2-vector bundles, loop spaces, and elliptic objects

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The project that I describe here aims at establishing a geometric perspective to elliptic cohomology, analogous to how vector bundles represent topological K-theory classes. The existence of such perspective is related to a famous conjecture of E. Witten and G. Segal, further established by S. Stolz and P. Teichner, and withstood already an attack led by N. Baas et al. almost twenty years ago. For this project, we consider a new promising candidate ("Schreiber 2-vector bundles") for geometric representatives of elliptic cohomology classes, developed in part by myself, in collaboration with M. Ludewig and P. Kristel, and also plan to use another approach to elliptic cohomology ("Tate K-theory"), due to N. Kitchloo and J. Morava; further investigated by N. Ganter and others. This new approach goes via a version of equivariant K-theory of loop spaces. Our aim is to combine these new ideas in order to carry out a new attack on the old problem of finding a geometric perspective to elliptic cohomology. The core idea is to develop the transgression of 2-vector bundles to lop spaces, in such a way that transgressed 2-vector bundles make up classes in Tate K-theory. We hope that progress in this direction will contribute to the solution of related, even deeper problems: what kind of differential operators act on sections of 2-vector bundles, do such operators have an index valued in elliptic cohomology, and what role plays the vanishing of these indices for the geometry of manifolds?