Extended Connection in Yang-Mills Theory

TL;DR

This paper introduces an extended connection in Yang-Mills theory that unifies gauge fields, gauge fixing, and ghost fields, providing a global framework that addresses Gribov obstructions in quantization.

Contribution

It generalizes gauge fixing via an extended connection, deriving BRST transformations without horizontality conditions and enabling path integral quantization with Gribov issues.

Findings

Derived global BRST transformations from extended connection curvature

Unified gauge field, gauge fixing, and ghost field in a single geometric object

Applied generalized gauge fixing to path integral quantization with Gribov obstruction

Abstract

The three fundamental geometric components of Yang-Mills theory -gauge field, gauge fixing and ghost field- are unified in a new object: an extended connection in a properly chosen principal fiber bundle. To do this, it is necessary to generalize the notion of gauge fixing by using a gauge fixing connection instead of a section. From the equations for the extended connection's curvature, we derive the relevant BRST transformations without imposing the usual horizontality conditions. We show that the gauge field's standard BRST transformation is only valid in a local trivialization and we obtain the corresponding global generalization. By using the Faddeev-Popov method, we apply the generalized gauge fixing to the path integral quantization of Yang-Mills theory. We show that the proposed gauge fixing can be used even in the presence of a Gribov's obstruction.