nonlinear sigma model fields, while $\Phi$ (the top-Higgs doublet) is a linear sigma model field.

![The gauge structure of the model in Moose notation. $g$ and $g^{\prime}$ are approximately the SM $SU(2)$ and hypercharge gauge couplings while $\tilde{g}$ represents the ‘bulk’ gauge coupling. The left (right) handed light fermions are mostly localized at site 0 (2) while their heavy counterparts are mostly at site 1. The links connecting sites 0 and 1 and sites 1 and 2 are nonlinear sigma model fields while the one connecting sites 0 and 2 is a linear sigma field. Site 2 is dotted to indicate that only the $\tau_3$ component is gauged.[]{data-label="fig:Triangle"}](triangle.pdf){width="2.4in"}

The kinetic energy terms of the link fields corresponding to these charge assignments are: $$\mathcal{L}_{gauge}=
   \frac{F^{2}}{4} 
   \textrm{Tr}[(D_{\mu}\Sigma_{01})^{\dagger}D^{\mu}\Sigma_{01}]
   + \frac{F^{2}}{4}
   \textrm{Tr}[(D_{\mu}\Sigma_{12})^{\dagger}D^{\mu}\Sigma_{12}]
   + (D_{\mu}\Phi)^{\dagger}D^{\mu}\Phi,
\label{eqn:Gauge L}$$ where the covariant derivatives are: $$\begin{aligned}
   D_{\mu}\Sigma_{01} &=& \partial_{\mu}\Sigma_{01} + igW_{0\mu} \Sigma_{01}
   - i\tilde{g}\Sigma_{01}W_{1\mu}, \nonumber \\
   D_{\mu}\Sigma_{12} &=& \partial_{\mu}\Sigma_{12} 
   + i\tilde{g}W_{1\mu} \Sigma_{12} 
   - ig^{\prime} \Sigma_{12}\tau^{3}B_{\mu}, \nonumber \\
   D_{\mu}\Phi &=& \partial_{\mu}\Phi + igW_{0\mu} \Phi
   - \frac{ig^{\prime}}{2}B_{\mu}\Phi.
\label{eqn:covariant}\end{aligned}$$ Here the gauge fields are represented by the matrices $W_{0\mu} =
W_{0\mu}^{a}\tau^{a}$ and $W_{1\mu} = W_{1\mu}^{a}\tau^{a}$, where $\tau^a = \sigma^a/2$ are the generators of SU(2). The nonlinear sigma model fields $\Sigma_{01}$ and 