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

Abstract

The purpose of this paper is to give a self-contained exposition of the Atiyah–Bott picture [AB83] 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 [Don4]. A central ingredient in our discussion is the Yang–Mills ow for which we assume longtime existence and convergence (see [Rad]).