The Asymptotics of Representations for Cyclic Opers

TL;DR

This paper analyzes the asymptotic behavior of holonomy representations for cyclic opers on Riemann surfaces, revealing their limits and associated geometric structures in the symmetric space.

Contribution

It provides explicit asymptotic formulas for holonomy of cyclic opers and constructs associated equivariant maps to symmetric spaces, connecting local data to global geometric limits.

Findings

Asymptotic formulas for holonomy in cyclic opers

Construction of equivariant maps to symmetric spaces

Limits of maps tend to a sub-building in the asymptotic cone

Abstract

Given a Riemann surface $X = (\Sigma, J)$ we find an expression for the dominant term for the asymptotics of the holonomy of opers over that Riemann surface corresponding to rays in the Hitchin base of the form $(0,0,\cdots,t\omega_n)$. Moreover, we find an associated equivariant map from the universal cover $(\tilde{\Sigma},\tilde{J})$ to the symmetric space SL$_n(\mathbb{C}) / \mbox{SU}(n)$ and show that limits of these maps tend to a sub-building in the asymptotic cone. That sub-building is explicitly constructed from the local data of $\omega_n$.