a degeneracy $M^2+1$ and 2, with a degeneracy $M^2-1$. The mass spectrum corresponds to the product of these different eigenvalues as follows:

-   $m=0$ with degeneracy $2M^2+(N-1)(M^2+1)$ ,

-   $m_k= 2 \sqrt{2} g v \sin \frac{k \pi}{N}$, for $1\leq k\leq N-1$, with degeneracy $M^2-1$.

However, this counting of the massless modes has not taken into account the Higgs mechanism or superpotential. First, $(N-1)(M^2-1)$ modes are eaten in the super-Higgs mechanism associated to the breaking of $N-1$ $SU(M)$ gauge groups, and these would-be Goldstone modes give the longitudinal components of the massive gauge bosons. Second, $2N$ scalars get a mass through the $F$–terms associated to the superpotential (\[suppot\]). Indeed $F_{S_i} = \mu^{-(M-2)} (\det Q_i - v^M) \supset  v (v/\mu)^{M-2} \,\mbox{tr} \phi_i$, such that the trace of $\phi$ at each site acquires a large mass, $v (v/\mu)^{M-2}$, and decouples from the low energy effective action. Thus we are left with only $2(M^2-1)$ real massless scalars.

### Fermion masses {#fermionmass-sec}

Finally, we consider the fermion fields. Our aim is to show that indeed the bifundamental fermions combine with the gaugino to give a supersymmetric spectrum which matches that of the gauge bosons and scalars. The 4D Kähler potential contains $$\sum_i \Phi_i^\dag\, e^{\sum_j V_j}\, 