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I have a basic question here:
Is the composite
S7⟶ℂP3⟶S4of
1) the canonical S1-fibration S7→ℂP3
followed by
2) the twistor fibration ℂP3→S4
equivalent to
3) the quaternionic Hopf fibration S7→S4
?
I expect it is, and probably it follows readily once the definitions are unwound suitably. But I don’t see it in detail yet.
Oh, I see it: It’s immediate from the reformulation of the Calanbi-Penrose fibrations given on this page.
Does an analogue occur for other Hopf fibrations? E.g., a composite
S15⟶ℍP3⟶S8as a composite of the S3-fibration S15→ℍP3 and an S4=ℍP1-fibration ℍP3→S8.
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Anonymous
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I don’t understand yet why many authors call it the “Calabi-Penrose fibration”. It seems that the first one to consider it was Atiyah 79, Sec III.1 (who credits Penrose for general inspiration, but otherwise seems to be conjuring the thing quite by himself).
What did Calabi do here, and what would be a relevant citation?
I think this paper may explain
Thanks! I have added that pointer:
A sorry, that’s another “Blaine Lawson”?! Am fixing it…
This page claims H is for Herbert. Presumably it’s all one person.
Thanks for the sanity check. With that out of the way, on to the article:
It’s interesting, but also mysterious:
On p. 4 we hear that
In one of his important works, E. Calabi has found…
but despite (or because of?) this alleged importance, we don’t get to see a citation for this. Instead, the only reference that follows is… to Atiyah el. al. (!) after the mysterious claim that (if I am translating this correctly?) this is the “only way by which we understand the link nowadays”.
But anyways it does look like this article of Lawson’s could be the origin of the term “Calabi-Penrose fibration”, thanks again.
According to B. Loo in jstor:2001066 Calabi’s fibration is in
E. Calabi, Quelques applications de l’analyse complexe aux surfaces d’aire minima, Topics in Complex Manifolds (Ed., H. Rossi), Les Presses de l’Univ. de Montreal, 1967, pp. 59-81. 5.
E. Calabi, Minimal immersions of surfaces in euclidean spheres, J. Differential Geometry 1 (1967), 111-125.
But I haven’t found a copy of the first one yet, and I haven’t recognized the construction in the second one yet.
“only way by which we understand the link nowadays”.
I think it’s rather
he provided a construction from which only now do we understand the connection to the one introduced by Penrose in GR…
or maybe in better English
he provided a construction whose connection to the one introduced by Penrose in GR only now do we understand…
AHS is relatively recent, hence the “seulement aujord’hui”.
Okay, thanks!
Meanwhile I have found this review
which gives a clear statement of what Calabi actually did here (around Lemma 2.31). It seems to require a bit of thought to see that this is equal (for m=2 in Calabi’s notation) to what Atiyah introduced as the twistor fibration.
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Eugenio Calabi, Minimal immersions of surfaces in euclidean spheres, J. Differential Geometry (1967), 111-125 (euclid:jdg/1214427884)
Eugenio Calabi, Quelques applications de l’analyse complexe aux surfaces d’aire minima, Topics in Complex Manifolds (Ed. H. Rossi), Les Presses de l’Universit ́e de Montr ́eal (1968), 59-81 (naid:10006413960)
Thanks, David.
What we are wondering about is whether there is a sensible generalization of the twistor fibration that would still be over the 4-sphere, but coming from attaching more than one 2-cell to it. Because it’s this 2-cell which gives the single gauge field F2 in Twistorial Cohomotopy.
What we’d probably need is something that would still factor the quaternionc Hopf fibration Sp(2)-equivariantly.
Thanks for the pointers, these are interesting articles.
I should say that what I am looking for here is likely not related to twistors anymore. (From the point of view of Cohomotopy theory, the appearance of twistor space in the factorization of the quaternionic Hopf fibration is a surprise, not something we asked for.)
More concretely, from the discussion on p. 5 we are naturally wondering if the construction of the combined Hopf/twistor fibration could be somehow modified such that instead of the gauge group S(U(1)2) a gauge group S(U(1)n) would appear, for small n.
For starters, for n=3 this would mean that the new fiber over S4 we are after has Sullivan model of this form:
ℝ[f(1)2,f(2)2,h3]/(df(1)2=0df(2)2=0dh3=−f(1)2∧f(1)2−f(1)2∧f(2)2−f(2)2∧f(2)2)That’s because the right hand side in the third line is the expression for the second Chern class on BS(U(1)3) in terms of the first Chern classes on the first two BU(1)-factors, and it’s these second Chern classes whose appearance is suggested by equation (6) on p .3.
So we are looking for an Sp(2)-equivariant factorization of the quaternionic Hopf fibration through a fibration over S4, whose fiber has the above Sullivan model.
This may or may not exist. And if it exists, it may or may not be related to twistors.
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(hat tip to Michael Murray here)
I am confused about the so-called twistor fibration mentioned here: https://www.e-periodica.ch/digbib/view?pid=ens-001:2003:49::488#488
My question is, in what sense is SO(5)U(2) equivalent to ℂℙ3? It is true that they both have six dimensions, but I believe they contain different representations with respect to different groups. As I understand, complex projective spaces as quotient spaces are defined as ℂℙn=SU(n+1)U(n). For instance, twistors are spinors of the conformal group SU(2,2), which motivates ℂℙ2,1, whose compact realization would be ℂℙ3=SU(4)U(3).
First, it is true that T=SO(5)U(2) admits a trivial fibration mapping to B=S4=SO(5)SO(4) with F=SO(4)U(2). The fibration is trivial because the total space T = F x B, the fiber space combined with the base space.
Second, it is true that both SU(4)U(3) and SO(5)U(2) lead to 3+3 representations, but they appear to differ. Consider so(5)→so(3)⊕so(2), giving representations 10=30⊕10⊕32⊕3−2. Additionally, consider su(4)→su(3)⊕u(1), giving representations 15=80⊕10⊕33⊕ˉ3−3. Both contain types of 3⊕3, but the former contains adjoint reps of so(3), while the latter contains fundamental reps of su(3), which seems to be different.
Third, we know that S7→ℂℙ3 with S1 fibers, which is a nontrivial fibration. However, I am not sure if ℂℙ3=SU(4)U(3) admits a nontrivial fibration that maps to S4. Are there any known references that mention this?
While I admit that I have much more to learn about these twistor fibrations, I currently don’t think that ℂℙ3 can be realized as SO(5)U(2), but accept that this admits a trivial fibration to S4. This trivial fibration seems to have no relation to twistors or twistor space. Am I missing something? Note how ℂℙn typically is embedded in ℂn+1, while SO(5)U(2) contains 6 real dimensions and does not seem to require embedding in ℂ4.
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