Noncommutative Generalization of Wilson Lines

TL;DR

This paper develops a noncommutative generalization of Wilson lines by leveraging covering projections and transformations, enabling their construction without traditional closed paths in noncommutative geometry.

Contribution

It introduces a novel framework for defining Wilson lines in noncommutative geometry using covering projections and transformations, extending classical concepts.

Findings

Noncommutative Wilson lines constructed via covering projections.

Global pure gauge fields on universal covers adapted to noncommutative setting.

Framework bridges classical Wilson lines and noncommutative geometry.

Abstract

A classical Wilson line is a cooresponedce between closed paths and elemets of a gauge group. However the noncommutative geometry does not have closed paths. But noncommutative geometry have good generalizations of both: the covering projection, and the group of covering transformations. These notions are used for a construction of noncommutative Wilson lines. Wilson lines can also be constructed as global pure gauge fields on the universal covering space. The noncommutative analog of this construction is also developed.