The geometry of loop groups
type: Item
instance of: scholarly article
title: The geometry of loop groups
zbMATH Open document ID: 0619.58003
author: Daniel S. Freed
published in: Journal of Differential Geometry
publication date: 1988-01-01T00:00:00Z
review text: The space \(\Omega\) G of based loops on a compact Lie group admits a Kähler metric. We find a formula for its curvature in terms of Toeplitz operators. By analogy with Chern-Weil theory in finite dimensions we use the curvature form to define Chern classes of \(\Omega\) G. These geometric Chern classes do not directly come from topology, as they do in finite dimensions. Extra geometric structure - a Fredholm structure - must be imposed before characteristic classes are defined topologically. There is a natural Fredholm structure on \(\Omega\) G induced from the family of Toeplitz operators. We use an index theorem for families of Fredholms parametrized by a group to show that the Chern classes of the Toeplitz family agree with the Chern classes defined by curvature. Explicit formulas for \(\Omega\) SU(n) are obtained. Extensions to more general groups of gauge transformations are considered.
Mathematics Subject Classification ID: 58B20
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 4005173
zbMATH Keywords: based loops on a compact Lie group
zbMATH Keywords: Kähler metric
zbMATH Keywords: Toeplitz operators
zbMATH Keywords: Chern-Weil theory in finite dimensions
zbMATH Keywords: Fredholm structure
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214442279
OpenAlex ID: W2100260105
Wikidata QID: Q115183538
DOI: 10.4310/JDG/1214442279
description: scientific article; zbMATH DE number 4005173