$\Sigma_{12}$ are 2$\times$2 special unitary matrix fields. To mimic the symmetry breaking caused by underlying technicolor and topcolor dynamics, we assume all link fields develop vacuum expectation values (vevs): $$\langle \Sigma_{01} \rangle = \langle \Sigma_{12} \rangle 
   = \mathbf{1}_{2\times 2}, \qquad \qquad
   \langle \Phi \rangle = 
   \left( \begin{array}{c} f/\sqrt{2} \\ 0 \end{array} \right).$$ In order to obtain the correct amplitude for muon decay, we parameterize the vevs in terms of a new parameter $\omega$, $$F=\sqrt{2}\,v\,\cos\,\omega, \qquad \qquad 
   f=v\,\sin\,\omega,
   \label{eq:Ff}$$ where $v = 246$ GeV is the weak scale. As a consequence of the vacuum expectation values, the gauge symmetry is broken all the way down to electromagnetism and we are left with massive gauge bosons (analogous to techni-resonances), top-pions and a top-Higgs. To keep track of how the degrees of freedom are partitioned after we impose the symmetry breaking, we expand $\Sigma_{01}$, $\Sigma_{12}$ and $\Phi$ around their vevs. The coset degrees of freedom in the bi-fundamental link fields $\Sigma_{01}$ and $\Sigma_{12}$ can be described by nonlinear sigma fields: $$\Sigma_{01}=\textrm{exp}\left(\frac{2i\pi_{0}^{a}\tau^{a}}{F}\right),
   \qquad \qquad 
   \Sigma_{12}=\textrm{exp}\left(\frac{2i\pi_{1}^{a}\tau^{a}}{F}\right),
\label{eq:vevs}$$ while the degrees of freedom in $\Phi$ fill out a linear representation, $$\Phi= \left( \begin{array}{c}
     (f + H_t + i\pi_{t}^{0})/\sqrt{2} \\
     i\pi^{-}_{t} \end{array} \right).
\label{eqn:phi representation}$$

The gauge-kinetic 