I added an example, that of Minkowski space.
Is this conformal geometry a development of what Weyl did in 1918 when he tried to unify general relativity and electromagnetism? Wasn’t the idea that there was extra freedom in the geometry so that scale was determined locally? And then Einstein pointed out that this couldn’t hold as otherwise particles arriving at the same place along different paths would have different spectra.
@David Roberts, please say where you added an example, I don’t see it.
@David Corfield, conformal geometry and conformal connections are a standard object in theoretical physics, whence “conformal field theory”, but they don’t serve the purpose that Weyl had envisioned. (But I am usure as to how what Weyl actually said in 1918 compares in detail to modern theory.) Weyl’s name has imprinted itself on all the relevant concepts, such as “Weyl transformation” for “conformal transformation”, “Weyl anomaly” for “conformal anomaly”, “Weyl tensor” for the conformally invariant part of the Riemann tensor, etc. All these concepts are of fundamental importance in modern physics, but Weyl transformations are not what unifies gravity with electromagnetism, as Weyl had proposed.
It is good to be aware how vastly the scope of theoretical physics has expanded since then: back then people thought fundamental physics consists of nothing but the field of gravity together with the field electromagnetism, and so Einstein, Weyl and friends were all after finding “the” unification of these two. Later the weak and the strong nuclear force were found, thus it was understood that generally there is Yang-Mills theory, then it was realized that there could be many more realizations of Yang-Mills theory than just as what appears like electromagnetism+nuclear forces, then finally it was found that gravity+YM-theory sets the pattern for a vast web of possible quantum field theories of various dimension and nature, all of them related by various transmutations of dimension, field content, gauge symmetry etc. Conformally invariant field theories are a cornerstone of one proposal for unifying all this, and that’s called string theory, as you know. But the way these CFTs ultimately connect to observation, if they do, is much less direct than what Weyl could possibly have envisioned.
There’s a nice paper on the early history - Early History of Gauge Theories and Weak Interactions - by Norbert Straumann.
Interesting to see a reversal as to which of Weyl and Einstein sees the need to rely on mathematical speculation.
Aus dem Jahre 1918 datiert der von mir unternommene erste Versuch, eine einheitliche Feldtheorie von Gravitation und Elektromagnetismus zu entwickeln, und zwar auf Grund des Prinzips der Eichinvarianz, das ich neben dasjenige der Koordinaten-Invarianz stellte. Ich habe diese Theorie selber l¨angst aufgegeben, nachdem ihr richtiger Kern: die Eichinvarianz, in die Quantentheorie her¨ubergerettet ist als ein Prinzip, das nicht die Gravitation, sondern das Wellenfeld des Elektrons mit dem elektromagnetischen verkn¨upft. — Einstein war von Anfang dagegen, und das gab zu mancher Diskussion Anlass. Seinen konkreten Einw¨anden glaubte ich begegnen zu k¨onen. Schliesslich sagte er dann: “Na, Weyl, lassen wir das! So — das heisst auf so spekulative Weise, ohne ein leitendes, anschauliches physikalisches Prinzip — macht man keine Physik!” Heute haben wir in dieser Hinsicht unsere Standpunkte wohl vertauscht. Einstein glaubt, dass auf diesem Gebiet die Kluft zwischen Idee und Erfahrung so gross ist, dass nur der Weg der mathematischen Spekulation, deren Konsequenzen nat¨urlich entwichelt und mit den Tatsachen konfrontiert werden m¨ussen, Aussicht auf Erfolg hat, w¨ahrend mein Vertrauen in die reine Spekulation gesunken ist und mir ein engerer Anschluss an die quanten-physikalischen Erfahrungen geboten scheint, zumal es nach meiner Ansicht nicht genug ist, Gravitation und Elektromagnetismus zu einer Einheit zu verschmelzen. Die Wellenfelder des Elektrons und was es sonst noch an unreduzierbaren Elementarteilchen geben mag, m¨ussen mit eigeschlossen werden.
In that the early Weyl took himself to be putting Husserlian ideas into practice, I guess it isn’t just mathematical speculation that’s becoming more dubious for Weyl, but perhaps also the role of philosophy in giving confidence to that speculation. On the other hand, there is important thinking going on in Weyl 1918, even if not quite as intended.
