On the topological solutions with vortices and antivortices for the Maxwell-Chern-Simons O(3) sigma model on a torus

In this paper, we consider the self-dual O(3) Maxwell-Chern-Simons-Higgs equations on a two-dimensional flat torus arising from the O(3) sigma gauge field model. Our main goal is to obtain the existence and uniqueness of topological solutions with vortices and antivortices. In order to achieve this goal, we show that the nondegeneracy of linearized operator for entire solution holds even when the symmetry is broken. Moreover, we also obtain the uniform bound of L¹ norm of nonlinearities with respect to large charge of electron. We expect that these results would play an significant role to get the asymptotic behavior of all possible solutions and count the total number of solutions.