On the Spectrum of Holonomy Algebras

TL;DR

This paper explores the relationships between different holonomy algebras derived from connections on trivial bundles, analyzing their spectra and the implications for gauge theories.

Contribution

It clarifies the connections and spectral properties of cylindrical and holonomy algebras associated with smooth connections on trivial bundles.

Findings

Relations between various holonomy algebras are established.

Spectral properties of these algebras are analyzed.

Implications for gauge theories are discussed.

Abstract

Connections on a trivial bundle MxG can be identified with their holonomy maps, i.e. with homomorphisms of a groupoid of paths in M into the gauge group G. For a connected compact G, various algebras depending on the set of the smooth connections through their holonomy maps have been introduced in the literature, called cylindrical and holonomy algebras. We discuss the relations between these algebras and the consistence of their spectra.