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Low energy dynamics of $U(1)$ vortices in systems with cholesteric vacuum structure

We discuss flux tubes in systems with $U(1)$ gauge, and spin-orbit locked $SO(3)_{S+L}$ symmetry. The spin-orbit locking is achieved explicitly in the Lagrangian by introducing a parity violating twist term which causes the spontaneous breaking of $SO(3)_{S+L} \rightarrow SO(2)$. Additionally, this term causes a spontaneous breaking of the translational symmetry along a particular direction. Thus, the system appears with a cholesteric vacuum under certain conditions of the parameter space. With this term, the system admits $U(1)$ topologically stable vortices with additional structure in the vortex cores. This added structure leads to additional moduli appearing in the low energy dynamics. We determine these solutions and their low energy theory.