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Tori detect invertibility of topological field theories

Christopher J Schommer-Pries

Geometry & Topology 22 (2018) 2713–2756
Abstract

A once extended d–dimensional topological field theory Z is a symmetric monoidal functor (taking values in a chosen target symmetric monoidal (,2)–category) assigning values to (d2)–manifolds, (d1)–manifolds, and d–manifolds. We show that if Z is at least once extended and the value assigned to the (d1)–torus is invertible, then the entire topological field theory is invertible, that is, it factors through the maximal Picard –category of the target. Similar results are shown to hold in the presence of arbitrary tangential structures.

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Keywords
topological field theory, dimensional reduction, invertible
Mathematical Subject Classification 2010
Primary: 18D05, 57R15, 57R56, 57R65
Secondary: 81T45
References
Publication
Received: 1 September 2016
Revised: 3 October 2017
Accepted: 31 October 2017
Published: 1 June 2018
Proposed: Mark Behrens
Seconded: Ralph Cohen, Ulrike Tillmann
Authors
Christopher J Schommer-Pries
Department of Mathematics
University of Notre Dame
Notre Dame, IN
United States
http://sites.nd.edu/chris-schommer-pries/