The effective action of a BPS Alice string

Recently a BPS Alice string has been found in a $U(1)\times SU(2)$ gauge theory coupled with a charged complex adjoint scalar field arXiv:1703.08971. It is a half BPS state preserving a half of supercharges when embedded into a supersymmetric gauge theory. In this paper, we study zero modes of a BPS Alice string. After presenting $U(1)$ and translational zero modes, we construct the effective action of these modes. In contrast to previous analysis of the conventional Alice string for which only large distance behaviors are known, we can perform calculation exactly in the full space thanks to BPS properties.