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](http://archive.numdam.org/article/SB_1987-1988__30__187_0.pdf), 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"?



What do loop groups and von Neumann algebras have to do with elliptic cohomology?
David Roberts mentioned in his comments the relationship


**K-theory : spin group  

TMF : string groups**


Let me recommend the first 6 pages of my [unfinished article](http://www.staff.science.uu.nl/%7Ehenri105/PDF/TringWP.pdf)
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](http://arxiv.org/pdf/1103.4187v1.pdf) of mine (joint with Chris Douglas).
> 
> 
> 


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


