Supersymmetrizing the Gorsky-Shifman-Yung soliton

We supersymmetrize the Hopfion studied by Gorsky et al. [Phys. Rev. D 88, 045026 (2013).PRVDAQ1550-799810.1103/PhysRevD.88.045026]. This soliton represents a closed semilocal vortex string in U(1) gauge theory. It carries nonzero Hopf number due to the additional winding of a phase modulus as one moves along the closed string. We study this solution in N=2 supersymmetric QED with two flavors. As a preliminary exercise, we compactify one space dimension and consider a straight vortex with periodic boundary conditions. It turns out to be 1/2-BPS saturated. An additional winding along the string can be introduced and it does not spoil the BPS nature of the object. Next, we consider a ringlike vortex in a non-compact space and show that the circumference of the ring L can be stabilized once the previously mentioned winding along the string is introduced. Of course, the ringlike vortex is not BPS but its energy becomes close to the BPS bound if L is large, which can be guaranteed in the case that we have a large value of the angular momentum J. Thus we arrive at the concept of asymptotically BPS-saturated solitons. BPS saturation is achieved in the limit J→.