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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeMay 7th 2015
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI made [[Cayley plane]] a little less stubby.

   I made Cayley plane a little less stubby.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 17th 2019
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded a couple of properties. <a href="https://ncatlab.org/nlab/revision/diff/Cayley+plane/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+plane/5">v5</a>, <a href="https://ncatlab.org/nlab/show/Cayley+plane">current</a>

   Added a couple of properties.

   diff, v5, current

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 27th 2020
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded Hisham's paper * [[Hisham Sati]], _$\mathbb{O}P^2$ bundles in M-theory_, ([arXiv:0807.4899](https://arxiv.org/abs/0807.4899)) Hmm, what's this about "We have provided evidence for some relations between M-theory and an octonionic version of Kreck-Stolz elliptic homology." <a href="https://ncatlab.org/nlab/revision/diff/Cayley+plane/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+plane/7">v7</a>, <a href="https://ncatlab.org/nlab/show/Cayley+plane">current</a>

   Added Hisham’s paper

   • Hisham Sati, bundles in M-theory, (arXiv:0807.4899)

   Hmm, what’s this about “We have provided evidence for some relations between M-theory and an octonionic version of Kreck-Stolz elliptic homology.”

   diff, v7, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 27th 2020
   • (edited Nov 27th 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexOh, I was offline and now just about to do the same. Okay, great. (I don't know about that octonionic elliptic cohomology. But I think we are beginning to see now how plain elliptic cohomological really fits in...)

   Oh, I was offline and now just about to do the same. Okay, great.

   (I don’t know about that octonionic elliptic cohomology. But I think we are beginning to see now how plain elliptic cohomological really fits in…)

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeNov 27th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave expanded this citation now as follows: *** Indications that [[M-theory]] in 10+1 dimensions may be understood as the [[KK-compactification]] on Cayley-plane [[fibers]] of some kind of [[bosonic M-theory]] in 26+1 dimensions: * [[Hisham Sati]], _$\mathbb{O}P^2$ bundles in M-theory_, Commun. Num. Theor. Phys 3:495-530,2009 ([arXiv:0807.4899](https://arxiv.org/abs/0807.4899)) * [[Hisham Sati]], _On the geometry of the supermultiplet in M-theory_, Int. J. Geom. Meth. Mod. Phys. 8 (2011) 1-33 ([arXiv:0909.4737](https://arxiv.org/abs/0909.4737)) * [[Michael Rios]], [[Alessio Marrani]], [[David Chester]], _Exceptional Super Yang-Mills in $D=27+3$ and Worldvolume M-Theory_, Phys. Lett. B, 808, (2020) ([arXiv:1906.10709](https://arxiv.org/abs/1906.10709)) <a href="https://ncatlab.org/nlab/revision/diff/Cayley+plane/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+plane/8">v8</a>, <a href="https://ncatlab.org/nlab/show/Cayley+plane">current</a>

   have expanded this citation now as follows:

   ---

   Indications that M-theory in 10+1 dimensions may be understood as the KK-compactification on Cayley-plane fibers of some kind of bosonic M-theory in 26+1 dimensions:

   • Hisham Sati, bundles in M-theory, Commun. Num. Theor. Phys 3:495-530,2009 (arXiv:0807.4899)

   • Hisham Sati, On the geometry of the supermultiplet in M-theory, Int. J. Geom. Meth. Mod. Phys. 8 (2011) 1-33 (arXiv:0909.4737)

   • Michael Rios, Alessio Marrani, David Chester, Exceptional Super Yang-Mills in and Worldvolume M-Theory, Phys. Lett. B, 808, (2020) (arXiv:1906.10709)

   diff, v8, current

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 28th 2020
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItex> But I think we are beginning to see now how plain elliptic cohomological really fits in… I look forward to seeing that. Out of interest, is F-theory a target? I see at [F-theory](https://ncatlab.org/nlab/show/F-theory#relation_to_elliptic_cohomology) > A series of articles arguing for a relation between the elliptic fibration of F-theory and elliptic cohomology... naming a paper by Hisham.

   But I think we are beginning to see now how plain elliptic cohomological really fits in…

   I look forward to seeing that.

