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A Non-Abelian Approach to Riemann Surfaces

Mehrzad Ajoodanian

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.

A Non-Abelian Approach to Riemann Surfaces

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 by replacing the non-canonical scalar Picard--Fuchs equation of order with a canonical second-order equation with 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.
Paper Structure (38 sections, 21 theorems, 285 equations, 1 figure)

This paper contains 38 sections, 21 theorems, 285 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Classical Schwarzian Derivative
  3. Second order scalar ODE
  4. Projective curvature and the Schwarzian anomaly
  5. Quantum vs classical
  6. Classical differentials and connections
  7. Quantum differentials and connections
  8. Curvature
  9. Example (Multiplication by an $\mathop{\mathrm{End}}\nolimits(E)$-valued $m$-differential).
  10. Example (Classical connections on $E\otimes K_X^{-e}$).
  11. Second order quantum ODE
  12. Main Lemma
  13. The gauge and $\star$ actions
  14. Main Theorem
  15. Modularity
  16. ...and 23 more sections

Key Result

Proposition 2.1

Let $y_1,y_2$ be two linearly independent solutions of eq:classical-second-order-pq. Then the Schwarzian derivative $S(y_1/y_2)$ is independent of the chosen ordered basis of solutions and satisfies

Figures (1)

  • Figure 1: Mass--spring system with $n=2$.

Theorems & Definitions (81)

  • Proposition 2.1
  • Example 4.1
  • Definition 5.1
  • Remark 5.2
  • Definition 5.3
  • Example 5.4
  • Example 5.5
  • Example 5.6: Wronskian for vector-valued functions
  • Example 5.7: Wronskian of a quantum differential
  • Lemma 5.8: Left Maurer--Cartan connection
  • ...and 71 more