Monopoles, Vortices, Domain Walls and D-Branes: The Rules of Interaction

TL;DR

The paper analyzes BPS solitons in a Higgs-phase, non-abelian gauge theory with ${ m N}=2$ supersymmetry, focusing on domain walls, vortex strings, confined monopoles, and D-brane–like configurations. It develops a unified framework: Bogomolny equations, a root-lattice classification of domain walls, and a detailed account of soliton interactions via 2D worldvolume dynamics and 4D/2D correspondences. A key result is the discovery of a finite, negative binding energy (a boojum) for strings ending on walls, together with an explicit 4D–2D equivalence for confined monopoles as kinks on vortex strings, $M_{ m kink}=M_{ m mono}$. The work also articulates precise interaction rules for which strings end on which walls and under what conditions walls and monopoles can pass through each other, linking field theory solitons to brane constructions and providing tools for analyzing composite solitons in non-abelian gauge theories.

Abstract

Non-abelian gauge theories in the Higgs phase admit a startling variety of BPS solitons. These include domain walls, vortex strings, confined monopoles threaded on vortex strings, vortex strings ending on domain walls, monopoles threaded on strings ending on domain walls, and more. After presenting a self-contained review of these objects, including several new results on the dynamics of domain walls, we go on to examine the possible interactions of solitons of various types. We point out the existence of a classical binding energy when the string ends on the domain wall which can be thought of as a BPS boojum with negative mass. We present an index theorem for domain walls in non-abelian gauge theories. We also answer questions such as: Which strings can end on which walls? What happens when monopoles pass through domain walls? What happens when domain walls pass through each other?

Figures (12)

• Figure 1: The profile of an elementary domain wall of type $\vec{g}={\vec{\alpha}}_i$ when $e^2v^2 \ll |m_i-m_{i+1}|^2$. The fundamental fields $q$ vary in the outer layer; the adjoint field $\phi$ varies in the inner layer. Notice that, with some of the $q$'s vanishing, the gauge group is partially restored in the inner layer.
• Figure 2: The two different orderings of the domain wall system $\vec{g}=\vec{\alpha}_1+2\vec{\alpha}_2+\vec{\alpha}_3$. The two orderings are related by interchanging the inner two domain walls, an operation which is allowed since ${\vec{\alpha}}_1\cdot{\vec{\alpha}}_3=0$. Notice that the vacuum in the middle of the system changes as the walls pass through each other.
• Figure 3: The $N_c$ different types of vortices, labelled by the element of the magnetic field or, equivalently, by the flavour $q_i$ which winds around the vortex. The $N_c$ different vortex strings are mutually BPS.
• Figure 4: The confined monopole which is a source for magnetic flux $\int d^2x B=2\pi \ {\rm diag}(1,-1,\ldots,0)$.
• Figure 5: For the ${\vec{\alpha}}_i$ elementary domain wall, the $q_i$ string may end on the left and the $q_{i+1}$ string may end on the right. All $q_j$ strings, for $j\neq i,i+1$ exist in both left and right vacua and pass right through the domain wall. The nodes represent the finite binding energy.