Tori detect invertibility of topological field theories

Schommer-Pries, Christopher¹

¹ Department of Mathematics, University of Notre Dame, Notre Dame, IN, United States

Geometry & topology, Tome 22 (2018) no. 5, pp. 2713-2756.

Résumé

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.

Détail

DOI : 10.2140/gt.2018.22.2713

Classification : 18D05, 57R15, 57R56, 57R65, 81T45
Keywords: topological field theory, dimensional reduction, invertible

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