A survey of the GIT picture for the Yang-Mills equation over Riemann surfaces

TL;DR

This paper provides a comprehensive survey of the Atiyah-Bott approach to the Yang-Mills equation on Riemann surfaces, emphasizing the analogy with geometric invariant theory and analyzing stability and flow properties.

Contribution

It offers a detailed exposition of the Atiyah-Bott picture, including new proofs and insights into the stability criteria and the Yang-Mills flow on Riemann surfaces.

Findings

Analysis of semistable and unstable orbits

Proof of the moment-weight inequality

Extension of the Ness and Kempf-Ness theorems

Abstract

The purpose of this paper is to give a self-contained exposition of the Atiyah-Bott picture for the Yang-Mills equation over Riemann surfaces with an emphasis on the analogy to finite dimensional geometric invariant theory. The main motivation is to provide a careful study of the semistable and unstable orbits: This includes the analogue of the Ness uniqueness theorem for Yang-Mills connections, the Kempf-Ness theorem, the Hilbert-Mumford criterion and a new proof of the moment-weight inequality following an approach outlined by Donaldson. A central ingredient in our discussion is the Yang-Mills flow for which we assume longtime existence and convergence.