Mass deformed world-sheet action of semi-local vortices

The mass deformed effective world-sheet theory of semi local vortices was constructed via the field theoretical method. By Euler-Lagrangian equations, the Ansatze for both the gauge field and the adjoint scalar were solved, this ensures that zero modes of vortices are minimal excitations of the system. Up to the $1/g^2$ order, all profiles are solved. The mass deformed effective action was obtained by integrating out the transverse plane of the vortex string. The effective theory interpolates between the local vortex and the lump. Respecting certain normalization conditions, the effective theory shows a Seiberg-like duality, which agrees with the result of the K\"ahler quotient construction.