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I have a basic question here:
Is the composite
$S^7 \longrightarrow \mathbb{C}P^3 \longrightarrow S^4$of
1) the canonical $S^1$-fibration $S^7 \to \mathbb{C}P^3$
followed by
2) the twistor fibration $\mathbb{C}P^3 \to S^4$
equivalent to
3) the quaternionic Hopf fibration $S^7 \to S^4$
?
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
$S^15 \longrightarrow \mathbb{H}P^3 \longrightarrow S^8$as a composite of the $S^3$-fibration $S^15 \to \mathbb{H}P^3$ and an $S^4 = \mathbb{H}P^1$-fibration $\mathbb{H}P^3 \to S^8$.
added pointer to this item (thanks to David C. For pointing it out):
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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 $F_2$ in Twistorial Cohomotopy.
What we’d probably need is something that would still factor the quaternionc Hopf fibration $\mathrm{Sp}(2)$-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 $S^4$ we are after has Sullivan model of this form:
$\mathbb{R} \left[ \array{ f_2^{(1)}, \\ f_2^{(2)}, \\ h_3 } \right] \Big/ \left( \begin{aligned} d\, f_2^{(1)} & = 0 \\ d\, f_2^{(2)} & = 0 \\ d\;\, h_3\; & = - f_2^{(1)} \wedge f_2^{(1)} - f_2^{(1)} \wedge f_2^{(2)} - f_2^{(2)} \wedge f_2^{(2)} \end{aligned} \right)$That’s because the right hand side in the third line is the expression for the second Chern class on $B S(U(1)^3)$ in terms of the first Chern classes on the first two $B U(1)$-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 $S^4$, 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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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 $\frac{SO(5)}{U(2)}$ equivalent to $\mathbb{CP}^3$? 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 $\mathbb{CP}^n = \frac{SU(n+1)}{U(n)}$. For instance, twistors are spinors of the conformal group $SU(2,2)$, which motivates $\mathbb{CP}^{2,1}$, whose compact realization would be $\mathbb{CP}^3 = \frac{SU(4)}{U(3)}$.
First, it is true that $T = \frac{SO(5)}{U(2)}$ admits a trivial fibration mapping to $B = S^4 = \frac{SO(5)}{SO(4)}$ with $F = \frac{SO(4)}{U(2)}$. 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 $\frac{SU(4)}{U(3)}$ and $\frac{SO(5)}{U(2)}$ lead to 3+3 representations, but they appear to differ. Consider $so(5) \rightarrow so(3) \oplus so(2)$, giving representations $10 = 3_0 \oplus 1_0 \oplus 3_2 \oplus 3_{-2}$. Additionally, consider $su(4) \rightarrow su(3) \oplus u(1)$, giving representations $15 = 8_0 \oplus 1_0 \oplus 3_3 \oplus \bar{3}_{-3}$. Both contain types of $3\oplus 3$, 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 $S^7 \rightarrow \mathbb{CP}^3$ with $S^1$ fibers, which is a nontrivial fibration. However, I am not sure if $\mathbb{CP}^3 = \frac{SU(4)}{U(3)}$ admits a nontrivial fibration that maps to $S^4$. 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 $\mathbb{CP}^3$ can be realized as $\frac{SO(5)}{U(2)}$, but accept that this admits a trivial fibration to $S^4$. This trivial fibration seems to have no relation to twistors or twistor space. Am I missing something? Note how $\mathbb{CP}^n$ typically is embedded in $\mathbb{C}^{n+1}$, while $\frac{SO(5)}{U(2)}$ contains 6 real dimensions and does not seem to require embedding in $\mathbb{C}^4$.
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