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

Firstly, we consider U($N_c$) Yang–Mills gauge theory on $R^{3,1}$ with $N_f>N_c$ flavours of scalar fields in the fundamental representation of U($N_c$). The moduli space of vacua is the Grassmannian manifold $\mathrm{Gr}(N_c,N_f)$. It is shown that for strong gauge coupling this 4d Yang–Mills–Higgs theory reduces to the Faddeev sigma model on $R^{3,1}$ with $\mathrm{Gr}(N_c,N_f)$ 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 consider a Yang–Mills–Higgs model with $N_f=2N_c$ and a Higgs potential breaking the flavour group $U\left(N_f\right)=U\left(2N_c\right)$ to $U_{+}\left(N_c\right)\times U_{-}\left(N_c\right)$, realizing the simplest $A_2\oplus A_2$-type quiver gauge theory. The vacuum moduli space of this model is the group manifold $U_h\left(N_c\right)$ which is the quotient of $U_{+}\left(N_c\right)\times U_{-}\left(N_c\right)$ by its diagonal subgroup. When the gauge coupling constant is large, this 4d Yang–Mills–Higgs model reduces to the Skyrme sigma model on $R^{3,1}$ with $U_h\left(N_c\right)$ as target. Thus, both the Skyrme and the Faddeev model arise as effective field theories in the infrared of Yang–Mills–Higgs models.