Skyrme-Faddeev model from 5d super-Yang-Mills

We consider 5d Yang-Mills-Higgs theory with a compact ADE-type gauge group $G$ and one adjoint scalar field on ${\mathbb{R}}^{3,1}\times {\mathbb{R}}_{+}$, where ${\mathbb{R}}_{+}=[0,\mathrm{\infty })$ is the half-line. The maximally supersymmetric extension of this model, with five adjoint scalars, appears after a reduction of 6d $\mathcal{N}=(2,0)$ superconformal field theory on ${\mathbb{R}}^{3,1}\times {\mathbb{R}}_{+}\times S^1$ along the circle $S^1$. We show that in the low-energy limit, when momenta along ${\mathbb{R}}^{3,1}$ are much smaller than along ${\mathbb{R}}_{+}$, the 5d Yang-Mills-Higgs theory reduces to a nonlinear sigma model on ${\mathbb{R}}^{3,1}$ with a coset $G\mathrm{/}H$ as its target space. Here $H$ is a closed subgroup of $G$ determined by the Higgs-field asymptotics at infinity. The 4d sigma model describes an infinite tower of interacting fields, and in the infrared it is dominated by the standard two-derivative kinetic term and the four-derivative Skyrme-Faddeev term.