Thanks for the link, I have added it to the list of articles on the history of gauge theory here.
(In that list I am saying that Weyl’s gauge theory is laid out in his “Raum, Zeit, Materie”, but looking at the table of contents there again, I am worried that this is not actually so. But I haven’t had a closer look. Do you know?)
Weyl took himself to be putting Husserlian ideas into practice
Could you expand on that? You may have to remind me of those Husserlian ideas, to start with.
According to Scholtz in H. Weyl’s and E. Cartan’s proposals for infinitesimal geometry in the early 1920s
The manuscript of Weyl’s first book on mathematical physics, Space – Time – Matter (STM) (Raum – Zeit – Materie), delivered to the publishing house (Springer) Easter 1918, did not contain Weyl’s new geometry and proposal for a UFT. It was prepared from the lecture notes of a course given in the Summer semester of 1917 at the Polytechnical Institute (ETH) Zürich. Weyl included his recent findings only in the 3rd edition (1919) of the book. The English and French versions (Weyl 1922b, Weyl 1922a), translated from the fourth revised edition (1921), contained a short exposition of Weyl’s generalized metric and the idea for a scale gauge theory of electromagnetism.
Interesting article. You get to see Cartan and Weyl’s interaction.
Thanks! I have added that, too.
By the way, just for completeness, next there is the proposal by Kaluza-Klein 1921/26 and that gets much closer… That in fact gets so close that today still there is a whole industry of people trying to make this work (namely to fix the remaining problem of moduli stabilization).
Naturally I’m particularly interested in those mathematicians/physicists who seem to have gained something from philosophy. It seems far fewer do as we go through the 20th century. Mac Lane and Lawvere were two exceptions.
Anyway, added pages for the people involved Theodor Kaluza, Oskar Klein, which got me to Klein-Gordon equation and so to Walter Gordon.
The idea that must have inspired Kaluza, Klein and maybe others (Wikipedia suggests also Nordström) is that fundamental physics should have non-baroque conceptual underpinnings. Here: if two classes of differential equations (Einstein’s and Maxwell’s) govern the world, then one should be subsumed by the other. This is a philosophical attitude, isn’t it?
I am fond of the fact that Kaluza and Klein’s idea applied to super-gravity realizes what one philosopher suggested much earlier, that matter arises as the unity of space and time, a suggestion that, independently but much later, Wheeler and his school made a big deal of in the 1960s.
This is a philosophical attitude, isn’t it?
I guess so, but then getting out of bed in the morning is one too in some sense.
I meant actually looking at some philosophy. You can include yourself, of course.
Weyl seems to have been very interested in Fichte, see hermann weyl as philosopher, and one of the outputs Husserlian and Fichtean Leanings: Weyl on Logicism, Intuitionism, and Formalism
Hermann Weyl was one of the greatest mathematicians of the last century. Within the last few years, his philosophical writings have also gained attention, in particular the way they were influenced by phenomenology as presented by its founder, Edmund Husserl. (See, for instance, [Ryckman 2005].) When he was a student of mathematics in Gottingen under David Hilbert, Weyl attended lectures by Husserl, who was then a professor of philosophy there.
However, after gaining his first chair in mathematics at the ETH in Zurich, Weyl, as he himself put it, got “deeply involved in Fichte” [Weyl 1946, 1]. (Translations throughout the paper are mine.) As further evidence of this involvement, Weyl attended a reading group on Fichte [Weyl 1948, 381] and also took extensive notes on Fichte’s major works [Weyl (undated I)]. Weyl’s strong involvement in the work of a German Idealist seems at first glance astonishing and, at least from the received view in contemporary philosophy, needs some explanation. Historically, Weyl’s interest in Fichte was initiated by his Zurich colleague, the philosopher and Fichte expert Fritz Medicus. As Weyl acknowledges, it was also through Medicus that “the theory of relativity, the problem of the infinite in mathematics, and finally quantum mechanics became the motivations for my attempts to help clarifying the methods of scientific understanding and the theoretical picture of reality as a whole” [Weyl 1946, 1].
Interesting, thanks.
