\mbox{tr} \left(
Q_i^\dagger\, T^a \, Q_i - Q_{i-1}\, T^a \, Q_{i-1}^\dagger \right)
\, ,$$ where $T^a$ are the generators of $SU(M)$ in the fundamental representation (thus $-T^{a\, *}=-T^{a\, t}$ are the generators in the antifundamental) and it is understood that we impose cyclic boundary conditions, [*i.e.*]{}, $Q_{0}\equiv Q_{N}$. When the $Q$’s develop a vev, the fluctuations around this vev get a mass. Decomposing $Q_{i}^{\alpha\beta}= v\,\delta^{\alpha\beta} + \phi_{i}^{\alpha\beta}$, we obtain the following mass term $$\begin{aligned}
&&
{\cal L}\supset - g^2 v^2
\sum_{i,a}
\left(
 (T^a \phi_i) (T^a \phi_i)
+  (T^a \phi_i^\dagger) (T^a \phi_i^\dagger)
+ 2 (T^a \phi_i) (T^a \phi_i^\dagger)
\right.
\nonumber\\
&&
\hskip.5cm
\left.
+(T^a \phi_i^\dagger) (T^a \phi_{i-1}^\dagger)
+(T^a \phi_i) (T^a \phi_{i-1})
+(T^a \phi_i^\dagger) (T^a \phi_{i-1})
+(T^a \phi_i) (T^a \phi_{i-1}^\dagger)
\right)
.
\label{eq:p2p1}\end{aligned}$$ where we have defined $(T^a\phi)=\mbox{tr} (T^a\phi)$. Using the Fierz identity for the fundamental representation of $SU(M)$ (see for example [@Cvitanovic]), $$\sum_a T^a_{\alpha\beta}T^a_{\gamma\delta} =
\left( \delta_{\alpha\delta}\delta_{\beta\gamma}-{{\textstyle\frac{1}{M}}} \delta_{\alpha\beta}\delta_{\gamma\delta}
\right),$$ we obtain $$\label{Fierz}
(T^a \phi)(T^a \psi) = \mbox{tr} (\phi \psi) - {{\textstyle\frac{1}{M}}} \mbox{tr} (\phi) \mbox{tr} (\psi)
\equiv (\phi\times\psi)
\, .$$ Thus the mass term becomes $$\begin{aligned}
&&
{\cal L}\supset - g^2 v^2 \sum_i \left(
(\phi_i \times \phi_i) + 2 (\phi_i \times \phi_i^\dagger)
+ (\phi_i^\dagger \times \phi_i^\dagger)
\right.
\nonumber\\
&&
\hskip2cm
\left.
- 