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Chromatic homotopy tends to mainly focus on $1$-dimensional (commutative) FGLs. From a geometric perspective, this is because line bundles form a group and n-plane bundles don't, so the first Chern class gives us a FGL and higher Chern classes do not. At any rate, this appears all over CHT. The Morava E-theories come from Artin-Mazur FGLs, which are $1$d (at least for Calabi-Yau varieties); the Landweber exact functor theorem only applies to $1$d FGLs; and the queen of all complex-oriented cohomology theories, $MU$, is considered to be a spectral version of the universal $1$d commutative FGL.

The closest thing to increasing dimension that I can find is the fact that the $r$th Artin-Mazur FGL $\Phi^r$, which gives rise to the $(r+1)$th Morava E-theory $E(r+1)$, is naturally considered as a formal $r$-group rather than a formal $1$-group. (Technically, it's the $r$-group of line $r$-bundles on a variety of dimension $r$.) But this is more of a progressively higher categorification than anything involving actual higher-dimensional group schemes. Indeed, in the $r\to\infty$ limit (if I'm not mistaken) it just goes to the spectral stack $\operatorname{Spet} MU$, which doesn't recover anything new dimension-wise: it's just the universal one-dimensional spectral formal group law.

Why am I asking about this, you may wonder? The reason comes from algebraic geometry and no small part mathematical physics. In dimensions $1$ and $2$ (and presumably in some fashion in higher dimensions), the Torelli theorem says that sufficiently nice varieties can be recovered from their principally polarized Jacobian/Picard variety. For genus $1$ curves, this principal polarization corresponds to choosing the basepoint of an elliptic curve, an operation which is crucial in relating "QFT over a complex torus" to "the cohomology theory associated to the formal completion of a complex torus". (Skipping over the details with torsors and whatnot, suffice to say that tori don't come with a basepoint, and you have to give them one to make sense of this stuff.) I believe that Stolz and Teichner's program implies that something similar should be true for QFT on K3 surfaces and K3 cohomology, and indeed for higher-dimensional Calabi-Yau varieties and corresponding Morava E-theories. But to generalize beyond this case (e.g. to consider worldsheets other than "a pair of strings spontaneously appear then annihilate") requires the consideration of higher-dimensional groups, since more complicated cohomology = more complicated Picard variety.

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    $\begingroup$ From the body of the question it seems like you might be interested in running a version of chromatic homotopy theory with n-dim FGLS, and it seems like the first step there would be to find a replacement for CP^infty, namely a (commutative?)-group-up-to-homotopy G such that H^*(G) is Z[x1,...,xn]...but from the title it seems like you're interested in whether n-dim FGLs play any role in the usual, 1-dim FGL-based chromatic story? $\endgroup$
    – kiran
    Jun 22, 2022 at 5:25
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    $\begingroup$ @kiran I'm looking for the first of those, but I would expect such a theory to be connected to traditional chromatic homotopy. $\endgroup$ Jun 22, 2022 at 8:21
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    $\begingroup$ No substantial answer, but two thoughts: (1) You might browse Stapleton and coauthors’ papers on field theories to see some adjacent concerns and a rather different approach to a resolution; (2) The connection between 1-dimensional formal groups and “classical” cohomology theories, whatever one takes “classical” to mean, is so strong that it’s hard to believe there’s room left over for any alternative connection that enjoys a similarly strong coupling. $\endgroup$ Jun 24, 2022 at 15:41

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