Higgs phase in a gauge $mathbf{U}(1)$ non-linear $mathbf{CP}^1$-model. Two species of BPS vortices and their zero modes

In this paper zero modes of fluctuation are dissected around the two species of BPS vortices existing in the critical Higgs phase, where the scalar and vector meson masses are equal, of a gauged $mathbb{U}(1)$ nonlinear $mathbb{CP}^1$-model. If $2pi n$, $nin mathbb{Z}$, is the quantized magnetic flux of the two species of BPS vortex solutions, $2n$ linearly independent vortex zero modes for each species are found and described. The existence of two species of moduli spaces of dimension $2n$ of these stringy topological defects is thus locally shown.