nForum - Discussion Feed (twistor fibration) 2023-09-24T03:31:18+00:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher David_Chester comments on "twistor fibration" (107432) https://nforum.ncatlab.org/discussion/11592/?Focus=107432#Comment_107432 2023-03-01T17:42:25+00:00 2023-09-24T03:31:16+00:00 David_Chester https://nforum.ncatlab.org/account/3339/ 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)} ...

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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Urs comments on "twistor fibration" (91010) https://nforum.ncatlab.org/discussion/11592/?Focus=91010#Comment_91010 2021-03-31T09:41:49+00:00 2023-09-24T03:31:16+00:00 Urs https://nforum.ncatlab.org/account/4/ 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, ...

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)

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Urs comments on "twistor fibration" (91009) https://nforum.ncatlab.org/discussion/11592/?Focus=91009#Comment_91009 2021-03-31T09:41:12+00:00 2023-09-24T03:31:16+00:00 Urs https://nforum.ncatlab.org/account/4/ added pointer to: Michael Atiyah, Nigel Hitchin, Isadore Singer, Example 2 on p. 438 in: Self-Duality in Four-Dimensional Riemannian Geometry, Proceedings of the Royal Society of London. Series A, ...

(hat tip to Michael Murray here)

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Urs comments on "twistor fibration" (87917) https://nforum.ncatlab.org/discussion/11592/?Focus=87917#Comment_87917 2020-11-21T19:29:48+00:00 2023-09-24T03:31:16+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

• 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)
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Urs comments on "twistor fibration" (86361) https://nforum.ncatlab.org/discussion/11592/?Focus=86361#Comment_86361 2020-08-20T08:06:18+00:00 2023-09-24T03:31:16+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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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David_Corfield comments on "twistor fibration" (86350) https://nforum.ncatlab.org/discussion/11592/?Focus=86350#Comment_86350 2020-08-19T19:45:16+00:00 2023-09-24T03:31:16+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ 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.

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.

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Urs comments on "twistor fibration" (86343) https://nforum.ncatlab.org/discussion/11592/?Focus=86343#Comment_86343 2020-08-19T16:18:19+00:00 2023-09-24T03:31:16+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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.

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David_Corfield comments on "twistor fibration" (86342) https://nforum.ncatlab.org/discussion/11592/?Focus=86342#Comment_86342 2020-08-19T16:05:24+00:00 2023-09-24T03:31:17+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ Added generalizations to all even spheres and beyond studied by Robert Bryant. diff, v14, current

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

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Urs comments on "twistor fibration" (86264) https://nforum.ncatlab.org/discussion/11592/?Focus=86264#Comment_86264 2020-08-14T06:21:08+00:00 2023-09-24T03:31:17+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

• 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)

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Urs comments on "twistor fibration" (86210) https://nforum.ncatlab.org/discussion/11592/?Focus=86210#Comment_86210 2020-08-07T08:35:39+00:00 2023-09-24T03:31:17+00:00 Urs https://nforum.ncatlab.org/account/4/ added pointer to: Bobby Acharya, Robert Bryant, Simon Salamon, Section 6 of: A circle quotient of a G 2G_2 cone (arXiv:1910.09518) diff, v12, current

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Urs comments on "twistor fibration" (86195) https://nforum.ncatlab.org/discussion/11592/?Focus=86195#Comment_86195 2020-08-06T13:46:02+00:00 2023-09-24T03:31:17+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

Okay, thanks!

Meanwhile I have found this review

• Jonas Nordstrom, Calabi’s construction of Harmonic maps from $S^2$ to $S^n$, 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 $m = 2$ in Calabi’s notation) to what Atiyah introduced as the twistor fibration.

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David_Corfield comments on "twistor fibration" (86193) https://nforum.ncatlab.org/discussion/11592/?Focus=86193#Comment_86193 2020-08-06T12:51:42+00:00 2023-09-24T03:31:17+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ AHS is relatively recent, hence the “seulement aujord’hui”.

