Narrow Results

We give formulae for minimal surfaces in ℝ³ deriving, via classical osculation duality, from elliptic curves in a line bundle over ℙ₁. Specialising to the case of charge 2 monopole spectral curves we find that the distribution of Gaussian curvature on the auxiliary minimal surface reflects the monopole's structure. This is elucidated by the behaviour of the surface's Gauss map.

In this paper we prove the existence of a smooth minimum for the Yang–Mills–Higgs functional over a disk in 3 dimensions among those configurations with monopoles with prescribed degree, which are covariant constant at the boundary. These boundary conditions come essentially from a 4-dimensional generalized Neumann problem for the pure Yang–Mills functional and dimensional reduction. This problem is well-posed only as a gauge theory in dimension 3. It extends analogous results on Ginzburg–Landau vortices in 2 dimensions.

This paper investigates discrete symmetries for dyon and the invariance of Maxwell’s field equations in the macroscopic media (material medium). Using advanced mathematical approaches, the paper derived electromagnetic duality and a unique form of Poynting theorem for dyon in macroscopic media. Here, we demonstrate that the combination of parity $(P)$, charge conjugation $(C)$, and time reversal $(T)$ symmetry: $\text{CPT}$, is an exact symmetry for dyon in macroscopic media. Furthermore, a comprehensive analysis of the electromagnetic wave equation for dyons in conducting media is also derived. These findings contribute significantly to the understanding of dyon behavior in electromagnetic fields.