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](http://ncatlab.org/nlab/show/Calabi-Yau+cohomology) in degree $n \geq 3$ (in particular $n =3$)?_

Here by _Calabi-Yau cohomology_ I mean [complex oriented cohomology theory](http://ncatlab.org/nlab/show/complex%20oriented%20cohomology%20theory) with [formal group](http://ncatlab.org/nlab/show/formal%20group) being (equivalent to) the [Artin-Mazur formal group](http://ncatlab.org/nlab/show/Artin-Mazur%20formal%20group) $\Phi^n_X$ of a given [Calabi-Yau variety](http://ncatlab.org/nlab/show/Calabi-Yau%20variety) 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](http://ncatlab.org/nlab/show/elliptic%20cohomology) it is traditional to think of the formal group associated with the [elliptic spectrum](http://ncatlab.org/nlab/show/elliptic%20spectrum) as "being" the formal completion of the given elliptic curve. But the [story](http://ncatlab.org/nlab/show/equivariant%20elliptic%20cohomology#InterpretationInQuantumFieldTheory) of [equivariant elliptic cohomology](http://ncatlab.org/nlab/show/equivariant%20elliptic%20cohomology) shows that more properly that formal group is identified with the [formal Picard group](http://ncatlab.org/nlab/show/formal%20Picard%20group) 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](http://ncatlab.org/nlab/show/K3-cohomology) is complex oriented cohomology theory with formal group being the [formal Brauer group](http://ncatlab.org/nlab/show/formal%20Brauer%20group) of a [K3 surface](http://ncatlab.org/nlab/show/K3%20surface). 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 complelling from the point of view of physics: while equivariant elliptic cohomology ("CY1-cohomology") [is essentially](http://ncatlab.org/nlab/show/equivariant%20elliptic%20cohomology#InterpretationInQuantumFieldTheory) the theory of the [modular functor](http://ncatlab.org/nlab/show/modular+functor) of [3d Chern-Simons theory](http://ncatlab.org/nlab/show/Chern-Simons%20theory)/[2d Wess-Zumino-Witten theory](http://ncatlab.org/nlab/show/Wess-Zumino-Witten+model), so the Artin-Mazur formal group $\Phi^3_X$ is just the formal approximation to the [intermediate Jacobian](http://ncatlab.org/nlab/show/intermediate%20Jacobian) which is the [phase space](http://ncatlab.org/nlab/show/phase%20space) of $U(1)$-[7d Chern-Simons theory](http://ncatlab.org/nlab/show/7d%20Chern-Simons%20theory). It should be quite interesting to ask for a (equivariant) CY3-cohomology theory here which similarly captures the [geometric quantization](http://ncatlab.org/nlab/show/geometric%20quantization) of this and hence yields the modular functor for the [infamous 6d theory](http://ncatlab.org/nlab/show/6d%20(2,0)-supersymmetric%20QFT)...

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](http://ncatlab.org/nlab/show/F%27-theory), and as such hypothetical Calabi-Yau cohomology was highlighted at the end of [Sati 05](http://ncatlab.org/nlab/show/Calabi-Yau+cohomology#Sati05).

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??