Gauge Invariant Geometric Variables For Yang-Mills Theory

TL;DR

This paper refines gauge invariant geometric variables for Yang-Mills theory, addressing subtleties in electric energy calculations, and explores specific configurations like instantons, monopoles, and merons.

Contribution

It introduces a small deformation approach to resolve zero mode issues in gauge invariant variables and analyzes their relation to known topological configurations.

Findings

Resolved calculation subtleties in electric energy involving inverse operators.

Explicit expressions for electric fields in spherical gauge configurations.

Identified geometries corresponding to instantons, monopoles, and merons.

Abstract

In a previous publication [1], local gauge invariant geometric variables were introduced to describe the physical Hilbert space of Yang-Mills theory. In these variables, the electric energy involves the inverse of an operator which can generically have zero modes, and thus its calculation is subtle. In the present work, we resolve these subtleties by considering a small deformation in the definition of these variables, which in the end is removed. The case of spherical configurations of the gauge invariant variables is treated in detail, as well as the inclusion of infinitely heavy point color sources, and the expression for the associated electric field is found explicitly. These spherical geometries are seen to correspond to the spatial components of instanton configurations. The related geometries corresponding to Wu-Yang monopoles and merons are also identified.