Dirac spinor that gets a mass $v(v/\mu)^{M-2}$ from the superpotential (\[suppot\]). The fermionic spectrum is thus:

-   $2(M^2-1)$ massless Weyl spinors,

-   $(M^2-1)$ Dirac spinors with mass $m_k= 2 \sqrt{2} g v \sin \frac{k \pi}{N},\ \ 0\leq k\leq N-1$,

showing the supersymmetric nature of the low energy spectrum. Indeed, the massless 5D ${\cal N}$=1 vector multiplet includes a gauge boson (3 components on-shell), a Dirac fermion (4 components) and a real scalar (1 component). Upon Kaluza–Klein reduction, we get a 4D ${\cal N}$=2 massless vector multiplet: a gauge boson (2 components), two Weyl fermions (2$\times$2 components) and a complex scalar (2 components); and massive vector multiplets: massive gauge boson (3 components), fermion (4 components) and real scalar (1 component). This decomposition agrees exactly with the spectrum we have found. Moreover each mass level transforms in the adjoint of the unbroken diagonal $SU(M)$ gauge group.

5D Lorentz invariance and supersymmetry
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Up to now we have shown that the mass spectrum of the theory indeed matches that of the higher dimensional supersymmetric theory. As we have mentioned above, the question of the existence of the full 5D Lorentz invariance is tightly related to the question of whether the full 5D ${\cal 