Gauge and Scalar Fields on $\mathbb{CP}^2$: A Gauge-invariant Analysis II. The measure for gauge fields and a 4d WZW theory

TL;DR

This paper analyzes the gauge orbit space volume for gauge fields on four-dimensional complex projective space, revealing a gauge-invariant measure involving a 4d WZW action and discussing implications for confinement and QCD.

Contribution

It introduces a gauge-invariant parametrization of gauge fields on P^2 and derives a measure involving a 4d WZW action, linking it to confinement and lattice results.

Findings

The measure includes a 4d WZW action for gauge fields.

A mass-like term relates to lattice and Schwinger-Dyson results.

The analysis suggests a regime where Yang-Mills approximates a 4d WZW theory.

Abstract

We consider the volume of the gauge orbit space for gauge fields on four-dimensional complex projective space. The analysis uses a parametrization of gauge fields where gauge transformations act homogeneously on the fields, facilitating a manifestly gauge-invariant analysis. The volume element contains a four-dimensional Wess-Zumino-Witten (WZW) action for a hermitian matrix-valued field. There is also a mass-like term for certain components of the gauge field. We discuss how the mass term could be related to results from lattice simulations as well as Schwinger-Dyson equations. We argue for a kinematic regime where the Yang-Mills theory can be approximated by the 4d-WZW theory. The result is suggestive of the instanton liquid picture of QCD. Further it is also indicative of the mechanism for confinement being similar to what happens in two dimensions.