This will take some digesting, but thanks!

]]>added pointer to:

- Gereon Quick,
*The $e$-invariant and the $J$-homomorphism*, lecture notes in:*Advanced algebraic topology*, 2014 (pdf)

added pointer to:

]]>added pointer to:

- Gereon Quick,
*The Hopf invariant one problem via K-theory*, lecture notes, 2014 (pdf)

added this pointer:

- Gereon Quick,
*The $e$-invariant*(pdf)

added this pointer:

- Christian Blanchet, Marco De Renzi,
*Modular Categories and TQFTs Beyond Semisimplicity*(arXiv:2011.12932)

Yes. 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.

]]>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.

]]>Entry to go with D=12 supergravity and bosonic M-theory, but for the moment just to record references

]]>renamed to better fit the naming convention of entries on SuGra/SYM

]]>just for completeness, as a further item in the disambiguation list at *signature*

brief `category:people`

-entry for hyoerlinking references at *octonionic projective plane*

Yes, I changed it already on the page itself.

]]>Added a redirect: tortile monoidal category (terminology of Joyal–Street–Verity).

]]>Deleted a dead link about “future visions”.

Added a remark that the project appears to be inactive since 2013.

]]>Hmm, is that meant to be $\mathbb{O}P^1$?

]]>minor

Valeria de Paiva

]]>added pointer to

- 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)

brief `category:people`

-entry for hyperlinking references at *12-dimensional supergravity*, *bosonic M-theory*, *Moonshine* and *Monster vertex operator algebra*

brief `category:people`

-entry for hyperlinking references at *12-dimensional supergravity*, *bosonic M-theory*, *Moonshine* and *Monster vertex operator algebra*

brief `category:people`

-entry for hyperlinking references at *12-dimensional supergravity*, *bosonic M-theory*, *Moonshine* and *Monster vertex operator algebra*

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,

*$\mathbb{O}P^2$ 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 $D=27+3$ and Worldvolume M-Theory*, Phys. Lett. B, 808, (2020) (arXiv:1906.10709)

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…)

]]>I have expanded this list of references here and completed the publication data:

Hisham Sati,

*$\mathbb{O}P^2$ 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 $D=27+3$ and Worldvolume M-Theory*, Phys. Lett. B, 808, (2020) (arXiv:1906.10709)

Will record this also at *Cayley plane* now.