# On the Monodromy of Meromorphic Cyclic Opers on the Riemann Sphere

## TL;DR

This paper investigates the monodromy of meromorphic cyclic SL(n,C)-opers on the Riemann sphere with a single pole, proving the monodromy map is an immersion under specific conditions using isomonodromic deformation theory.

## Contribution

It introduces a new method based on Jimbo-Miwa-Ueno theory, adapted to the Lie algebra decomposition, to analyze the monodromy of cyclic opers and their deformation properties.

## Key findings

- Monodromy map is an immersion when pole order is a multiple of n.
- Deformations of certain cyclic SL(n,C)-opers are not infinitesimally isomonodromic.
- Develops a system of equations aligned with Lie algebra decomposition for analysis.

## Abstract

We study the monodromy of meromorphic cyclic $\mathrm{SL}(n,\mathbb{C})$-opers on the Riemann sphere with a single pole. We prove that the monodromy map, sending such an oper to its Stokes data, is an immersion in the case where the order of the pole is a multiple of $n$. To do this, we develop a method based on the work of M. Jimbo, T. Miwa, and K. Ueno from the theory of isomonodromic deformations. Specifically, we introduce a system of equations that is equivalent to the isomonodromy equations of Jimbo-Miwa-Ueno, but which is adapted to the decomposition of the Lie algebra $\mathfrak{sl}(n,\mathbb{C})$ as a direct sum of irreducible representations of $\mathfrak{sl}(2,\mathbb{C})$. Using properties of some structure constants for $\mathfrak{sl}(n,\mathbb{C})$ to analyze this system of equations, we show that deformations of certain families of cyclic $\mathrm{SL}(n,\mathbb{C})$-opers on the Riemann sphere with a single pole are never infinitesimally isomonodromic.