Can you trace Weyl’s idea of “conformal gauge” back to his “looking at some philosophy”? I am asking now because I would have guessed that Weyl had arrived at that proposal in just the same way that Kaluza arrived at his: by trying to see if a concept that already worked in one place also works more generally.
If there is something in Fichte that may be identified as motivating Weyl’s conformal gauge theory, I’d be rather interested in a pointer.
Sorry, in conformal compactification.
Erhard Scholz has done some of the best work on Weyl, see at the arXiv. Taking one of these - Philosophy as a cultural resource and medium of reflection for Hermann Weyl - look at pp. 8-9. Scholz makes a case that there Weyl found affinities with three themes of Fichte’s work Grundrisse des Eigenthümlihen der Wissenshaftslehre
construction of a spacelike continuum from infinitesimal parts (not as a set of points, endowed only in an additional step with a continuous, differentiable, etc. structure, as intended - not yet achieved - by mainstream mathematics of the time),
characterization of the space-filling entities as forces, the actions of which were initially specified only in the infinitesimal parts (Weyl’s postulate to build geometrical structures purely infinitesimally),
formation of matter as a form of appearance of space-filling forces (dynamical theory of matter, going back to Kant and mathematically rejuvenated by Mie and Hilbert)
You will find the bottom of p. 13 and top of p.14 amusing, Hilbert and “Hegelian physics”.
Thanks, that’s really interesting, especially the bit about space built out of infinitesimal entities. I should go and look at some of this more…
I have started a stubby category:reference-entry for Grundriss des Eigenthümlichen der Wissenschaftslehre, for the moment just so as to record that neat statement
Der unendlich kleinste Theil des Raumes ist immer ein Raum, etwas, das Continuität hat, nicht aber ein blosser Punct, oder die Grenze zwischen bestimmten Stellen im Raume;
I took the liberty of adding that as a prefacing quote in the entries infinitesimal neighbourhood and synthetic differential geometry.
Quite a claim, no?
Weyl geometric gravity theory, with modified scale invariant Hilbert-Einstein action coupled to the scalar field, sheds new light on cosmological redshift. In this frame, the famous expanding space explanation of the Hubble redshift appears only as one possible perspective among others. From a theoretical point of view, it even need not be considered as the most convincing one. With such questions we enter a terrain which physicists usually consider as morass; but Weyl geometry gives these investigations a safe conceptual framework. E. Scholz, Weyl geometry in late 20th century physics.
Taken on further in
Morass is a good word, one needs to be really careful here. Scholz cites some C.C. a lot, which maybe he shouldn’t.
David, let me see if I understand:
So in Fichte’s Grundriss there is the idea of space and geometry being determined by infinitesimal neighbourhoods of points, and Weyl took that as motivation or inspiration or confirmation of the idea of expressing theories of physics in terms of connection data aka infinitesimal parallel transport, and hence looked for further variants or generalizations of the Levi-Civita-Cartan connection of Einstein gravity to see what else in nature there may be that may be captured by the concept of connection (hence of infinitesimal paralle transport).
Is that right? Is that the relation of thoughts that you are highlighting to me? I like this (since, indeed, from just positing synthetic infinitesimals in differential cohesion it is possible to formalize the concepts of infinitesimally integrable Cartan geometries).
But now how does the scale degree of freedom enter the story? My understanding is, or was, that this just so happens to be one additional degree of freedom potentially present in a connection, and that therefore Weyl went for it and tried to see if that is the bit one has to include in order to unify Einstein gravity with Maxwell electromagnetism.
Is that the case, or is there an argument that the idea specifically of a scale degree of freedom has a motivation maybe from thoughts of Fichte, too?
Not wanting to be stubborn, but to put previous comments into this context, let me highlight that if the answer to this last question is “no”, then we are still not all that far from what must have motivated Kaluza and Klein (I haven’t checked their history of thoughts, though). Also their approach proceeds from the thought that the parallel transport that the tangent vector along the trajectory of a particle goes through in a gravitational background might subsume the parallel transport that the electromagnetic phase goes through in an electromagnetic background, and that hence both effects might be unified by one single parallel transport taking place in a bigger space whose tangent spaces in turn unify the degrees of freedom of a tangent vector to spacetime and of an electromagnetic phase. This is the 5-dimensional space which they considered. With its tangent spaces being, as it were, an -extension of the 4-dimensional tangent spaces of spacetime, this is rather close to Weyl’s proposal of looking for -extensions of the Lie algebra of linear isometries of these tangent spaces.
