fermions is computed by the index theorem as in [@Maldacena:2000mw; @Benini:2012cz; @Benini:2013cda] $$\label{nrnldef}
n_r^{(\sigma)}-n_{\ell}^{(\sigma)} = -t_{\frak{b}}^{(\sigma)}\eta_{\Sigma}\;, \qquad \sigma=\lambda, \chi_2, \chi_3 \;,$$ where $\eta_{\Sigma}=2|\mathfrak{g}-1|$ for $\mathfrak{g}\neq1$, $\eta_{\Sigma} =1$ for $\mathfrak{g}=1$, and $t_{\frak b}^{(\sigma)}$ is the charge of each of the three species of four-dimensional fermion fields under the background gauge field . The values of $t_{\frak{b}}^{(\sigma)}$ are given in Table \[table:two\].

The two-dimensional (right-moving) trial central charge is computed using the fact that in two-dimensional $(0,2)$ SCFTs the conformal anomaly is proportional to the quadratic ’t Hooft anomaly of the unique superconformal $U(1)$ $R$-symmetry [@AlvarezGaume:1983ig] $$\label{ctr1}\begin{split}
c_{r}^{\text{tr}} &= 3 \,d_{G}\ds\sum_{\sigma}(n_r^{(\sigma)}-n_{\ell}^{(\sigma)}) (q_{\rm tr}^{(\sigma)})^2\\ & = -3 \,\eta_{\Sigma}\, d_{G} \left[ t_{\frak{b}}^{(\lambda)}(q_{\rm tr}^{(\lambda)})^2 +t_{\frak{b}}^{(\chi_2)}(q_{\rm tr}^{(\chi_2)})^2 + t_{\frak{b}}^{(\chi_3)}(q_{\rm tr}^{(\chi_3)})^2 \right]\;,
\end{split}$$ where $d_{G}$ is the dimension of the gauge group and $q_{\rm tr}^{(\sigma)}$ are the charges of the four-dimensional fermions under the trial $R$-symmetry given in Table \[table:two\]. Plugging these charges in , we find $$\label{ctr2}
c_{r}^{\text{tr}}(\epsilon,\frak b) = \frac{3}{8}\, \eta_{\Sigma}\, d_{G}\,(3-\epsilon^2+4\,\mathfrak{b}\,\epsilon)\;.$$ Next, we extremize $c_{r}^{\text{tr}}(\epsilon,\frak b)$ with respect to $\epsilon$, which gives $$\epsilon=2\mathfrak{b}\;,$$ so that the two-dimensional right-moving central charge is $$\label{cc2db}
c_{r}(\frak b) = \frac{3}{8}\,\eta_{\Sigma}\,d_G\, (3+ 4 \mathfrak{b}^2)\;.$$ The central charge in is always positive and this suggests that, for fixed $G$ and 