Two types of domain walls in $\mathcal{N}=1$ super-QCD: how they are classified and counted

TL;DR

This paper classifies and counts two types of BPS domain walls in $ =1$ super-QCD, analyzing their multiplicities, junctions, and behavior under varying quark masses using advanced supersymmetric techniques.

Contribution

It introduces a comprehensive classification of domain walls in $ =1$ SQCD, including their multiplicities and how they evolve with quark mass, unifying previous results and exploring new wall junction candidates.

Findings

Two classes of domain walls identified: locally distinguishable and topologically distinguishable.

The overall multiplicity of walls remains constant across different flavor numbers and mass regimes.

Locally distinguishable walls only become topologically distinguishable at infinite quark mass.

Abstract

We study multiplicities and junctions of BPS domain walls interpolating between different chiral vacua in $\mathcal{N}=1$ supersymmetric QCD (SQCD) with the SU$(N)$ gauge group and a varying number of fundamental quarks. Depending on the number of flavors $F$, two distinct classes of {\em degenerate} domain walls emerge: (i) locally distinguishable, i.e., those which differ from each other locally, in local experiments; and (ii) those which have identical local structure and are differentiated only topologically, through judiciously chosen compactifications. In the first class, two-wall junctions exist, while in the second class, such junctions do not exist. Acharya and Vafa counted {\em topologically distinguishable} walls in pure super-Yang-Mills. Ritz, Shifman, and Vainshtein counted the {\em locally distinguishable} walls in $F=N$ SQCD. In both cases, the multiplicity of $k$ walls was the same, $\nu_{N,k}^\text{walls}= N!/\big[(N-k)!k!\big]$. We study the general case $0\leqslant F\leqslant N$, with mixed sets of walls from both classes (i) and (ii) simultaneously, and demonstrate that the above overall multiplicity remains intact. We argue that the growth of the quark masses exhibits no phase transition at any finite mass. The locally distinguishable walls can turn into topologically distinguishable ones only at $m=\infty$. The evolution of the low-energy wall worldsheet theory in the passage from small to large $m$ is briefly discussed. We also propose a candidate for the low-energy description of wall junctions. The tools used are localization of instantons, supersymmetry enhancement on the walls, and circle compactification.