Hitchin Pairs for non-compact real Lie groups

TL;DR

This paper explores Hitchin pairs, a generalization of Higgs bundles, for non-compact real Lie groups, focusing on their theoretical foundations and recent developments for groups like U(p,q).

Contribution

It extends the theory of Hitchin pairs to non-compact real Lie groups and reviews recent advances, including work on U(p,q) Hitchin pairs.

Findings

Basic theory of Hitchin pairs for real Lie groups outlined.

Recent results on Hitchin pairs for U(p,q) discussed.

Connections to Higgs bundles and line bundle twisting explained.

Abstract

Hitchin pairs on Riemann surfaces are generalizations of Higgs bundles, allowing the Higgs field to be twisted by an arbitrary line bundle. We consider this generalization in the context of $G$-Higgs bundles for a real reductive Lie group $G$. We outline the basic theory and review some selected results, including recent results by Nozad and the author arXiv:1602.02712 [math.AG] on Hitchin pairs for the unitary group of indefinite signature $\mathrm{U}(p,q)$.