Calabi-Yau cohomology?

Question

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

Comments

A naive question: I chased your link to the nforum page on Artin-Mazur fg. Looking at it, I don't see what is special about Calabi-Yau varieties, over arbitrary ones of dim n. I'm guessing the CY condition is what's need to make the AMfg one dimensional, yes?

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

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To guarantee that $\Phi^n$ be prorepresentable by a formal group, it suffices to have $\Phi^{n-1}$ be formally smooth. The condition that $H^{n-1}(X, \mathcal{O}_X)=0$ is an overly strong way to guarantee that. Then yes, $\dim H^n(X, \mathcal{O}_X)$ gives the dimension of the formal group $\Phi^n$, so for a Calabi-Yau it is one-dimensional.

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

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@CharlesRezk, true, I should have mentioned this. For completeness I have now added a brief remark on that to the relevant nLab entry: ncatlab.org/nlab/show/… (just as Matt says, of course).

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

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can I ask where the term "calabi-yau cohomology" was first coined?

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

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(Szymik 10) ($\leftarrow$ mouseover) points in his reference item [19] to a "report" by Jack Morava on Mike Hopkins’ work on Calabi-Yau cohomology, which apparently, as I mentioned above, was presented in 1992 at the Midwest topology Seminar. But I haven't seen this report. The earliest mentioning that I have seen in print (so far) is (Sati 05), where the terminology is proposed on the last page.

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

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@UrsSchreiber: thanks!

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

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Is this interesting in characteristic 0, or is it only non-trivial in positive characteristic?

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Only in positive characteristic. That's as expected: Also elliptic curves contribute to elliptic cohomology genuinely at height 2 only in positive characteristic.

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The Artin-Mazur type formal groups, hence any associated CY-cohomology, is non-trivial only in positive characteristic, yes. Let me just amplify that this does not mean that the CY-schenes appearing are not possibly spacetime fibers: we gould have an CY defined over the integers (say the standard quintic) and then regard its points either over the complex numbers, or else over a prime field etc. In the first case we get the underlying "physical space", in the second we get something that may be fed into the Artin-Mazur machine to produce CY-cohomology.

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Thanks. If I understand you correctly, the physical significance is that it can be expected that the CY-cohomology over a finite field somehow reflects the geometry or physics of the fiber over $\Bbb C$, in a way that may not be immediately apparent in the characteristic 0 fiber, right? Is it easy to give an example of that (e.g. relating the height of $\Phi_X^3$ for a given $p$ to the cohomology in characteristic 0)? (Sorry, I won't be able to answer your question but now I am asking you things)

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Yes, that's the right idea. If we have a complex manifold that we realize as the complex points of an arithmetic scheme, then we may remember this extra information, and it is like an extra structure on the complex manifold. You may think of this as a analog to the simpler situation where we have a real manifold and then remember that it actually underlies a complex manifold (if it does).

Something like this is known to happen for elliptic cohomology: A priori the Witten genus as defined from the physics produces modular forms that are forms on the moduli space of elliptic curves over the complex numbers (the string worldsheets). But then as the Witten genus is realized as the homotopy groups of the universal elliptic cohomology theory tmf then this gets replaced by the moduli space of all elliptic curves, over all base rings (not just the complex numbers):in the construction of the Witten genus via the sigma-orientation of tmf complex numbers never ever appear.