Studying Boojums in N=2 Theory with Walls and Vortices

TL;DR

This work analyzes 1/4-BPS boojums in ${\cal N}=2$ SQED with walls and vortices, showing that the energy of a flux tube ending on a domain wall is infrared log-divergent. The authors compute the boojum energy from both the (2+1)D wall world-volume theory and the (3+1)D bulk theory, and verify the results via central-charge decompositions of BPS objects. They further study interactions of two boojums on a wall, finding configurations where gauge, Higgs, and string-length contributions cancel, implying a moduli space for their separation, while other flux configurations yield attractive or repulsive forces and a finite central-charge term in the total energy. The paper also probes the self-energy of a single boojum through boojum–antiboojum configurations, but concludes that a definitive finite binding energy remains definition-dependent and likely requires numerical analysis. Overall, the results illuminate the delicate balance between wall and bulk dynamics in 1/4-BPS composites and outline scenarios where finite contributions can be isolated, guiding future numerical studies.

Abstract

We study 1/2 BPS domain walls, 1/2 BPS flux tubes (strings) and their 1/4 BPS junctions. We consider the simplest example of N=2 Abelian gauge theory with two charged matter hypermultiplets which contains all of the above-listed extended objects. In particular, we focus on string-wall junctions (boojums) and calculate their energy. It turns out to be logarithmically divergent in the infrared domain. We compute this energy first in the (2+1)-dimensional effective theory on the domain wall and then, as a check, obtain the same result from the point of view of (3+1)-dimensional bulk theory. Next, we study interactions of boojums considering all possible geometries of string-wall junctions and directions of the string magnetic fluxes.

Figures (11)

• Figure 1: Internal structure of the domain wall: two edges (domains $E_{1,2}$) of the width $\sim \xi^{-1/2}$ are separated by a broad middle band (domain $M$) of the width $R \sim \Delta m /(g^2\xi)$.
• Figure 2: Bending of the wall due to the string-wall junction. The flux tube extends to the right infinity. The wall profile is logarithmic at transverse distances larger than $g^{-1}\xi^{-1/2}$ from the string axis. At smaller distances the adiabatic approximation fails.
• Figure 3: Geometry of the boojum configuration
• Figure 4: Direction of the magnetic flux inside the wall.
• Figure 5: Vortices on opposite sides of the wall which carry the same positive U$(1)$ flux in the $z$ direction. This is a deformation with the same energy of the configuration of a vortex crossing the wall.