Real Slices of ${\rm SL}(r,\mathbb{C})$-Opers

TL;DR

This paper investigates the fixed points of anti-holomorphic involutions acting on ${\rm SL}(r,\mathbb{C})$-opers on Riemann surfaces, providing a parametrization of real slices and exploring their geometric properties.

Contribution

It introduces a natural parametrization of the real slices of ${\rm SL}(r,\mathbb{C})$-opers fixed by anti-holomorphic involutions using differentials on Riemann surfaces.

Findings

Parametrization of real slices via differentials.

Analysis of geometric properties of fixed point loci.

Construction of involutions for various descriptions of opers.

Abstract

Through the action of an anti-holomorphic involution $\sigma$ (a real structure) on a Riemann surface $X$, we consider the induced actions on ${\rm SL}(r,\mathbb{C})$-opers and study the real slices fixed by such actions. By constructing this involution for different descriptions of the space of ${\rm SL}(r,\mathbb{C})$-opers, we are able to give a natural parametrization of the fixed point locus via differentials on the Riemann surface, which in turn allows us to study their geometric properties.