   Out of interest, is F-theory a target? I see at F-theory

   A series of articles arguing for a relation between the elliptic fibration of F-theory and elliptic cohomology…

   naming a paper by Hisham.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeNov 28th 2020
   • (edited Nov 28th 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes. So here is the idea: The equivariance group of the quaternionic Hopf fibration is $Sp(2).Sp(1) \subset Spin(8)$, hence J-twists for Cohomotopy in joint degree 7 and 4 requires $Sp(2).Sp(1)$-structure folds $X^8$. Up to homotopy this may be thought of as stand-ins for 11-manifolds $ \mathbb{R}^{2,1} \times X^8$, as considered in [[M-theory on 8-manifolds]]. But can we see any non-trivial longitudinal geometry other than $\mathbb{R}^{2,1} $? Consider, for simplicity, the case that $X^8 = (\mathbb{R}^8 \setminus \{0\})/G$, for $G \subset Sp(2).Sp(1)$ a finite subgroup, the near horizon geometry of a black M2-brane. Then J-twisted 4-Cohomotopy on $X^8$ is $G$-equivariant 4-Cohomotopy of $S^7$. To analyze this, choose a multiplicative $G$-equivariant cohomology theory $E$ to detect the resulting C-field fluxes. The latter trivialize on the $S^7$, but so we record how it trivializes (i.e. the $H_3$-flux). This is equivalently the choice of a quaternionic orientation to-second-stage on $E$. In this way second-stage quaternionic orientations on cohomology theories $E$ serve to define $E$-valued character maps on twistorial Cohomotopy in which to detect C-field flux in the vicinity of M2-branes. Now one way to obtain quaternionic orientations $p_1^E$ is to choose a complex orientation $c_1^E$ and set $p_1^E \coloneqq c_2^E$. Notice that, via Hypothesis H, this equates the shifted C-field flux $G_4^E$ as seen in $E$-theory with the second $E$-Chern class of a gauge bundle -- which "is" again the Horava-Witten Green-Schwarz mechanism (since in the present situation of finite $G$-quotients of flat space, no gravitational instanton contribution is present). In conclusion then, the C-field flux seen in $E$-theory after a choice of complex orientation is the second Conner-Floyd $E$-Chern class of the complex vector bundle underlying the tautological quaternionic line bundle over $\mathbb{H}P^2$ (all $G$-equivariantly). If we take $E$ to be Landweber exact, then the resulting flux is a function on a group scheme. If it's moreover a [[Calabi-Yau cohomology]] theory this function is a wave function of a higher self-dual gauge theory on some "hidden" variety (hidden in that it's not part of the 8-manifold $X^8$ we started with). In particular, if $E$ is elliptic, the flux we measure depends, besides the visible 8-geometry, on an elliptic curve $\Sigma$. Hence this way we end up cohomologically detecting an effective spacetime geometry of the form $\mathbb{R}^{0,1} \times X^8 \times \Sigma$. This "is" the F-theory setup of M-theory (on 8-manifolds) compactified on an elliptic curve. Much as in Connes's NCG formulation of [[KK-compactification]], the fiber space $\Sigma$ has shrunken to the point that it is classically invisible, but reveals itself through cohomological effects. Something like this.

   Yes. So here is the idea:

   The equivariance group of the quaternionic Hopf fibration is , hence J-twists for Cohomotopy in joint degree 7 and 4 requires -structure folds .

   Up to homotopy this may be thought of as stand-ins for 11-manifolds , as considered in M-theory on 8-manifolds. But can we see any non-trivial longitudinal geometry other than ?

   Consider, for simplicity, the case that , for a finite subgroup, the near horizon geometry of a black M2-brane. Then J-twisted 4-Cohomotopy on is -equivariant 4-Cohomotopy of .

   To analyze this, choose a multiplicative -equivariant cohomology theory to detect the resulting C-field fluxes. The latter trivialize on the , but so we record how it trivializes (i.e. the -flux). This is equivalently the choice of a quaternionic orientation to-second-stage on .

   In this way second-stage quaternionic orientations on cohomology theories serve to define -valued character maps on twistorial Cohomotopy in which to detect C-field flux in the vicinity of M2-branes.

