On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters

TL;DR

This paper investigates the number and symmetry properties of solutions to the $SU(3)$ Toda system on flat tori with singular sources, establishing upper bounds, conditions for even solutions, and connections to algebraic geometry.

Contribution

It provides the first sharp upper bound on the number of solutions, characterizes even solutions for specific cases, and introduces new proof techniques based on integrability.

Findings

Maximum number of solutions is explicitly bounded.

Even solutions characterized by parity of parameters.

Finiteness of solutions for certain parameter regimes.

Abstract

We study the $SU(3)$ Toda system with singular sources \[ \begin{cases} \Delta u+2e^{u}-e^v=4\pi\sum_{k=0}^m n_{1,k}\delta_{p_k}\quad\text{ on }\; E_{\tau},\\ \Delta v+2e^{v}-e^u=4\pi \sum_{k=0}^m n_{2,k}\delta_{p_k}\quad\text{ on }\; E_{\tau}, \end{cases} \] where $E_{\tau}:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ with $\operatorname{Im}\tau>0$ is a flat torus, $\delta_{p_k}$ is the Dirac measure at $p_k$, and $n_{i,k}\in\mathbb{Z}_{\geq 0}$ satisfy $\sum_{k}n_{1,k}\not\equiv \sum_k n_{2,k} \mod 3$. This is known as the non-critical case and it follows from a general existence result of \cite{BJMR} that solutions always exist. In this paper we prove that (i) The system has at most \[\frac{1}{3\times 2^{m+1}}\prod_{k=0}^m(n_{1,k}+1)(n_{2,k}+1)(n_{1,k}+n_{2,k}+2)\in\mathbb{N}\] solutions. We have several examples to indicate that this upper bound should be sharp. Our proof presents a nice combination of the apriori estimates from analysis and the classical B\'{e}zout theorem from algebraic geometry. (ii) For $m=0$ and $p_0=0$, the system has even solutions if and only if at least one of $\{n_{1,0}, n_{2,0}\}$ is even. Furthermore, if $n_{1,0}$ is odd, $n_{2,0}$ is even and $n_{1,0}<n_{2,0}$, then except for finitely many $\tau$'s modulo $SL(2,\mathbb{Z})$ action, the system has exactly $\frac{n_{1,0}+1}{2}$ even solutions. Differently from \cite{BJMR}, our proofs are based on the integrability of the Toda system, and also imply a general non-existence result for even solutions of the Toda system with four singular sources.