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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 27th 2012
   • (edited May 14th 2014)
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   Author: Urs
   Format: MarkdownItexI am starting something on [U-duality](http://ncatlab.org/nlab/show/supergravity#UDuality) at _[[supergravity]]_. But all still very skeletal at the moment.

   I am starting something on U-duality at supergravity.

   But all still very skeletal at the moment.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 14th 2014
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   Author: Urs
   Format: MarkdownItexI have split off _[[U-duality]]_ to a stand-alone entry, added some more references and more hyperlinks (but otherwise the text is as before) and pointed to this from all of the relevant entries [[E6]], [[E7]], [[E8]], [[E9]], [[E10]], [[E11]]

   I have split off U-duality to a stand-alone entry, added some more references and more hyperlinks (but otherwise the text is as before) and pointed to this from all of the relevant entries E6, E7, E8, E9, E10, E11

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 15th 2014
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   Author: David_Corfield
   Format: MarkdownItexSo this relates to your Google+ comments on exceptional objects and M-theory. We heard from John McKay [once](http://golem.ph.utexas.edu/category/2006/09/mathematical_kinds.html#c023440) >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](http://golem.ph.utexas.edu/category/2008/10/john_mckay_visits_kent.html) relate >2. 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](http://www.math.lsa.umich.edu/~rlg/researchandpublications/pdffiles/moonshinepath13oct09.pdf) and [Arithmetic groups and the affine E8 Dynkin diagram](http://arxiv.org/abs/0810.1465). 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.

   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

   1. 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.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 15th 2014
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   Author: Urs
   Format: MarkdownItexThanks 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-compactification|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)

   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)

5. • CommentRowNumber5.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 15th 2014
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   Author: David_Corfield
   Format: MarkdownItexWikipedia believes the series continues [En](http://en.wikipedia.org/wiki/En_%28Lie_algebra%29). Qualifiers like affine, hyperbolic, Lorentzian seem to stop at 11 though.

   Wikipedia believes the series continues En. Qualifiers like affine, hyperbolic, Lorentzian seem to stop at 11 though.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMay 15th 2014
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   Author: Urs
   Format: MarkdownItexOh,right. Thanks.

   Oh,right. Thanks.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMay 28th 2014
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   Author: Urs
   Format: MarkdownItexadded pointers to two review talks from last week,on the $E_{10}$-case: [by Thibault Damour](http://ncatlab.org/nlab/show/U-duality#Damour14) and [by Hermann Nicolai](http://ncatlab.org/nlab/show/U-duality#Nicolai14)

   added pointers to two review talks from last week,on the $E_{10}$-case: by Thibault Damour and by Hermann Nicolai

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeAug 27th 2024
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   Author: Urs
   Format: MarkdownItexadded pointer to today's * [[Hisham Sati]], [[Alexander A. Voronov]]: *Mysterious Triality and the Exceptional Symmetry of Loop Spaces* &lbrack;[arXiv:2408.13337](https://arxiv.org/abs/2408.13337)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/U-duality/39">diff</a>, <a href="https://ncatlab.org/nlab/revision/U-duality/39">v39</a>, <a href="https://ncatlab.org/nlab/show/U-duality">current</a>

   added pointer to today’s

   • Hisham Sati, Alexander A. Voronov: Mysterious Triality and the Exceptional Symmetry of Loop Spaces [arXiv:2408.13337]

   diff, v39, current