On the Gauge-invariant Functional Measure for Gauge Fields on CP^2

TL;DR

This paper develops a gauge-invariant measure for nonabelian gauge fields on CP^2, revealing a divergence linked to a higher-dimensional Wess-Zumino-Witten action and proposing a nonperturbative dimensional parameter akin to dimensional transmutation.

Contribution

It introduces a parametrization for gauge fields on CP^2 and analyzes the volume element of the gauge-orbit space, connecting divergences to a higher-dimensional Wess-Zumino-Witten action and proposing a nonperturbative measure.

Findings

Identified the divergence in the gauge orbit volume element.

Connected the divergence to a higher-dimensional Wess-Zumino-Witten action.

Proposed the introduction of a dimensional parameter for the measure.

Abstract

We introduce a general parametrization for nonabelian gauge fields on the four-dimensional space ${\mathbb{CP}}^2$. The volume element for the gauge-orbit space or the space of physical configurations is then investigated. The leading divergence in this volume element is obtained in terms of a higher dimensional Wess-Zumino-Witten action, which has previously been studied in the context of K\"ahler-Chern-Simons theories. This term, it is argued, implies that one needs to introduce a dimensional parameter to specify the integration measure, a step which is a nonperturbative version of the well-known dimensional transmutation in four-dimensional gauge theories.