BPS boojums in $N=2$ supersymmetric gauge theories II

We continue our study of 1/4 Bogomol'nyi-Prasad-Sommerfield (BPS) composite solitons of vortex strings, domain walls, and boojums in $N=2$ supersymmetric Abelian gauge theories in four dimensions. In this work, we numerically confirm that a boojum appearing at an endpoint of a string on a thick domain wall behaves as a magnetic monopole with a fractional charge in three dimensions. We introduce a "magnetic" scalar potential whose gradient gives magnetic fields. The height of the magnetic potential has a geometrical meaning that is the shape of the domain wall. We find a semilocal extension of a boojum that has an additional size moduli at an endpoint of a semilocal string on the domain wall. Dyonic solutions are also studied and we numerically confirm that the dyonic domain wall becomes an electric capacitor storing opposite electric charges on its skins. At the same time, the boojum becomes a fractional dyon whose charge density is proportional to $\overrightarrow{E}\cdot \overrightarrow{B}$. We also study dual configurations with an infinite number of boojums and anti-boojums on parallel lines and analyze the ability of domain walls to store magnetic charge as magnetic capacitors. In understanding these phenomena, the magnetic scalar potential plays an important role. We study the composite solitons from the viewpoints of the Nambu-Goto and Dirac-Born-Infeld actions, and find the semilocal BIon as the counterpart of the semilocal boojum.