Triangular isomonodromic solutions to a Fuchsian system from superelliptic curves

2025/07/14 by Anwar Al Ghabra, Ghabra, Anwar Al, Benjamin Piché +3
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Algebraic and Geometric Analysis #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph)

paper · pdf · doi:10.48550/arxiv.2507.10692

openalex publication_date 2025/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

We give fundamental solutions of arbitrarily sized matrix Fuchsian linear systems, in the case where the coefficients B⁽ⁱ⁾ of the systems are matrix solutions of the Schlesinger system that are upper triangular, and whose eigenvalues follow an arithmetic progression of a rational difference. The values on the superdiagonals of the matrices B⁽ⁱ⁾ are given by contour integrals of meromorphic differentials defined on Riemann surfaces obtained by compactification of superelliptic curves. We show that our fundamental solutions are isomonodromic by obtaining their monodromy matrices.

Citations

• The Wiener--Hopf Technique, its Generalisations and Applications: Constructive and Approximate Methods
• Triangular Schlesinger systems and superelliptic curves
• Note on algebro-geometric solutions to triangular Schlesinger systems
• Algebro-geometric solutions of the Schlesinger systems and the\n Poncelet-type polygons in higher dimensions
• On the Reductions and Classical Solutions of the Schlesinger equations

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