Tori detect invertibility of topological field theories
type: Item
instance of: scholarly article
title: Tori detect invertibility of topological field theories
zbMATH Open document ID: 1405.18009
author: Christopher J. Schommer-Pries
published in: Geometry \& Topology
publication date: 2018-06-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1511.01772
review text: A topological field theory in the sense of [\textit{M. Atiyah}, Publ. Math., Inst. Hautes Étud. Sci. 68, 175--186 (1988; Zbl 0692.53053); \textit{M. Atiyah}, Turk. J. Math. 21, No. 1, 1--7 (1997; Zbl 0890.57019); \textit{G. B. Segal}, NATO ASI Ser., Ser. C 250, 165--171 (1988; Zbl 0657.53060); \textit{G. Segal}, Lond. Math. Soc. Lect. Note Ser. 308, 421--577 (2004; Zbl 1372.81138)] is a symmetric monoidal functor \[ \mathcal{Z}:\mathrm{Cod}_{d}\rightarrow\mathrm{Vect} \] with \(\mathrm{Cod}_{d}\) being the symmetric monoidal category of closed compact \((d-1)\)-dimensional manifolds as objects and equivalence classes of \(d\)-dimensional bordisms as morphisms (the monoidal structure is simply the disjoint union) and \(\mathrm{Vect}\) being the category of vector spaces with its standard tensor products as its monoidal structure). The dimension of the vector space \(\mathcal{Z}(M)\) for each closed \((d-1)\)-dimensional manifold \(M\) is a very coarse measure of the complexity of the topological field theory, and invertible field theories assigning a one-dimensional vector space to each \((d-1)\)-dimensional manifold are by no means the simplest theories. Chao-Ming Jian [\url{http://mathoverflow.net/q/218982}] asked whether a \(d\)-dimensional theory assigning one-dimensional vector spaces \(\mathcal{Z}(S^{d-1})\) and \(\mathcal{Z}(T^{d-1})\) to the \((d-1)\)-dimensional sphere \(S^{d-1}\) and the \((d-1)\)-dimensional torus \(T^{d-1}\) must also assign one-dimensional vector space to all other \((d-1)\)-dimensional manifolds. This paper settles the question affirmatively in the sense that, as far as a topological field theory \(\mathcal{Z}\) is so extended that it assigns data to \(d\)-, \((d-1)\)- and \((d-2)\)-manifolds, \(\mathcal{Z}(T^{d-1})\) is one-dimensional if and only if the theory assigns one-dimensional vector space to all closed \((d-1)\)-dimensional manifolds. The principal result in this paper is a far-reaching generalization of this result: Theorem 1…
reviewed by: Hirokazu Nishimura
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 57R15
Mathematics Subject Classification ID: 57R56
Mathematics Subject Classification ID: 57R65
Mathematics Subject Classification ID: 81T45
zbMATH DE Number: 6882289
zbMATH Keywords: topological field theory
zbMATH Keywords: dimensional reduction
zbMATH Keywords: invertible
MaRDI profile type: Publication
OpenAlex ID: W2251500530
arXiv ID: 1511.01772
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DOI: 10.2140/GT.2018.22.2713
description: scientific article