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

TL;DR

This paper explores how anti-holomorphic involutions on Riemann surfaces induce real slices in the moduli space of parabolic SL(r,C)-opers, unifying different descriptions of these structures.

Contribution

It establishes a natural anti-holomorphic involution on the space of parabolic SL(r,C)-opers and proves the equivalence of involutions across various descriptions.

Findings

Fixed-point locus defines real slices of opers.

Involutions on differential operators coincide.

Provides a unified framework for real structures in opers.

Abstract

Let $X$ be a Riemann surface equipped with an anti-holomorphic involution $\sigma_X$. We show that this induces a natural anti-holomorphic involution on the space of parabolic $\mathrm{SL}(r,\mathbb{C})$-opers. The fixed-point locus of this involution is defined as real slice. We further study the induced involutions on different descriptions of parabolic $\mathrm{SL}(r,\mathbb{C})$-opers, in particular differential operators, and prove that these involutions coincide.