   Now one way to obtain quaternionic orientations is to choose a complex orientation and set . Notice that, via Hypothesis H, this equates the shifted C-field flux as seen in -theory with the second -Chern class of a gauge bundle – which “is” again the Horava-Witten Green-Schwarz mechanism (since in the present situation of finite -quotients of flat space, no gravitational instanton contribution is present).

   In conclusion then, the C-field flux seen in -theory after a choice of complex orientation is the second Conner-Floyd -Chern class of the complex vector bundle underlying the tautological quaternionic line bundle over (all -equivariantly).

   If we take to be Landweber exact, then the resulting flux is a function on a group scheme. If it’s moreover a Calabi-Yau cohomology theory this function is a wave function of a higher self-dual gauge theory on some “hidden” variety (hidden in that it’s not part of the 8-manifold we started with). In particular, if is elliptic, the flux we measure depends, besides the visible 8-geometry, on an elliptic curve .

   Hence this way we end up cohomologically detecting an effective spacetime geometry of the form .

   This “is” the F-theory setup of M-theory (on 8-manifolds) compactified on an elliptic curve. Much as in Connes’s NCG formulation of KK-compactification, the fiber space has shrunken to the point that it is classically invisible, but reveals itself through cohomological effects.

   Something like this.

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 28th 2020
   • (edited Nov 28th 2020)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThis will take some digesting, but thanks!

   This will take some digesting, but thanks!

9. • CommentRowNumber9.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 30th 2020
   • (edited Nov 30th 2020)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAt [[M-theory on 8-manifolds]] we have > If the 8-dimensional fibers themselves are elliptic fibrations, then M-theory on these 8-manifolds is supposedly T-dual to F-theory KK-compactified to 3+1 spacetime-dimensions. But at the end of #7 you're talking about a further compactification on an elliptic curve, so in total on the 10-manifold $X^8 \times \Sigma$? I don't recall mention of compactification on a 10-manifold, but there is a mention in [Twisted K-Theory from Monodromies](https://arxiv.org/abs/hep-th/0302081)

   At M-theory on 8-manifolds we have

   If the 8-dimensional fibers themselves are elliptic fibrations, then M-theory on these 8-manifolds is supposedly T-dual to F-theory KK-compactified to 3+1 spacetime-dimensions.

   But at the end of #7 you’re talking about a further compactification on an elliptic curve, so in total on the 10-manifold ?

   I don’t recall mention of compactification on a 10-manifold, but there is a mention in Twisted K-Theory from Monodromies

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2020
    • PermaLink
    Author: Urs
    Format: MarkdownItexWe were lacking a page on that, true. Starting one here now: _[[F/M-theory on elliptically fibered Calabi-Yau 5-folds]]._

    We were lacking a page on that, true. Starting one here now: F/M-theory on elliptically fibered Calabi-Yau 5-folds.

11. • CommentRowNumber11.
    • CommentAuthorSamuel Adrian Antz
    • CommentTimeFeb 2nd 2024
    • (edited Feb 2nd 2024)
    • PermaLink
    Author: Samuel Adrian Antz
    Format: MarkdownItexAdded homotopy groups and cohomology of octonionic projective plane, already stated in Lackman 19 (https://arxiv.org/abs/1909.07047) already present on the page, and also in Mimura 67 (https://doi.org/10.1215/kjm/1250524375) now added as a new reference as well. Replaced 15-sphere by 16-disk in Proposition 2.2. as stated by both of those sources. (The same edit was done on [[octionionic projective space]].) <a href="https://ncatlab.org/nlab/revision/diff/Cayley+plane/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+plane/11">v11</a>, <a href="https://ncatlab.org/nlab/show/Cayley+plane">current</a>

    Added homotopy groups and cohomology of octonionic projective plane, already stated in Lackman 19 (https://arxiv.org/abs/1909.07047) already present on the page, and also in Mimura 67 (https://doi.org/10.1215/kjm/1250524375) now added as a new reference as well. Replaced 15-sphere by 16-disk in Proposition 2.2. as stated by both of those sources. (The same edit was done on octionionic projective space.)

    diff, v11, current