We argue that the world sheet dynamics of magnetic $k$ strings in the Higgs phase of the mass-deformed $\mathcal{N}=4$ theory is controlled by a bosonic $O(3)$ sigma model with anisotropy and a topological $\theta$ term. The theory interpolates between a massless $O(2)$ symmetric regime, a massive $O(3)$ symmetric phase, and another massive phase with a spontaneously broken ${\mathbb{Z}}_2$ symmetry. The first two phases are separated by a Kosterlitz-Thouless transition. When $\theta =\pi$, the $O(3)$ symmetric phase flows to an interacting fixed point; sigma model kinks and their dyonic partners become degenerate, mirroring the behavior of monopoles in the parent gauge theory. This leads to the identification of the kinks with monopoles confined on the string.

• Received 4 September 2009

DOI:https://doi.org/10.1103/PhysRevLett.103.231601