Volume 22, issue 5 (2018)

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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 $\mathsc{Z}$ 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 $\mathsc{Z}$ 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.

Keywords
topological field theory, dimensional reduction, invertible
Mathematical Subject Classification 2010
Primary: 18D05, 57R15, 57R56, 57R65
Secondary: 81T45
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/