The domain geometry and the bubbling phenomenon of rank two Gauge theory

Let $\Omega$ be a flat torus and $G$ be the green's function of $-\Delta$ on $\Omega$. One intriguing mystery of $G$ is how the number of its critical points is related to blowup solutions of certain PDEs. In this article we prove that for the following equation that describes a Chern-Simons model in Gauge theory: \begin{equation}\label{e103} \left\{ \begin{array}{ll} \Delta u_1+\frac{1}{\varepsilon^2}e^{u_2}(1-e^{u_1})=8\pi\delta_{p_{1}} \Delta u_2+\frac{1}{\varepsilon^2}e^{u_1}(1-e^{u_2})=8\pi\delta_{p_{2}} \end{array} \text{ in }\quad \Omega\right., \quad p_1-p_2 \mbox{ is a half period}, \end{equation} if fully bubbling solutions of Liouville type exist, $G$ has exactly three critical points. In addition we establish necessary conditions for the existence of fully bubbling solutions with multiple bubbles.