Conformal limits in Cayley components and $\Theta$-positive opers

TL;DR

This paper investigates the conformal limit of Gaiotto's Hitchin equations for certain real Lie groups, introducing a new class of opers called $ heta$-positive opers that generalize Hitchin's integrable system.

Contribution

It identifies a family of flat connections with conformal limits forming $ heta$-positive opers, extending the understanding of Hitchin systems for non-split real groups.

Findings

Existence of conformal limits for specific solutions

Introduction of $ heta$-positive opers as a new class

Parameterization by a generalized base space

Abstract

We study Gaiotto's conformal limit for the $G^{\mathbb{R}}$-Hitchin equations, when $G^{\mathbb{R}}$ is a simple real Lie group admitting a $\Theta$-positive structure. We identify a family of flat connections coming from certain solutions to the equations for which the conformal limit exists and admits the structure of an oper. We call this new class of opers appearing in the conformal limit $\Theta$-positive opers. The two families involved are parameterized by the same base space. This space is a generalization of the base of Hitchin's integrable system in the case when the structure group is a split real group.