Monopoles Can be Confined by 0, 1 or 2 Vortices

TL;DR

This work shows that in ${\cal N}=2$ gauge theories with fundamental matter softly broken by a superpotential (and optionally an FI term), monopoles come in three varieties—0, 1, and 2—that can be confined by one or two vortices attached to distinct unbroken gauge components. A central concept, color-flavor locking, governs which vortices exist and how many confine a given monopole; the vortex tensions are given by $T_i=4\pi\sqrt{|W'(m_i)|^2+r^2}$ in the appropriate regime, and flux matching ties monopole flux to vortex flux. The authors derive selection rules for monopole–vortex interactions, show when vortices are mutually BPS, and provide a unified field-theory and MQCD (brane) perspective that extends to ${\rm SO}(N)$, ${\rm SP}(N)$, and quiver theories, where two-colored vortex pictures classify monopoles and their bound states. These results illuminate the structure of confinement mechanisms in supersymmetric theories and offer a quiver-theoretic language for analyzing monopole spectra and their dynamics. The work also clarifies limitations in pure SU($N$) cases and suggests directions for studying non-BPS corrections and marginal locking in broader gauge-theory contexts.

Abstract

There are three types of monopole in gauge theories with fundamental matter and N=2 supersymmetry broken by a superpotential. There are unconfined 0-monopoles and also 1 and 2-monopoles confined respectively by one or two vortices transforming under distinct components of the unbroken gauge group. If a Fayet-Iliopoulos term is added then there are only 2-monopoles. Monopoles transform in the bifundamental representation of two components of the unbroken gauge symmetry, and if two monopoles share a component they may form a boundstate. Selection rules for this process are found, for example vortex number is preserved modulo 2. We find the tensions of the vortices, which are in general distinct, and also the conditions under which vortices are mutually BPS. Results are derived in field theory and also in MQCD, and in quiver theories a T-dual picture may be used in which monopoles are classified by quiver diagrams with two colors of vertices.

Figures (13)

• Figure 1: Here are brane cartoons for the 0,1,2-monopoles in a $U(2)$ gauge theory. The 't Hooft-Polyakov monopoles are D2-branes bounded by the D4 and NS5-branes, while the vortices are the continuations of these D2-branes that must exist if one of the D4-NS5 corners is ruptured, which corresponds to a superpotential or FI term.
• Figure 2: A 't Hooft-Polyakov monopole in a theory with two Colors and three flavors.
• Figure 3: This is another view of the previous figure.
• Figure 4: We have turned on an FI term, which corresponds to displacing the NS5 on the right. The boundary of the monopole has now been broken in two places because the NS5 brane on the right no longer touches either of the color-flavor locked branes. Each of these gaps in the D2-brane (the monopole) boundary is the beginning of a vortex, since it would be illegal (RR current conservation) for the 2-brane to end anywhere but on another brane. This is the 1/4 BPS 2-monopole configuration of Ref. tong-monopolo in which the vortices leave the monopole in opposite directions.
• Figure 5: This is Figure \ref{['FI']} viewed from another perspective.