Pseudo-real principal $G$-bundles over a real curve

TL;DR

This paper studies the moduli spaces of pseudo-real principal G-bundles over real algebraic curves, establishing their connectedness and analyzing fixed point varieties under involutions, thus advancing the understanding of real bundle structures.

Contribution

It introduces the concept of pseudo-real principal bundles over real curves, proves the connectedness of their moduli spaces, and describes fixed point varieties under involutions.

Findings

Moduli spaces of pseudo-real bundles are connected.

Fixed point varieties under involutions are characterized.

A gauge theory framework is developed for these bundles.

Abstract

We consider stable and semistable principal bundles over a smooth projective real algebraic curve, equipped with a real or pseudo-real structure in the sense of Atiyah. After fixing suitable topological invariants, one can build a suitable gauge theory, and show that the resulting moduli spaces of pseudo-real bundles are connected. This in turn allows one to describe the various fixed point varieties on the complex moduli spaces under the action of the real involutions on the curve and the structure group.