An ArtinMazur formal group is, when it exists, the deformation theory of ordinary cohomology of some degree, on some algebraic variety. My question here is:
Has the generalization of the theory of ArtinMazur formal groups been considered when ordinary cohomology is replaced by some generalized cohomology theory, notably by some flavor of Ktheory?
(I am thinking of the hopefully obvious definition: form the kernel of the restriction map from the Ktheory group (of the chosen flavor) of any formal thickening of the given variety to that of the underlying variety and ask if the presheaf given as the thickenings vary is prorepresentable by a formal group).
This question is motivated by the analogy suggested by the following pattern:

For $X$ a 1dimensional CalabiYau variety  an elliptic curve  the 1dimensional ArtinMazur formal group $\Phi^1_X$ induces elliptic cohomology, and its equivariant version essentially encodes the (modular functor of the) 2d CFT boundary theory of 3d ChernSimons theory.

For $X$ a 3dimesional CalabiYau variety the 1dimensional ArtinMazur formal group $\Phi^3_X$ is the deformation theory of the intermediate Jacobian curve and this is the phase space of $U(1)$7d ChernSimons theory (which in view of the previous case makes one wonder about the relation of the complex oriented cohomology theory, if any, associated with $\Phi^3_X$ and how it relates to the 6d SCFT, that was the thrust of a related recent question of mine here, titled CalabiYau cohomology?);

But now for $X$ a 5dimensional variety one might be tempted to ask the previous question again, just with the degrees increased  but in fact the string theory story here suggests that the 11dimensional ChernSimons theory in question is not defined on cocycles in ordinary (differential) cohomology, but on cocycles in (differential) Ktheory.
The "Ktheoretic Jacobian" involved here was considered in this context in (DiaconescuMooreWitten 00) and explicitly considered as the phase space of an 11d ChernSimonstype theory given by fiber integration over the cup square of differential Kcycles in (BelovMoore 06).
The general mechanism of quantization of selfdual higher gauge theory which is at play in all these dimensions was originally observed/found/discussed in (Witten 96) for ordinary (differential) cohomology and for this case it was then turned into theorems in (HopkinsSinger 02). One may ask generally what happens here when cup products in ordinary differential cohomology are replaced by cup products in differential Ktheory. But here I am wondering just about one specific aspect of this: do we get ArtinMazur type formal groups from deformation of Ktheory classes on suitable varieties? Does there exist any work on this?
This post imported from StackExchange MathOverflow at 20140810 09:07 (UCT), posted by SEuser Urs Schreiber