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I made Cayley plane a little less stubby.
Added Hisham’s paper
Hmm, what’s this about “We have provided evidence for some relations between M-theory and an octonionic version of Kreck-Stolz elliptic homology.”
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…)
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)
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.
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.
This will take some digesting, but thanks!
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 $X^8 \times \Sigma$?
I don’t recall mention of compactification on a 10-manifold, but there is a mention in Twisted K-Theory from Monodromies
We were lacking a page on that, true. Starting one here now: F/M-theory on elliptically fibered Calabi-Yau 5-folds.
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