We treat the geometrical optics as an Abelian $U(1)$ local gauge theory the same as the Abelian $U(1)$ Maxwell's gauge theory. We propose there exists a knot in a 3-dimensional Euclidean (flat) space of the geometrical optics (the eikonal equation) as a consequence there exists a knot in the Maxwell's theory in a vacuum. We formulate the Chern-Simons integral using an eikonal. We obtain the relation between the knot (the geometric optical helicity, an integer number) and the refractive index. We propose that the nature of the singularities of the phase is determined by the fact that the gauge potential is a smooth single-valued function of its variables.
We study nonlinear sigma model, especially Skyrme model without twist and Skyrme model with twist: twisted Skyrmion string. Twist term, mkz, is indicated in vortex solution. Necessary condition for stability of vortex solution has consequence that energy of vortex is minimum and scale-free (vortex solution is neutrally stable to changes in scale). We find numerically that the value of vortex minimum energy per unit length for twisted Skyrmion string is 20.37 × 10 60 eV/m.
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