Tori detect invertibility of topological field theories

Abstract

A once extended $d$–dimensional topological field theory $\mathcal{Z}$ is a symmetric monoidal functor (taking values in a chosen target symmetric monoidal $\left(\infty ,2\right)$–category) assigning values to $\left(d-2\right)$–manifolds, $\left(d-1\right)$–manifolds, and $d$–manifolds. We show that if $\mathcal{Z}$ is at least once extended and the value assigned to the $\left(d-1\right)$–torus is invertible, then the entire topological field theory is invertible, that is, it factors through the maximal Picard $\infty$–category of the target. Similar results are shown to hold in the presence of arbitrary tangential structures.