Breakdown of pointwise convergence in Maxwell-Chern-Simons O(3) Sigma Model

Abstract

In this paper, we consider the self-dual O(3) Maxwell–Chern–Simons-Higgs equation, a semilinear elliptic system, defined on a flat two torus. Singular points in the equation are classified as either vortex or anti-vortex points depending on the sign of their associated weighted mass. Our primary contribution is to clarify significant distinctions between scenarios involving both vortex and anti-vortex points and those characterized by singularities of only one type. Specifically, we observe the potential breakdown of some pointwise convergence result, which represents the Chern-Simons limit behavior of our system, near singularities when both vortex points and anti-vortex points coexist. Building upon this observation, we establish the existence of solutions that exhibit a loss of the prescribed pointwise convergence near certain singular points. Our rigorous arguments provide an example where the formal limit behavior fails to hold at a certain point. It is noteworthy that such solutions are not attainable in systems with singularities of only one type.