Tori Detect Invertibility of Topological Field Theories

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.

33 pages, 4 figures; added application to Crane-Yetter TQFTs

References in corpus (2)

• Short-range entanglement and invertible field theories
• On dualizable objects in monoidal bicategories, framed surfaces and the Cobordism Hypothesis

Cited by in corpus (14)

• On the classification of topological orders
• Symmetry Protected Topological phases and Generalized Cohomology
• On 2-form gauge models of topological phases
• Gapped boundary theories in three dimensions
• Ground state degeneracy on torus in a family of ${\mathbb{Z}}_N$ toric code
• Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
• Anomalies in (2+1)D fermionic topological phases and (3+1)D path integral state sums for fermionic SPTs
• Invertible braided tensor categories
• Topological orders and factorization homology
• Bordism for the 2-group symmetries of the heterotic and CHL strings
• What bordism-theoretic anomaly cancellation can do for U
• Semisimple 4-dimensional topological field theories cannot detect exotic smooth structure
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• Semisimple Field Theories Detect Stable Diffeomorphism