Algebraic methods in periodic singular Liouville equations

TL;DR

This paper explores how algebraic geometry and monodromy theory are applied to analyze solutions of singular Liouville equations on tori, extending previous work from single to multiple singular sources.

Contribution

It introduces new algebraic and monodromy-based methods for studying multi-source singular Liouville equations, including counting formulas and solution parametrizations.

Findings

Derived an exact algebraic degree counting formula for odd total singularity order.

Proposed the existence of generalized Lamé curves for even total singularity order.

Extended the algebraic geometric framework from single to multiple singular sources.

Abstract

We explain how algebraic geometry comes into play in the study of non-linear mean field (singular Liouville) equations $$ \triangle u + e^u = 4\pi \sum_{i = 1}^N \ell_i \delta_{p_i} $$ on a flat torus $E = \Bbb C/\Lambda$, where $N, \ell_1, \ldots, \ell_N \in \Bbb N$, $p_i \in E$ are distinct points, and $\delta_{p_i}$ is the Dirac measure at $p_i$. The case with one singular source ($N = 1$) had been studied extensively in recent years. We start with a survey of this case with emphasizes on the constructions of Lam\'e curves $\overline X_n$ and pre-modular forms $Z_n(\sigma, \tau)$ which encodes the structure of solutions of the PDE. We then discuss extensions to the case of general $N$. The basic tool is the monodromy theory for generalized Lam\'e equations. Two aspects are discussed: (1) For $\ell := \sum_{i = 1}^N \ell_i$ being odd, an exact counting formula of \emph{algebraic degree} is proved. (2) For $\ell$ being even, the existence of generalized Lam\'e curves parametrizing logarithmic-free solutions is proposed.