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

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David_Corfield comments on "twistor fibration" (86192) https://nforum.ncatlab.org/discussion/11592/?Focus=86192#Comment_86192 2020-08-06T12:49:26+00:00 2023-09-24T03:31:17+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ “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 ...

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…

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Urs comments on "twistor fibration" (86189) https://nforum.ncatlab.org/discussion/11592/?Focus=86189#Comment_86189 2020-08-06T12:07:26+00:00 2023-09-24T03:31:17+00:00 Urs https://nforum.ncatlab.org/account/4/ 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), ...

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.

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Urs comments on "twistor fibration" (86188) https://nforum.ncatlab.org/discussion/11592/?Focus=86188#Comment_86188 2020-08-06T11:25:18+00:00 2023-09-24T03:31:17+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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.

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David_Corfield comments on "twistor fibration" (86187) https://nforum.ncatlab.org/discussion/11592/?Focus=86187#Comment_86187 2020-08-06T11:05:13+00:00 2023-09-24T03:31:17+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ This page claims H is for Herbert. Presumably it’s all one person.

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

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Urs comments on "twistor fibration" (86186) https://nforum.ncatlab.org/discussion/11592/?Focus=86186#Comment_86186 2020-08-06T10:56:35+00:00 2023-09-24T03:31:17+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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”.

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Urs comments on "twistor fibration" (86185) https://nforum.ncatlab.org/discussion/11592/?Focus=86185#Comment_86185 2020-08-06T10:53:16+00:00 2023-09-24T03:31:17+00:00 Urs https://nforum.ncatlab.org/account/4/ A sorry, that’s another “Blaine Lawson”?! Am fixing it…

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

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Urs comments on "twistor fibration" (86183) https://nforum.ncatlab.org/discussion/11592/?Focus=86183#Comment_86183 2020-08-06T10:48:52+00:00 2023-09-24T03:31:17+00:00 Urs https://nforum.ncatlab.org/account/4/ Thanks! I have added that pointer: H. Blaine Lawson, Surfaces minimales et la construction de Calabi-Penrose, Séminaire Bourbaki : volume 1983/84, exposés 615-632, Astérisque no. 121-122 ...

Thanks! I have added that pointer:

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David_Corfield comments on "twistor fibration" (86182) https://nforum.ncatlab.org/discussion/11592/?Focus=86182#Comment_86182 2020-08-06T09:36:18+00:00 2023-09-24T03:31:17+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ I think this paper may explain Harlan P. Blaine, Jr. Lawson, Surfaces minimales et la construction de Calabi-Penrose, pdf

I think this paper may explain

• Harlan P. Blaine, Jr. Lawson, Surfaces minimales et la construction de Calabi-Penrose, pdf
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Urs comments on "twistor fibration" (86181) https://nforum.ncatlab.org/discussion/11592/?Focus=86181#Comment_86181 2020-08-06T09:15:49+00:00 2023-09-24T03:31:17+00:00 Urs https://nforum.ncatlab.org/account/4/ 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, ...

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?

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Urs comments on "twistor fibration" (86180) https://nforum.ncatlab.org/discussion/11592/?Focus=86180#Comment_86180 2020-08-06T09:00:47+00:00 2023-09-24T03:31:17+00:00 Urs https://nforum.ncatlab.org/account/4/ 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

• Bonaventure Loo, The space of harmonic maps of $S^2$ into $S^4$, Transactions of the American Mathematical Society Vol. 313, No. 1 (1989) (jstor:2001066)
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nLab edit announcer comments on "twistor fibration" (86166) https://nforum.ncatlab.org/discussion/11592/?Focus=86166#Comment_86166 2020-08-05T13:33:56+00:00 2023-09-24T03:31:17+00:00 nLab edit announcer https://nforum.ncatlab.org/account/1691/ added this pointer: John Armstrong, Simon Salamon, Twistor Topology of the Fermat Cubic, SIGMA 10 (2014), 061, 12 pages (arXiv:1310.7150) Anonymous diff, v7, current