On the Monodromy of Meromorphic Cyclic Opers on the Riemann Sphere

DOI10.1093/IMRN/RNAA047zbMATH Open1483.32012arXiv1906.04004OpenAlexW3016605193MaRDI QIDQ5022015

Charles Lebarron Alley

Publication date: 18 January 2022

Published in: IMRN. International Mathematics Research Notices (Search for Journal in Brave)

Abstract: We study the monodromy of meromorphic cyclic $mathrmSL(n,mathbbC)$-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 $mathfraksl(n,mathbbC)$ as a direct sum of irreducible representations of $mathfraksl(2,mathbbC)$. Using properties of some structure constants for $mathfraksl(n,mathbbC)$ to analyze this system of equations, we show that deformations of certain families of cyclic $mathrmSL(n,mathbbC)$-opers on the Riemann sphere with a single pole are never infinitesimally isomonodromic.

Full work available at URL: https://arxiv.org/abs/1906.04004

zbMATH Keywords

Riemann sphereholomorphic principal bundles\(\mathrm{SL}(n,\mathbb C)\)-opers

Mathematics Subject Classification ID

Holomorphic bundles and generalizations (32L05)

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