Differential geometry on the space of connections via graphs and projective limits
type: Item
instance of: scholarly article
title: Differential geometry on the space of connections via graphs and projective limits
zbMATH Open document ID: 0851.53014
DOI: 10.1016/0393-0440(95)00028-G
author: Abhay Ashtekar
author: Jerzy Lewandowski
published in: Journal of Geometry and Physics
publication date: 1996-05-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/hep-th/9412073
review text: The authors consider a certain completion \(\overline {{\mathcal A}/{\mathcal G}}\) of the space \({\mathcal A}/{\mathcal G}\) of gauge equivalent connections, characterized as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs. Using this characterization, differential geometry is developed through algebraic methods; thus, in \(\overline {{\mathcal A}/{\mathcal G}}\), are introduced: differential forms, exterior derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and associated heat kernels and heat kernel measures. The described approach does not require a background metric and provides a geometrical framework well suited for diffeomorphism invariant theories such as quantum relativity.
reviewed by: Vladimir Balan
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 58D15
Mathematics Subject Classification ID: 81V17
zbMATH DE Number: 881727
zbMATH Keywords: diffeomorphism invariance
zbMATH Keywords: Wilson loop
zbMATH Keywords: loop quantization
zbMATH Keywords: heat equation
MaRDI profile type: Publication
Wikidata QID: Q58879892
arXiv ID: hep-th/9412073
cites work: Q4872106
cites work: Q4508083
cites work: Representations of the holonomy algebras of gravity and nonAbelian gauge theories
cites work: Projective techniques and functional integration for gauge theories
cites work: Quantization of diffeomorphism invariant theories of connections with local degrees of freedom
cites work: Generalized measures in gauge theory
cites work: Q4357183
cites work: Q3954541
cites work: A construction of two-dimensional quantum chromodynamics
cites work: Q3238703
cites work: Group of loops, holonomy maps, path bundle and path connection
cites work: On the support of the Ashtekar-Lewandowski measure
cites work: Discreteness of area and volume in quantum gravity
cites work: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. (AM-63)
OpenAlex ID: W2037266697
description: scientific article