Existence theorems for a general \(2\times 2\) non-abelian Chern-Simons-Higgs system over a torus (Q526024)

This work considers a general \(2\times 2\) non-abelian Chern-Simons-Higgs system in the case when the matrix \[ K= \begin{pmatrix} 1+a&-a\\ -b&1+b \end{pmatrix} \] of coefficients multiplied by exponential terms contains negative parameters \(a\) and \(b\). The approach used here is based on the crucial feature of the non-positivity of the off-diagonal matrix elements in the case of a flat 2-torus. For this case, the existence of solutions for this system is proven over \(\mathbb T^2\) when \(|a|\) and \(|b|\) are small enough