Configurations of vortex strings stretched between or ending on domain walls were previously found to be $1/4$ Bogomol’nyi-Prasad-Sommerfield (BPS) states in $\mathcal{N}=2$ supersymmetric gauge theories in $3+1$ dimensions. Among zero modes of string positions, the center of mass of strings in each region between two adjacent domain walls is shown to be non-normalizable whereas the rests are normalizable. We study dynamics of vortex strings stretched between separated domain walls by using two methods, the moduli space (geodesic) approximation of full $1/4$ BPS states and the charged particle approximation for string end points in the wall effective action. In the first method we explicitly obtain the effective Lagrangian in the strong coupling limit, which is written in terms of hypergeometric functions, and find the 90° scattering for head-on collision. In the second method the domain wall effective action is assumed to be $U(1{)}^N$ gauge theory, and we find a good agreement between two methods for well-separated strings.

• Received 24 October 2008

DOI:https://doi.org/10.1103/PhysRevD.79.045015