Skyrme and Faddeev models in the low-energy limit of 4d Yang-Mills-Higgs theories

TL;DR

This paper demonstrates how low-energy limits of 4d Yang-Mills-Higgs theories with specific symmetry breaking patterns lead to effective Skyrme and Faddeev sigma models, connecting gauge theories to topological soliton models.

Contribution

It shows the derivation of Skyrme and Faddeev models from Yang-Mills-Higgs theories with particular symmetry breaking, including the stabilization terms and specific target manifolds.

Findings

Faddeev model arises from G/H coset with a four-derivative term.

Skyrme model emerges when Higgs field is in bi-fundamental representation.

Both models are effective theories in the infrared limit of Yang-Mills-Higgs theories.

Abstract

Firstly, we consider Yang-Mills theory on ${\mathbb R}^{3,1}$ with an adjoint Higgs field spontaneously breaking a compact gauge group $G$ to a subgroup $H$, so that the Higgs vacuum manifold forms the coset $G/H$. It is shown that in the low-energy limit, when the Higgs vacuum value is large, the 4d Yang-Mills-Higgs theory reduces to the Faddeev sigma model on ${\mathbb R}^{3,1}$ with $G/H$ as target. Its action contains the standard two-derivative sigma-model term as well as the four-derivative Skyrme-type term, which stabilizes solutions against scaling. Secondly, we put the Higgs field in the bi-fundamental representation of $G=\textrm{U}_+(N)\times\textrm{U}_-(N)$, realizing the simplest $A_2$-type quiver gauge theory. Breaking $G$ to $H{=}\,\textrm{diag}(G)$, the vacuum manifold $G/H\cong\textrm{U}(N)$ is a group. In this case, when the Higgs vacuum value is large, the 4d $A_2$-quiver gauge theory reduces to the Skyrme sigma model on ${\mathbb R}^{3,1}$ with U$(N)$ as target. Thus, both the Skyrme and the Faddeev model arise as effective field theories in the infrared of Yang-Mills-Higgs models.