The String Coffee Table

3) $K(\mathbf{V})$ and Elliptic Cohomology

Warning. This third part is extremely sketchy, due to three reasons. 1) It is the technically most demaning part, but 2) was dealt with only near the end of our seesion when 3) I was not paying due attention to some details. This part is planned to be dealt with in more detail in future sessions. So if the following is about as mysterious to you as it is to me, just read it like you would read a horoscope. ;-)

Question. What is elliptic cohomology, anyway??

Two answers:

1) Homotopy Theorist’s answer

Theorem [Hopkins, Miller, Goerss, and others]

There is a spectrum (a generalized cohomology theory) called TMF = “topological modular forms” that is universal with respect to all elliptic cohomology theories.

So if $E$ are homology theories that send elliptic curves to formal group laws, then

(Or something along these lines. As I have said, this is sketchy. More than sketchy, actually.)

Here is another vague statement of this sort.

“TMF gives a topological version of the Witten genus.”

Whatever TMF is, BDR, have proven the following:

Theorem. [BDR]

$K(\mathbf{V})$ is not quite the same as TMF, but it “sees as much” as the stable homotopy groups of spheres.

2) Segal’s answer

“Elliptic cohomology should have something to do with CFTs”.

This is where 2-vector bundles enter the game. We can regard parallel transport in an ordinary vector bundle with connection as a form of a 1-dimensional field theory, one which assigns “Hilbert spaces of states” (fibers of the bundle) to points and whose “propagator” is the parallel transport.

Hence, in as far as K-theory is about classes of vector bundles, it is about classes of 1-dimensional field theories. (This is explained in much more detail in section 3 of What is an elliptic object?).

So one might expect that there is something like parallel surface transport in a 2-vector bundle such that this is similarly related to elliptic cohomology.

Reminder.

For ordinary vector bundles we have

which says that the homotopy classes of maps from a space $X$ to the K-theory space of unitary vector bundles form the Grothedieck group completion of classes of vector bundles over $X$.

Question. What do maps $X \to K(\mathbf{V})$ see?

Theorem. [BDR] For “nice” $X$ (for instance CW complexes) the homotopy classes of maps $[X \to K(\mathbf{V})]$ are given by the colimit over cyclic Serre fibrations $Y\to X$ over the Grothedieck group completion of classes of 2-vector bundles over $Y$.

But what is a 2-vector bundle?

That’s easy to describe once some general 2-bundle language is established. A BDR 2-vector bundle is simply a 2-bundle with structure “2-group” $\mathrm{GL}_n(\mathbf{B})$. Only that this is not a 2-group, but just a 2-monoid.

So a 2-vector bundle in the sense of BDR is defined in terms of local trivializations as a 2-bundle whose transition functions take values in objects of $\mathrm{GL}_n(\mathcal{B})$ and whose transition modifications on triple overlaps take values in morphisms of that category. Note that $\mathrm{GL}_(\mathcal{B})$ is not a 2-group, but just a 2-monoid. This means that transitions in BDR 2-vector bundles have a direction. If we can make a transition in a BDR 2-bundle from $U_i$ to $U_j$ we cannot in general make a transition from $U_j$ to $U_i$.

Our first session endet with the following statement

Hope. [BDR] 2-vector bundles with 2-connection might realize parallel surface transport that realizes elliptic objects.