\Phi_i$$ where $V_j$ is the vector superfield associated to the gauge group $SU(M)_j$ and $\Phi_i$ is the link chiral superfield that transforms as $({\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}},\overline{{\raisebox{-.5pt}{{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}}})$ under $SU(M)_i\times SU(M)_{i+1}$. When expanded in components, this gives the gaugino–scalar–fermion interaction $$\label{fermioninteract}
{\cal L}\supset
i\sqrt{2} g \sum_i \,\mbox{tr} \left(
Q_i^\dagger T^a (q_i \lambda^a_i)
- (\bar{q}_i^t \bar{\lambda}^a_i) T^a Q_i
- (q_i \lambda_{i+1}^a) T^a Q_i^\dagger
+ Q_i T^a (\bar{q}_i^t \bar{\lambda}_{i+1}^a)
\right)$$ where $\lambda$ is the gaugino, $q$ is the two component Weyl fermion in the bifundamental, while $Q$ is the scalar component in the bifundamental. Note that again these terms only give mass to the traceless parts of the bifundamental fermions as a result of the tracelessness of the generators $T^a$, or equivalently because the gauginos transform in the adjoint representation of the gauge group. Putting in the expectation values of the $Q$’s will give us the fermion mass terms, which are then given by $${\cal L}\supset
i \sqrt{2} g v\sum_i \, \mbox{tr} \left(
\lambda_i (q_i-q_{i-1})-\bar{\lambda}_i (\bar{q}_i-\bar{q}_{i-1}) \right)
\ ,$$ where we have defined $\lambda_{i\, \alpha\beta} = \lambda^a_i T^a_{\alpha\beta}$ (note that our normalization of the Casimir of $SU(M)$ in the fundamental representation, [*i.e.*]{}, ${\rm tr}\, T^a T^b = \delta^{ab}$, ensures that $\lambda_{\alpha\beta}$ are $M^2-1$ Weyl fermions with 