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.
Categories: math.DG math.CA