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I am starting something on U-duality at supergravity.
But all still very skeletal at the moment.
So this relates to your Google+ comments on exceptional objects and M-theory.
We heard from John McKay once
Study of the sporadics leads me to conclude that we ought to examine the role of the Schur multiplier. For the three sporadics M, B, F24’ the multipliers are L*/L, L = root lattice for E8,7,6 respectively.
And I heard him once relate
- Symplectic geometry: There’s a correspondence between (E8, E7, E6) and (Monster, Baby monster, Fischer group Fi24) and with
(120 tritangents on sextic of genus 4, 28 bitangents on quartic, 27 lines on cubic).
And 360 cusps link to the 120 tritangents.
Others have looked to relate E8 and the Monster, e.g., A moonshine path from E8 to the monster and Arithmetic groups and the affine E8 Dynkin diagram. Makes you wonder what might be possible for the higher Es.
By the way, is there a reason at E11, why the series stops
… E6, E7, E8, E9, E10, E11, ?
What happens? It can continue to E12 and beyond, but that’s no longer nice in some sense.
Thanks for all the pointers and links. I’d need to further look into this.
Regading the last question:
I don’t know what happens mathematically, but in terms of physics the sequence terminates because $E_{n(n)}$ is the U-duality group of the maximal 11-dimensional supergravity KK-compactified to dimension $11-n$. So $E_{8(8)}$ is the gauge group of maximal 3d supergravity and I guess $E_{9(9)}$ as the “current algebra” in 2d supergravity is still pretty clear, but then beyond that it gets speculative: Hermann Nicolai conjectures that for 1-dimensional sugra we get the superparticle propagating on the “group manifold” $E_{10(10)}$ (which is mathematically not really under control, I gather, but is at least a thoroughly plausible object that ought to exist) and then West went all the way and conjectures that KK-compactification all the way to the point yields $E_{11(11)}$ as U-duality. This is curious now, since it means that a configuration of all of (conjecturally M-completed, with all non-perturbative M-brane effects and everything) 11d-Sugra is given by just picking one point in $E_{11(11)}$.
West speaks about this in his textbook Introduction to Strings and Branes.
I’d say this is enough speculation to chew on for the moment, that nobody would be well advised to speculate that “KK-compatification to (-1)-dimensions” (whatever that might mean) yields U-duality $E_{12(12)}$ (whatever that might mean)
Wikipedia believes the series continues En. Qualifiers like affine, hyperbolic, Lorentzian seem to stop at 11 though.
Oh,right. Thanks.
added pointers to two review talks from last week,on the $E_{10}$-case: by Thibault Damour and by Hermann Nicolai
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