A Non-Abelian Approach to Riemann Surfaces

TL;DR

This paper introduces a non-abelian, gauge-theoretic framework for analyzing Riemann surfaces and their periods, extending classical Schwarzian theory to higher-genus curves and higher-dimensional varieties with applications in mechanics.

Contribution

It develops a novel non-abelian gauge-theoretic approach to the Schwarzian derivative, generalizing classical methods to complex algebraic curves and higher-dimensional varieties.

Findings

Extended Schwarzian approach to elliptic periods for genus g curves

Replaced scalar Picard--Fuchs equations with matrix-valued equations

Applied non-abelian Schwarzian concepts to mechanics systems

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.