Loop groups and holomorphic bundles

TL;DR

This paper explores the geometric relationships between loop groups, flag manifolds, and bundles over rational curves, establishing stability results for moduli spaces of G-bundles and their applications to instantons and calorons.

Contribution

It introduces a stability theorem for G-bundles with flag structures on rational ruled surfaces, linking loop group geometry to moduli space stability and automorphism-induced Hecke transforms.

Findings

Proves stability of G-bundles with flag structures on rational ruled surfaces.

Establishes stability for moduli of K-instantons and K-calorons.

Analyzes Hecke transforms induced by outer automorphisms of loop groups.

Abstract

This paper considers the links between the geometry of the various flag manifolds of loop groups and bundles over families of rational curves. Aa an application, a stability result for the moduli on a rational ruled surface of G-bundles with additional flag structure along a line is proven for any reductive group; this gives the corresponding stability statement for any compact group K for the moduli of K-instantons over the four-sphere, and for the moduli of K-calorons over the three-sphere times the circle. The paper also considers the Hecke transforms on bundles induced by outer automorphisms of the loop groups.