But now how does the scale degree of freedom enter the story?
No doubt it is no easy matter to know how someone has been influenced by some rather vague philosophical thoughts. I suppose in the way those ’spheres of activity’ are described, they might go to make up a space with that scale degree of freedom.
Scholz at least imagines that Fichte might assist with an interpretation of the scale freedom:
Weyl was convinced of important consequences of his new gauge geometry for physics. The infinitesimal neighbourhoods understood as spheres of activity, as Fichte might have said, suggested looking for interpretations of the length connection as a field representing physically active quantities.
When it comes to the matter of attributing influences, one way to see how hard it is is to think of someone in the future working out how you yourself came to certain positions. Perhaps you’re not such a good example, since you leave a detailed trace of you ideas. Even so, will people 100 years hence know what’s due to Hegel?
I have no trouble with that problem, indeed that is in part the point which I am trying to make: why not grant that in Kaluza and Klein there is a similar kind of philosophical impetus as in Weyl, even if we don’t seem to have a trace of them making it as explicit as he did.
But the problem of vagueness and arbitrariness remains in either case, clearly. The point I see in Hegel (via Lawvere) is the systematics. Hegel asks to derive the concepts from the dialectics, that’s why he talks about "science". One may disagree about how successful he was with deducing the results of his method (or with communicating them), but the key point is that there is an attempt at a method at all. For instance in his lectures on the history of philosophy he criticizes Plato for talking about his Ideas without deriving them, he criticizes Spinoza for stating definitions where he should have tried to state propositions instead, and I’d imagine he might have asked Fichte to go into more detail with the would-be derivation of the nature of the geometry of space. Though of course he attributes the very idea of trying to intoduce a concept of systematic derivation into philosopyhy at all to Fichte:
Die Fichtesche Philosophie hat den großen Vorzug und das Wichtige, aufgestellt zu haben, daß Philosophie Wissenschaft aus höchstem Grundsatz sein muß, woraus alle Bestimmungen notwendig abgeleitet sind. (Lect Hist Phil)
This is, I think, exactly what we should be following here, here on this unique discussion forum which we subtitled "on Mathematics, Physics and Philosophy", while still having to live up to that trinity, properly.
The remaining problem with Hegel is that it requires intuition and tolerance to see how he followed the system. Is it actually true that measure is the unity of quantity and quality? Maybe he made a mistake there in this derivation? (He writes himself (709) that he found "measure" the hardest part to derive) Can we check? But since Lawvere has roughly explained how to go about checking it, that’s what we should be doing now.
I think that if we do this carefully, then the method takes us to the kind of differential geometric structures under discussion here. In fact just these days I have been struggling with figuring out this kind of question (whence this very thread here): I know now an essentially first-principle construction of Poincare-symmetry (like so: posit a 3-stage system of oppositions and resolutions, ask for faithful models, observe that super-formal-smooth infinity-stacks provide one, check for the smallest objects in there which exhibit purely negative moments with respect to the topmost modalities, check that these are the superpoints, base the bouquet on these, find by Majorana spinor representation theory that the universal extensions in low dimension are the super-Minkowski spacetimes). What I don’t have is a similar essentially-first-principles derivation of conformal symmetry. I can of course find it in the model by hand, but I don’t get nearly as close as characterizing it by the method. So currently the method takes me closer to Kaluza-Klein than to Weyl. But then, I am certain that there are many aspects that haven’t become clear to me yet. I am hoping we could talk more about it to jointly make progress.
I think in principle we could put all old texts aside and do the Arbeit am Begriff systematically ourselves now: work out which system of adjoint modalities may be sensibly posited, and work out what it means.
Ok, good.
Something that would be helpful would be to know “What’s left to do?”, if this makes any sense. I mean, I had no idea that conformal symmetry was going to appear on your agenda once you felt happy enough with Poincare symmetry. But then I wouldn’t be able to predict once conformal symmetry has been derived nicely, whether you’ll raise a new symmetry to consider.
