Uniqueness of topological multi-vortex solutions for a skew-symmetric Chern-Simons system

Consider the following skew-symmetric Chern-Simons system \begin{equation*}\left { \begin{split} &\Delta u_{1}+\frac{1}{\varepsilon^2} e^{u_{2}}(1-e^{u_{1}})=4\pi \sum^{N_1}{j=1}\delta{p_{j,1}}\ &\Delta u_{2}+\frac{1}{\varepsilon^2} e^{u_{1}}(1-e^{u_{2}})=4\pi \sum^{N_2}{j=1}\delta{p_{j,2}} \end{split}\right.\quad$\text{ in }$\quad\Omega, \end{equation*} where $\mathrm{\Omega }$ is a flat 2-dimensional torus ${\mathbb{T}}^2$ or ${\mathbb{R}}^2$, $\epsilon >0$ is a coupling parameter, and ${\delta }_p$ denotes the Dirac measure concentrated at $p$. In this paper, we prove that, when the coupling parameter $\epsilon$ is small, the topological type solutions to the above system are uniquely determined by the location of their vortex points. This result follows by the bubbling analysis and the non-degency of linearized equations.