Field theory configuration spaces for connective ko-theory
type: Item
instance of: scholarly article
title: Field theory configuration spaces for connective ko-theory
zbMATH Open document ID: 1195.19002
author: Elke K. Markert
published in: Algebraic \& Geometric Topology
publication date: 2010-07-06T00:00:00Z
review text: This paper is based on the author's thesis [``Connective one-dimensional Euclidean field theories'', Ph.D. Thesis, University of Notre Dame (2005)]. The author presents a new \(\Omega\)-spectrum for connective \(ko\)-theory by constructing connective covers (denoted by \(\text{Inf}_n\) below) of the spaces \(\mathcal{EFT}_n\) of certain types of euclidean fileld theories defined by \textit{S. Stolz} and \textit{P. Teichner} [London Mathematical Society Lecture Note Series 308, 247--343 (2004; Zbl 1107.55004)], which are weakly homotopy equivalent to the \((-n)\)-th space in an \(\Omega\)-spectrum associated to \(KO\). The key to the author's approach is reformulating these spaces \(\mathcal{EFT}_n\) in terms of their generating operators, or certain configurations associated to these operators. Let \(\mathcal{C}_n\) be the standard Clifford algebra associated to the positive definite inner product on \({\mathbb R}^n\) with generators \(e_1, \cdots, e_n\). For \(n > 0\), define \(\mathcal{C}_{-n}\) to be the Clifford algebra with respect to the negative definite inner product with generators \(f_1, \cdots, f_n\) which anti-commute pairwise and square to \(+1\). Also fix a separable real Hilbert space \(H_n\) with compatible right \(\mathcal{C}_{n-1}\)-module structure, which contains all irreducible \(\mathcal{C}_{n-1}\)-modules infinitely many times. Then when viewing \(\mathcal{C}_n\) as a \({\mathbb Z}/2\)-graded algebra such that the even part \(\mathcal{C}^{\text{ev}}_n\) is isomorphic to \(\mathcal{C}_{n-1}\), one obtains a \({\mathbb Z}/2\)-graded \(\mathcal{C}_n\)-module \(\mathcal{H}_n= H_n\otimes_{\mathcal{C}^{\text{ev}}_n}\mathcal{C}_n\). Denote by \(\mathrm{Inf}_n\) the set of all odd, self-adjoint operators on the closures of their domains in \(\mathcal{H}_n\) which have compact resolvent and are \(\mathcal{C}_n\)-linear. Firstly it is shown that for all \(n\), \(\mathcal{EFT}_n\) become homotopy equivalent to \(\mathrm{Inf}_n\) (nicely topologized) (Lem…
reviewed by: Haruo Minami
Mathematics Subject Classification ID: 19L41
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 81Q60
Mathematics Subject Classification ID: 81T08
Mathematics Subject Classification ID: 81T60
zbMATH DE Number: 5731285
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: connective
zbMATH Keywords: field theory
zbMATH Keywords: Euclidean field theory
MaRDI profile type: Publication
full work available at URL: agt.2010.10.1187
OpenAlex ID: W2088647518
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DOI: 10.2140/AGT.2010.10.1187
description: scientific article; zbMATH DE number 5731285