D-brane Construction for Non-Abelian Walls

TL;DR

This work constructs non-Abelian domain walls in a supersymmetric U(N_C) gauge theory with N_F fundamental hypermultiplets using a brane setup of Nc fractional Dp-branes at a Z2 orbifold on Nf separated D(p+4)-branes. Walls arise as kinky fractional Dp-branes interpolating between D(p+4)-branes, and the analysis reveals how wall solutions and the Nc ↔ Nf−Nc duality extend the Hanany-Witten effect and s-rule to these configurations, including brane reconnection phenomena. The moduli space of walls is linked to the field-theory Grassmann base G_{N_F,N_C} and the massless vacua to the hyper-Kähler space T^*G_{N_F,N_C}, with precise counting of topological sectors via an index that matches field-theory results. Overall, the brane picture provides a transparent and unifying framework for understanding the rich dynamics of non-Abelian walls, their tensions, dualities, and moduli.

Abstract

Supersymmetric U(Nc) gauge theory with Nf massive hypermultiplets in the fundamental representation is given by the brane configuration made of Nc fractional Dp-branes stuck at the Z_2 orbifold singularity on Nf separated D(p+4)-branes. We show that non-Abelian walls in this theory are realized as kinky fractional Dp-branes interpolating between D(p+4)-branes. Wall solutions and their duality between Nc and Nf - Nc imply extensions of the s-rule and the Hanany-Witten effect in brane dynamics. We also find that the reconnection of fractional D-branes occurs in this system. Diverse phenomena in non-Abelian walls found in field theory can be understood very easily by this brane configuration.

Figures (25)

• Figure 1: A kinky D-brane configuration for Abelian walls. The theory is realized on a single D$p$-brane in the background of $N_{\rm F}$ D($p+4$)-branes with separated positions and the $B$-field. Multiple walls in SUSY $U(1)$ gauge theory are realized as a single kinky D-brane interpolating between the separated D($p+4$)-branes LTGTT2. The extra coordinate is denoted by $y = x^1$.
• Figure 2: A D-brane configuration for massless hypermultiplets in the fundamental representation.
• Figure 3: The T-dualized brane configuration for massless hypermultiplets. The triplet of the FI-parameters correspond to separations $\Delta x^{3,4,5}$ of the D$2$-branes along the D$4$-branes. Separation $\Delta x^{2}$ of the two NS$5$-branes along the $x^2$-coordinate is proportional to the gauge coupling constant squared on the effective theory on the D$2$-branes.
• Figure 4: The moduli space for vacua. Its dimension can be counted as the D$2$-brane degree of freedom in the $x^{3,4,5}$-positions and the position on $S^1$ in the M-theory. The s-rule plays a crucial role for this counting.
• Figure 5: The brane configuration for massive hypermultiplets. Hypermultiplets coming from strings stretched between D$2$- and D$4$-branes become massive by placing D$4$-branes with distances in the $x^{6,7,8,9}$-coordinates. $\Delta m$ and $\Delta m'$ denote mass differences for the hypermultiplets. The relation with coordinates is $\Delta \vec{m} = l_s^{-2} (\Delta x^6,\Delta x^7,\Delta x^8,\Delta x^9)$.