Supersymmetrising the GSY Soliton

TL;DR

This paper supersymmetrises the GSY soliton, a Hopfion in $U(1)$ gauge theory, demonstrating how a ring-like vortex can be stabilized with large angular momentum, approaching BPS saturation asymptotically.

Contribution

It introduces a supersymmetric extension of the Hopfion soliton and shows how winding and angular momentum stabilize ring-like vortices near BPS saturation.

Findings

Ring-like vortex stability depends on winding and angular momentum.

Large angular momentum makes the energy approach the BPS bound.

Supersymmetrisation preserves BPS properties in certain limits.

Abstract

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 $J\rightarrow \infty$.