Deformation classes of invertible field theories and the Freed--Hopkins conjecture

TL;DR

This paper proves a conjecture linking deformation classes of certain invertible field theories with a generalized cohomology, advancing understanding of the mathematical structure underlying quantum field theories.

Contribution

It establishes the Freed-Hopkins conjecture, connecting deformation classes of invertible field theories to Thom spectrum cohomology, and constructs a smooth Brown--Comenetz dual of the sphere spectrum.

Findings

Proved the Freed--Hopkins conjecture.

Constructed a smooth variant of the Brown--Comenetz dual.

Calculated the deformation type of the extended geometric bordism category.

Abstract

We prove a conjecture of Freed and Hopkins, which relates deformation classes of reflection positive, invertible, $d$-dimensional extended field theories with fixed symmetry type to a certain generalized cohomology of a Thom spectrum. Along the way, we establish several results, including the construction of a smooth variant of the Brown--Comenetz dual of the sphere spectrum and a calculation of the "deformation type" of the extended geometric bordism category.