On the geometry of free loop spaces
type: Item
instance of: scholarly article
title: On the geometry of free loop spaces
zbMATH Open document ID: 1011.58006
DOI: 10.1155/S0161171202011560
author: R. Smith
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2002-08-12T00:00:00Z
full work available at URL: https://eudml.org/doc/49995
full work available at URL: S0161171202011560.html
review text: Let \(M\) be a finite-dimensional path-connected Riemannian manifold. The infinite-dimensional space of all smooth maps from the circle group \(S^1\) to \(M\) is the Fréchet manifold \(\Lambda M\) called the free loop space. The points in \(\Lambda M\) can be described as free loops in \(M\). \textit{M. F. Atiyah} [Colloq. Honneur L. Schwartz, Ec. Polytech. 1983, Vol. 1, Astérisque 131, 43-59 (1985; Zbl 0578.58039)] showed that there is a fundamental closed 2-form \(\omega\) on \(\Lambda M\) which, unlike the finite-dimensional case, can be degenerate at some points. In this paper, the author proves that the set of all points where \(\omega\) is nondegenerate is open. In modern physical theories such as string theory, the idea of representing particles not as points but rather as loops on some manifold \(M\) has resulted in the formulation of spinors on the loop space \(\Lambda M\) of \(M\). A string structure is defined as a lifting of the structural group to an \(S^1\)-central extension of the loop group. If \(\widetilde G\to \widetilde P\to X\) is a lifting of a principal Fréchet bundle \(G\to P\to X\) over a Fréchet manifold \(X\) and if \(S^1\to \widetilde G\to G\) is an \(S^1\)-central extension of \(G\), then the author shows that every connection on the principal bundle \(G\to P\to X\) together with a \(\widetilde G\)-invariant connection on \(S^1\to \widetilde P\to P\) defines a connection on \(\widetilde G\to \widetilde P\to X\), and, as a consequence of this result, that there exist connections on the string structure of \(\Lambda M\). Finally, the author defines Christoffel symbols on \(\Lambda M\) which are described in terms of the Christoffel symbols on \(M\) and compute the corresponding curvature.
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 58D15
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 53C07
zbMATH DE Number: 1779095
zbMATH Keywords: free loop spaces
zbMATH Keywords: Fréchet bundle
zbMATH Keywords: string structure
MaRDI profile type: Publication
OpenAlex ID: W2024643126
description: scientific article; zbMATH DE number 1779095