Supersymmetrising the GSY Soliton

We supersymmetrise the Hopfion studied in a previous work. 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 $\mathcal{N}= 2$ supersymmetric QED with two flavours. 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 ring-like vortex in a non-compact space and show that the circumference of the ring $L$ can be stabilised once the previously mentioned winding along the string is introduced. Of course the ring-like 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 $J\rightarrow \infty$.