Is there any sense of a largest symmetry? I see Wikipedia has
The largest possible (global) symmetry group of a non-supersymmetric interacting field theory is a direct product of the conformal group with an internal group.[3] Such theories are known as conformal field theories.
Is there something similar for the supersymmetric case?
Something that would be helpful would be to know “What’s left to do?”
Great, I have been trying to gather forces on this, but I may have failed to express myself.
The open problem is this — and while it is ultimately to inform the mathematics, it is of a curious nature that requires input from philosophically and linguistically trained minds:
add a system of adjoint modal operators to homotopy type theory, then what do they mean? We know that their presence determines on all the types some kind of extra qualities. But we need to figure out which qualities these are. And what the systematic way is to answer this in the first place.
For instance, just recently there was something like a breakthrough here, to my mind at least, regarding the meaning of adding an opposition such that . We understood (turns out Lawvere had said this since long ago, but the world spirit proceeds so very slowly) that with suitable minimality condition added, this is actually fully determined, it’s the ground topos of double negation sheaves, when essential. We need more of this kind of Arbeit am Begriff. Say next we force three more such levels, what will the resulting formal substance feel like, what will the dozen or so new qualities determined thereby mean? Of course we should do this one-by-one, but this just to indicate what the grand picture is.
More specifically, this highlights the following open problem: any moment/modal operator may induce not only “opposite determinations” but also “negative determination”. It remains to be understood how these are to be related and if and when one wants extra axioms forcing that they indeed are related. There is now a parallel thead on this open question. It feels like I am finally making a little progress there, but we’ll need to go much further.
You see maybe what I mean with needing Arbeit am Begriff here: this is about trying to begreifen what it means to take the left or right adjoint of a moment, or its cofiber, and how they combine. Turns out one such combination is actually equivalent to the extra axioms that Lawvere threw in ad hoc (or so it seems) into “Axiomatic cohesion”, so this is about understanding what the axioms really mean, why and when one wants to impose them, and which. One thing we can do is try to construct loads of models and see what all this comes down to in these models. But that is slow and piecemeal and will only help so much. Better if we got a more intrinsic grasp. And curiously the point is that while this is about making progress in mathematics, it is of the curious kind that it helps to get some linguistic and philosophical grip on in. The more I re-read Lawvere’s manifesto the more I find it spot-on. He said this 23 years ago, and nobody reacted. If you look at what it takes to react to that, it seems to me that we here, the nLab/nForum regulars are the unique little community (besides Lawvere himself) that has the expertise to go for this. We should do so.
Here is a more specific open question: so I mentioned that characterization of space-time and fermions from just a process of oppositions. But I kept saying it’s “essentially” a first-principles derivation. There are gaps. Notably where I mention the model of super-formal-smooth infinity-stacks, that’s precisely a problem, I go external because I can’t help myself otherwise. What we eventually need is instead a fully internal way to arrive at the same conclusion.
As a first step in this direction, it might pay to try to think about what it means that characterizes these superpoints. So in the model stands out as being the unique (I think) type which
is 0-truncated:
is non-degenerate;
has purely negative bosonic moment;
has purely opposite bosonic moment.
That’s an observation in the model, but what does it mean? In the model I see that this characterizes the smallest superpoint, and that looking for the emanation of extensions that this gives the transmutation of the point into higher dimensional spacetime. But is this already purely in the axioms somehow? What could it generally, abtractly, mean to consider three levels of opposititions and then check for those types which have maximally negative and opposite determination with respect to the qualities implicit in that third level? Presently I have no idea, but I feel like this is a good problem for the categorically inclined philosopher to chew on. I gather there is a small industry of philosophers claiming to derive facts in metaphysics from modal predicate logic (had dinner with one, once), but I think this is doomed to be empty exercise, there is nothing to be found there. Here instead in modal homotopy type theory we see we have struck the metaphysical gold mine, it’s here that something is actually going on. But it needs more work to mine for it.
Much of what I am alluding to here I am currently writing dedicated notes on here.
Regarding finally the conformal group: I don’t really know yet, but of course given that we see the brane boquet appear in the axiomatics, then the conformal groups are not far away, the physics and also the Lie theory connect them closely to the brane bouquet (in fact people speak of the “near horizon brane scan” for the superconformal case). On a very model-based level this is all pretty clear, but on a more general abstract level it needs work. I have a little bit of an idea, but maybe nothing concrete enough to be reall helpful. I am currently trying to discuss this further with John Huerta. For the moment my strategy here would be to instead of aiming for big jumps at the target instead try to make little steps from where one already is, to explore if there is a path that may eventually lead to the target.
