Counting Domain Walls in N=1 Super Yang-Mills Theory

TL;DR

The paper addresses counting BPS domain walls between the $h=T_G$ vacua of ${\cal N}=1$ SYM by deforming to a weakly coupled Higgs phase via $N_f=N$ fundamental flavors, where the wall worldvolume becomes an ${\cal N}=1$ Grassmannian sigma model. The reduced moduli space of a $k$-wall is the complex Grassmannian $G(k,N)$, leading to a worldvolume Witten index that equals the Euler characteristic $\chi(G(k,N))=\binom{N}{k}$; this result is shown to be regulator- and holomorphy-protected and agrees with the Acharya–Vafa D4-brane picture. Key steps include deriving the BPS equations, identifying Goldstone modes from broken flavor symmetry, and verifying the index in regulated 2D/1D compactifications on $T^2$ and $S^1$. The work provides a robust, deformation-insensitive count of wall degeneracies and connects field-theoretic domain walls to string-theoretic constructions, with implications for large-$N$ behavior and wall dynamics. Overall, the paper establishes $\nu_k=\binom{N}{k}$ as the exact multiplicity of $k$-walls in SU($N$) ${\cal N}=1$ SYM under suitable infrared regulation.

Abstract

We study the multiplicity of BPS domain walls in N=1 super Yang-Mills theory, by passing to a weakly coupled Higgs phase through the addition of fundamental matter. The number of domain walls connecting two specified vacuum states is then determined via the Witten index of the induced worldvolume theory, which is invariant under the deformation to the Higgs phase. The worldvolume theory is a sigma model with a Grassmanian target space which arises as the coset associated with the global symmetries broken by the wall solution. Imposing a suitable infrared regulator, the result is found to agree with recent work of Acharya and Vafa in which the walls were realized as wrapped D4-branes in IIA string theory.

Figures (3)

• Figure 1: Possible winding trajectories $\eta_1$ and $\eta_2$ for the eigenvalues.
• Figure 2: Wall trajectory on the resolved $S^3$ at the tip of the deformed conifold.
• Figure 3: A schematic representation of the flavor moduli space realized topologically as $S^2 \subset {\mathbb{R}}^3$, showing the dependence of the induced metric on the hierarchical structure of the masses: (a) equal masses implying SO(3) isometry; (b) $\xi\neq 0$ implying SO(2) isometry; and (c) the decoupling limit where $\xi \gg 1$. The overall scale has been chosen for convenience. The bold arrows indicate an example of the dominant flavor mode profile in the wall.