General composite non-Abelian strings and flag manifold sigma models

Edwin Ireson

doi:10.1103/physrevresearch.2.013038

Abstract

We fully investigate the symmetry breaking patterns occurring upon creation of composite non-Abelian strings: vortex strings in non-Abelian theories where different sets of colours have different amounts of flux. After spontaneous symmetry breaking, there remains some internal colour degrees of freedom attached to these objects, which we argue must exist in a Flag manifold, a more general kind of projective space than both $\mathbb{CP}(N)$ and the Grassmannian manifold. These strings are expected to be BPS, since its constituents are. We demonstrate that this is true and construct a low-energy effective action for the fluctuations of the internal Flag moduli, which we then re-write it in two different ways for the dynamics of these degrees of freedom: a gauged linear sigma model with auxiliary fields and a non-linear sigma model with an explicit target space metric for the Flag Manifolds, both of which $\mathcal{N}=(2,2)$ supersymmetric. We finish by performing some groundwork analysis of the resulting theory.

Highlights

The CP (N ) nonlinear sigma model has undergone much analysis in many contexts, in particular because it provides a very tractable theory in which confinement occurs [1]
We have introduced the notion of a fully composite nonAbelian string: a more complex version of the Grassmannian string, it can be viewed as the admixture of several Grassmannian strings with overlapping but unequal sets of color fluxes running through them, such that different groups of colors have different amounts of flux or winding number
The symmetry breaking that the existence of such an object enforces endows it with internal degrees of freedom, and we argued that they must exist in a flag manifold

Summary

INTRODUCTION

The CP (N ) nonlinear sigma model has undergone much analysis in many contexts, in particular because it provides a very tractable (in its simplest formulation, exactly solvable) theory in which confinement occurs [1]. Because the models we are studying are Kähler manifolds, the study of (1,1)-supersymmetric manifolds is trivial: the complex structure for these spaces automatically provides a supersymmetry (SUSY) enhancement to (2,2) In this context we will show how flag manifolds arise on the world sheet of generic, highly composite non-Abelian vortex strings. There are N equivalent ways of choosing this field which experiences winding, and this produces an additional selection rule due to the following nontrivial homotopy structure: dividing through by the center group The objects that this construction produces, the ZN string, has minimal tension [18]. We will perform the construction of the string via the fields that compose it

Setting up the radial ansatz

Varying the gauge moduli

Gauged linear sigma model

Nonlinear sigma model

PHYSICAL PROPERTIES OF THE MODEL

CONCLUSIONS