Morse theory for the space of Higgs G-bundles

TL;DR

This paper studies the gradient flow of the Yang--Mills--Higgs functional on Higgs G-bundles over a Riemann surface, proving flow convergence and identifying limit points, extending previous results for Higgs vector bundles.

Contribution

It generalizes the analysis of the Yang--Mills--Higgs flow to principal G-bundles, establishing flow preservation and convergence in this broader setting.

Findings

Flow preserves the subset of Higgs G-bundles.

Flow from any Higgs G-bundle has a well-defined limit.

Limit points of the flow are explicitly identified.

Abstract

Fix a $C^\infty$ principal $G$--bundle $E^0_G$ 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 $E^0_G$. 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.