On the Monodromy of Meromorphic Cyclic Opers on the Riemann Sphere

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.