I am reading Ravenel's *Localization with Respect to Certain Periodic Homology Theories* where he states;



> 
> For $n\ge2$, the spectra E(n) 
>  represent periodic homology theories which at present have no known geometric interpretation comparable to the description of K-theory in terms of 
>  vector bundles.
> 
> 
> 


This is the paper where he gives his seven conjectures, all but one of which have since been proven. That would lead me to believe that this interpretation has been found in the process, but I am not aware of it. 


Q: Is there a geometric interpretation of the Johnson-Wilson E(n) analogous to the vector bundle description of K-theory? If so, where I could I read about it in the literature?



Is there a geometric interpretation of Johnson-Wilson E(n) analogous to vector bundles for K-theory?
The answer is no, at least as far as I am aware of. For height 2 case, you can
consider elliptic cohomology as some sort of geometric interpretation of $E(2)$
theory, but this is far less straight-forward than the vector bundle description
of K-theory. Furthermore, the discovery by Igor Kriz of a group G with
$K(n)^{odd}(BG)\neq 0$ makes it difficult to believe that one can find an
$E(n)$-analogue of Atiyah-Segal completion theorem that relates $K$-theory
and the representation ring.${}{}$


