Tori Detect Invertibility of Topological Field Theories

TL;DR

This paper establishes that for once-extended topological field theories, invertibility of the (d-1)-torus value implies the entire theory's invertibility, extending to theories with various tangential structures.

Contribution

It proves that invertibility of the (d-1)-torus value ensures the invertibility of the whole topological field theory, generalizing previous results to arbitrary tangential structures.

Findings

Invertibility of the (d-1)-torus value implies the entire theory is invertible.

The result applies to theories with arbitrary tangential structures.

The theory factors through the maximal Picard infty-category of the target.

Abstract

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