Freed–Hopkins classification conjecture for reflection-positive invertible QFTs

Let $G=\{G_d,s_d,\rho_d\}_{d\in\mathbb{Z}_{\geq 0}}$ be a tangential structure, and let $I\mathbb{C}^{\times}$ denote the Brown–Comenetz dual of the sphere. Write $I\Omega^G$ for the corresponding generalized cohomology theory, and let $n$ be the spacetime dimension. A Freed–Hopkins classification conjecture asserts that deformation classes of reflection-positive invertible $n$-dimensional fully extended field theories with symmetry type $G$ are in one-to-one correspondence with

$$(I\Omega^G)^{n+1}(\operatorname{pt}).$$

This conjecture proposes a generalized-cohomological classification of invertible topological QFTs and, consequently, of anomalies. The symmetry types in the original Freed–Hopkins formulation satisfy additional conditions beyond the tangential structures described here; the general status of the proposed classification is not resolved in the supplied source.