An Artin-Mazur 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 Artin-Mazur formal groups been considered when ordinary cohomology is replaced by some generalized cohomology theory, notably by some flavor of K-theory?
(I am thinking of the hopefully obvious definition: form the kernel of the restriction map from the K-theory 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 pro-representable by a formal group).
This question is motivated by the analogy suggested by the following pattern:
For $X$ a 1-dimensional Calabi-Yau variety -- an elliptic curve -- the 1-dimensional Artin-Mazur 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 Chern-Simons theory.
For $X$ a 3-dimesional Calabi-Yau variety the 1-dimensional Artin-Mazur formal group $\Phi^3_X$ is the deformation theory of the intermediate Jacobian curve and this is the phase space of $U(1)$-7d Chern-Simons 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 Calabi-Yau cohomology?);
But now for $X$ a 5-dimensional 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 11-dimensional Chern-Simons theory in question is not defined on cocycles in ordinary (differential) cohomology, but on cocycles in (differential) K-theory.
The "K-theoretic Jacobian" involved here was considered in this context in (Diaconescu-Moore-Witten 00) and explicitly considered as the phase space of an 11d Chern-Simons-type theory given by fiber integration over the cup square of differential K-cycles in (Belov-Moore 06).
The general mechanism of quantization of self-dual 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 (Hopkins-Singer 02). One may ask generally what happens here when cup products in ordinary differential cohomology are replaced by cup products in differential K-theory. But here I am wondering just about one specific aspect of this: do we get Artin-Mazur type formal groups from deformation of K-theory classes on suitable varieties? Does there exist any work on this?