Yang-Mills connections on surfaces and representations of the path group

TL;DR

This paper characterizes Yang-Mills connections on surfaces via holonomy properties related to the area between paths, providing new proofs and insights into their structure and classification.

Contribution

It offers an alternative proof of Atiyah-Bott's theorem and characterizes Yang-Mills connections on surfaces through holonomy and conjugacy classes.

Findings

Yang-Mills connections are characterized by area-dependent holonomy.

On the sphere, Yang-Mills connections are isolated and linked to conjugacy classes of geodesics.

Provides an alternative proof of a fundamental theorem relating connections and fundamental group representations.

Abstract

We prove that Yang-Mills connections on a surface are characterized as those with the property that the holonomy around homotopic closed paths only depends on the oriented area between the paths. Using this we have an alternative proof for a theorem of Atiyah and Bott that the Yang-Mills connections on a compact orientable surface can be characterized by homomorphisms to the structure group from an extension of the fundamental group of the surface. In addition, we obtain the results that the Yang-Mills connections on the sphere are isolated and correspond with the conjugacy classes of closed geodesics through the identity in the structure group.