What do loop groups and von Neumann algebras have to do with elliptic cohomology?

Question

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"?

Answer 1

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 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 $TMF$.

Answer 2

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.

You probably know most of this by now, but here are some thoughts. As usual I will be working at a very heuristic level throughout this answer.

Von Neumann Algebras. One place to start is the observation is that for $H$ an infinite-dimensional Hilbert space, the von Neumann algebra $B(H)$ has automorphism group the projective unitary group $PU(H)$. $PU(H)$ fits into a short exact sequence

$$1 \to U(1) \to U(H) \to PU(H) \to 1$$

and $U(H)$ is contractible by Kuiper's theorem; thus $PU(H)$ is a $B^2 \mathbb{Z}$, and $BPU(H)$ is a $B^3 \mathbb{Z}$. Hence $H^3(X, \mathbb{Z})$ is, in a suitable sense, a "Brauer group" of $X$ describing bundles of von Neumann algebras (isomorphic to $B(H)$) over $X$.

The significance of this observation is that $H^3(X, \mathbb{Z})$ is a natural cohomology group parameterizing twists of K-theory over $X$; equivalently, there is a natural map from $B^3 \mathbb{Z}$ to $BGL_1(KU)$. Given a bundle of von Neumann algebras over $X$, the corresponding twisted K-theory groups are something like the K-theory of module bundles of Hilbert modules over the bundle of von Neumann algebras, but don't trust me to have the specifics right here.

Now it's also known (see e.g. Ando-Blumberg-Gepner) that $H^4(X, \mathbb{Z})$ is a natural cohomology group parameterizing twists of tmf over $X$; equivalently, there is a natural map from $B^4 \mathbb{Z}$ to $BGL_1(tmf)$. If you could build a (higher) category which deloops von Neumann algebras in a suitable sense, you might hope to realize $BPU(H) \cong B^3 \mathbb{Z}$ as the automorphisms of an object in this category, and then families of those objects over $X$ could be a geometric avatar of these twists of tmf in the same way as above. I believe that conformal nets is explicitly intended to be such a delooping.

Loop group representations. This is the analogue of $G$-equivariant K-theory having something to do with the representation theory of $G$. One picture of tmf whose accuracy I can't comment on is that it should look heuristically like $K(ku)$, the cohomology theory presented by (Baas-Dundas-Rognes) 2-vector bundles. So $G$-equivariant tmf should look heuristically like $G$-equivariant 2-vector bundles, which over a point should look heuristically like representations of $G$ on (suitably dualizable) 2-vector spaces.

Whatever that means, such a thing ought to have a "character" which, rather than being a class function on $G$, or equivalently a function on the adjoint quotient $G/G$, is instead an equivariant vector bundle on $G/G$. Now $G/G$ looks heuristically like the classifying stack $BLG$ of the loop group $LG$, and hence an equivariant vector bundle on $G/G$ looks heuristically like a representation of $LG$. Freed-Hopkins-Teleman makes this precise. The non-equivariant version of this story is Witten's story about tmf having something to do with ($S^1$-equivariant?) K-theory of the free loop space $LX$.

Conformal field theories. This is the analogue of K-theory being presentable by vector bundles with connection. One way to think about a vector bundle with connection on a manifold $X$ is that it is a "$1$-dimensional topological field theory over $X$": that is, it assigns a vector space to a finite set of points equipped with signs in $X$ (the tensor product of either the fibers of the vector bundle or their dual depending on the orientation), and it assigns a map of vector spaces to every oriented cobordism between such points in $X$ (the tensor product of either the holonomies or the evaluation or coevaluation maps).

The historical motivation for generalizing this to $2$-dimensional conformal rather than topological field theories is, I think, to explain the modularity properties of the Witten genus. But again, don't trust me to have the specifics right here. (I guess it's $2$-dimensional topological rather than conformal field theories over $X$ that look more like $2$-vector bundles.)