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arXiv · 2603.04153

A Non-Abelian Approach to Riemann Surfaces

Mehrzad Ajoodanian
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Abstract

We develop a non-abelian, gauge-theoretic framework for the Schwarzian derivative and for second-order differential equations on Riemann surfaces. As applications, we extend Dedekind's Schwarzian approach to elliptic periods to generic one-parameter families of curves of genus $g$ by replacing the non-canonical scalar Picard--Fuchs equation of order $2g$ with a canonical second-order equation with $g\times g$ matrix coefficients on the Hodge bundle. In higher dimensions, we discuss periods of a one-parameter family of cubic threefolds via the intermediate Jacobian. Finally, we discuss mass--spring systems in mechanics as a natural testing ground for the non-abelian Schwarzian viewpoint.

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Mehrzad Ajoodanian. 2026-03-14. A Non-Abelian Approach to Riemann Surfaces. https://arxiv.org/abs/2603.04153

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