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

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 $G\mathrm{/}H$. 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 $G={\text{U}}_{+}(N)\times {\text{U}}_{-}(N)$, realizing the simplest $A_2$-type quiver gauge theory. Breaking $G$ to $H=\text{diag}(G)$, the vacuum manifold $G\mathrm{/}H\cong \text{U}(N)$ 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.