Loop space index theory detecting exotic smooth structures

Develop a loop space index theory—generalizing the signature operator as suggested by Witten—that can detect exotic smooth structures on free loop spaces, providing analytic tools compatible with the geometry of field theories envisioned in the TMF conjecture.

Background

The authors cite Hopkins’ sketch of detecting exotic smooth structures on free loop spaces via a TMF-generalization of the signature operator suggested by Witten.

They explicitly state that this kind of loop space index theory remains out of reach without a robust field-theoretic framework, making it a concrete analytic-geometric problem linked to Conjecture 1.1.