Just a few snatched minutes between lectures (3 hours in a row now!)
Is the suggestion that the reason we turn to conformal symmetries is the fact that they’ve appeared anyway in our models? And no other extensions appear like this?
near horizon brane scan
Time for a near horizon brane bouquet?
The big picture that was found in string theory is the following beautiful one:
First we find sigma-model -branes on -dimensional spacetimes with -supersymmetries whenever there is an exceptional cocycle on .
Then we generalize to branes on Cartan geoemtries modeled on – Lorentzian supergemetry.
Then we find that the equations of motion of these Cartan geometries contain black branes that match precisely the sigma-model branes that we started with. (This is somehow one of the most mysterious and beautiful effects in this whole story, to my mind.) And the near-horizon geometry of these black branes turns out to always be anti-de Sitter spacetimes times orthogonal direction.
So then we consider the perturbation theory of these sigma-model -branes around asymptotic embeddings into AdS-spaces, i.e. we embed the sigma model branes right there where their non-perturbative black avatars sit (or don’t sit, being just the remaining naked singularities) and study their perturbation theory.
Result: one finds that the worldvolume theories on the p-branes thus found are precisely those given by super Cartan geometries for the (few) superconformal groups.
So it’s a beautiful grand story of which the first steps have been absorbed by the method, while the rest is still waiting to be absorbed. Eventually it will, but it needs a bit more work.
“trying to begreifen” :) It’s very interesting to read up on the “old thoughts” you post here along the discussion.
Regarding every “super” and “string” and contemporary physics like: What is the full range of existing math you want to try to embed in the formalism? I could imagine much of the prevalent structure in contemporary theories just to be baggage, so how to gauge what present notions should be sought to be included.
Arguing in the other direction: I think in case you want to go beyond the “this unique discussion forum” on “on Mathematics, Physics and Philosophy”, you might need a selling point that is the solution of an unsolved issue, and one more plastic than “If you make the effort to learn about this perspective, then you understand how everything comes together”. That raises the question (to me at least) what such problems actually are (in physics!).
first an addendum to #30: I have added now a similar paragraph with hyperlinks and references here:
Nikolaj,
what exactly is accidental and what is essential in contemporary physics I don’t know beforehand, I find out by trying what naturally emerges by itself out of cohesive homotopy theory, and what doesn’t. For instance it turns out that everything of Chern-Simons-type and of Wess-Zumino-Witten type is essence, while, from what I can tell, old friends such as are accident.
Large chunks of string theory turn out to be essence. Just look at this miracle we just discussed: all of the dimensions that enter the AdS-CFT correspondence may be read off right there from the just the classification of simply super Lie algebras. Lie theorist would eventually have found the structures in AdS-CFT even if they had never heard of string theory otherwise.
Regarding open questions: there are plenty of open questions which are so open that traditional theory struggles to even phrase them. Since the last weeks I have been touring around giving talks about such a problem and its solution in cohesive homotopy theory: the open problem is that backgrounds in string theory generically need not just those super-flux forms to be globalized from Minkowski to curved spacetime, but the full global higher WZW terms that pre-quantize the forms need eventually to be carried along in order to have a genuine (“classical anomaly free”) sigma model for the given branes on that target. This is the content of
(which in turn is a digest of material in dcct, see there to get the material with working cross-links). I am inclined to think that without the cohesive homotopy theory these results are not tractable.
Closely related to this is the following issue: what now are the super-symmetries of such super-spacetimes? Traditional lore has it that these are those BPS charge extensions of the supersymmetry algebras, but closer inspection (it follows from the above…) shows that these must be super Lie n-algebra extensions instead. This I will be speaking about, too, to anyone who is interested, currently the next occasion for that is in June at ISQS-23
There are many more open problems. Recently the arXiv saw Seiberg et al reinvent higher differential form symmetries in field theory. After a while they will see the need for higher gauge theory proper. But, yeah, it’s frustratingly slow.
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