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Recall that complex $K$-theory is a cohomology theory on topological spaces, which can be described in several equivalent ways:

  • Given a finite complex $X$, $K^0(X)$ is the Grothendieck group of vector bundles on $X$. $K^*$ is even-periodic, and this determines the entire cohomology theory. Using the tensor product of vector bundles, $K$ becomes a multiplicative cohomology theory. There is a corresponding ring spectrum.
  • The classifying space $BU \times \mathbb{Z}$ for $K^0$ is, by a theorem of Atiyah, the space of Fredholm operators on a countably-dimensional Hilbert space. So we can think of classes in $K^0(X)$ as "families of Fredholm operators" parametrized by $X$: the "index" of such a family should be a virtual vector bundle, which connects to the previous definition.
  • $K$-theory is an even-periodic theory, so it is complex-orientable and corresponds to a formal group on $K^0(\ast) = \mathbb{Z}$. This formal group is the multiplicative one, which turns out to be Landweber-exact. Consequently, one can construct $K$-theory directly from the formal multiplicative group (once one has the spectrum $MU$) via $K_\bullet(X) = MU_\bullet(X) \otimes_{MU_\bullet} K_\bullet$.
  • The spectrum for $K$-theory can be obtained by taking the ring spectrum $\Sigma^\infty \mathbb{CP}^\infty_+$ (which is a ring spectrum as $\mathbb{CP}^\infty$ is a topological abelian monoid) and inverting the natural element in $\pi_2$. (This is a theorem of Snaith.)

It's sort of remarkable that $K$-theory can be described both geometrically (via vector bundles or operators) or algebraically (via formal groups or Snaith's theorem). The only explanation that I can think of for this is that the correspondence between (complex-orientable) ring spectra and formal groups is given more or less in terms of Chern classes of vector bundles, so a cohomology theory built directly from vector bundles would have a good chance of furnishing a fairly simple formal group law. (One can use this sort of argument to prove Snaith's theorem, for instance.)

A much less formal example of a formal group is that associated to an elliptic curve. If $E/\mathrm{Spec} R$ is an elliptic curve, then under appropriate hypothesis (Landweber exactness, or flatness of the map $\mathrm{Spec} R \to M_{1,1} \to M_{FG}$, or more concretely that $R$ is torsion-free and for each $p$, the Hasse invariant $v_1$ is a nonzerodivisor in $R/pR$) we can construct an "elliptic cohomology" theory $\mathrm{Ell}^*$ which is even-periodic and whose formal group is that of $E/R$. The associated formal group can have height up to $2$, so we get something much more complicated than $K$-theory.

It has been suggested that there should be a geometric interpretation of elliptic cohomology. This seems a lot more difficult, because the formal group law associated to an elliptic curve is less elementary than $\hat{\mathbb{G}_m}$. There are various programs (which start with Segal's survey, I believe), all of which I know nothing about, to interpret elliptic cohomology classes in terms of von Neumann algebras, loop group representations, conformal field theories, ...

I can understand why a geometric interpretation of elliptic cohomology would be desirable, but it's mystifying to me why researchers in this area are concentrating on these specific objects. Is there a "high-concept" explanation for this, and motivation (to someone without a background in geometry) for how one might "believe" in these visions? Is there a reason loop groups should be "height two" where the unitary group is "height one"?

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There is the relation between the group $String$ and tmf on one hand, and String as a 3-connected cover of $Spin$ and the canonical central extension (=2-connected cover) of $\Omega Spin$. String structures were first considered via loop groups, as there were no smooth constructions of String (we do as of about a year ago). Things like Witten genus should be mentioned. von Neumann algebras (and bimodules, and morphisms of bimodules) come up because they seem to be a good candidate for constructing categorified $L^2$ (see closing comments in arxiv.org/abs/0812.4969) for reps of 2-groups –  David Roberts Jul 6 '12 at 1:49
...most smooth models of $String$ being 2-groups, including the the only finite-dimensional model so far, this is clearly something that needs further study. (the recent smooth construction of $String$ I mentioned earlier is a vanilla Frechet-Lie group, there were earlier Frechet-Lie 2-group models). Andre Henriques' research proposal, available from his website, has a lot of good material on looking at conformal nets and how they relate to tmf. –  David Roberts Jul 6 '12 at 2:20
Very interesting. One thing that I've heard is that "chromatic level" is supposed to correspond to "categorical level" in some sense: i.e., if $K$-theory is supposed to come from vector bundles (a category), elliptic cohomology is supposed to come from a 2-category. (The place where I read that also said that ordinary cohomology was 0-categorical, which didn't make much sense to me: $H^1$, for instance, classifies torsors, which are a category and not a set.) –  Akhil Mathew Jul 6 '12 at 16:21
(Your mention of "categorified $L^2$" reminded me of this.) Anyway, it'll take me some time to digest this article, but I look forward to reading it! –  Akhil Mathew Jul 6 '12 at 16:23
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up vote 16 down vote accepted

David Roberts mentioned in his comments the relationship

K-theory : spin group
TMF : string groups

Let me recommend the first 6 pages of my unifinished article for a uniform construction of $SO(n)$, $Spin(n)$, and $String(n)$, which suggests the existence of a similar uniform construction of $H\mathbb R$, $KO$ (or $KU$), and $TMF$. You'll see that von Neumann algebras appear in the construction. More precisely, von Neumann algebras appear in the definition of conformal nets. The latter are functors from 1-manifolds to von Neumann algberas.

For a summary of the conjectural relationship between conformal nets and $TMF$, have a look at page 8 of this other paper of mine (joint with Chris Douglas).

Loop groups yield non-trivial examples of conformal nets. Those conformal nets are related (conjecturally) to equivariant $TMF$.

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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. –  Urs Schreiber Jul 6 '12 at 12:40
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. –  André Henriques Jul 6 '12 at 13:20
@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? –  Dmitri Pavlov Jul 6 '12 at 16:45
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? –  Akhil Mathew Jul 6 '12 at 21:30
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. –  André Henriques Jul 6 '12 at 22:31
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