Morse theory for the space of Higgs G-bundles

Abstract

Fix a C^∞ principal G-bundle E⁰_G on a compact connected Riemann surface X, where G is a connected complex reductive linear algebraic group. We consider the gradient flow of the Yang-Mills-Higgs functional on the cotangent bundle of the space of all smooth connections on E⁰_G. We prove that this flow preserves the subset of Higgs G-bundles, and, furthermore, the flow emanating from any point of this subset has a limit. Given a Higgs G-bundle, we identify the limit point of the integral curve passing through it. These generalize the results of the second named author on Higgs vector bundles.