Timeline for What is operator tmf?
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9 events
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| Dec 12, 2014 at 18:10 | comment | added | Qiaochu Yuan | @Andreas: thanks for the suggestions! In fact this theory should not assign anything to an arbitrary C*-algebra, the problem being that the category of modules (whatever one means by this) over an arbitrary C*-algebra lacks a natural tensor structure. This theory should at best assign things to commutative C*-algebras and maybe Hopf C*-algebras. | |
| Dec 12, 2014 at 17:39 | comment | added | Qiaochu Yuan | @Ronnie: yes, as a special case this construction needs to produce a "higher group C*-algebra" of, at the very least, a finite 2-group. In the case of a finite group $G$ I know a simple version of this construction: just take the tensor category of $G$-graded vector spaces. I have also been told a candidate for what the tmf of this thing should be but I don't remember the details quite well enough to communicate it. | |
| Dec 12, 2014 at 11:34 | comment | added | Ronnie Brown | To throw out a possibly wild idea, this question might be related to this one: mathoverflow.net/questions/86617/… | |
| Dec 12, 2014 at 9:25 | comment | added | Andreas Thom | I have shown in my dissertation that any cohomology theory for $C^{\ast}$-algebras that has the property that the inclusion $A \subset M_2(A)$ induces an isomorphism must be a $bu$-module, i.e., it admits some self-map of degree two. So any generalization of $tmf$ cannot be nicely compatible with notions like Morita invariance etc. However, there is unpublished work using unitary vertex superalgebras as some sort of categorification of complex Clifford algebras, this goes in a new and promising direction, see maths.gla.ac.uk/~cvoigt/research.xhtml | |
| Dec 12, 2014 at 3:48 | comment | added | David Roberts♦ | If vertex algebras are "the" answer, then mathoverflow.net/questions/124637 should be relevant. | |
| Dec 12, 2014 at 1:59 | comment | added | Paul Siegel | Some people seem to think that vertex algebras should play the role of C*-algebras for TMF. Vertex algebras are known to be related to lots of things that are known to be related to TMF - the Witten genus, conformal field theory, string structures, and so on - but I'm not sure there's any more evidence than that. | |
| Dec 12, 2014 at 0:30 | comment | added | Qiaochu Yuan | @David: tmf stands for "topological modular forms." It is a "higher" version of K-theory. For a pointer to the literature see, for example, ms.unimelb.edu.au/~nganter/talbot. | |
| Dec 12, 2014 at 0:14 | comment | added | David Handelman | What's a tmf? [That was my question, but for some reason comments require at least fifteen characters.] | |
| Dec 11, 2014 at 22:34 | history | asked | Qiaochu Yuan | CC BY-SA 3.0 |