Timeline for What do loop groups and von Neumann algebras have to do with elliptic cohomology?
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12 events
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| Dec 19, 2020 at 3:37 | history | edited | Allen Knutson | CC BY-SA 4.0 |
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| Jul 7, 2012 at 21:39 | comment | added | André Henriques | @Dmitri: The reference is an unfinished paper of mine from a couple a years ago that I just posted on my webpage: staff.science.uu.nl/~henri105/PDF/BottPer.pdf | |
| Jul 6, 2012 at 22:31 | comment | added | André Henriques | Unfortunately, I don't know how to explain "why"... See also mathoverflow.net/questions/62654/… for more unanswered questions about the relationship between Clifford algebras and KO. | |
| Jul 6, 2012 at 21:35 | comment | added | Akhil Mathew | I like the analogy between Clifford algebras corresponding to Spin groups and free fermions corresponding to the String group (though I don't yet know what a free fermion is). But OK --- say I believe that Clifford algebras and Morita equivalence should lead to the Spin group (as you write in one of the articles). Is there an a priori reason the spin group should naturally related to $KO$-theory, before the Spin-orientation of $KO$ (which uses Clifford modules)? | |
| Jul 6, 2012 at 21:30 | comment | added | Akhil Mathew | Awesome. Thank you! There's a lot to digest here, and maybe I'll comment after I've understood a bit more. But for starers I am curious if there is a "good" reason for the appearance of Clifford modules in $KO$-theory, especially the fact that Clifford modules can be used to produce $\pi_* KO$. (See this question mathoverflow.net/questions/85516/…). I take it general von Neumann algebras are supposed to be some sort of extension of Clifford algebras? | |
| Jul 6, 2012 at 21:28 | vote | accept | Akhil Mathew | ||
| Jul 6, 2012 at 19:00 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Jul 6, 2012 at 16:45 | comment | added | Dmitri Pavlov | @André: On page 8 you mention that elements in K-theory can modeled by quasibundles (finite dimensional not necessarily locally trivial bundles) of Clifford modules. Do you have a reference for that? | |
| Jul 6, 2012 at 13:20 | comment | added | André Henriques | Hi Urs. No, that remains to be done. That model of the string group doesn't look very smooth. Making sense of a smooth family of Hilbert spaces is already quite a tricky buisness: that's already something that I don't know how to do... So making sense of a smooth family of von Neumann algebras (or defects between conformal nets -- same issues) is much more tricky, and I also don't know how to do it. | |
| Jul 6, 2012 at 12:40 | comment | added | Urs Schreiber | By the way: in that note you obtain String as a topological 2-group. Have you, or can you, construct an equivalence to the topological 2-group underlying any one of the smooth 2-group models? (That would be good to have, since it would allow to associate fermionic net 2-bundles to smooth String-principal 2-bundles.) Last time that I looked into this with colleages we found a natural candidate homomorphism from the strict smooth 2-group version of String to that 2-group of net automorphisms in your writeup. But we didn't quite check a bunch of operator algebra things that one needs to check. | |
| Jul 6, 2012 at 9:55 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Jul 6, 2012 at 3:28 | history | answered | André Henriques | CC BY-SA 3.0 |