A framework for geometric field theories and their classification in dimension one
type: Item
instance of: scholarly article
title: A framework for geometric field theories and their classification in dimension one
zbMATH Open document ID: 1482.57035
DOI: 10.3842/SIGMA.2021.072
author: Matthias Ludewig
author: Augusto Stoffel
published in: SIGMA. Symmetry, Integrability and Geometry: Methods and Applications
publication date: 2021-10-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/2001.05721
review text: This paper introduces a framework in terms of stacks to define bordisms with geometric structures in quantum field theory. A ``geometry'' on \(d\)-manifolds is a stack \(\mathcal{G}\rightarrow \operatorname{Man}^d\) on the site \(\operatorname{Man}^d\) of manifolds of dimension \(d\). The geometric bordism category \(\mathcal{G}\operatorname{Bord}\) has as objects \(X\in \operatorname{Man}^d\) together with a \(\mathcal{G}\)-structure on \(X\), that is, an object of the groupoid \(\mathcal{G}(X)\) (the subcategory lying over \(\operatorname{id}_X\)). The geometric bordism category becomes ``smooth'' considering simplicial stacks or a complete Segal object in the category of stacks. Therefore, \(\mathcal{G}Bord_n\) consists of \(d\)-dimensional manifolds \(X\) together with a collection of marked hypersurfaces \(Y_k\subset X\), for \(k=0,\ldots,n\), where the composition is encoded by forgetting the marked hypersurfaces. The site \(\operatorname{Man}^d\) is promoted to a category of submersions \(X\rightarrow S\) and morphisms of submersions in order to have a suitable Grothendieck topology. A \(d\)-dimensional field theory with geometry \(\mathcal{G}\) is a symmetric monoidal smooth functor \(Z:\mathcal{G}\operatorname{Bord}\rightarrow \operatorname{Vect}\), where \(\operatorname{Vect}\) is the smooth category of finite dimensional vector bundles on \(\operatorname{Man}\). In dimension one, the papers shows that the groupoid of \(1\)-dimensional oriented topological field theories over \(M\) is equivalent to the groupoid of finite-dimensional vector bundles with connections over \(M\). A similar result is proved for the unoriented case.
reviewed by: Carlos Segovia
Mathematics Subject Classification ID: 57R56
Mathematics Subject Classification ID: 14D21
Mathematics Subject Classification ID: 57R22
zbMATH DE Number: 7413745
zbMATH Keywords: field theory
zbMATH Keywords: vector bundles
zbMATH Keywords: bordism
MaRDI profile type: Publication
arXiv ID: 2001.05721
cites work: Topological quantum field theories
cites work: Smooth one-dimensional topological field theories are vector bundles with connection
cites work: Gerbes and homotopy quantum field theories
cites work: Classical Chern-Simons theory. I
cites work: Espaces analytiques rélatifs et théorème de finitude
cites work: Q4662870
cites work: Local and analytic cyclic homology
cites work: A model for the homotopy theory of homotopy theory
cites work: Parallel transport and functors
cites work: Classifying spaces and spectral sequences
cites work: Q4905314
cites work: Supersymmetric field theories and generalized cohomology
cites work: Q5557339
cites work: Q5704403
cites work: Topological quantum field theory
description: scientific article