Vacua, walls and junctions in $G_{N_F,N_C}$

TL;DR

This paper explores the structure of vacua, walls, and junctions in mass-deformed nonlinear sigma models on Grassmann manifolds, introducing polyhedral diagrams for analysis and confirming consistency with prior Plücker embedding results.

Contribution

It extends polyhedral diagram methods to non-Abelian sigma models on Grassmann manifolds, providing a new visualization and analysis tool for junctions and vacua.

Findings

Polyhedral diagrams effectively describe non-Abelian junctions.

Results align with previous Plücker embedding analyses.

New visualization aids understanding of complex vacuum structures.

Abstract

We discuss vacua, walls and three-pronged junctions of the mass-deformed nonlinear sigma models on the Grassmann manifold $G_{N_F,N_C}=\frac{SU(N_F)}{SU(N_C)\times SU(N_F-N_C)\times U(1)}$, which are non-Abelian gauge theories for $N_C\geq 2$. Polyhedra are proposed in \cite{Eto:2005cp} to describe Bogomol'nyi-Prasad-Sommerfield objects of the mass-deformed nonlinear sigma models on the complex projective space, which are Abelian gauge theories. We show that we can produce similar polyhedra for the mass-deformed nonlinear sigma models on the Grassmann manifold by applying the moduli matrix formalism \cite{Isozumi:2004jc} and the pictorial representation \cite{Lee:2017kaj}. Non-Abelian junctions can be analysed by making use of the polyhedra instead of the Pl\"{u}cker embedding. We present diagrams for vacua, walls and three-pronged junctions, and compute three-pronged junction positions of the mass-deformed nonlinear sigma models on the Grassmann manifold. We show that the results are consistent with the known results of \cite{Eto:2005fm}, which are worked out by using the Pl\"{u}cker embedding.