Non-Abelian Walls in Supersymmetric Gauge Theories

TL;DR

This work constructs 1/2-BPS non-Abelian domain walls in five-dimensional ${ m SUSY}$ ${U(N_{ m C})}$ gauge theories with ${N_{ m F}}>{N_{ m C}}$ flavors, using a moduli-matrix framework that reveals a world-volume $GL(N_{ m C},{f C})$ redundancy. At infinite gauge coupling, exact multi-wall solutions are obtained for generic moduli, while finite coupling yields solutions on special subspaces via a factorization into Abelian sectors; the full moduli space of walls is the complex Grassmann manifold ${G}_{N_{ m F},N_{ m C}}$ endowed with a deformed metric due to hypermultiplet masses. The authors derive gauge-invariant observables, construct the world-volume effective action as a Kähler sigma model on the moduli space, and identify Nambu-Goldstone and quasi-Nambu-Goldstone modes. They provide explicit infinite-coupling solutions for ${N_{ m C}}=2$ with ${N_{ m F}}=3,4$, uncover rich wall-interaction phenomena such as penetrable and compressed walls, and develop an algebraic framework using wall-creating operators to classify topological sectors. The results illuminate the geometry of non-Abelian wall moduli, dualities between theories with ${N_{ m C}}$ and ${N_{ m F}-N_{ m C}}$ colors, and pave the way for localized gauge dynamics on wall world-volumes with potential links to gravity and integrable systems.

Abstract

The Bogomol'nyi-Prasad-Sommerfield (BPS) multi-wall solutions are constructed in supersymmetric U(N_C) gauge theories in five dimensions with N_F(>N_C) hypermultiplets in the fundamental representation. Exact solutions are obtained with full generic moduli for infinite gauge coupling and with partial moduli for finite gauge coupling. The generic wall solutions require nontrivial configurations for either gauge fields or off-diagonal components of adjoint scalars depending on the gauge. Effective theories of moduli fields are constructed as world-volume gauge theories. Nambu-Goldstone and quasi-Nambu-Goldstone scalars are distinguished and worked out. Total moduli space of the BPS non-Abelian walls including all topological sectors is found to be the complex Grassmann manifold SU(N_F) / [SU(N_C) x SU(N_F-N_C) x U(1)] endowed with a deformed metric.

Figures (16)

• Figure 1: Diagrammatic representation of vacua for $N_{\rm C}=2$ case. Dashed lines are defined by $\Sigma^0+\Sigma^3 =2m_A$ and $\Sigma^0 - \Sigma^3 =2m_A$. Vacua are given as intersection points of these lines except for $\Sigma ^3=0$, because of Eq. (\ref{['D-term-cond-b']}).
• Figure 2: ${\bf C}P^1$ 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 3: Walls for $N_{\rm C}=2$ and $N_{\rm F}=3$. This model admits three single walls. Two of them are elementary walls and the other is a compressed wall. The latter is obtained in a particular limit of the double wall configuration. Meaning of arrows is explained in Fig. \ref{['yajirusi']}.
• Figure 4: An arrow with a single arrowhead denotes an elementary wall. An arrow with duplicate ($l$-uninterrupted) arrowhead denotes a compressed wall of level $1$ ($l-1$). An arrow with two separate arrowheads denotes a double wall consisting of two single walls with the relative distance as a moduli.
• Figure 5: Configurations for $\Sigma ^0,\Sigma ^3$ and $(H^1)^{22},(H^1)^{23}$, in the case of $(m_1,\,m_2,\,m_3)=(1,0,-1)$ and $r_1=0$.