Solitons in the Higgs phase -- the moduli matrix approach --

TL;DR

This work provides a comprehensive, moduli-matrix–based framework for classifying and constructing topological solitons in the Higgs phase of $U(N_C)$ gauge theories with $N_F$ fundamental flavors. By solving the hypermultiplet (half) BPS sector through the moduli matrix and treating the vector-multiplet sector via master equations, the authors obtain complete moduli spaces for 1/2 BPS domain walls and vortices, along with their rich 1/4 BPS composites such as wall webs, wall–vortex–monopole systems, and instanton–vortex configurations. They establish a deep correspondence with Kähler and hyper-Kähler quotients, reveal dualities at strong coupling, and derive effective worldvolume Lagrangians for walls and vortices. The paper also develops geometric tools (Plücker coordinates, grid diagrams) to analyze wall webs, junctions, and the energetics of complex soliton networks, including boojums and Hitchin-type junctions, thereby providing a unified, highly geometric picture of soliton dynamics in supersymmetric gauge theories with potential brane-theoretic interpretations. The techniques enable exact results in the strong coupling limit and offer a versatile platform for exploring interactions and effective theories of composite solitons in high-energy and mathematical physics contexts.

Abstract

We review our recent work on solitons in the Higgs phase. We use U(N_C) gauge theory with N_F Higgs scalar fields in the fundamental representation, which can be extended to possess eight supercharges. We propose the moduli matrix as a fundamental tool to exhaust all BPS solutions, and to characterize all possible moduli parameters. Moduli spaces of domain walls (kinks) and vortices, which are the only elementary solitons in the Higgs phase, are found in terms of the moduli matrix. Stable monopoles and instantons can exist in the Higgs phase if they are attached by vortices to form composite solitons. The moduli spaces of these composite solitons are also worked out in terms of the moduli matrix. Webs of walls can also be formed with characteristic difference between Abelian and non-Abelian gauge theories. We characterize the total moduli space of these elementary as well as composite solitons. Effective Lagrangians are constructed on walls and vortices in a compact form. We also present several new results on interactions of various solitons, such as monopoles, vortices, and walls. Review parts contain our works on domain walls (hep-th/0404198, hep-th/0405194, hep-th/0412024, hep-th/0503033, hep-th/0505136), vortices (hep-th/0511088, hep-th/0601181), domain wall webs (hep-th/0506135, hep-th/0508241, hep-th/0509127), monopole-vortex-wall systems (hep-th/0405129, hep-th/0501207), instanton-vortex systems (hep-th/0412048), effective Lagrangian on walls and vortices (hep-th/0602289), classification of BPS equations (hep-th/0506257), and Skyrmions (hep-th/0508130).

Figures (33)

• Figure 1: Internal structures of the domain walls.
• Figure 2: Comparison of the profile of $\langle {\cal W}\rangle,{\cal W}^{\langle 1 \rangle}, {\cal W}^{\langle 2 \rangle}$ as functions of $y$. Linear functions ${\cal W}^{\langle A \rangle}$ are good approximations in their respective dominant regions.
• Figure 3: ${\bf C}P^1$ and the potential $V$. The base space of $T^*{\bf C}P^1$, ${\bf C}P^1 \simeq S^2$, is displayed. This model contains two discrete vacua denoted by $N$ and $S$. The potential $V$ is also displayed on the right of the ${\bf C}P^1$. It admits a single wall solution connecting these two vacua expressed by a curve. The $U(1)$ isometry around the axis connecting $N$ and $S$ is spontaneously broken by the wall configuration.
• Figure 4: Comparison of the profile of $\langle {\cal W}\rangle,{\cal W}^{\langle 1 \rangle}, {\cal W}^{\langle 2 \rangle}$ as functions of $y$. $\langle {\cal W}\rangle$ connects smoothly dominant linear functions ${\cal W}^{\langle A \rangle}$ in respective regions.
• Figure 5: Multiple non-Abelian walls as kinky D-branes.