My question here is going to be this -- but I'll give a bit of background to explain myself in a moment:

*What has been done/what results are available on Calabi-Yau cohomology in degree $n \geq 3$ (in particular $n =3$)?*

Here by *Calabi-Yau cohomology* I mean complex oriented cohomology theory with formal group being (equivalent to) the Artin-Mazur formal group $\Phi^n_X$ of a given Calabi-Yau variety of dimension $n$. (So in particular this here is *not* about cohomology *of* Calabi-Yau varieties. I am aware that this will now make 90% of all readers frown, but I can't help it.)

To put this in perspective:

In discussion of elliptic cohomology it is traditional to think of the formal group associated with the elliptic spectrum as "being" the formal completion of the given elliptic curve. But the story of equivariant elliptic cohomology shows that more properly that formal group is identified with the formal Picard group of the elliptic curve, namely the perturbation/deformation theory of the moduli of line bundles over it. It just so happens that elliptic curves are self-dual abelian varieties so that both these perspectives are equivalent, but the latter is the fundamental one that generalizes.

In the next step, K3-cohomology is complex oriented cohomology theory with formal group being the formal Brauer group of a K3 surface. This has been discussed.

But there is an obvious continuation of this story to higher $n$, and I am looking for whatever results exist for $n \geq 3$. My understanding is that Michael Hopkins talked about this "Calabi-Yau cohomology" at the Midwest Topology Seminar in 1992, but I haven't seen anything except this pointer.

Notice that the case $n = 3$ is quite compelling from the point of view of physics: while equivariant elliptic cohomology ("CY1-cohomology") is essentially the theory of the modular functor of 3d Chern-Simons theory/2d Wess-Zumino-Witten theory, so the Artin-Mazur formal group $\Phi^3_X$ is just the formal approximation to the intermediate Jacobian which is the phase space of $U(1)$-7d Chern-Simons theory. It should be quite interesting to ask for a (equivariant) CY3-cohomology theory here which similarly captures the geometric quantization of this and hence yields the modular functor for the infamous 6d theory...

There seems to be fairly strong motivation on the physics side for looking at CY3-cohomology, also if one looks at it from the perspective of F'-theory, and as such hypothetical Calabi-Yau cohomology was highlighted at the end of Sati 05.

But so the **question** is: what is actually already known about CY3-cohomology? For instance: what are sufficient conditions for the Artin-Mazur formal group $\Phi^3_{CY3}$ to be Landweber exact??

This post imported from StackExchange MathOverflow at 2016-07-18 16:09 (UTC), posted by SE-user Urs Schreiber