Elliptic cohomology and 2-vector bundles

Currently I'm interested in the geometric construction of elliptic cohomology theories and how to relate it to the algebro-topological aspect of the theories. Other than a spectrum, we may consider constructing the geometric representing objects of them, i.e. a higher version of vector bundles. I constructed a model of 2-representation and 2-vector bundles. Next, I want to relate it (or any reasonable model of 2-vector bundles) to elliptic cohomology, which may be interpreted as a relation between the classical Redshift conjecture and the geometric version of it, i.e. a n-vector bundle should be a "loop space" of a (n-1)-vector bundle.