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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 4th 2020

    starting something

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 4th 2020
    • (edited Aug 4th 2020)

    I have a basic question here:

    Is the composite

    S 7P 3S 4 S^7 \longrightarrow \mathbb{C}P^3 \longrightarrow S^4

    of

    1) the canonical S 1S^1-fibration S 7P 3S^7 \to \mathbb{C}P^3

    followed by

    2) the twistor fibration P 3S 4\mathbb{C}P^3 \to S^4

    equivalent to

    3) the quaternionic Hopf fibration S 7S 4S^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.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 4th 2020

    Oh, I see it: It’s immediate from the reformulation of the Calanbi-Penrose fibrations given on this page.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 5th 2020

    added statement of and references for the definition via sending complex line to quaternionic lines

    diff, v4, current

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 5th 2020
    • (edited Aug 5th 2020)

    Does an analogue occur for other Hopf fibrations? E.g., a composite

    S 15P 3S 8 S^15 \longrightarrow \mathbb{H}P^3 \longrightarrow S^8

    as a composite of the S 3S^3-fibration S 15P 3S^15 \to \mathbb{H}P^3 and an S 4=P 1S^4 = \mathbb{H}P^1-fibration P 3S 8\mathbb{H}P^3 \to S^8.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 5th 2020

    added pointer to this item (thanks to David C. For pointing it out):

    • B. Loo and Alberto Verjovsky, On quotients of Hopf fibrations, in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 26 (1994), pp. 103-108 (hdl:10077/4637, pdf)

    diff, v6, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 5th 2020

    added this pointer:

    • Simon Brain, Giovanni Landi, Differential and Twistor Geometry of the Quantum Hopf Fibration, Commun. Math. Phys. 315 (2012):489-530 (arXiv:1103.0419)

    diff, v6, current

  1. added this pointer:

    Anonymous

    diff, v7, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2020

    added pointer to

    • Bonaventure Loo, The space of harmonic maps of S 2S^2 into S 4S^4, Transactions of the American Mathematical Society Vol. 313, No. 1 (1989) (jstor:2001066)

    diff, v8, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2020

    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?

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 6th 2020

    I think this paper may explain

    • Harlan P. Blaine, Jr. Lawson, Surfaces minimales et la construction de Calabi-Penrose, pdf
    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2020

    Thanks! I have added that pointer:

    diff, v9, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2020

    A sorry, that’s another “Blaine Lawson”?! Am fixing it…

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2020

    No, wait, how many “H. Blaine Lawson, Jr.” can there be, on this planet?

    It must be that the “H = Harlan” on SemanticScholar here is a glitch of automated reference-crawling? The article it self (here) doesn’t claim that “H = Harlan”.

    • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 6th 2020

    This page claims H is for Herbert. Presumably it’s all one person.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2020
    • (edited Aug 6th 2020)

    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.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2020

    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.

    • CommentRowNumber18.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 6th 2020
    • (edited Aug 6th 2020)

    “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…

    • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 6th 2020
    • (edited Aug 6th 2020)

    AHS is relatively recent, hence the “seulement aujord’hui”.

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeAug 6th 2020
    • (edited Aug 6th 2020)

    Okay, thanks!

    Meanwhile I have found this review

    • Jonas Nordstrom, Calabi’s construction of Harmonic maps from S 2S^2 to S nS^n, Lund University 2008 (pdf)

    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=2m = 2 in Calabi’s notation) to what Atiyah introduced as the twistor fibration.

    diff, v11, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeAug 7th 2020

    added pointer to:

    diff, v12, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeAug 14th 2020

    finally added pointer to

    • 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)

    diff, v13, current

    • CommentRowNumber23.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 19th 2020

    Added generalizations to all even spheres and beyond studied by Robert Bryant.

    diff, v14, current

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2020
    • (edited Aug 19th 2020)

    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 2F_2 in Twistorial Cohomotopy.

    What we’d probably need is something that would still factor the quaternionc Hopf fibration Sp(2)\mathrm{Sp}(2)-equivariantly.

    • CommentRowNumber25.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 19th 2020

    I haven’t seen anything else mentioned other than the Penrose fibration composed with the antipodal map, as in Example 1 of this and p. 5 of this.

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2020
    • (edited Aug 20th 2020)

    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)S(U(1)^2) a gauge group S(U(1) n)S(U(1)^n) would appear, for small nn.

    For starters, for n=3n =3 this would mean that the new fiber over S 4S^4 we are after has Sullivan model of this form:

    [f 2 (1), f 2 (2), h 3]/(df 2 (1) =0 df 2 (2) =0 dh 3 =f 2 (1)f 2 (1)f 2 (1)f 2 (2)f 2 (2)f 2 (2)) \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 BS(U(1) 3)B S(U(1)^3) in terms of the first Chern classes on the first two BU(1)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)Sp(2)-equivariant factorization of the quaternionic Hopf fibration through a fibration over S 4S^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.

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeNov 21st 2020

    added this pointer:

    • Angel Cano, Juan Pablo Navarrete, José Seade, Section 10.1 in: Kleinian Groups and Twistor Theory, In: Complex Kleinian Groups, Progress in Mathematics, vol 303. Birkhäuser 2013 (doi:10.1007/978-3-0348-0481-3_10)

    diff, v17, current

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2021
    • (edited Mar 31st 2021)

    added pointer to:

    (hat tip to Michael Murray here)

    diff, v18, current

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2021

    am finally changing the page name from “Calabi-Penrose fibration” to “twistor fibration” (which does less unjustice to Atiyah et al while still highlighting Penrose’s contribution, clearly)

    diff, v18, current

    • CommentRowNumber30.
    • CommentAuthorDavid_Chester
    • CommentTimeMar 1st 2023

    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)\frac{SO(5)}{U(2)} equivalent to ℂℙ 3\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 ℂℙ n=SU(n+1)U(n)\mathbb{CP}^n = \frac{SU(n+1)}{U(n)}. For instance, twistors are spinors of the conformal group SU(2,2)SU(2,2), which motivates ℂℙ 2,1\mathbb{CP}^{2,1}, whose compact realization would be ℂℙ 3=SU(4)U(3)\mathbb{CP}^3 = \frac{SU(4)}{U(3)}.

    First, it is true that T=SO(5)U(2)T = \frac{SO(5)}{U(2)} admits a trivial fibration mapping to B=S 4=SO(5)SO(4)B = S^4 = \frac{SO(5)}{SO(4)} with F=SO(4)U(2)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 SU(4)U(3)\frac{SU(4)}{U(3)} and SO(5)U(2)\frac{SO(5)}{U(2)} lead to 3+3 representations, but they appear to differ. Consider so(5)so(3)so(2)so(5) \rightarrow so(3) \oplus so(2), giving representations 10=3 01 03 23 210 = 3_0 \oplus 1_0 \oplus 3_2 \oplus 3_{-2}. Additionally, consider su(4)su(3)u(1)su(4) \rightarrow su(3) \oplus u(1), giving representations 15=8 01 03 33¯ 315 = 8_0 \oplus 1_0 \oplus 3_3 \oplus \bar{3}_{-3}. Both contain types of 333\oplus 3, but the former contains adjoint reps of so(3)so(3), while the latter contains fundamental reps of su(3)su(3), which seems to be different.

    Third, we know that S 7ℂℙ 3S^7 \rightarrow \mathbb{CP}^3 with S 1S^1 fibers, which is a nontrivial fibration. However, I am not sure if ℂℙ 3=SU(4)U(3)\mathbb{CP}^3 = \frac{SU(4)}{U(3)} admits a nontrivial fibration that maps to S 4S^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 ℂℙ 3\mathbb{CP}^3 can be realized as SO(5)U(2)\frac{SO(5)}{U(2)}, but accept that this admits a trivial fibration to S 4S^4. This trivial fibration seems to have no relation to twistors or twistor space. Am I missing something? Note how ℂℙ n\mathbb{CP}^n typically is embedded in n+1\mathbb{C}^{n+1}, while SO(5)U(2)\frac{SO(5)}{U(2)} contains 6 real dimensions and does not seem to require embedding in 4\mathbb{C}^4.